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: 30

1990 Romania Team Selection Test, 2
Tags: algebra , binomial , sum , combinatorics
Prove the following equality for all positive integers $m,n$: $$\sum_{k=0}^{n} {m+k \choose k} 2^{n-k} +\sum_{k=0}^m {n+k \choose k}2^{m-k}= 2^{m+n+1}$$

2020 Czech and Slovak Olympiad III A, 6
Tags: inequalities , binomial , divisible , number theory
For each positive integer $k$, denote by $P (k)$ the number of all positive integers $4k$-digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$. Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$ and determine all $k$ for which equality occurs. (Note: A positive integer cannot begin with a digit of $0$.) (Jaromir Simsa)

2014 Hanoi Open Mathematics Competitions, 2
Tags: binomial , inequalities , combination , combinatorics
How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ? (A): $15$, (B): $16$, (C): $17$, (D): $18$, (E) None of the above. Note: $C^{m}_{n}$ stands for $\binom {m}{n}$

2014 Indonesia MO Shortlist, C5
Tags: combinatorics , binomial
Determine all pairs of natural numbers $(m, r)$ with $2014 \ge m \ge r \ge 1$ that fulfill $\binom{2014}{m}+\binom{m}{r}=\binom{2014}{r}+\binom{2014-r}{m-r} $

2008 Thailand Mathematical Olympiad, 4
Tags: number theory , binomial , divisible , divide
Let $n$ be a positive integer. Show that $${2n+1 \choose 1} -{2n+1 \choose 3}2008 + {2n+1 \choose 5}2008^2- ...+(-1)^{n}{2n+1 \choose 2n+1}2008^n $$ is not divisible by $19$.