美国数学题——精选推荐

美国数学题
Math IA1/IA2/IA3/IAB1/IAB2 Placement Test
Spring 2013 – March 2 & 3, 2013
Please check & correct your grade and the school name on the Admit Card. Make sure that you are taking the test indicated on the Admit Card. Indicate any schedule preferences on the Admit Card. The placement test will be posted on the web as Week #1 homework on March 13th. Math IAB students requesting advancement to Math IB must also take the Math IB test as a second test. You must score at least 15 (50%) to take the Math IB test. Please staple the placement test along with your answers and the Admit Card.
1.Different figures represent the different digits. Find the digit corresponding to the square.
2.In a trunk there are 5 chests, in each chest there are 3 boxes, and in each box there are 10
gold coins. The trunk, the chests, and the boxes are locked. How many locks must be
opened in order to get 50 coins?
3. A caterpillar starts from his home and move directly on a ground, turning after each hour
at 90° to the left or to the right. In the first hour he moved 1 m, in the second hour 2 m,
and so on. At what minimum distance from his home the caterpillar would be after six
hours traveling?
4.The sum of ten distinct positive numbers is 100. The largest of these numbers is?
5. A rod of length 15 dm was divided into the greatest possible number of pieces of
different integer lengths in dm. The number of cuts is:
6. A river goes through a city and there are two islands. There are also six bridges as shown
in the figure. How many paths there are going out of a shore of the river (point A) and
come back (to point B) after having spent one and only one time for each bridge?
7.In which of triples the central number is strictly in the middle between two others.
A) 1/2, 1/3, 1/4 B) 12, 21, 32 C) 3, 7, 13 D) 1/3, 1/2, 2/3
8.What is the smallest number of dots that need to be removed from the pattern shown, so
that no three of the remaining dots are at the vertices of an equilateral triangle?
9.Iqra is 10 years old. Her mother Asma is 4 times as old. How old will Asma be when Iqra
is twice as old as she is now?
10.To the right side of a given 2-digit number we write the same number obtaining 4-digit
number. How many times the 4-digit number is greater than the 2-digit number?
11.Ahmed thought of an integer. Umar multiplied it either by 5 or by 6. Ali added to the
Umar’s result either 5 or 6. Tahir subtracted from Ali’s result either 5 or 6. The obtained result was 73. What number did Ahmed think of?
12.The multiplication uses each of the digits 1 to 9 exactly once. What is digit Y?
13.One of the cube faces is cut along its diagonals (see the fig.). Which of the following nets
are impossible?
14.Seven cards lie in a box. Numbers from 1 to 7 are written on these cards (exactly one
number on the card). Two persons take the cards as follows: The first person takes, at
random, 3 cards from the box and the second person takes 2 cards (2 cards are left in the box). Then the first person tells the second one: “I know that the sum of the numbers of your cards is even”. The sum of card’s numbers of the first person is equal to
15.For each 2-digit number from 30 to 50, the digit of units was subtracted from the digit of
tens. What is the sum of all the results?
16.How many digits can be at most erased from the 1000-digit number 20082008…2008,
such that the sum of the remaining digits is 2008?
17.Basil wants to paint the slogan VIVAT KANGAROO on a wall. He wants different
letters to be colored differently, and the same letters to be colored identically. How many colors will he need?
18.When it is 4 o'clock in the afternoon in London, it is 5 o'clock in the afternoon in Madrid
and it is 8 o'clock in the morning on the same day in San Francisco. Ann went to bed in San Francisco at 9 o'clock yesterday evening. What was the time in Madrid at that
moment?
19.The picture shows a pattern of hexagons. We draw a new pattern by connecting all the
midpoints of any neighboring hexagons. What pattern do we get?
20.The upper coin is rotated without slipping around the fixed lower coin to a position
shown on the picture. Which is the resulting relative position of kangaroos?
21.Vivien and Mike were given some apples and pears by their grandmother. They had 25
pieces of fruit in their basket altogether. On the way home Vivien ate 1 apple and 3 pears, and Mike ate 3 apples and 2 pears. At home they found out that they brought home the
same number of pears as apples. How many pears were they given by their grandmother?
22.Which three of the numbered puzzle pieces should you add to the picture to complete the
square?
23.There are 4 gearwheels on fixed axles next to each other, as shown. The first one has 30
gears, the second one 15, the third one 60 and the last one 10. How many revolutions
does the last gearwheel make, when the first one turns through one revolution?
24. A regular octagon is folded in half exactly three times until a triangle is obtained, as
shown. Then the apex is cut off at right angles, as shown in the picture. If the paper is
unfolded what will it look like?
25.Winnie's vinegar-wine-water marinade contains vinegar and wine in the ratio 1 to 2, and
wine and water in the ratio 3 to 1. Which of the following statements is true?
A.There is more vinegar than wine.
B.There is more wine than vinegar and water together.
C.There is more vinegar than wine and water together.
D.There is more water than vinegar and wine together.
E.There is less vinegar than either water or wine.
26.Kangaroos Hip and Hop play jumping by hopping over a stone, then landing across so
that the stone is in the middle of the segment traveled during each jump. Picture 1 shows how Hop jumped three times hopping over stones marked 1, 2 and 3. Hip has the
configuration of stones marked 1, 2 and 3 (to jump over in this order), but starts in a
different place as shown on Picture 2. Which of the points A, B, C, D or E is his landing point?
27.There were twelve children at a birthday party. Each child was either 6, 7, 8, 9 or 10
years old, with at least one child of each age. Four of them were 6 years old. In the group the most common age was 8 years old. What was the average age of the twelve children?
28.Rectangle ABCD is cut into four smaller rectangles, as shown in the figure. The four
smaller rectangles have the properties: (a) the perimeters of three of them are 11, 16 and
19, (b) the perimeter of the fourth is neither the biggest nor the smallest of the four. What is the perimeter of the original rectangle ABCD?
29.Peter wants to cut a rectangle of size 6 × 7 into squares with integer sides. What is the
minimal number of squares he can get?
30.Abid's house number has 3 digits. Removing the first digit of Abid's house number, you
obtain the house number of Ben. Removing the first digit of Ben's house number, you get the house number of Chiara. Adding the house numbers of Abid, Ben and Chiara gives
912. What is the second digit of Abid's house number?。