2018年欧洲女子数学奥林匹克答案
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The EGMO 2015 Problem Committee thanks the following countries for submitting problem proposals: • Ireland • Japan • Luxembourg • Macedonia • Mexico • Republic of Moldova • Netherlands • Poland • Romania • Slovenia • Turkey • Ukraine • United States of America
The Members of the Problem Committee: Yauheni Barabanov Mikhail Karpuk Aliaksei Vaidzelevich Igor Voronovich
PROBLEMS
Day 1 Problem 1. Let ABC be an acute-angled triangle, and let D be the foot of the altitude from C. The angle bisector of ∠ABC intersects CD at E and meets the circumcircle ω of triangle ADE again at F. If ∠ADF = 45◦ , show that CF is tangent to ω. Problem 2. A domino is a 2 × 1 or 1 × 2 tile. Determine in how many ways exactly n2 dominoes can be placed without overlapping on a 2n × 2n chessboard so that every 2 × 2 square contains at least two uncovered unit squares which lie in the same row or column. Problem 3. Let n, m be integers greater than 1, and let a1 , a2 , . . . , am be positive integers not greater than nm . Prove that there exist positive integers b1 , b2 , . . . , bm not greater than n, such that gcd(a1 + b1 , a2 + b2 , . . . , am + bm ) < n, where gcd(x1 , x2 , . . . , xm ) denotes the greatest common divisor of x1 , x2 , . . . , xm .
4
Problem 1. Let ABC be an acute-angled triangle, and let D be the foot of the altitude from C. The angle bisector of ∠ABC intersects CD at E and meets the circumcircle ω of triangle ADE again at F. If ∠ADF = 45◦ , show that CF is tangent to ω. (Luxembourg) C F Solution 1: Since ∠CDF = 90◦ − 45◦ = 45◦ , the line DF bisects ∠CDA, and so F lies on the perpendicular bisector of segment AE, which meets AB at G. Let ∠ABC = 2β. Since ADEF is cyclic, ∠AF E = 90◦ , and hence ∠F AE = 45◦ . Further, as BF bisects ∠ABC, we have ∠F AB = 90◦ − β, and thus ∠EAB = ∠AEG = 45◦ − β, and ∠AED = 45◦ + β,
Day 2 Problem 4. Determine whether there exists an infinite sequence a1 , a2 , a3 , . . . of positive integers which satisfies the equality √ an+2 = an+1 + an+1 + an for every positive integer n. Problem 5. Let m, n be positive integers with m > 1. Anastasia partitions the integers 1, 2, . . . , 2m into m pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to n. Problem 6. Let H be the orthocentre and G be the centroid of acute-angled triangle ABC with AB = AC. The line AG intersects the circumcircle of ABC at A and P. Let P be the reflection of P in the line BC. Prove thaቤተ መጻሕፍቲ ባይዱ ∠CAB = 60◦ if and only if HG = GP .
The Members of the Problem Committee: Yauheni Barabanov Mikhail Karpuk Aliaksei Vaidzelevich Igor Voronovich
PROBLEMS
Day 1 Problem 1. Let ABC be an acute-angled triangle, and let D be the foot of the altitude from C. The angle bisector of ∠ABC intersects CD at E and meets the circumcircle ω of triangle ADE again at F. If ∠ADF = 45◦ , show that CF is tangent to ω. Problem 2. A domino is a 2 × 1 or 1 × 2 tile. Determine in how many ways exactly n2 dominoes can be placed without overlapping on a 2n × 2n chessboard so that every 2 × 2 square contains at least two uncovered unit squares which lie in the same row or column. Problem 3. Let n, m be integers greater than 1, and let a1 , a2 , . . . , am be positive integers not greater than nm . Prove that there exist positive integers b1 , b2 , . . . , bm not greater than n, such that gcd(a1 + b1 , a2 + b2 , . . . , am + bm ) < n, where gcd(x1 , x2 , . . . , xm ) denotes the greatest common divisor of x1 , x2 , . . . , xm .
4
Problem 1. Let ABC be an acute-angled triangle, and let D be the foot of the altitude from C. The angle bisector of ∠ABC intersects CD at E and meets the circumcircle ω of triangle ADE again at F. If ∠ADF = 45◦ , show that CF is tangent to ω. (Luxembourg) C F Solution 1: Since ∠CDF = 90◦ − 45◦ = 45◦ , the line DF bisects ∠CDA, and so F lies on the perpendicular bisector of segment AE, which meets AB at G. Let ∠ABC = 2β. Since ADEF is cyclic, ∠AF E = 90◦ , and hence ∠F AE = 45◦ . Further, as BF bisects ∠ABC, we have ∠F AB = 90◦ − β, and thus ∠EAB = ∠AEG = 45◦ − β, and ∠AED = 45◦ + β,
Day 2 Problem 4. Determine whether there exists an infinite sequence a1 , a2 , a3 , . . . of positive integers which satisfies the equality √ an+2 = an+1 + an+1 + an for every positive integer n. Problem 5. Let m, n be positive integers with m > 1. Anastasia partitions the integers 1, 2, . . . , 2m into m pairs. Boris then chooses one integer from each pair and finds the sum of these chosen integers. Prove that Anastasia can select the pairs so that Boris cannot make his sum equal to n. Problem 6. Let H be the orthocentre and G be the centroid of acute-angled triangle ABC with AB = AC. The line AG intersects the circumcircle of ABC at A and P. Let P be the reflection of P in the line BC. Prove thaቤተ መጻሕፍቲ ባይዱ ∠CAB = 60◦ if and only if HG = GP .