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| l...
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.
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