2018年世界各地数学竞赛试题汇集(PDF版)
2018年全国初中数学联合竞赛试题(含解答)
2018年全国初中数学联合竞赛试题(含解答)2018年全国初中数学联合竞赛试题参考答案及评分标准说明：评阅试卷时，请依据本评分标准。
第一试，选择题和填空题只设7分和0分两档；第二试各题，请严格按照本评分标准规定的评分档次给分，不要再增加其他中间档次。
如果考生的解答方法和本解答不同，只要思路合理，步骤正确，在评卷时请参照本评分标准划分的档次，给予相应的分数。
第一试一、选择题（本题满分42分，每小题7分）1.已知$x,y,z$满足$\frac{2355x-y}{y+2z}=\frac{x}{z-z^2}$，则$\frac{y+2z}{3x-y-z}$的值为（）A) 1.(B) $\frac{5}{3}$。
(C) $-\frac{1}{3}$。
(D) $-\frac{3}{5}$.答】B.解：由$\frac{2355x-y}{y+2z}=\frac{x}{z-z^2}$，得$5x-3y=3xz-3xz^2$，即$y=\frac{5}{3}x-\frac{3}{3}z+\frac{3}{3}xz^2$，所以$\frac{y+2z}{3x-y-z}=\frac{\frac{5}{3}x+\frac{1}{3}z}{\frac{4}{3}x-\frac{2}{3}z}=\frac{5}{3}$，故选（B）。
注：本题也可用特殊值法来判断。
2.当$x$分别取值$1,\frac{1}{2},\frac{1}{3},\cdots,\frac{1}{2005},\frac{1}{2006}, \frac{1}{2007}$时，计算$\frac{1}{2007}+\frac{x}{21+x^2}$代数式的值，将所得的结果相加，其和等于（）A) $-1$。
(B) $1$。
(C) $0$。
(D) $2007$.答】C.解：$\frac{1}{2007}+\frac{x}{21+x^2}=\frac{1}{21}\left(\frac{21}{ 2007}+\frac{21x}{21+x^2}\right)=\frac{1}{21}\left(\frac{1}{1+x ^{-2}}\right)$，所以当$x=1,\frac{1}{2},\frac{1}{3},\cdots,\frac{1}{2005},\frac{1}{200 6},\frac{1}{2007}$时，计算所得的代数式的值之和为$0$，故选（C）。
加拿大国际袋鼠数学竞赛试题 及答案-2018年
I N T E R N A T I O N A L C O N T E S T-G A M EM A T H K A N G A R O OC A N AD A,2018I N S T R U C T I O N SG R A D E1-21.You have 45 minutes to solve 18 multiple choice problems. For eachproblem, circle only one of the proposed five choices. If you circle more than one choice, your response will be marked as wrong.2.Record your answers in the response form. Remember that this is the onlysheet that is marked, so make sure you have all your answers transferred to the response form before giving it back to the contest supervisor.3.The problems are arranged in three groups. A correct answer of the first 6problems is worth 3 points. A correct answer of the problems 7-12 is worth4 points. A correct answer of the problems 13-18 is worth5 points. Foreach incorrect answer, one point is deducted from your score. Each unanswered question is worth 0 points. To avoid negative scores, you start from 18 points. The maximum score possible is 90.4.The use of external material or aid of any kind is not permitted.5.The figures are not drawn to scale. They should be used only for illustrationpurposes.6.Remember, you have about 2 to 3 minutes for each problem; hence, if aproblem appears to be too difficult, save it for later and move on to another problem.7.At the end of the allotted time, please give the response form to thecontest supervisor.8.Do not forget to pick up your Certificate of Participation on your way out!Good luck!Canadian Math Kangaroo Contest teamCanadian Math Kangaroo ContestPart A: Each correct answer is worth 3 points1.Which shape cannot be formed using and ?(A) (B) (C) (D) (E)2.At least how many 4-ray stars like this are glued together tomake this shape ?(A) 5 (B) 6 (C) 7 (D) 8 (E) 93.This pizza was divided into equal slices.How many slices are missing?(A) 1 (B) 2 (C) 3 (D) 4 (E) 54.How many kangaroos must be moved from one park to the other in order toget the same number of kangaroos in each park?(A) 4 (B) 5 (C) 6 (D) 8 (E) 95.Which of these ladybugs has to fly away so that the rest of them have 20dots in total?(A) (B) (C) (D) (E)6.Emilie builds towers in the following patternWhich one will be the tower number 6?(A) (B) (C) (D) (E)Part B: Each correct answer is worth 4 points7.If ◊+ ◊ = 4 and ∆ + ∆ + ∆ = 9, what is the value of ◊ + ∆ = ?(A) 2 (B) 3 (C) 4 (D) 5 (E) 68.Lisa has 4 pieces , but she only needs 3 forcompleting her puzzle frame . Which piece will be left over?(A)(B)(C)(D) (E)or9.How many right hands are in this picture?(A) 3 (B) 4 (C) 5 (D) 6 (E) 710.The dog went to its food following a path. In total it made 3 right turns and2 left turns. Which path did the dog follow?(A) (B) (C)(D) (E)11.What number is in the box marked "?" ?(A) 6 (B) 13 (C) 24 (D) 29 (E) Some other number12.Charles cut a rope in three equal pieces and then made some equal knotswith them. Which figure correctly shows the three pieces with the knots?(A) (B)(C) (D)(E)Part C: Each correct answer is worth 5 points13.How many circles and how many squares are covered by the blot in thepicture?(A) 1 circle and 3 squares(B) 2 circles and 1 square(C) 3 circles and 1 square(D) 1 circles and 2 squares(E) 2 circles and 2 squares14.Diana shoots three arrows at a target.On her first try, she gets 6 points and the arrows land like this: 6 pointsOn her second try, she gets 8 points and the arrows land like this: 8 pointsOn her third try, the arrows land like this:? points How many points will she get the third time?(A) 8 (B) 10 (C) 12 (D) 14 (E) 1615.How many different numbers greater than 10 and smaller than 25 with distinct digits can we make by using any two of the digits 2, 0, 1, and 8?(A) 4 (B) 5 (C) 6 (D) 7 (E) 816.Mark had some sticks of length 5 cm and width 1 cm.With the sticks he constructed the fence below.What is the length of the fence?(A) 20 cm(B) 21 cm(C) 22 cm (D) 23 cm (E) 25 cmlength17.The road from Anna's house to Mary's house is 16 km long.The road from Mary's house to John's house is 20 km long.The road from the crossroad to Mary's house is 9 km long.How long is the road from Anna’s house to John's house?(A) 7 km (B) 9 km (C) 11 km (D) 16 km (E) 18 km18.There are four ladybugs on a 4×4 board. Two are asleep and do not move.The other two ladybugs move one square every minute (up, down, left, or right). Here are pictures of the board for the first four minutes:Minute 1 Minute 2 Minute 3 Minute 4Which of these is a picture of the fifth minute (Minute 5)?(A) (B) (C) (D) (E)International Contest-GameMath Kangaroo Canada, 2018Answer KeyGrade 1-21 A B C D E 7 A B C D E 13 A B C D E2 A B C D E 8 A B C D E 14 A B C D E3 A B C D E 9 A B C D E 15 A B C D E4 A B C D E 10 A B C D E 16 A B C D E5 A B C D E 11 A B C D E 17 A B C D E6 A B C D E 12 A B C D E 18 A B C D EI N T E R N A T I O N A L C O N T E S T-G A M EM A T H K A N G A R O OC A N AD A,2018I N S T R U C T I O N SG R A D E3-41.You have 60 minutes to solve 24 multiple choice problems. For each problem,circle only one of the proposed five choices. If you circle more than one choice, your response will be marked as wrong.2.Record your answers in the response form. Remember that this is the only sheetthat is marked, so make sure you have all your answers transferred to the response form before giving it back to the contest supervisor.3.The problems are arranged in three groups. A correct answer of the first 8problems is worth 3 points. A correct answer of the problems 9-16 is worth 4 points. A correct answer of the problems 17-24 is worth 5 points. For each incorrect answer, one point is deducted from your score. Each unanswered question is worth 0 points. To avoid negative scores, you start from 24 points. The maximum score possible is 120.4.The use of external material or aid of any kind is not permitted.5.The figures are not drawn to scale. They should be used only for illustrationpurposes.6.Remember, you have about 2 to 3 minutes for each problem; hence, if a problemappears to be too difficult, save it for later and move on to another problem.7.At the end of the allotted time, please give the response form to the contestsupervisor.8.Do not forget to pick up your Certificate of Participation on your way out!Good luck!Canadian Math Kangaroo Contest teamGrade 3-42018 Canadian Math Kangaroo ContestPart A: Each correct answer is worth 3 points1.Lea has 10 rubber stamps. Each stamp has one of the digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9.She prints the date of St. Patrick’s Day 2018:How many different stamps does she use?(A) 5(B) 6 (C) 7 (D) 9 (E) 102.The picture shows three flying arrows and nine fixedballoons. When an arrow hits a balloon, it bursts,and the arrow flies further in the same direction.How many balloons will be hit by the arrows?(A) 2 (B) 3 (C) 4(D) 5 (E) 63.Susan is six years old. Her sister is one year younger, and her brother is one yearolder. What is the sum of the ages of the three siblings?(A) 10 (B) 15 (C) 18 (D) 21 (E) 304.Here is a picture of Sophie the ladybug. She turns around. Which picture ofthe ladybugs below is not Sophie?