Turkish IMO Team Selection Test 2012

Day 1, March 24, 2012

Problem 1
 

Let \( A=\{1,2,\dots, 2012\} \), \( B=\{1,2,\dots, 19\} \), and let \( S \) be the set of all subsets of \( A \). Determine the number of functions \( f:S\to B \) that satisfy the condition \( f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\} \) for all \( A_1 \), \( A_2\in S \).

Problem 2
 

In an acute triangle \( ABC \) let \( D \) be a point on the side \( BC \) different from vertices. Let \( M_1 \), \( M_2 \), \( M_3 \), \( M_4 \), \( M_5 \) be the midpoints of the segments \( AD \), \( AB \), \( AC \), \( BD \), \( CD \), respectively; \( O_1 \), \( O_2 \), \( O_3 \), \( O_4 \) the circumcenters of \( \triangle ABD \), \( \triangle ACD \), \( \triangle M_1M_2M_4 \), \( \triangle M_1M_3M_5 \), respectively; and let \( S \) and \( T \) be the midpoints of the segments \( AO_1 \) and \( AO_2 \), respectively. Prove that \( SO_3O_4T \) is an isosceles trapezoid.

Problem 3
 

Prove that the inequality \[ a+b+c+\sqrt 3\geq 8abc\left(\frac1{1+a^2}+\frac1{1+b^2}+\frac1{1+c^2}\right)\] holds for all positive real numbers \( a \), \( b \), and \( c \) that satisfy \( ab+bc+ca\leq 1 \).

Day 2, March 25, 2012

Problem 4
 

The incircle of \( \triangle ABC \) touches the sides \( BC \), \( CA \), and \( AB \) at the points \( D \), \( E \), and \( F \), respectively. The circle passing through \( A \) that is tangent to \( BC \) at \( D \) intersects the segments \( BF \) and \( CE \) at \( K \) and \( L \) respectively. Let \( P \) be the intersection point of the line through \( E \) parallel to \( DL \) and line through \( F \) parallel to \( DK \). Let \( R_1 \), \( R_2 \), \( R_3 \), and \( R_4 \) be the circumradii of the triangles \( AFD \), \( AED \), \( FPD \), and \( EPD \), respectively. Prove that \( R_1R_4=R_2R_3 \).

Problem 5
 

Find all positive integers \( n \) with the following property: If there exists an integer \( K \) that can be expressed as a sum of squares of \( n \) integers divisible by \( n \), then \( K \) can be expressed as a sum of squares of \( n \) integers none of which is divisible by \( n \).

Problem 6
 

Alice and Bob play a game on a \( 1\times m \) board using 2012 cards numbered from 1 through 2012. At each step, Alice chooses a card and Bob places it on an empty square of the board. Bob wins the game if the numbers on the cards on the board are in an increasing order after k steps where 1 ≤ k ≤ 2012, otherwise Alice wins. Find all pairs \( (k, m) \) for which Bob has a winning strategy.

Day 3, March 26, 2012

Problem 7
 

Let \( S_r(n)=1^r+2^r+\cdots + n^r \) where \( r \) is a rational number and \( n \) a positive integer. Find all triples \( (a,b,c)\in\mathbb Q_+\times\mathbb Q_+\times \mathbb N \) for which there exist infinitely many positive integers \( n \) satisfying \( S_a(n)=\left(S_b(n)\right)^c \).

Problem 8
 

Let \( A \), \( B \), \( C \), \( A^{\prime} \), \( B^{\prime} \), \( C^{\prime} \) be distinct points in the plane that satisfy \( \triangle ABC\cong \triangle A^{\prime}B^{\prime}C^{\prime} \). Let \( G \) be the centroid of \( \triangle ABC \). Assume that the circle with center \( A^{\prime} \) passing through \( G \) and the circle of diameter \( AA^{\prime} \) intersect at point \( A_1 \). Similarly, let \( B_1 \) be the intersection point of the circle with center \( B^{\prime} \) passing through \( G \) with the circle with diameter \( BB^{\prime} \), and let \( C_1 \) be the intersection point of the circle with center \( C^{\prime} \) passing through \( G \) with the circle with diameter \( CC^{\prime} \). Prove that \[ AA_1^2+BB_1^2+CC_1^2\leq AB^2+BC^2+CA^2.\]

Problem 9
 
For two sets \( A, S\subseteq\mathbb N \), we say that \( A \) is \( S \)-proper if there exists a positive integer \( N \) such that for all \( a\in A \) and all \( b\in\{0,1,\dots, a-1\} \) there exists a sequence \( s_1 \), \( \dots \), \( s_n \) of (not necessarily distinct) elements of \( S \) such that \( b\equiv s_1+s_2+\cdots + s_n \) (mod \( a \)) and \( n\in\{1,2,\dots, N\} \). Find a subset \( S\subseteq N \) for which the set \( \mathbb P \) of all prime numbers is \( S \)-proper, but \( \mathbb N \) is not.

Acknowledgements

Material received from Boš Beleš.


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