MathDB

8

Part of 2018 Online Math Open Problems

Problems(2)

2017-2018 Spring OMO Problem 8

Source:

4/3/2018
Compute the number of ordered quadruples (a,b,c,d)(a,b,c,d) of distinct positive integers such that ((ab)(cd))=21\displaystyle \binom{\binom{a}{b}}{\binom{c}{d}}=21.
Proposed by Luke Robitaille
Online Math Open
2018-2019 Fall OMO Problem 8

Source:

11/7/2018
Let ABCABC be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let N=2017N = 2017, and define the AA-ntipodes to be the points A1,…,ANA_1,\dots, A_N to be the points on segment BCBC such that BA1=A1A2=⋯=AN−1AN=ANCBA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC, and similarly define the BB, CC-ntipodes. A line ℓA\ell_A through AA is called a qevian if it passes through an AA-ntipode, and similarly we define qevians through BB and CC. Compute the number of ordered triples (ℓA,ℓB,ℓC)(\ell_A, \ell_B, \ell_C) of concurrent qevians through AA, BB, CC, respectively.
Proposed by Brandon Wang