(A)(B)(C)(D)(E)5.Lucy folds a sheet of paper in half. Then she cuts a piece out of it. What willshe see when she unfolds the paper?(A)(B)(C) (D)(E)1 70320186. A table is set for 8 people.How many settings have the fork to the left of the plate and the knife to the right of the plate?(A) 5(B) 4 (C) 6 (D) 2 (E) 3 7.Emily added two 2-digit numbers correctly on paper. Then she painted out two cells,as shown below.What is the sum of two digits in the painted cells?(A) 5(B) 7 (C) 8 (D) 9 (E) 13 8.First, Diana scores 12 points in total with three arrows. On her second turn shescores 15 points.How many points does she score on her third turn?(A) 18 (B) 19 (C) 20 (D) 21 (E) 22 Part B: Each correct answer is worth 4 points9.How many different numbers greater than 12 and smaller than 58 with distinct digitscan we make by using any two of the digits 0, 1, 2, 5, and 8?(A) 3(B) 5 (C) 7(D) 8 (E) 912 points15 points ? points10.Roberto makes designs using tiles like this .How many of the following five designs can he make?(A) 1 (B) 2 (C) 3 (D) 4 (E) 511.Each of these five figures ,, , , , appears exactly once in everycolumn and every row of the given table.Which figure must we put in the cell with the question mark?(A) (B) (C) (D) (E)12.Toby glues 10 cubes together to make the structure shown.He paints the whole structure, even the bottom.How many cubes are painted on exactly four of their faces?(A) 6 (B) 7 (C) 8 (D) 9 (E) 1013.The opposite faces of a cube are identical, being dark, bright or patterned.Which picture below is the unfolded net of this cube?(A)14.Tom cuts two types of pieces out of grid paper.What is the smallest number of pieces identical to the ones shown that Tom needs to build the boat in the picture?(A) 5 (B) 6 (C) 7 (D) 8 (E) 915.The rooms in Kanga's house are numbered. Baby Roo entersthe main door, passes through some rooms and leaves thehouse. The numbers of the rooms that he visits are alwaysincreasing. Through which door does he leave the house?(A) A (B) B (C) C (D) D (E) E16.Peta rabbit had 20 carrots. She ate two carrots every day. She ate the twelfth carroton Wednesday. On which day did she start eating the carrots?(A) Monday (B) Tuesday (C) Wednesday (D) Thursday (E) FridayPart C: Each correct answer is worth 5 points17.The belt shown in the drawing can be fastened in five ways.How much longer is the belt fastened in one hole than the belt fastened in all five holes?(A) 4 cm (B) 8 cm (C) 10 cm (D) 16 cm (E) 20 cm18.In an ancient writing the symbols represent thenumbers 1, 2, 3, 4, and 5. Nobody knows which symbol represents which number.We know thatWhich symbol represents the number 3?(A)(B) (C) (D) (E)19. A stained-glass tile is flipped along the black line. The figure shows the tile after thefirst flip.What will the stained-glass tile look like after the third flip (at the far right)?(A)(B)(C)(D)(E)20.The large rectangle is made up of squares of varied sizes. The three smallest squareseach have an area of 1, as shown.What is the area of the largest square?(A) 81 (B) 100 (C) 110 (D) 121 (E) 14421.Five ducklings walk behind the mother duck in a row from the oldest to the youngestlike this: Dina and Becca walk right one after the other, Mingo walks behind Lisa but in front of Becca, Becca walks directly in front of Pip. What is the name of theyoungest duckling?(A) Dina (B) Pip (C) Becca (D) Lisa (E) Mingo22.Four balls each weigh 10, 20, 30 and 40 grams. Which ball weighs 30 grams?(A) A (B) B (C) C (D) D (E) it could be A or B23.Lois wants to write the numbers from 1 to 7 in the grid shown.Two consecutive numbers cannot be written in two neighbouringcells. Neighbouring cells meet at the edge or at a corner. Whatnumbers can she write in the cell marked with a question mark?(A) all seven numbers (B) only odd numbers(C) only even numbers (D) only number 4(E) only the numbers 1 or 7 24.The distance from Anna's to Mary's house is 16 kilometers along the shown road.The distance from Mary's to Nick's house is 20 kilometers.The distance from Nick's to John's house is 19 kilometers.How far is Anna's house from John's?(A) 15 (B) 16(C) 18(D) 19 (E) 20 ?International Contest-GameMath Kangaroo Canada, 2018Answer KeyGrade 3-41 A B C D E 9 A B C D E17 A B C D E2 A B C D E10 A B C D E 18 A B C D E3 A B C D E 11 A B C D E 19 A B C D E4 A B C D E 12 A B C D E 20 A B C D E5 A B C D E 13 A B C D E21 A B C D E6 A B C D E 14 A B C D E 22 A B C D E7 A B C D E 15 A B C D E 23 A B C D E8 A B C D E 16 A B C D E24 A B C D EI N T E R N A T I O N A L C O N T E S T-G A M EM A T H K A N G A R O OC A N AD A,2018I N S T R U C T I O N SG R A D E5-121.You have 75 minutes to solve 30 multiple choice problems. For eachproblem, circle only one of the proposed five choices. If you circle more than one choice, your response will be marked as wrong.2.Record your answers in the response form. Remember that this is theonly sheet that is marked, so make sure you have all your answers transferred to that form before giving it back to the contest supervisor. 3.The problems are arranged in three groups. A correct answer of the first10 problems is worth 3 points. A correct answer of problems 11 -20 isworth 4 points. A correct answer of problems 21-30 is worth 5 points. For each incorrect answer, one point is deducted from your score. Each unanswered question is worth 0 points. To avoid negative scores, you start from 30 points. The maximum score possible is 150.4.The use of external material or aid of any kind is not permitted.5.The figures are not drawn to scale. They should be used only forillustration purposes.6.Remember, you have about 2 to 3 minutes for each problem; hence, if aproblem appears to be too difficult, save it for later and move on to another problem.7.At the end of the allotted time, please give the response form to thecontest supervisor.8.Do not forget to pick up your Certificate of Participation on your way out!Good luck!Canadian Math Kangaroo Contest teamCanadian Math Kangaroo ContestPart A: Each correct answer is worth 3 points1.The drawing shows 3flying arrows and 9fixed balloons. Whenan arrow hits a balloon, it bursts, and the arrow flies further inthe same direction. How many balloons will not be hit byarrows?(A) 3 (B) 2(C) 6(D) 5(E) 42.The image shows a structure made of three objects.What does Peter see if he looks at the structure from above?(A)(B)(C) (D) (E)3.Diana played darts throwing arrows toward a target with three sections. First she got 14 points with twoarrows on the target. The second time she got 16 points. How many points did she get the third time?(A) 17(B) 18(C) 19 (D) 20 (E) 22 4. A garden is divided into identical squares. A fast snail and a slow snail move along the perimeter of thegarden starting simultaneously from the corner S but in different directions. The slow snail moves at the speed of 1 metre per hour (1 m/h) and the fast one at 2 metres per hour (2 m/h).At what point will the two snails meet?(A) A (B) B (C) C (D) D(E) E 14 points16 points ? A B CDE S 1 m/h2 m/h5.In which of the four squares is the fraction of the black area the largest?(A) A (B) B (C) C (D) D (E) they are all the same6. A star is made out of four equilateral triangles and a square. The perimeter of thesquare is 36 cm. What is the perimeter of the star?(A) 144 cm (B) 120 cm (C) 104 cm (D) 90 cm (E) 72 cm7.From the list 3, 5, 2, 6, 1, 4, 7 Masha chose 3 different numbers whose sum is 8. From the same list Dashachose 3 different numbers whose sum is 7. How many common numbers have been chosen by both girls?(A) none (B) 1 (C) 2 (D) 3 (E) impossible to determine8.We move a bead along a piece of wire. What shall we see when the beadcomes to the end of the wire?(A) (B) (C)(D) (E)9.There are 3squares in the figure. The side length of the smallest square is 6 cm.What is the side length of the biggest square?(A) 8(B) 10(C) 12(D) 14(E) 1610.In the following figure, the circles are light bulbs connected to some other lightbulbs. Initially, all light bulbs are off. When you touch a light bulb, this light bulband all its neighbours (e.g., the light bulbs connected to it) are lit.At least how many light bulbs do you have to touch to turn on all the light bulbs?