Subcontests
(4)Lots of Cyclic Quads
In convex cyclic quadrilateral ABCD, we know that lines AC and BD intersect at E, lines AB and CD intersect at F, and lines BC and DA intersect at G. Suppose that the circumcircle of △ABE intersects line CB at B and P, and the circumcircle of △ADE intersects line CD at D and Q, where C,B,P,G and C,Q,D,F are collinear in that order. Prove that if lines FP and GQ intersect at M, then ∠MAC=90∘.Proposed by Kada Williams Sad Combinatorics II
Let an be the number of permutations (x1,x2,…,xn) of the numbers (1,2,…,n) such that the n ratios kxk for 1≤k≤n are all distinct. Prove that an is odd for all n≥1.Proposed by Richard Stong Sad Number Theory
For a given integer n≥2, let {a1,a2,…,am} be the set of positive integers less than n that are relatively prime to n. Prove that if every prime that divides m also divides n, then a1k+a2k+⋯+amk is divisible by m for every positive integer k. Proposed by Ivan Borsenco FEel the burn
Find all functions f:(0,∞)→(0,∞) such that f(x+y1)+f(y+z1)+f(z+x1)=1 for all x,y,z>0 with xyz=1.