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

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Found problems: 2

2014 IMAC Arhimede, 5
Let $p$ be a prime number. The natural numbers $m$ and $n$ are written in the system with the base $p$ as $n = a_0 + a_1p +...+ a_kp^k$ and $m = b_0 + b_1p +..+ b_kp^k$. Prove that $${n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)$$
binomial system number theory modulo product numerical systems prime number prime numbers

2016 Finnish National High School Mathematics Comp, 2
Suppose that $y$ is a positive integer written only with digit $1$, in base $9$ system. Prove that $y$ is a triangular number, that is, exists positive integer $n$ such that the number $y$ is the sum of the $n$ natural numbers from $1$ to $n$.
triangular number number theory numerical systems algebra number base