Olympiad problems

2018 CMI B.Sc. Entrance Exam, 4

Let $ABC$ be an equilateral triangle of side length $2$. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k<1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$. with $CB'=AC'=k$. Line segments are drawn from points $A',B',C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Prove that $\Delta PQR$ is an equilateral triangle with side length ${4(1-k) \over \sqrt{k^2-2k+4}}$.

Kvant 2024, M2778

Tags: bash , geometry

A parabola and a hyperbola are drawn on the coordinate plane. The graphs intersect at three points $A, B, C$ and the axis of the parabola is the asymptote of the hyperbola. Prove that the intersection point of the medians of the triangle $ABC$ lies on the axis of the parabola. [i]From the folklore[/i]

2018 AIME Problems, 11

Tags: bash , lifting the exponent

Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.