2016 USAJMO Problems真题及答案
The isosceles triangle , with , is inscribed in the circle . Let be a variable point on the arc that does not contain , and let and denote the incenters of triangles and , respectively.
Prove that as varies, the circumcircle of triangle passes through a fixed point.
Prove that there exists a positive integer such that has six consecutive zeros in its decimal representation.
Let be a sequence of mutually distinct nonempty subsets of a set . Any two sets and are disjoint and their union is not the whole set , that is, and , for all . Find the smallest possible number of elements in .
Find, with proof, the least integer such that if any elements are removed from the set , one can still find distinct numbers among the remaining elements with sum .
Let be an acute triangle, with as its circumcenter. Point is the foot of the perpendicular from to line , and points and are the feet of the perpendiculars from to the lines and , respectively.
Given thatprove that the points and are collinear.
Find all functions such that for all real numbers and ,