(A) 2 (B) 3 (C) 4 (D) 5 (E) 6Part B: Each correct answer is worth 4 points11.Each square contains one of the numbers 1, 2, 3, 4, or 5, so that both of thecalculations following the arrows are correct. A number may be used morethan once. What number goes into the box with the question mark?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5 12. Nine cars arrive at a crossroads and drive off as indicated by the arrows. Which figure shows these cars after leaving the crossroads?(A)(B) (C) (D) (E) 13. The faces of a cube are painted black, white or grey so that opposite faces are of different colour. Which of the following is not a possible net of this cube?(A)(B) (C) (D) (E)14.In a box there are many one-euro, two-euro and five-euro coins. A dispenser draws coins out of the box – one at a time, and stops when three identical coins are taken out. What is the largest possible amount that can be withdrawn? (A) 24 (B) 23 (C) 22 (D) 21 (E) 1515.Two girls, Eva and Olga and three boys, Adam, Isaac and Urban play with a ball. When a girl has the ball, she throws it to the other girl or to a boy. When a boy has the ball, he throws it to another boy but never to the boy from whom he just received it. Eva starts by throwing the ball to Adam. Who will do the fifth throw?(A) Adam (B) Eva (C) Isaac (D) Olga (E) Urban16.Emily wants to enter a number into each cell of the triangular table. The sum of thenumbers in any two cells with a common edge must be the same. She has alreadyentered two numbers. What is the sum of all the numbers in the table?(A) 18 (B) 20 (C) 21 (D) 22 (E) impossible to determine17.John coded a correct addition calculation naming the digits AA , BB , CC and DD .Which digit is represented by BB ?(A) 0 (B) 2 (C) 4 (D) 5(E) 6 + A B C C B A D D DD18.On Monday Alexandra shares a picture with 5 friends. For several days, everybody who receives thepicture, sends it once on the next day to two friends. On which day does the number of people who have seen the picture (including Alexandra) become greater than 75, if it is known that no one receives the picture more than once?(A) Wednesday (B) Thursday (C) Friday (D) Saturday (E) Sunday 19.The sum of the ages of Kate and her mother is 36, and the sum of the ages of her mother and her grandmother is 81. How old was the grandmother when Kate was born? (A) 28 (B) 38 (C) 45 (D) 53 (E) 56 20.Annie replaced the letters with numbers in the word KANGAROO (identical letters with the same digits, different letters with different digits) so that she got the largest possible 8-digit number, which is not a multiple of 4. What is the sum of the last three digits replacing the word ROO? (A) 13 (B) 14 (C) 12 (D) 15 (E) 11Part C: Each correct answer is worth 5 points21.Captain Hook has plundered a safe that contains 2520 gold coins. During the night, each of his pirates secretly took out some coins just for themselves. The first one took out �12�of the coins, the second one�13�of the remaining coins, the third one �14�of the remaining coins and so on. When Captain Hook opened the safe in the morning, he found only 252 coins inside. How many pirates are commanded by Captain Hook?(A) 8 (B) 9 (C) 10 (D) 11 (E) 12 22.In the figure on the right, the five balls A, B, C, D and E weigh 30, 50, 50, 50 and 80 grams, but not necessarily in this order. Which ball weighs 30 grams? (A) A (B) B (C) C (D) D (E) E23.If A, B, C are distinct digits, which of the following numbers cannot be the largest possible 6-digit number written using three digits A, two digits B, and one digit C? (A) AAABBC (B) CAAABB (C) BBAAAC (D) AAABCB (E) AAACBB 24.In the World of Numbers, there are many number-machines, which work in the following way: the machine adds the two beginning digits of the number and replaces them by their sum. For example, beginning with the number 87312 and using six such machines we obtain:How many such machines should be used in order to get the number times509...9 from the numbertimes1009...9? (A) 50(B) 60(C) 100(D) 80(E) Not possible to obtain this number8731215312 6312 91210212 3Page 525.Nick wants to arrange the numbers 2, 3, 4, ..., 10 into several groups such that the sum of the numbers in each group is the same. What is the largest number of groups he can get?(A) 2 (B) 3 (C) 4 (D) 6 (E) other answer 26.Peter cut an 8-cm wide wooden plank with a saw into 9 parts across the width of the plank.One piece was a square, the other were rectangles. Then he arranged all the pieces together as shown in the picture. What was the length of the plank?(A) 150 cm (B) 168 cm (C) 196 cm (D) 200 cm (E) 232 cm 27.Write 0 or 1 in each cell of the 5×5 table so that each 2×2 square of the 5×5 table contains exactly 3 equal numbers. What is the largest possible sum of all the numbers in the table?(A) 22 (B) 21 (C) 19 (D) 17 (E) 1528.14 people are seated at a round table.Each person is either a liar or tells the truth. Everybody says: "Both my neighbours are liars". What is themaximum number of liars at the table?(A) 7 (B) 8 (C) 9(D) 10(E) 1429.There are eight domino tiles on the table (pic 1). One half of one tile is covered. The 8 tiles can be arranged into a 4×4 square (pic 2), so that the number of dots in each row and column is the same.How many dots are on the covered part? (A) 1 (B) 2 (C) 3 (D) 4(E) 530.Four ladybugs sit on different cells of a 4×4 grid.One of them is sleeping and does not move. Each time you whistle, the other three ladybugs move toa free neighbouring cell. They can move up, down,right or left but they are not allowed to go back tothe cell they just came from. Which of the following images might show the result after the fourth whistle?(A)(B)(C)(D)(E)pic 1pic 2initial position after firstwhistleafter second whistle after third whistle Both my neighboursare liars.International Contest-GameMath Kangaroo Canada, 2018Answer KeyGrade 5-61 A B C D E 11 A B C D E21 A B C D E2 A B C D E 12 A B C D E 22 A B C D E3 A B C D E 13 A B C D E23 A B C D E4 A B C D E 14 A B C D E 24 A B C D E5 A B C D E15 A B C D E 25 A B C D E6 A B C D E16 A B C D E 26 A B C D E7 A B C D E 17 A B C D E 27 A B C D E8 A B C D E 18 A B C D E 28 A B C D E9 A B C D E 19 A B C D E 29 A B C D E10 A B C D E 20 A B C D E30 A B C D EI N T E R N A T I O N A L C O N T E S T-G A M EM A T H K A N G A R O OC A N AD A,2018I N S T R U C T I O N SG R A D E5-121.You have 75 minutes to solve 30 multiple choice problems. For eachproblem, circle only one of the proposed five choices. If you circle more than one choice, your response will be marked as wrong.2.Record your answers in the response form. Remember that this is theonly sheet that is marked, so make sure you have all your answers transferred to that form before giving it back to the contest supervisor. 3.The problems are arranged in three groups. A correct answer of the first10 problems is worth 3 points. A correct answer of problems 11 -20 isworth 4 points. A correct answer of problems 21-30 is worth 5 points. For each incorrect answer, one point is deducted from your score. Each unanswered question is worth 0 points. To avoid negative scores, you start from 30 points. The maximum score possible is 150.4.The use of external material or aid of any kind is not permitted.5.The figures are not drawn to scale. They should be used only forillustration purposes.6.Remember, you have about 2 to 3 minutes for each problem; hence, if aproblem appears to be too difficult, save it for later and move on to another problem.7.At the end of the allotted time, please give the response form to thecontest supervisor.8.Do not forget to pick up your Certificate of Participation on your way out!Good luck!Canadian Math Kangaroo Contest teamPage 1Canadian Math Kangaroo ContestPart A: Each correct answer is worth 3 points1.When the letters of the word MAMA are written vertically above one another, the word has a vertical line of symmetry. Which of these words also has a vertical line of symmetry when written in the same way?