Subcontests
(5)Two variable function
Let T be a positive integer. Find all functions f:Z+×Z+→Z+, such that there exists integers C0,C1,…,CT satisfying:
(1) For any positive integer n, the number of positive integer pairs (k,l) such that f(k,l)=n is exactly n.
(2) For any t=0,1,…,T, as well as for any positive integer pair (k,l), the equality f(k+t,l+T−t)−f(k,l)=Ct holds. Filling an infinite square grid
Let n be an odd positive integer, and consider an infinite square grid. Prove that it is impossible to fill in one of 1,2 or 3 in every cell, which simultaneously satisfies the following conditions:
(1) Any two cells which share a common side does not have the same number filled in them.
(2) For any 1×3 or 3×1 subgrid, the numbers filled does not contain 1,2,3 in that order be it reading from top to bottom, bottom to top, or left to right, or right to left.
(3) The sum of numbers of any n×n subgrid is the same. Iterating function and fixed point
Let S={1,2,…,999}. Consider a function f:S→S, such that for any n∈S,
fn+f(n)+1(n)=fnf(n)(n)=n.
Prove that there exists a∈S, such that f(a)=a. Here fk(n)=kf(f(…f(n)…)). Equal lengths and concurrency on circle
Given a scalene triangle △ABC, D,E lie on segments AB,AC respectively such that CA=CD,BA=BE. Let ω be the circumcircle of △ADE. P is the reflection of A across BC, and PD,PE meets ω again at X,Y respectively. Prove that BX and CY intersect on ω.