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: 3

2018 Saint Petersburg Mathematical Olympiad, 7
Tags: combinatorics , catalan numbers , Saint Petersburg
The checker moves from the lower left corner of the board $100 \times 100$ to the right top corner, moving at each step one cell to the right or one cell up. Let $a$ be the number of paths in which exactly $70$ steps the checker take under the diagonal going from the lower left corner to the upper right corner, and $b$ is the number of paths in which such steps are exactly $110$. What is more: $a$ or $b$?

2018 Saint Petersburg Mathematical Olympiad, 6
Tags: algebra , inequalities , Saint Petersburg
Let $a,b,c,d>0$ . Prove that $a^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}$

2021 Saint Petersburg Mathematical Olympiad, 7
Tags: Saint Petersburg , number theory
For a positive integer $n$, prove that $$\sum_{n \le p \le n^4} \frac{1}{p} < 4$$ where the sum is taken across primes $p$ in the range $[n, n^4]$ [i]N. Filonov[/i]