MathDB

28

Part of 2016 Online Math Open Problems

Problems(2)

2015-2016 Spring OMO #28

Source:

3/29/2016
Let NN be the number of polynomials P(x1,x2,…,x2016)P(x_1, x_2, \dots, x_{2016}) of degree at most 20152015 with coefficients in the set {0,1,2}\{0, 1, 2 \} such that P(a1,a2,⋯ ,a2016)≡1(mod3)P(a_1,a_2,\cdots ,a_{2016}) \equiv 1 \pmod{3} for all (a1,a2,⋯ ,a2016)∈{0,1}2016.(a_1,a_2,\cdots ,a_{2016}) \in \{0, 1\}^{2016}.
Compute the remainder when v3(N)v_3(N) is divided by 20112011, where v3(N)v_3(N) denotes the largest integer kk such that 3k∣N.3^k | N.
Proposed by Yang Liu
Online Math Open
2016-2017 Fall OMO Problem 28

Source:

11/16/2016
Let ABCABC be a triangle with AB=34,BC=25,AB=34,BC=25, and CA=39CA=39. Let O,H,O,H, and ω \omega be the circumcenter, orthocenter, and circumcircle of △ABC\triangle ABC, respectively. Let line AHAH meet ω\omega a second time at A1A_1 and let the reflection of HH over the perpendicular bisector of BCBC be H1H_1. Suppose the line through OO perpendicular to A1OA_1O meets ω\omega at two points QQ and RR with QQ on minor arc ACAC and RR on minor arc ABAB. Denote H\mathcal H as the hyperbola passing through A,B,C,H,H1A,B,C,H,H_1, and suppose HOHO meets H\mathcal H again at PP. Let X,YX,Y be points with XH∥AR∥YP,XP∥AQ∥YHXH \parallel AR \parallel YP, XP \parallel AQ \parallel YH. Let P1,P2P_1,P_2 be points on the tangent to H\mathcal H at PP with XP1∥OH∥YP2XP_1 \parallel OH \parallel YP_2 and let P3,P4P_3,P_4 be points on the tangent to H\mathcal H at HH with XP3∥OH∥YP4XP_3 \parallel OH \parallel YP_4. If P1P4P_1P_4 and P2P3P_2P_3 meet at NN, and ONON may be written in the form ab\frac{a}{b} where a,ba,b are positive coprime integers, find 100a+b100a+b.
Proposed by Vincent Huang
geometryOmo