MathDB

Problem 5

Part of 2000 Mongolian Mathematical Olympiad

Problems(2)

system of inequalities, number of integer solutions

Source: Mongolia MO 2000 Grade 10 P5

4/22/2021
Given a natural number nn, find the number of quadruples (x,y,u,v)(x,y,u,v) of integers with 1≤x,y,y,v≤n1\le x,y,y,v\le n satisfy the following inequalities: \begin{align*} &1\le v+x-y\le n,\\ &1\le x+y-u\le n,\\ &1\le u+v-y\le n,\\ &1\le v+x-u\le n. \end{align*}
inequalitiescombinatorics
inequality in positive integers

Source: Mongolia MO 2000 Teachers P5

4/22/2021
Let m,n,km,n,k be positive integers with m≥2m\ge2 and k≥log⁡2(m−1)k\ge\log_2(m-1). Prove that ∏s=1nms−1ms<12n+12k+1.\prod_{s=1}^n\frac{ms-1}{ms}<\sqrt[2^{k+1}]{\frac1{2n+1}}.
inequalitiesnumber theory