Subcontests
(5)Geometry of Rhombus and Four Incenters
Let ABCD be a rhombus with center O. P is a point lying on the side AB. Let I, J, and L be the incenters of triangles PCD, PAD, and PBC, respectively. Let H and K be orthocenters of triangles PLB and PJA, respectively. Prove that OI⊥HK.Proposed by buratinogigle Hard Graph Theory Problem
Let n and k be positive integers, with n≥k+1. There are n countries on a planet, with some pairs of countries establishing diplomatic relations between them, such that each country has diplomatic relations with at least k other countries. An evil villain wants to divide the countries, so he executes the following plan:(1) First, he selects two countries A and B, and let them lead two allies, A and B, respectively (so that A∈A and B∈B). (2) Each other country individually decides wether it wants to join ally A or B. (3) After all countries made their decisions, for any two countries with X∈A and Y∈B, eliminate any diplomatic relations between them.Prove that, regardless of the initial diplomatic relations among the countries, the villain can always select two countries A and B so that, no matter how the countries choose their allies, there are at least k diplomatic relations eliminated.Proposed by YaWNeeT. Prime Factors of Square Part
Let n be a given positive integer. We say that a positive integer m is n-good if and only if there are at most 2n distinct primes p satisfying p2∣m. (a) Show that if two positive integers a,b are coprime, then there exist positive integers x,y so that axn+byn is n-good.(b) Show that for any k positive integers a1,…,ak satisfying gcd(a1,…,ak)=1, there exist positive integers x1,…,xk so that a1x1n+a2x2n+⋯+akxkn is n-good.(Remark: a1,…,ak are not necessarily pairwise distinct)Proposed by usjl. Easy Mixtilinear Incircle Problem
Let ABC be a triangle with AB<AC, and let Ia be its A-excenter. Let D be the projection of Ia to BC. Let X be the intersection of AIa and BC, and let Y,Z be the points on AC,AB, respectively, such that X,Y,Z are on a line perpendicular to AIa. Let the circumcircle of AYZ intersect AIa again at U. Suppose that the tangent of the circumcircle of ABC at A intersects BC at T, and the segment TU intersects the circumcircle of ABC at V. Show that ∠BAV=∠DAC.Proposed by usjl. Blackening the area
There are 2020 points on the coordinate plane {Ai=(xi,yi):i=1,...,2020}, satisfying
0=x1<x2<...<x2020
0=y2020<y2019<...<y1Let O=(0,0) be the origin, OA1A2...A2020 forms a polygon C. Now, you want to blacken the polygon C. Every time you can choose a point (x,y) with x,y>0, and blacken the area {(x′,y′):0≤x′≤x,0≤y′≤y}. However, you have to pay xy dollars for doing so.Prove that you could blacken the whole polygon C by using 4∣C∣ dollars. Here, ∣C∣ stands for the area of the polygon C.Proposed by me