MathDB

20

Part of 2018 Online Math Open Problems

Problems(2)

2017-2018 Spring OMO Problem 20

Source:

4/3/2018
Let ABCABC be a triangle with AB=7,BC=5,AB = 7, BC = 5, and CA=6CA = 6. Let DD be a variable point on segment BCBC, and let the perpendicular bisector of ADAD meet segments AC,ABAC, AB at E,F,E, F, respectively. It is given that there is a point PP inside △ABC\triangle ABC such that APPC=AEEC\frac{AP}{PC} = \frac{AE}{EC} and APPB=AFFB\frac{AP}{PB} = \frac{AF}{FB}. The length of the path traced by PP as DD varies along segment BCBC can be expressed as mnsin⁡−1(17)\sqrt{\frac{m}{n}}\sin^{-1}\left(\sqrt \frac 17\right), where mm and nn are relatively prime positive integers, and angles are measured in radians. Compute 100m+n100m + n.
Proposed by Edward Wan
Online Math Open
2018-2019 Fall OMO Problem 20

Source:

11/7/2018
For positive integers k,nk,n with k≤nk\leq n, we say that a kk-tuple (a1,a2,…,ak)\left(a_1,a_2,\ldots,a_k\right) of positive integers is tasty if
[*] there exists a kk-element subset SS of [n][n] and a bijection f:[k]→Sf:[k]\to S with ax≤f(x)a_x\leq f\left(x\right) for each x∈[k]x\in [k], [*] ax=aya_x=a_y for some distinct x,y∈[k]x,y\in [k], and [*] ai≤aja_i\leq a_j for any i<ji < j.
For some positive integer nn, there are more than 20182018 tasty tuples as kk ranges through 2,3,…,n2,3,\ldots,n. Compute the least possible number of tasty tuples there can be.
Note: For a positive integer mm, [m][m] is taken to denote the set {1,2,…,m}\left\{1,2,\ldots,m\right\}.
Proposed by Vincent Huang and Tristan Shin