(A) ROOT (B) BOOM (C) BOOT (D) LOOT (E) TOOT2.A triangle has sides of length 6, 10 and 11. An equilateral triangle has the same perimeter. What is the length of each side of the equilateral triangle?(A) 6 (B) 9 (C) 10 (D) 11 (E) 273.Which number should replace ∗in the equation 2 ∙ 18 ∙ 14 = 6 ∙ ∗ ∙ 7to make it correct?(A) 8 (B) 9 (C) 10 (D) 12 (E) 154.The panels of Fergus' fence are full of holes. One morning, one of the panels fell flat on the floor.Which of the following could Fergus see as he approaches his fence?(A) (B) (C) (D) (E)5.How many possible routes are there to go from A to B in the direction indicated by the arrows?(A) 2 (B) 3 (C) 4 (D) 5 (E) 66.Martha multiplied two 2-digit numbers correctly on a piece of paper.Then she scribbled out three digits as shown.What is the sum of the three digits she scribbled out? (A) 5 (B) 6 (C) 9 (D) 12 (E) 14 7.A large rectangle is made up of nine identical rectangles whose longest sides are 10 cm long. What is the perimeter of the large rectangle?(A) 40 cm(B) 48 cm(C) 76 cm(D) 81 cm(E) 90 cm8. A hotel on an island in the Caribbean advertises using the slogan "350 days of sun every year!''. According tothe advert, what is the smallest number of days Willi Burn has to stay at the hotel in 2018 to be certain of having two consecutive days of sun?(A) 17 (B) 21 (C) 31 (D) 32 (E) 359.The diagram shows a rectangle of dimensions 7 × 11 containing two circles eachtouching three of the sides of the rectangle. What is the distance between the centres of the two circles?(A) 1 (B) 2(C) 3(D) 4 (E) 510.Only one of the digits in the year 2018 is a prime number. How many years will pass till the next year whenall of the digits in the year number are prime numbers?(A) 201 (B) 202 (C) 203 (D) 204 (E) 205Part B: Each correct answer is worth 4 points11.Square AAAAAAAA has sides of length 3 cm. The points MM and NN lie on AAAA and AAAA so that AAMMand AANN split the square into three pieces of the same area. What is the length of AAMM?(A) 0.5 cm (B) 1 cm (C) 1.5 cm (D) 2 cm (E) 2.5 cm12.A rectangle is divided into 40 identical squares. The rectangle contains more than one row of squares. Avacoloured the middle row. What is the largest possible number of squares that remain uncoloured?(A) 20 (B) 30 (C) 32 (D) 35 (E) 3913.A lion is hidden in one of three rooms. A note on the door of room 1 reads "The lion is here". A note on thedoor of room 2 reads "The lion is not here". A note on the door of room 3 reads "2+3=2×3". Only one of these statements is true. In which room is the lion hidden?(A) In room 1 (B) In room 2 (C) In room 3 (D) It may be in any room(E) It may be in either room 1 or room 214.Valeriu draws a zig-zag line inside a rectangle, creating angles of 10°,14°,33°, and 26°as shown.What is the size of angle θθ?(A) 11°(B) 12°(C) 16°(D) 17°(E)33°。
2017-2018全国大学生数学竞赛试题及答案（最完整版）.pdf
=0
绕 y 轴旋转形成的椭球面的上半
部分（ z ≥ 0 ）取上侧，Π 是 S 在 P ( x, y, z ) 点处的切平面， ρ ( x, y, z ) 是原点到切平面Π
的距离， λ, μ,ν 表示 S 的正法向的方向余弦。计算：
（1）
∫∫
S
ρ
(
z x, y,
z
)
dS
；
（2） ∫∫ z (λx + 3μ y +ν z)dS 。 S 165
L
2
五、（本题满分 10 分）已知 y1 = xex + e2x ， y2 = xex + e−x ， y3 = xe x + e2x − e−x 是某二
阶常系数线性非齐次微分方程的三个解，试求此微分方程。
六、（本题满分 10 分）设抛物线 y = ax2 + bx + 2 ln c 过原点。当 0 ≤ x ≤ 1 时, y ≥ 0 ,又已
2
f (x)dx − 2 , 则 f (x) =
0
；
3．曲面 z = x2 + y2 − 2 平行平面 2x + 2 y − z = 0 的切平面方程是
；
2
4．设函数 y = y(x) 由方程 xe f ( y) = e y ln 29 确定，其中 f 具有二阶导数，且 f ′ ≠ 1 ，则
d2y =
an Snα
收敛；
∑ （2）当α ≤ 1且 sn
→
∞(n
→
∞)
时，级数
+∞ n=1
an Snα
发散。
五、（本题满分 15 分）设 l 是过原点、方向为 (α , β ,γ ) ，（其中α 2 + β 2 + γ 2 = 1) 的直线，
2018年度国外数学竞赛试题翻译汇编
(升级版)
赵力 2019 - 06 - 19
时间，就像高铁，一眨眼，就过站了……
人生很简单 总有一些风景，注定要错过 与其执着，不如随缘 只要懂得“珍惜、知足、感恩”就可以了！
笑看世事繁华，淡定人生心态 不索不可取，不求不可得 学会感恩，做人做事，无憾我心 不再奢望浮华之梦，不再……
v
纯属公益, 免费使用分享, 只送不卖
2018 年亚太地区数学奥林匹克试题
时间，一点不像高铁，过了站，居然买不到回来的车票！
生命，不就如一场雨吗 你曾无知地在其间雀跃，曾痴迷地在其间沉吟 但更多时候 你得忍受那些寒冷与潮湿，那些无奈与寂寞 并且以晴日的幻想来度日
当你握紧双手，里面什么也没有 当你打开双手，世界就在你手中
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目录
2018 年亚太地区数学奥林匹克 ……………………………………… 1 2018 年波罗的海地区数学奥林匹克 ………………………………… 2 2018 年第 10 届 Benelux 数学奥林匹克 ……………………………… 5 2018 年巴尔干地区数学奥林匹克 …………………………………… 6 2018 年巴尔干地区数学奥林匹克预选题…………………………… 7 2018 年巴尔干地区初中数学奥林匹克 ……………………………… 10 2018 年高加索地区数学奥林匹克 …………………………………… 11 2018 年中美洲及加勒比地区数学奥林匹克 ………………………… 13 2018 年 Cono Sur 数学奥林匹克 ……………………………………… 14 2018 年捷克-波兰-斯洛伐克联合数学竞赛 ………………………… 15 2018 年捷克和斯洛伐克数学奥林匹克 ……………………………… 16 2018 年多瑙河地区数学奥林匹克 …………………………………… 17 2018 年欧洲女子数学奥林匹克 ……………………………………… 19 2018 年欧洲数学杯奥林匹克 ………………………………………… 21 2018 年拉丁美洲数学奥林匹克 ……………………………………… 23 2018 年国际大都市数学竞赛(IOM) ………………………………… 24 2018 年第 2 届 IMO 复仇赛 …………………………………………… 25 2018 年第 5 届伊朗几何奥林匹克 …………………………………… 26 2018 年第 17 届基辅数学节竞赛 …………………………………… 30 2018 年地中海地区数学竞赛 ………………………………………… 32 2018 年中欧数学奥林匹克 …………………………………………… 33 2018 年北欧数学奥林匹克 …………………………………………… 35 2018 年泛非数学奥林匹克 …………………………………………… 36 2018 年泛非数学奥林匹克预选题 …………………………………… 38 2018 年罗马尼亚大师杯数学奥林匹克 ……………………………… 42
2018年美国数学竞赛 AMC 试题
2018 AIME I ProblemsProblem 1Let be the number of ordered pairs ofintegers with and such that thepolynomial can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when is divided by .Problem 2The number can be written in base as , can be written inbase as , and can be written in base as , where . Find the base- representation of .Problem 3Kathy has red cards and green cards. She shuffles the cards and laysout of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy,but RRRGR will not. The probability that Kathy will be happy is ,where and are relatively prime positive integers. Find . Problem 4In and . Point lies strictlybetween and on and point lies strictly between and on sothat . Then can be expressed in the form ,where and are relatively prime positive integers. Find .Problem 5For each ordered pair of real numbers satisfyingthere is a real number such thatFind the product of all possible values of .Problem 6Let be the number of complex numbers with the propertiesthat and is a real number. Find the remainder when is divided by .Problem 7A right hexagonal prism has height . The bases are regular hexagons with side length . Any of the vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).Problem 8Let be an equiangular hexagon suchthat , and . Denote the diameter of the largest circle that fits inside the hexagon. Find .Problem 9Find the number of four-element subsets of with the propertythat two distinct elements of a subset have a sum of , and two distinct elements of a subset have a sum of . Forexample, and are two such subsets.Problem 10The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along thepath , which has steps. Let be the number of paths with steps that begin and end at point . Find the remainder when is divided by .Problem 11Find the least positive integer such that when is written in base , its two right-most digits in base are .Problem 12For every subset of , let be the sum of the elements of , with defined to be . If is chosen at random among allsubsets of , the probability that is divisible by is , where and are relatively prime positive integers. Find .Problem 13Let have side lengths , , and .Point lies in the interior of , and points and are the incentersof and , respectively. Find the minimum possible areaof as varies along .Problem 14Let be a heptagon. A frog starts jumping at vertex . From any vertex of the heptagon except , the frog may jump to either of the two adjacentvertices. When it reaches vertex , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than jumps that end at .Problem 15David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius . Let denote the measure of the acute angle made by the diagonals of quadrilateral , and define and similarly. Supposethat , , and . All three quadrilaterals have thesame area , which can be written in the form , where and are relatively prime positive integers. Find .2018 AMC 8 ProblemsProblem 1An amusement park has a collection of scale models, with ratio , of buildings and other sights from around the country. The height of the United States Capitol is 289 feet. What is the height in feet of its replica to the nearest whole number?Problem 2What is