This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2

2023 Serbia Team Selection Test, P5
Tags: modular arithmetic , factorial , nt , weird , number theory
For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\] Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.

2019 USA EGMO Team Selection Test, 5
Tags: geometry , weird
Let the excircle of a triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively. Denote by $\gamma$ the circumcircle of triangle $A_1B_1C_1$ and assume that $\gamma$ passes through vertex $A$. [list = a] [*] Show that $\overline{AA_1}$ is a diameter of $\gamma$. [*] Show that the incenter of $\triangle ABC$ lies on line $B_1C_1$. [/list]