2004-2005斯洛伐克数学奥林匹克

Slovakia, 2004/2005, olympiad problems 1) Find number of infinite arithmetic progressions of integers a ,a ,a , …such that there exists two distinct integers 1 2 3 i，j for which 1≤i, j≤10 such that ai=1,aj=2005. 2) In parallelogram ABCD, |AB|>|BC| let K,L,M,N be points of touch of incircles of triangles ACD, BCD, ABC, ABD (respectively) to one of diagonals (so K is the point where incircle of ACD touches AC). Prove that KLMN is rectangle. 3) For what positive integers k has 2 1( 2) ( ) ( 3k k k x k k k )− ≤ + ≤ + exactly 2( 1)k + soluitions in integers? 4) For positive reals a,b,c with product 1 prove and find when it attains equality: 3 ( 1)( 1) ( 1)( 1) ( 1)( 1) 4 a b c a b b c c a + + ≥+ + + + + + . 5) Solve this system of equalities in integers: 2 2 2 2 2 2 ( 1) ( 1) ( 1) x y z y z y z x z x z x y x y ⎧ 5 5 5 + + = + −⎪ + + = + −⎨⎪ + + = + −⎩ 6) In a plane there's triangle KLM with |KM|=|LM|. Let circle k be tangent to line KM in point K and let l be tangent to line LM in point L. Let k and l be externally tangent in point T. Find locus of all points T. 7) Find all pairs of positive integers a,b such that a+b has last digit 3, |a-b| is prime and ab is square.