the value of the productProblem 3Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?Problem 4The twelve-sided figure shown has been drawn on graph paper. What is the area of the figure in ?Problem 5What is the valueof ?Problem 6On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?Problem 7The -digit number is divisible by . What is the remainder when this number is divided by ?Problem 8Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?Problem 9Tyler is tiling the floor of his 12 foot by 16 foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?Problem 10The of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?Problem 11Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?Problem 12The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins shopping he notes that his car clock and his watch (which is accurate) both say 12:00 noon. When he is done shopping, his watch says 12:30 and his car clock says 12:35. Later that day, Sri loses his watch. He looks at his car clock and it says 7:00. What is the actual time?Problem 13Laila took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?Problem 14Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of ?Problem 15In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of square unit, then what is the area of the shaded region, in square units?Problem 16Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?Problem 17Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distancebetween their houses is miles, which is feet, and Bella covers feet with each step. How many steps will Bella take by the time she meets Ella?Problem 18How many positive factors does have?Problem 19In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?Problem 20In a point is on with and Point ison so that and point is on so that What is the ratio of the area of to the area ofProblem 21How many positive three-digit integers have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11?Problem 22Point is the midpoint of side in square and meets diagonal at The area of quadrilateral is What is the areaofProblem 23From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?Problem 24In the cube with opposite vertices and and are the midpoints of edges and respectively. Let be the ratio of the area of the cross-section to the area of one of the faces of the cube. What isProblem 25How many perfect cubes lie between and , inclusive?2018 AMC 10A ProblemsProblem 1What is the value ofProblem 2Liliane has more soda than Jacqueline, and Alice has more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Liliane has more soda than Alice.Problem 3A unit of blood expires after seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?Problem 4How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)Problem 5Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?Problem 6Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?Problem 7For how many (not necessarily positive) integer values of is the value of an integer?Problem 8Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins?Problem 9All of the triangles in the diagram below are similar to iscoceles triangle , inwhich . Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?Problem 10Suppose that real number satisfies. What is the valueof ?Problem 11When fair standard -sided die are thrown, the probability that the sum of the numbers on the top faces is can be written as, where is a positive integer. What is ?Problem 12How many ordered pairs of real numbers satisfy the following system ofequations?Problem 13A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?Problem 14What is the greatest integer less than or equal toProblem 15Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points and , as shown in the diagram. The distance can be written in the form , where and are relatively prime positive integers. What is ?Problem 16Right triangle has leg lengths and . Including and , how many line segments with integer length can be drawn from vertex to a point on hypotenuse ?Problem 17Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible values of an element inProblem 18How many nonnegative integers can be written in theformwhere for ?Problem 19A number is randomly selected from the set , and a number is randomly selected from . What is the probabilitythat has a units digit of ?Problem 20A scanning code consists of a grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?Problem 21Which of the following describes the set of values of for which thecurves and in the real -plane intersect at exactly points?Problem 22Let and be positive integers suchthat , , ,and . Which of the following must be a divisor of ?Problem 23Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplantedsquare so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?Problem 24Triangle with and has area . Let be the midpointof , and let be the midpoint of . The angle bisectorof intersects and at and , respectively. What is the area of quadrilateral ?Problem 25For a positive integer and nonzero digits , , and , let be the -digit integer each of whose digits is equal to ; let be the -digit integer each of whose digits is equal to , and let be the -digit (not -digit) integer each of whose digits is equal to . What is the greatest possiblevalue of for which there are at least two values of such that ?2018 AMC 10B ProblemsProblem 1Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?Problem 2Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?Problem 3In the expression each blank is to be filled in with one of the digits or with each digit being used once. How many different values can be obtained?Problem 4A three-dimensional rectangular box with dimensions , , and has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is ?Problem 5How many subsets of contain at least one prime number?Problem 6A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required?Problem 7In the figure below, congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let be the combined area of the small semicircles and be the area of the region inside the large semicircle but outside the small semicircles. The ratio is 1:18. What is ?Problem 8Sara makes a staircase out of toothpicks as shown:This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks?Problem 9The faces of each of 7 standard dice are labeled with the integers from 1 to 6. Let be the probability that when all 7 dice are rolled, the sum of the numbers on the top faces is 10. What other sum occurs with the same probability ?Problem 10In the rectangular parallelepiped shown, , , and . Point is the midpoint of . What is the volume of the rectangular pyramid with base and apex ?Problem 11Which of the following expressions is never a prime number when is a prime number?Problem 12Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?Problem 13How many of the first numbers in the sequence are divisible by ?Problem 14A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?Problem 15A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length and height . What is the area of the sheet of wrapping paper?Problem 16Let be a strictly increasing sequence of positive integers suchthat What is the remainderwhen is divided by ?Problem 17In rectangle , and . Points and lie on ,points and lie on , points and lie on , and points and lie on so that and the convex octagon is equilateral. The length of a side of this octagon can be expressed in the form , where , , and are integers and is not divisible by the square of any prime. What is ?Problem 18Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?Problem 19Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?Problem 20A function is defined recursivelyby and for allintegers . What is ?Problem 21Mary chose an even -digit number . She wrote down all the divisors of in increasing order fromleft to right: . At some moment Mary wrote as a divisor of . What is the smallest possible value of the next divisor written to the right of ?Problem 22Real numbers and are chosen independently and uniformly at random from the interval . Which