MathDB

15

Part of 2023 LMT Fall

Problems(2)

2023 Fall LMT Team #15 area of PQYX , (BCM), (BCN)

Source:

2/4/2024
In triangle ABCABC with AB=26AB = 26, BC=28BC = 28, and CA=30C A = 30, let MM be the midpoint of ABAB and let NN be the midpoint of CAC A. The circumcircle of triangle BCMBCM intersects ACAC at X≠CX\ne C, and the circumcircle of triangle BCNBCN intersects ABAB at Y≠BY\ne B. Lines MXMX and NYNY intersect BCBC at PP and QQ, respectively. The area of quadrilateral PQYXPQY X can be expressed as pq\frac{p}{q} for positive integers pp and qq such that gcd(p,q)=1(p,q) = 1. Find qq.
geometry
2023 Fall LMT Speed p15 , 7^2 x11 divides n^3 -n^2

Source:

3/2/2024
Find the least positive integer nn greater than 11 such that n3−n2n^3 -n^2 is divisible by 72×117^2 \times 11.
Proposed by Jacob Xu
number theory