Subcontests
(6)All good sequences are degenerate
An infinite sequence of positive integers a1,a2,… is called good if
(1) a1 is a perfect square, and
(2) for any integer n≥2, an is the smallest positive integer such that na1+(n−1)a2+⋯+2an−1+an is a perfect square.
Prove that for any good sequence a1,a2,…, there exists a positive integer k such that an=ak for all integers n≥k.
(reposting because the other thread didn't get moved) f this \8char
Let N={1,2,3,…} be the set of all positive integers. Find all functions f:N→N such that for any positive integers a and b, the following two conditions hold:
(1) f(ab)=f(a)f(b), and
(2) at least two of the numbers f(a), f(b), and f(a+b) are equal. n \neq N
Given a positive integer n≥2, determine the largest positive integer N for which there exist N+1 real numbers a0,a1,…,aN such that
(1) a0+a1=−n1, and
(2) (ak+ak−1)(ak+ak+1)=ak−1−ak+1 for 1≤k≤N−1. Odd Tilings
For all positive integers n, k, let f(n,2k) be the number of ways an n×2k board can be fully covered by nk dominoes of size 2×1. (For example, f(2,2)=2 and f(3,2)=3.) Find all positive integers n such that for every positive integer k, the number f(n,2k) is odd. OP circles
Let ABCD be a cyclic quadrilateral with circumcenter O. Let the internal angle bisectors at A and B meet at X, the internal angle bisectors at B and C meet at Y, the internal angle bisectors at C and D meet at Z, and the internal angle bisectors at D and A meet at W. Further, let AC and BD meet at P. Suppose that the points X, Y, Z, W, O, and P are distinct.
Prove that O, X, Y, Z, W lie on the same circle if and only if P, X, Y, Z, and W lie on the same circle.