of the following numbers is closest to the probability that and are the side lengths of an obtuse triangle?Problem 23How many ordered pairs of positive integers satisfy theequation where denotes the greatest common divisor of and , and denotes their least common multiple?Problem 24Let be a regular hexagon with side length . Denote by , , and the midpoints of sides , , and , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of and ?Problem 25Let denote the greatest integer less than or equal to . How many real numbers satisfy the equation ?2018 AMC 12A ProblemsProblem 1A large urn contains balls, of which are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be ? (No red balls are to be removed.)Problem 2While exploring a cave, Carl comes across a collection of -pound rocks worth each, -poundrocks worth each, and -pound rocks worth each. There are at least of each size. He can carry at most pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?Problem 3How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)Problem 4Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of ?Problem 5What is the sum of all possible values of for which the polynomials andhave a root in common?Problem 6For positive integers and such that , both the mean and the median ofthe set are equal to . What is ?Problem 7For how many (not necessarily positive) integer values of is the value of an integer?Problem 8All of the triangles in the diagram below are similar to iscoceles triangle , in which. Each of the 7 smallest triangles has area 1, and has area 40. What is the area of trapezoid ?Problem 9Which of the following describes the largest subset of values of within the closed interval forwhich for every between and , inclusive?How many ordered pairs of real numbers satisfy the following system of equations?Problem 11A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?Problem 12Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element inProblem 13How many nonnegative integers can be written in the formwherefor ?Problem 14The solutions to the equation , where is a positive real number other thanor , can be written as where and are relatively prime positive integers. What is ?A scanning code consists of a grid of squares, with some of its squares colored black and therest colored white. There must be at least one square of each color in this grid of squares. A scanning code is called if its look does not change when the entire square is rotated by a multiple of counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?Problem 16Which of the following describes the set of values of for which the curves andin the real -plane intersect at exactly points?Problem 17Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted squareso that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?Triangle with and has area . Let be the midpoint of, and let be the midpoint of . The angle bisector of intersects and atand , respectively. What is the area of quadrilateral ?Problem 19Let be the set of positive integers that have no prime factors other than , , or . The infinite sumof the reciprocals of the elements of can be expressed as , where and are relatively primepositive integers. What is ?Problem 20Triangle is an isosceles right triangle with . Let be the midpoint ofhypotenuse . Points and lie on sides and , respectively, so thatand is a cyclic quadrilateral. Given that triangle has area , the length canbe written as , where , , and are positive integers and is not divisible by the square of any prime. What is the value of ?Problem 21Which of the following polynomials has the greatest real root?Problem 22The solutions to the equations and whereform the vertices of a parallelogram in the complex plane. The area of thisparallelogram can be written in the form where and are positive integersand neither nor is divisible by the square of any prime number. What isProblem 23In and Points and lie on sidesand respectively, so that Let and be the midpoints of segmentsand respectively. What is the degree measure of the acute angle formed by linesandProblem 24Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all thenumbers between and Armed with this information, what number should Carol choose to maximize her chance of winning?Problem 25For a positive integer and nonzero digits , , and , let be the -digit integer each of whosedigits is equal to ; let be the -digit integer each of whose digits is equal to , and let bethe -digit (not -digit) integer each of whose digits is equal to . What is the greatest possible value of for which there are at least two values of such that ?2018 AMC 12B ProblemsProblem 1Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?Problem 2Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?Problem 3A line with slope 2 intersects a line with slope 6 at the point . What is the distance between the -intercepts of these two lines?Problem 4A circle has a chord of length , and the distance from the center of the circle to the chord is . What is the area of the circle?Problem 5How many subsets of contain at least one prime number?Suppose cans of soda can be purchased from a vending machine for quarters. Which of the following expressions describes the number of cans of soda that can be purchased for dollars, where 1 dollar is worth 4 quarters?Problem 7What is the value ofProblem 8Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?Problem 9What isProblem 10A list of positive integers has a unique mode, which occurs exactly times. What is the least number of distinct values that can occur in the list?Problem 11。
高中数学竞赛分类：集合-1981-2018年历年数学联赛48套真题分类汇编含详细答案
1981年--2018年全国高中数学联赛一试试题分类汇编1、集合部分2018A1、设集合{}99,,3,2,1 =A ，集合{}A x x B ∈=|2，集合{}A x x C ∈=2|，则集合C B 的元素个数为24★解析：由条件知，{}48,,6,4,2 =C B ，故C B 的元素个数为24。
2018B1、设集合{}8,1,0,2=A ，集合{}A a a B ∈=|2，则集合B A 的所有元素之和是◆答案：31★解析:易知{}16,2,0,4=B ，所以{}16,8,4,2,1,0=B A ，元素之和为31.2018B 三、（本题满分50分）设集合{}n A ,,2,1 =，Y X ,均为A 的非空子集（允许Y X =）．X 中的最大元与Y 中的最小元分别记为Y X min ,max ．求满足Y X min max >的有序集合对),(Y X 的数目。
★解析:先计算满足Y X min max ≤的有序集合对),(Y X 的数目.对给定的X m max =，集合X 是集合{}1,,2,1-m 的任意一个子集与{}m 的并，故共有12-m 种取法.又Y m min ≤，故Y 是{}n m m m ,,2,1, ++的任意一个非空子集，共有121--+m n 种取法.因此，满足Y X min max ≤的有序集合对),(Y X 的数目是：()[]()12122122111111+⋅-=-=-∑∑∑=-==-+-n nm m n m n nm mn m n 由于有序集合对),(Y X 有()()()2121212-=--nnn个，于是满足Y X min max >的有序集合对),(Y X 的数目是()()124122122+-=-+⋅--n n n n n n n 2017B 二、（本题满分40分）给定正整数m ，证明：存在正整数k ，使得可将正整数集+N 分拆为k 个互不相交的子集k A A A ,,,21 ，每个子集i A 中均不存在4个数d c b a ,,,（可以相同），满足m cd ab =-．★证明：取1k m =+，令{(mod 1),}i A x x i m x N +=≡+∈，1,2,,1i m =+ 设,,,i a b c d A ∈，则0(mod 1)ab cd i i i i m -≡∙-∙=+，故1m ab cd +-，而1m m +，所以在i A 中不存在4个数,,,a b c d ，满足ab cd m-=2017B 四、（本题满分50分）。
(完整版)2018年六年级数学竞赛试题及答案,推荐文档
2018年度六年级数学才艺展示题一、填空：（ 前7题每题5分，后3题每题6分，共53分 ）1、如果x ÷y=z （x 、y 、z 均为整数，且y 不等于0），那么x 和y 的最大公因数是（ y ），最小公倍数是（ x ）。
2、已知x+=y+=z+，( z )＜( x ) ＜( y )2014201320132012201520143、☆、○、◎各代表一个数，已知:☆+◎=46, ☆+○=91, ○+◎=63 , ☆=(37 ),○=( 54 )◎= （ 9 ）。
4、学校买来历史、文艺、科普三种图书各若干本，每个学生从中任意借两本。
那么，至少（ 7 ）个学生中一定有两人所借的图书属于同一种。
5、李伟和王刚两人大学毕业后合伙创业，李伟出资1.6万元，王刚出资1.2万元，一年后盈利1.4万，如果按照出资多少来分配利润，李伟分得（ 8000 ）元，王刚分得（ 6000 ）元。
6、某商场由于节日效应一月份的营业额是150万元，二月份的营业额延续节日需求，比一月份增长了10%，三月份和一月份相比增长率为－9%，一季度营业额（ 451.5 ）万元。
7、庆“六一”，学校决定进行现场绘画比赛吗，按照如下摆放桌子和椅子，如果每个椅子坐一位同学，1张桌子可以坐6人，2张桌子可以10人，……，n 张桌子可以做（ 4n+2 ）人。
如果像这样摆20张桌子，最多可以坐（ 82 ）人。
8、数学小组的同学在一次数学比赛中成绩统计如左下图。
如果得优良和及格的同学都算达标。
达标同学的平均成绩是80分，而全体同学的平均成绩是70分，则不及格同学的平均成绩（ 40 ）分。
9、如右上图，已知长方形的面积是28,阴影部分的面积（9.44 ）。
2cm 2cm 10、“重阳节”那天，延龄茶社来了25位老人品茶。
他们的年龄恰好是25个连续自然数，两年以后，这25位老人的年龄之和正好是2000。
其中年龄最大的老人今年（ 90 ）岁。
二、用自己喜欢的方法计算：（每题5分，共15分）1、0.78×7－＋4×2、12.5×8÷12.5×850395039（7） （64）543、（88－）×＋（78－）×＋（68－）×＋……＋（18－）×8181818181818181（ 52）87三、应用题：（每题8分，共32分）1、中国北部地区严重缺水，节约用水是美德，某地生活用水收费标准规定如下：用水数X （吨）X≤77＜X≤10X≥10价格／吨（元） 2.4 3.2 3.6已知大伟家在本月应交水费33.6元，算一算他家这个月用了多少吨水？（12吨）2、王大妈买了一套售价为32万元的普通商品房。
(完整版)(完整版)2018年(第59届)国际数学奥林匹克(IMO)竞赛试题及答案图片版
