MathDB

2015 Azerbaijan IMO TST

Part of Azerbaijan Team Selection Test

Subcontests

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Azerbaijan IMO TST

Let nn and kk be two positive integers such that n>kn>k. Prove that the equation xn+yn=zkx^n+y^n=z^k has a solution in positive integers if and only if the equation xn+yn=zn−kx^n+y^n=z^{n-k} has a solution in positive integers.

2015 Azerbaijan IMO TST

Consider a trapezoid ABCDABCD with BC∣∣ADBC||AD and BC<ADBC<AD. Let the lines ABAB and CDCD meet at XX. Let ω1\omega_1 be the incircle of the triangle XBCXBC, and let ω2\omega_2 be the excircle of the triangle XADXAD which is tangent to the segment ADAD . Denote by aa and dd the lines tangent to ω1\omega_1 , distinct from ABAB and CDCD, and passing through AA and DD, respectively. Denote by bb and cc the lines tangent to ω2\omega_2 , distinct from ABAB and CDCD, passing through BB and CC respectively. Assume that the lines a,b,ca,b,c and dd are distinct. Prove that they form a parallelogram.
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Azerbaijan IMO TST

Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.

2015 Azerbaijan IMO TST

Find all functions f:[0,1]→Rf:[0,1] \to \mathbb{R} such that the inequality (x−y)2≤∣f(x)−f(y)∣≤∣x−y∣(x-y)^2\leq|f(x) -f(y)|\leq|x-y| is satisfied for all x,y∈[0,1]x,y\in [0,1]