Subcontests
(9)f: A \to A, f(k)\ne g(k) and f(f(f(k))= g(k)$ for k \in A
For a positive integer n denote A={1,2,...,n}. Suppose that g:A→A is a fixed function with g(k)=k and g(g(k))=k for k∈A. How many functions f:A→A are there such that f(k)=g(k) and f(f(f(k))=g(k) for k∈A? xy=- 1 (mod z), yz = 1 (mod x), zx = 1 (mod y).
Consider the system of congruences ⎩⎨⎧xy≡−1(modz)yz≡1(modx)zx≡1(mody)
Find the number of triples (x,y,z) of distinct positive integers satisfying this system such that one of the numbers x,y,z equals 19. x^2+y^2+z^2 + xy+yz + zx> =2(\sqrt{x} +\sqrt{y}+ \sqrt{z}) if xyz = 1, x,y,z>0
If x,y,z are arbitrary positive numbers with xyz=1, prove the inequality
x2+y2+z2+xy+yz+zx≥2(x+y+z). P_1P_2,P_2P_3,...,P_nP_1 have equal lengths and midpoints A_1, A_2, A_3, A_1,..
Given two distinct points A1,A2 in the plane, determine all possible positions of a point A3 with the following property: There exists an array of (not necessarily distinct) points P1,P2,...,Pn for some n≥3 such that the segments P1P2,P2P3,...,PnP1 have equal lengths and their midpoints are A1,A2,A3,A1,A2,A3,... in this order.