岁马尼亚克卢日蜻沐卡第一天«1. itΓ<HΛ三角砒4〃C的外44圈・点D和EAru殳/CAC上∙^nAD ≈ AEφ BI)^CE的•克羊分线⅛Γ上劣弧AB AC分別文于点FG im ADE⅜FG1 ⅛A÷*t•⅛ 2.求所有的整4⅛□23∙便俗存在实软5皿2.・・・.<¼+2∙滿足"*ι = <M∙ 5∙2 Ua2异且<≡∙<<∙⅛1 + 1 = α∣÷3— 1.2. - - ■” 戍立・題3・反忖斷卡三蔦砒是由铁俎戎的一个正三角外障•港足除了鬟下方一行.孕个敦是它下方相你两金铁之屋的绘对值•例*\下而是一金四忡的反恤浙卡三角耐・由Hl MlO tt⅛.42 65 7 18 3 10 9请MΛ5 4Λ2018fτ的反帕浙卡三 E 包含IMl +2十・∙∙ + 2018所亦的蹩典？鈿二夭« 4.我们呀谓一个（IJL是斯d角坐栋丰而上的一个A（X.,V）∙乳中工・"需足不雄述20的正史软.最初时•所有400个位豆那是空的.甲乙两人轮濃霖放石子•由甲先遗ft∙毎次伦刘甲时.他41 一个空的住I±Λ±-¼*的化也若子•要求任急两金红己石子舸息<1 Jt之问的距离都不#于％・每次伦刘乙片•他/1任直一个空的CiJt上崔上一个M6⅛2Lt>&子.（Jl色石子所在位直与戻它石于所在位直之问雎禹可以是任倉值・）4此UAitfTT去直至某金人无法再霖放石子•试确岌遥大的位再无论乙知何报就這色若予.Y⅛*Ef⅛Ui∙>∙4X⅛K个红已若子・« 5. Ha i.a2.…走一个>LfPil正整软斥列.已知4在於敦N>l∙使碍对每个^Kn > .V t Oi i o2 . I Q*1“ I OH――+ — + ・• • + ・■■■・ + —。
2018年全国高中数学联赛试题及答案详解(B卷)_PDF压缩
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2018年世界少年奥林匹克数学竞赛八年级海选赛试题含答案
绝密★启用前世界少年奥林匹克数学竞赛（中国区）选拔赛地方海选赛选手须知：1、本卷共三部分，第一部分：填空题，共计50分；第二部分：计算题，共计12分；第三部分：解答题，共计58分。
2、答题前请将自己的姓名、学校、赛场、参赛证号码写在规定的位置。
3、比赛时不能使用计算工具。
4、比赛完毕时试卷和草稿纸将被收回。
八年级试题（Ａ卷）（本试卷满分120分 ，考试时间90分钟 ）一、填空题。
（每题5分，共计50分）1、36的平方根是 。
2、若方程mx+ny=6的两个解是⎩⎨⎧==11y x 及⎩⎨⎧-==12y x ，则m= ，n = 。
3、已知1=-b a ，=+=+b a b a ,2522。
4、已知x=y+z=2，则=+++xyz z y x 333223。
5、如果实数a ，b 满足条件,12|21|,12222a b a b a b a -=+++-=+则a+b= 。
6、某班级春游时48人到杭州西湖划船。
已知每只小船坐3个人，租金16元；每只大船坐5个人，租金24元，则这个班级租金至少花 元。
7、在△ABC 中，∠B=60°，∠C ＞∠A ，且222B A )C (）（）（∠+∠=∠,则△ABC 的形状是 。
8、观察下列式子：181092+⨯=；198100992+⨯=；199810009992+⨯=，……，按规律写出=2999999 。
（填写具体数字）9、如图，韩梅梅从A 点出发，沿直线前进10米后向左转30°，再沿直线前进10米，又向左转30°，照这样子走下去，他第一次回到起点A 时走了 米。
10、如图直线L 与∠A 的两边相交于点B 、C ，则图中以A 、B 、C 为端点的射线有 条。
二、计算题。
（每题6分，共计12分）11、 1+3+5+7+9+…+2017+201912、 1+5+52+53+…+5100省 市 学校 姓名 赛场 参赛证号∕∕∕∕∕〇∕∕∕∕∕∕〇∕∕∕∕∕∕∕〇∕∕∕∕∕∕ 密 〇 封 〇 装 〇 订 〇 线 ∕∕∕∕∕∕〇∕∕∕∕∕∕〇∕∕∕∕∕∕〇∕∕∕∕∕∕〇∕∕∕∕∕∕密 封 线 内 不 要 答 题三、解答题。
三角-1981-2018年历年数学联赛48套真题WORD版分类汇编含详细答案.pdf
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目 录2018年亚太地区数学奥林匹克 (1)2018年波罗的海地区数学奥林匹克 (2)2018年第10届Benelux数学奥林匹克 (5)2018年巴尔干地区数学奥林匹克 (6)2018年巴尔干地区初中数学奥林匹克 (7)2018年高加索地区数学奥林匹克 (8)2018年中美洲及加勒比地区数学奥林匹克 (10)2018年Cono Sur数学奥林匹克 (11)2018年捷克-波兰-斯洛伐克联合数学竞赛 (12)2018年捷克和斯洛伐克数学奥林匹克 (13)2018年多瑙河地区数学奥林匹克 (14)2018年欧洲女子数学奥林匹克 (16)2018年欧洲数学杯奥林匹克 (18)2018年拉丁美洲数学奥林匹克 (20)2018年国际大都市数学竞赛(IOM) (21)2018年第2届IMO复仇赛 (22)2018年第5届伊朗几何奥林匹克 (23)2018年第17届基辅数学节竞赛 (27)2018年地中海地区数学竞赛 (29)2018年中欧数学奥林匹克 (30)2018年北欧数学奥林匹克 (32)2018年泛非数学奥林匹克 (33)2018年罗马尼亚大师杯数学奥林匹克 (35)2018年第14届Sharygin几何奥林匹克 (36)2018年丝绸之路数学奥林匹克 (42)2018年Tuymaada国际数学奥林匹克 (43)2018年乌克兰几何奥林匹克 (45)2018年第14届Zhautykov国际数学奥林匹克 (47)2018年ARML数学竞赛 (48)2018年美国数学邀请赛(AIME) I (57)2018年美国数学邀请赛(AIME) II (60)2018年美国数学奥林匹克 (63)2018年美国初中数学奥林匹克 (64)2018年美国IMO代表队选拔考试 (65)2018年美国TSTST (67)2018年美国第20届ELMO (69)2018年第20届美国旧金山湾区数学奥林匹克 (71)2017-2018年度USAMTS (74)2018年美国女子数学奖学金竞赛(决赛) (79)2017-2018年度威斯康星数学、科学与工程学人才选拔 (80)2018年奥地利数学奥林匹克 (84)2018年澳大利亚、英国IMO国家队联合训练考试 (87)2018年波黑数学奥林匹克(地区级) (88)2018年波黑EGMO代表队选拔考试 (90)2018年波黑JBMO代表队选拔考试 (91)2018年巴西数学奥林匹克 (92)2018年巴西数学奥林匹克复仇赛 (94)2017/2018英国数学竞赛 (95)2018年保加利亚数学奥林匹克 (97)2018年保加利亚JBMO代表队选拔考试 (98)2018年加拿大数学奥林匹克 (99)2018年塞浦路斯IMO代表队选拔考试 (100)2018年塞浦路斯JBMO代表队选拔考试 (102)2018年丹麦数学奥林匹克(第二轮) (105)2018年德国数学奥林匹克(12年级决赛) (106)2018年希腊数学奥林匹克 (107)2018年香港数学奥林匹克 (109)2018年香港IMO代表队选拔考试 (110)2018年匈牙利库尔沙克数学竞赛 (112)2018年印度全国数学奥林匹克 (113)2018年印度IMO代表队选拔考试 (114)2018年伊朗数学奥林匹克 (117)2018年伊朗IMO代表队选拔考试 (120)2018年爱尔兰数学奥林匹克 (123)2018年意大利数学奥林匹克 (125)2018年哈萨克斯坦数学奥林匹克(11年级决赛) (126)2018韩国数学奥林匹克 (127)2018年韩国数学冬令营训练题 (130)2018年科索沃IMO培训考试 (131)2018年马其顿数学奥林匹克 (132)2018年墨西哥数学奥林匹克 (133)2018年摩尔多瓦EGMO代表队选拔考试 (135)2018年摩尔多瓦IMO代表队选拔赛 (136)2018年摩尔多瓦JBMO代表队选拔考试 (138)2018年摩洛哥IMO代表队选拔考试 (139)2017-2018年度挪威数学奥林匹克(决赛) (140)2017-2018年度波兰数学奥林匹克 (141)2017-2018年度波兰初中数学奥林匹克 (145)2018年罗马尼亚数学奥林匹克 (147)2018年罗马尼亚IMO代表队选拔考试 (149)2018年罗马尼亚JBMO代表队选拔考试 (150)2018年全俄数学奥林匹克 (154)2018年圣彼得堡数学奥林匹克 (158)2018年塞尔维亚数学奥林匹克 (161)2018年塞尔维亚JBMO代表队选拔考试 (162)2018年斯洛文尼亚IMO代表队选拔考试 (163)2018年南非数学奥林匹克 (164)2018年西班牙数学奥林匹克 (165)2018年塔吉克斯坦IMO代表队选拔考试 (166)2018年土耳其数学奥林匹克 (168)2018年乌克兰数学奥林匹克 (169)2018年越南数学奥林匹克 (171)2018年越南IMO代表队选拔考试 (173)2018年国际大学生数学竞赛(IMC) (175)2018年V ojtěch Jarník国际大学生数学竞赛 (177)2018年Putnam数学竞赛 (179)2018年哈佛大学-麻省理工学院数学竞赛春季赛 (181)2018年哈佛大学-麻省理工学院数学邀请赛 (189)2018年哈佛大学-麻省理工学院数学竞赛冬季赛 (190)2018年Berkeley数学竞赛 (197)2018年卡内基梅隆大学数学竞赛 (213)2018年普林斯顿大学数学竞赛 (226)2018年斯坦福大学数学竞赛 (237)2018年哈维穆德学院数学竞赛 (254)2018年MMATHS数学竞赛 (259)2018年Duke大学数学竞赛 (264)2018年亚太地区数学奥林匹克试题比赛时间: 2018年3月13日1. 设H 为△ABC 的垂心. 点M , N 分别为边AB , AC 的中点, 点H 位于四边形BMNC 的内部. △BMH 与△CNH 的外接圆相外切. 过H 作BC 的平行线, 与△BMH 与△CNH 的外接圆分别相交于点K , L (均不同于点H ). 直线MK 与NL 相交于点F . 设△MNH 的内心为J . 证明: FJ = F A .2. 对实数x , 定义函数f (x ), g (x )如下:20181...41211)(-++-+-+=x x x x x f , 20171...513111)(-++-+-+-=x x x x x g . 证明: 对任意满足0 < x < 2018的非整数的实数x , 有|f (x ) – g (x )| > 2成立.3. 我们称平面上n 个正方形的摆放方式为"三足鼎立"的, 如果它们同时满足以下三个条件:i) 所有正方形均全等.ii) 如果两个正方形有公共点P , 则P 同时为这两个正方形的顶点.iii) 每一个正方形都恰好与其他三个正方形有公共点.求在2018 ≤ n ≤ 3018范围内, 有多少个整数n , 使得存在n 个正方形为"三足鼎立"的.4. 一束光线从正△ABC 的顶点A 出发, 在三角形内部遵循光反射定律(即入射角等于出射角)不断反射, 但当光线到达△ABC 的任一顶点处时, 反射停止. 求所有可能的正整数n , 使得光线在△ABC 内经过n 次反射后, 恰在顶点A 处停止.5. 求所有的整系数多项式P (x ), 使得对任意的实数s , t , 如果P (s ), P (t )均为整数, 则P (st )也是整数.2018年波罗的海地区数学奥林匹克试题1. 称一个由有限个正实数(不必互异)构成的集合为"平衡"的, 如果其中每一个数都小于其余各数之和. 求所有的整数m ≥ 3, 使得任何由m 个正实数构成的平衡集均可被划分为3个无公共元素的子集, 满足每个子集的各元素之和均小于另两个子集的各元素的总和.2. 考虑一个100 ⨯ 100的表格. 对每一个整数1 ≤ k ≤ 100, 该表格的第k 行填有按自左向右递增顺序排列的数1, 2, …, k (但不一定位于连续的格子内); 而该行其余的100 – k 个格子均填0. 证明: 该表格中存在两列, 使得其中一列的各数之和至少是另一列各数之和的19倍.3. 设正实数a , b , c , d 满足abcd = 1. 证明:110321≤+++∑cyc c b a . 4. 求所有具有下述特点的函数f : [0, ∞) → [0, ∞): 对所有的正整数n 及非负实数x 1, x 2, …, x n , 有2222122221)(...)()()...(n n x f x f x f x x x f +++=+++成立. 5. 称一个实系数多项式f (x )为"生成"的, 如果对每一个实系数多项式ϕ(x ), 均存在正整数k 及实系数多项式g 1(x ), g 2(x ), …, g k (x ), 使得ϕ(x ) = f (g 1(x )) + f (g 2(x )) + … + f (g k (x ))成立. 求所有的生成多项式.6. 设n 为正整数. 精灵Elfie 从原点(0, 0, 0)开始, 在三维空间里旅行. 每一步, 她可以瞬移至距她当前所在点距离恰为n 的任意整点. 但是, 瞬移是一件复杂的事情: Elfie 最初处于正常状态, 但是第一次瞬移后变为怪异状态, 第二次瞬移后恢复为正常状态, 以后则如此交替变化. 求所有的n , 使得对所有整点, Elfie 都能够以正常状态访问过该点.7. 一个16 ⨯ 16圆环体有512条边(如图), 将每条边染为红色或蓝色之一. 称一种染色方式为"好"的, 如果每一个顶点都是偶数条红色边的顶点. 定义一步"转换"为将任一格的四条边均改变颜色(红变蓝, 蓝变红). 问最多有多少个"好"的染色方式, 使得其中任意一个染色方式都不能够通过一系列的"转换"而变为另一个.8. 一个图具有N个顶点. 在某一顶点处有一只不可见的兔子. 一群猎人计划猎杀这只兔子. 在每一步, 每个猎人都瞄准某一个顶点同时开枪射击, 他们可以事先商量好每人瞄准哪一个顶点. 如果兔子恰在被瞄准射击的顶点之一, 则打猎活动结束. 否则, 兔子在接下来的一步中可以选择继续停留在原顶点处或跳至某个相邻顶点处. 假设已知有一种方案可以使猎人至多经N!步就可以猎杀兔子. 证明: 存在一种方案, 可以使得猎人至多经2N步就可以猎杀兔子.9. Olga和Sasha在一个无限六边形网格上玩游戏. 他们轮流选择一个空的六边形,并在其上放置一张骨牌, 由Olga先行. 恰在第2018张骨牌放置之前, 一条新规则开始起效: 从此时起, 只能在和至少两个已被放置骨牌的六边形相邻的空六边形上放置骨牌. 如果一个玩家无法继续放置骨牌, 或者放置骨牌后会出现呈菱形分布的四个相邻六边形均被放置骨牌的情况(如图所示, 但方向可以不同), 则判该玩家输. 确定是否某个玩家有获胜策略; 如果有, 赢家是谁?10. 将整数1, 2, …, n写在n张卡片上, 每张上写一个不同的数. 首先, 由玩家1取走一张卡片. 接下来, 玩家2取走写有连续正整数的两张卡片. 然后, 再由玩家1取走写有连续正整数的三张卡片. 最后, 由玩家2取走写有连续正整数的四张卡片. 求最小的n, 使得玩家2能确保完成他的两次取卡片的操作. 11. 给定一圆w及圆上依A, B, C, D顺序排列的四点, 且AD为圆w的直径. 假设AB = BC = a , CD = c , 其中a 和c 为互质正整数. 证明: 如果圆w 的直径长d 也是正整数, 则d 及2d 中必有一个完全平方数.12. 锐角△ABC 的高BB 1, CC 1相交于点H . 点B 2, C 2分别位于线段BH , CH 上, 且BB 2 = B 1H , CC 2 = C 1H . △B 2HC 2的外接圆与△ABC 的外接圆相交于点D 和E . 证明: △DEH 为直角三角形.13. 在△ABC 中, ∠A 的内角平分线与直线BC 交于点D , 与△ABC 的外接圆交于点E . 设K , L , M , N 分别为线段AB , BD , CD , AC 的中点. 点P , Q 分别为△EKL , △EMN 的外心. 证明: ∠PEQ = ∠BAC .14. 设四边形ABCD 有内切圆w . 令圆w 与AC 的交点中较靠近点A 的那个为E . 设F 为E 关于圆w 的对径点. 经点F 作圆w 的切线, 分别交直线AB , BC 于A 1, C 1, 并与直线AD , CD 分别交于A 2, C 2. 证明: A 1C 1 = A 2C 2.15. 考虑平面内相离的两个圆. 分别选取两个圆的直径A 1B 1和A 2B 2, 使得线段A 1A 2与B 1B 2相交于点C . 设A 1A 2, B 1B 2的中点分别为A , B . 证明: 不管如何选取直径A 1B 1和A 2B 2, △ABC 的垂心总位于一条直线上.16. 设p 为奇质数. 求所有的正整数n , 使得np n -2为正整数.17. 证明: 对所有满足q p >11的正整数p , q , 不等式pqq p 2111>-成立. 18. 设整数n ≥ 3满足4n + 1为质数. 证明: 4n + 1整除12-n n .19. 设无限正整数集合B 满足以下条件: 对任意的a , b ∈ B 且a > b , 有),gcd(b a b a - ∈ B . 证明: B 是由所有正整数构成的集合.20. 求所有的正整数(a , b , c ), 使得ba c a cbc b a 444)()()(+++++为整数, 且a + b + c 为质数.2018年第10届Benelux 数学奥林匹克试题比赛时间: 2018年4月28日1. a) 设x , y 为正实数. 求⎪⎭⎫ ⎝⎛-+⎪⎭⎫ ⎝⎛++⎪⎪⎭⎫ ⎝⎛-+⎪⎪⎭⎫ ⎝⎛+201811201811x y x y y x y x 的最小值. b) 设x , y 为正实数. 求⎪⎭⎫ ⎝⎛++⎪⎭⎫ ⎝⎛++⎪⎪⎭⎫ ⎝⎛++⎪⎪⎭⎫ ⎝⎛+201811201811x y x y y x y x 的最小值. 2. 在七星岛上, 共使用4种不同的硬币和3种不同的纸币, 它们的面额分别为7个不同的正整数, 且最小额纸币的面额大于4种不同硬币的面额之和. 一位游客恰好有不同面额的硬币各1枚及不同面额的纸币各1张, 但是这些钱的总额不够支付他想购买的一本关于钱币学的书. 幸运的是, 爱好数学的书店老板同意将此书按这位游客所提出的价格卖给他, 但前提是该游客可以用超过一种方式支付此价格.(游客可以用超过一种方式支付某价格, 指的是在由他的硬币与纸币构成的集合中, 存在两个不同的子集, 每个子集里钱的面额之和均等于该价格.) a) 证明: 如果每张纸币的面额均小于49, 则该游客能够购买到这本书. b) 证明: 如果最大额纸币的面额等于49, 则该游客有可能空手而归.3. 设H 为三角形ABC 的垂心, D , E , F 分别为AB , AC , AH 的中点. 点B , C 关于点F 的对称点分别为P , Q .a) 证明: 直线PE 和QD 的交点位于三角形ABC 的外接圆上.b) 证明: 直线PD 和QE 的交点位于AH 上.4. 我们称一个恰有s 个正因数1 = d 1 < d 2 < … < d s = n 的正整数n > 2为"好"的, 如果存在整数2 ≤ k ≤ s , 满足d k > 1 + d 1 + … + d k –1.如果一个整数n > 2不是"好"的, 则称之为"坏"的.a) 证明: 存在无穷多个"坏"的整数.b) 证明: 在均大于2的任意7个连续整数中, 至少有4个整数为"好"的. c) 证明: 存在无穷多个由连续7个整数构成的序列, 其中每个序列里的数都是"好"的.2018年巴尔干地区数学奥林匹克试题比赛时间: 2018年5月9日1. 凸四边形ABCD内接于圆k, 其中AB > CD, 且AB不平行于CD. 点M为对角线AC与BD的交点, 自M作AB的垂线, 与AB相交于点E.如果EM平分 CED, 证明: AB为圆k的一条直径.2. 设q为正有理数. 两只蚂蚁最初均位于平面上的同一点X处. 在第n分钟(n = 1, 2, …), 每只蚂蚁各自在北, 东, 南, 西四个方向中选择一个方向, 并沿此方向移动q n米. 经过整数个分钟后, 它们再次位于平面上的同一点处(不一定是点X), 但是在此时间段内它们的移动路径并不完全相同.求q所有可能的取值.3. Alice和Bob一起玩如下的移硬币游戏. 他们从两堆均非空的硬币开始, 首先由Alice开始, 轮流进行以下操作: 该轮玩家选择数目为偶数的一堆硬币, 将该堆的一半硬币移到另一堆里. 如果某位玩家无法进行上述操作, 则游戏结束, 并判对方获胜.求所有的正整数对(a, b), 使得如果最初两堆分别有a和b枚硬币, 则Bob有获胜策略.4. 求所有的素数对(p, q), 使得3p q–1 + 1整除11p + 17p.2018年巴尔干地区初中数学奥林匹克(JBMO)试题比赛时间: 2018年6月21日1. 求满足方程m 5 – n 5 = 16mn 的所有整数(m , n ).2. 设有n 个三位正整数同时满足以下条件:i) 每个数均不含数字0;ii) 每个数的数字和为9;iii) 任意两个数的个位数均不同;iv) 任意两个数的十位数均不同;v) 任意两个数的百位数均不同.求n 的最大可能值.3. 设k > 1为正整数, n > 2018为正奇数. 不全相等的非零有理数x 1, x 2, …, x n 满足如下关系式:11433221...x k x x k x x k x x k x x k x n n n +=+==+=+=+-. i) 求乘积n x x x ⋅⋅⋅...21(用关于k 和n 的函数表示).ii) 求最小的k , 使得存在满足所给条件的n , x 1, x 2, …, x n .4. 设A', B', C'分别为△ABC 的顶点A , B , C 关于对边的对称点. △ABB'的外接圆交△ACC'的外接圆于A 1 (A 1 ≠ A ). 类似定义点B 1和C 1. 证明: AA 1, BB 1, CC 1三线共点.2018年高加索地区数学奥林匹克试题初级组第1天(2018年3月16日)1. 设a, b, c为不全为0的实数. 证明: a + b + c = 0的充分必要条件为a2 + ab + b2 = b2 + bc + c2 = c2 + ca + a2.2. 在8 ⨯ 8国际象棋棋盘上, 放置了n > 6只马, 使得对其中任意6只马, 均存在2只马可以互相攻击. 求n的最大可能值.3. 正整数a, b, c满足条件: a b整除b c, a c整除c b. 证明: a2整除bc.4. 我们定义四边形的重心为连接对边中点的两条直线的交点. 设六边形ABCDEF内接于以O为圆心的圆Ω, 且AB = DE, BC = EF. 令X, Y, Z分别为四边形ABDE, BCEF, CDFA的重心. 证明: O为△XYZ的垂心.第2天(2017年3月17日)5. Munсhausen男爵发现了如下"定理": 对任意正整数a和b, 总存在正整数n, 使得an为完全平方数, 而bn为完全立方数. 请确定该男爵的"定理"是否正确.6. 凸四边形ABCD中, ∠BCD = 90o, E为AB的中点. 证明: 2EC≤AD + BD.7. 给定正整数n > 1. 考虑一个n⨯n棋盘. 最初棋盘上没有玻璃球, 按照以下规则逐个地往棋格里放入玻璃球: 如果一个空棋格与至少2个空棋格相邻(指有一公共边), 则该棋格内可以放入一个玻璃球. 问在此规则下, 棋盘内最多可以放入多少个玻璃球?8. 设a, b, c为一个三角形的三边长. 证明:2)()()()(2cbacaacbccbabba++≥+++++.高级组第1天(2018年3月16日)1. 给定一个四面体. 是否能够将10个连续正整数分别放置在该四面体的四个顶点及六条棱的中点上, 使得每条棱中点上的数等于该棱两端点上的数的算术平均值?2. 设I为锐角△ABC的内心. 点P, Q, R分别在边AB, BC, CA上, 满足AP = AR, BP = BQ, ∠PIQ = ∠BAC. 证明: QR⊥AC.3. 我们称2n个正整数的一个匹配(即分成n对)为"非平方"的, 如果每一对中的2个数之积均不是完全平方数. 证明: 如果存在一个"非平方"匹配, 则至少存在n!个"非平方"匹配.4. Morteza在n⨯n棋盘的每一个棋格内放置一个[0, 1] → [0, 1]的函数(即定义域为[0, 1], 值域为[0, 1]的函数). Pavel计划在棋盘每一行的左边及每一列的下边分别放置一个[0, 1] → [0, 1]的函数(共放置2n个函数), 使得棋盘的每一格均满足以下条件:如果h为该棋格内的函数, f为该棋格所在列下边的函数, g为该棋格所在行左边的函数, 则h(x) = f(g(x))对所有的x∈ [0, 1]成立.证明: Pavel总是可以实现他的计划.第2天(2018年3月17日)5. Munсhausen男爵发现了如下"定理": 对任意正整数a和b, 总存在正整数n, 使得an为完全立方数, 而bn为完全五次方数. 请确定该男爵的"定理"是否正确.6. 在坐标平面内, 两个二次多项式的图像G1, G2的交点为A, B. 设O为G1的顶点. 直线OA, OB分别与G2再次相交于点C, D. 证明: CD平行于x轴.7. 锐角△ABC中, 经过顶点A, B, C的高分别交对边于A1, B1, C1, 并分别交△ABC的外接圆于A2, B2, C2. 直线A1C1分别交△AC1C2, △CA1A2的外接圆于点P, Q (P≠C1, Q≠A1). 证明: △PQB1的外接圆与AC相切.8. 考虑一个8 ⨯8棋盘. 最初棋盘上没有玻璃球, 按照以下规则逐个地往棋格里放入玻璃球: 如果一个空棋格与至少3个空棋格相邻(指有一公共边), 则该棋格内可以放入一个玻璃球. 问在此规则下, 棋盘内最多可以放入多少个玻璃球?2018年中美洲及加勒比地区数学奥林匹克试题第1天1. 在2018张卡片上分别标记数1, 2, …, 2018, 每张卡片上标记一个数. 卡片上的数始终可见. Tito 和Pepe 一起玩游戏. 由Tito 首先开始, 他们轮流选取一张卡片, 已选过的卡片不能再选, 直到所有卡片均被选取. 然后, 每个人计算自己选取卡片上所标记数的和, 判和为偶数者获胜. 确定谁有获胜策略, 并描述该策略.2. △ABC 的外接圆为w , 外心为O . 设T 为C 关于点O 的对称点, T'为T 关于直线AB 的对称点. 直线BT'与圆w 再次相交于点R . 过O 作CT 的垂线, 交直线AC 于点L . 直线TR 与AC 相交于点N . 证明: CN = 2AL .3. 设x , y 为实数, 使得x – y , x 2 – y 2, x 3 – y 3均为素数. 证明: x – y = 3.第2天4. 求所有的3元正整数组(p , q , r ), 其中p , q 为素数, 满足215222=--p q r . 5. 设1 < n < 2018为正整数. 对i = 1, 2, …, n , 定义多项式S i (x ) = x 2 – 2018x + l i , 其中l 1, l 2, …, l n 为互不相同的正整数. 证明: 如果多项式S 1(x ) + S 2(x ) + … + S n (x )至少有一个整数根, 则l 1, l 2, …, l n 中至少有一个数不小于2018.6. 2018对夫妻参加在哈瓦那举行的一场舞会. 舞会中, 将一个圆周上2018个互异的点分别标记为0, 1, …, 2017, 每一对夫妻位于一个点上(不同夫妻位于不同的点). 对整数i ≥ 1, 令s i ≡ i (mod 2018), r i ≡ 2i (mod 2018). 舞会从第0分钟开始, 在第i 分钟, 位于点s i 的夫妻(如果存在的话)移至点r i , 而位于点r i 的夫妻(如果存在的话)则退场, 舞会由剩下的夫妻继续进行. 在20182分钟后, 舞会结束. 请确定舞会结束时还剩下多少对夫妻在场上注: 如果r i = s i , 则位于点s i 的夫妻留在原位, 不退场.2018年Cono Sur 数学奥林匹克试题第1天1. 设ABCD 为凸四边形, 点R , S 分别位于边DC , AB 上, 且满足AD = RC , BC = SA . 点P , Q , M 分别是RD , BS , CA 的中点. 设∠MPC + ∠MQA = 90o . 证明: ABCD 为圆内接四边形.2. 证明: 每一个正整数都可以表示成3, 4和7的若干幂的和, 其中同一个数不允许重复出现相同的幂次.例如: 2 = 70 + 70和22 = 32 + 32 + 41就是不允许出现的表示方式; 但是, 2= 30 + 70和22 = 32 + 30 + 41 + 40 + 71则是允许出现的表示方式.3. 考虑乘积P n = 1!⋅2!⋅3!⋅…⋅n !.i) 求所有的正整数m , 使得!2020m P 为完全平方数. ii) 证明: 存在无穷多个正整数n , 使得至少对2个正整数m ,!m P n 为完全平方数.第2天4. 对每一个正整整n ≥ 4, 考虑{1, 2, …, n }的m 个子集A 1, A 2, …, A m , 使得A 1恰含1个元素, A 2恰含2个元素, …, A m 恰含m 个元素; 且这些子集中没有一个子集是另一个子集的子集. 求m 可能取的最大值.5. 锐角△ABC 中, ∠BAC = 60o , I 为内心, O 为外心. 设H 为O 在△BOC 外接圆上的对径点. 证明: IH = BI + IC .6. 称正整数序列a 1, a 2, …, a n 为"好"的, 如果对所有的正整数n , 以下两个条件同时成立:i) n n a a a a a ...321!=.ii) a n 为某个正整数的n 次幂.求所有"好"的序列.2018年捷克-波兰-斯洛伐克联合数学竞赛试题第1天 (2018年6月25日)1. 求所有的函数f : R → R , 使得对所有的实数x , y , 成立等式:)()()()()(2y x xf x yf y f x f xy x f +++=+.2. 设△ABC 为锐角非等边三角形. 点D , E 分别在边AB , AC 上, 满足BD = CE . 设O 1, O 2分别为△ABE , △ACD 的外心. 证明: △ABC , △ADE 及△AO 1O 2的外接圆有一个异于点A 的公共点.3. 2018个玩家围桌而坐. 在游戏开始时, 我们将一摞共K 张牌任意地分发给玩家(有些玩家可能没有得到牌). 定义一轮操作如下: 如果一名玩家的左右邻居的牌数均非零, 则选他为这一轮的幸运玩家(如果有多名玩家符合条件, 则由我们任意选取一个), 让他从左右邻居那儿各拿一张牌给自己. 如果找不出这样的玩家, 则游戏结束. 求K 的最大可能值, 使得无论我们如何发牌及如何挑选幸运玩家, 该游戏总能在有限轮次后结束.第2天 (2018年6月26日)4. 设锐角△ABC 的周长为2s . 分别以A , B , C 为圆心, 作3个两两之间无公共内点的圆(不包括边界). 证明: 存在一个半径为s 的圆, 将上述三个圆同时覆盖.5. 在一个2 ⨯ 3矩形的内部, 有一个长度为36的折线(允许折线自交). 证明: 存在一条平行于矩形两边的直线, 与矩形的另两条边的内部相交, 且与折线的交点数少于10个.6. 我们称正整数n 为"奇妙"的, 如果存在正有理数a 和b , 使得bb a a n 11+++=. a) 证明: 存在无穷多个质数p , 使得p 的倍数均不是"奇妙"的.b) 证明: 存在无穷多个质数p , 使得p 的某个倍数是"奇妙"的.2018年捷克和斯洛伐克数学奥林匹克试题第1天1. 在一群人中, 存在一些两人对, 这两人相互为朋友. 对正整数k ≥ 3, 我们称该群人为"k -佳"的, 如果该群人中每k 个人(不计顺序)组成的一组人都可以围桌而坐, 使得每个人的邻座均为其朋友. 证明: 如果一群人是"6-佳"的, 则该群人必是"7-佳"的.2. 设x , y , z 为实数, 且数|2|12yz x +, |2|12zx y +, |2|12xy z +构成一非退化三角形的三边长. 求xy + yz + zx 的所有可能值.3. 三角形ABC 中, 点D 为∠A 内角平分线与边BC 的交点. 点E , F 分别是三角形ABD , ACD 的外心. 设三角形AEF 的外心位于直线BC 上. 求∠BAC 的所有可能值.第2天4. 设整数a , b , c 为某一三角形的三边长, 满足gcd(a , b , c ) = 1, 且c b a c b a -+-+222, a c b a c b -+-+222, ba cb ac -+-+222的值也均为整数. 证明: (a + b – c )(b + c – a )(c + a – b )和2(a + b – c )(b + c – a )(c + a – b )中, 至少有一个为完全平方数.5. 设ABCD 为等腰梯形, AB 为较长的底边. 令I △ABC 的内心, J 为△ACD 对应于顶点C 的旁心. 证明: IJ // AB .6. 求具有下述性质的最小正整数n : 无论用三种颜色对整数1, 2, …, n 如何染色(每个数染三种颜色之一), 从中总能够找到互异的两个数a , b , 它们染有相同的颜色, 并且|a – b |为完全平方数.2018年多瑙河地区数学奥林匹克试题比赛时间: 2018年10月27日初级组1. 求所有同时满足以下条件的正整数对(n , m ):i) n 是合数;ii) 如果d 1, d 2, …, d k (k ∈ Z +)为n 的所有真因数, 则d 1 + 1, d 2 + 1, …, d k + 1为m 的所有真因数.2. 设在△ABC 内部存在一点D , 使得∠DAC = ∠DCA = 30o , ∠DBA = 60o . E 为BC 的中点. 点F 位于线段AC 上, 且AF = 2FC . 证明: DE ⊥EF .3. 求所有具有下述性质的正整数n : 存在正整数k ≥ 2及正有理数a 1, a 2, …, a k , 使得a 1 + a 2 + … + a k = a 1a 2…a k = n 成立.4. 设M 为由全体正奇数构成的集合. 对每一个正整数n , 定义A (n )为满足元素和为n 的M 的子集的个数. 例如, A (9) = 2, 因为恰有M 的两个子集满足其元素和为9, 分别是{9}, {1, 3, 5}.a) 证明: 对每一个正整数n ≥ 2, A (n ) ≤ A (n + 1).b) 求满足A (n ) = A (n + 1)的所有正整数n ≥ 2.高级组1. 假设我们有一个由n 颗珍珠构成的项链. 在每一颗珍珠上标记一个整数, 使得所有珍珠上的数之和为n – 1. 证明: 我们可以将此项链从某处切断, 形成一根所标记整数依次为x 1, x 2, …, x n 的珍珠链, 满足11-≤∑=k x ki i 对所有k = 1, 2, …,n 成立.2. 证明: 存在无穷多组正整数(m , n )同时满足以下条件: m 整除n 2 + 1, n 整除m 2 + 1.3. 设△ABC 为非等腰锐角三角形. ∠A 的内角平分线与△ABC 的外接圆再次相交于点D . 设O 为△ABC 的外心. ∠AOB , ∠AOC 的角平分线分别与以AD 为直径的圆γ相交于点P , Q . 直线PQ 与AD 的垂直平分线相交于点R . 证明: AR // BC .4. 设n≥ 3为奇数. 将n⨯n方格纸的每一单元格都染为红色或蓝色之一. 称两个单元格为"相邻"的, 如果它们同色且至少有一个公共顶点. 称两个单元格a, b 为"连通"的, 如果存在若干个单元格c1, c2, …, c k, 满足c1 = a, c k = b, 且对每一个i = 1, 2, …, k – 1, c i与c i+1均相邻; 否则, 就称a, b为"不连通"的. (例如, 两个染色不同的单元格就是不连通的). 求最大的正整数M, 使得存在一种染色方案, 其中有M个两两不连通的单元格.2018年欧洲女子数学奥林匹克试题第1天 (2018年4月11日)1. 三角形ABC 中, CA = CB , ∠ACB = 120o , M 为AB 的中点. 设P 为三角形ABC 外接圆上一动点, Q 为线段CP 上一点, 且满足QP = 2QC . 已知经过点P 且垂直于AB 的直线与直线MQ 相交于唯一的一点N . 证明: 对点P 的所有可能位置, 点N 均位于一个固定圆上.2. 考虑集合A = ⎭⎬⎫⎩⎨⎧=+,...3,2,1:11k k . a) 证明: 每一个整数x ≥ 2均可以表示成A 中至少1个元素之积(各元素不必互异).b) 对每一个整数x ≥ 2, 设f (x )为最小的整数, 使得x 可以表示成A 中f (x )个元素之积(各元素不必互异). 证明: 存在无穷多组整数对(x , y ), 满足x ≥ 2, y ≥ 2, 且f (xy ) < f (x ) + f (y ).(如果x 1 ≠ x 2或y 1 ≠ y 2, 则认为整数对(x 1, y 1)与(x 2, y 2)是不同的.)3. 设某一届EGMO 的n 个参赛者为C 1, C 2, …, C n . 在比赛结束后, 所有参赛者在餐厅门口按照以下规则排成一个队列候餐:i) 由组委会确定各位参赛者在队列中的最初位置.ii) 每一分钟, 组委会选择一个整数i , 其中1 ≤ i ≤ n .-- 如果在参赛者C i 前面至少有i 名其他参赛者, 她将付给组委会1欧元, 并在队列中向前移动i 个位置.-- 如果在参赛者C i 前面的其他参赛者少于i 名, 则餐厅门打开, 候餐结束. a) 证明: 不管组委会如何选择, 上述候餐过程总会结束.b) 对每一个n , 求在经过精巧地选择最初位置及移动顺序下, 组委会能够得到的欧元数的最大值.第2天 (2018年4月12日)4. 定义多米诺骨牌指的是1 ⨯ 2或2 ⨯ 1的骨牌. 设n ≥ 3为整数. 在n ⨯ n 棋盘内放置若干多米诺骨牌, 使得每一个多米诺骨牌恰好覆盖两个棋格, 且多米诺骨。