Found problems: 6530
1966 IMO Longlists, 13
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality
\[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]
1992 Baltic Way, 18
Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.
2018 Belarusian National Olympiad, 9.6
For all positive integers $m$ and $n$ prove the inequality
$$
|n\sqrt{n^2+1}-m|\geqslant \sqrt{2}-1.
$$
1969 Yugoslav Team Selection Test, Problem 1
Given real numbers $a_i,b_i~(i=1,2,\ldots,n)$ such that
\begin{align*}
&a_1\ge a_2\ge\ldots\ge a_n>0,\\
&b_1\ge a_1,\\
&b_1b_2\ge a_1a_2,\\
&\vdots\\
&b_1b_2\cdots b_n\ge a_1a_2\cdots a_n,
\end{align*}prove that $b_1+b_2+\ldots+b_n\ge a_1+a_2+\ldots+a_n$.
2010 Today's Calculation Of Integral, 556
Prove the following inequality.
\[ \sqrt[3]{\int_0^{\frac {\pi}{4}} \frac {x}{\cos ^ 2 x\cos ^ 2 (\tan x)\cos ^ 2(\tan (\tan x))\cos ^ 2(\tan (\tan (\tan x)))}dx}<\frac{4}{\pi}\]
Last Edited.
Sorry, I have changed the problem.
kunny
2009 Brazil Team Selection Test, 2
Be $x_1, x_2, x_3, x_4, x_5$ be positive reais with $x_1x_2x_3x_4x_5=1$. Prove that
$$\frac{x_1+x_1x_2x_3}{1+x_1x_2+x_1x_2x_3x_4}+\frac{x_2+x_2x_3x_4}{1+x_2x_3+x_2x_3x_4x_5}+\frac{x_3+x_3x_4x_5}{1+x_3x_4+x_3x_4x_5x_1}+\frac{x_4+x_4x_5x_1}{1+x_4x_5+x_4x_5x_1x_2}+\frac{x_5+x_5x_1x_2}{1+x_5x_1+x_5x_1x_2x_3} \ge \frac{10}{3}$$
2005 Germany Team Selection Test, 3
Let $ABC$ be a triangle with area $S$, and let $P$ be a point in the plane. Prove that $AP+BP+CP\geq 2\sqrt[4]{3}\sqrt{S}$.
2013 China Girls Math Olympiad, 1
Let $A$ be the closed region bounded by the following three lines in the $xy$ plane: $x=1, y=0$ and $y=t(2x-t)$, where $0<t<1$. Prove that the area of any triangle inside the region $A$, with two vertices $P(t,t^2)$ and $Q(1,0)$, does not exceed $\frac{1}{4}.$
2005 Taiwan TST Round 1, 1
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
2014 Contests, 3
Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]
1987 Tournament Of Towns, (132) 1
Prove that for all values of $a$, $3(1+a^2+a^4) \ge (1+a+a^2)^2$ .
2010 Contests, 1
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$;
(2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$;
(2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$.
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.
1968 Dutch Mathematical Olympiad, 2
It holds: $N,a > 0$. Prove that $\frac12 \left(\frac{N}{a}+a \right) \ge \sqrt{N}$, and if $N \ge 1$ and $a = [\sqrt{N}]$.
Prove that if $a \ne \sqrt{N}: \frac12 \left(\frac{N}{a}+a \right)$ is a better approximation for $\sqrt{N}$ than $a$.
2013 Uzbekistan National Olympiad, 1
Let real numbers $a,b$ such that $a\ge b\ge 0$. Prove that \[ \sqrt{a^2+b^2}+\sqrt[3]{a^3+b^3}+\sqrt[4]{a^4+b^4} \le 3a+b .\]
2016 Iran Team Selection Test, 4
Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
1998 China National Olympiad, 3
Let $x_1,x_2,\ldots ,x_n$ be real numbers, where $n\ge 2$, satisfying $\sum_{i=1}^{n}x^2_i+ \sum_{i=1}^{n-1}x_ix_{i+1}=1$ . For each $k$, find the maximal value of $|x_k|$.
2009 China Second Round Olympiad, 2
Let $n$ be a positive integer. Prove that
\[-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}\]
2017 India IMO Training Camp, 1
Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$
2017 Hanoi Open Mathematics Competitions, 13
Let $a, b, c$ be the side-lengths of triangle $ABC$ with $a+b+c = 12$.
Determine the smallest value of $M =\frac{a}{b + c - a}+\frac{4b}{c + a - b}+\frac{9c}{a + b - c}$.
1996 Czech And Slovak Olympiad IIIA, 1
A sequence $(G_n)_{n=0}^{\infty}$ satisfies $G(0) = 0$ and $G(n) = n-G(G(n-1))$ for each $n \in N$. Show that
(a) $G(k) \ge G(k -1)$ for every $k \in N$;
(b) there is no integer $k$ for which $G(k -1) = G(k) = G(k +1)$.
V Soros Olympiad 1998 - 99 (Russia), 11.5
Find the smallest value of the expression
$$(x -y)^2 + (z - u)^2,$$
if $$(x -1)^2 + (y -4)^2 + (z-3)^2 + (u-2)^2 = 1.$$
2004 Singapore MO Open, 4
If $0 <x_1,x_2,...,x_n\le 1$, where $n \ge 1$, show that
$$\frac{x_1}{1+(n-1)x_1}+\frac{x_2}{1+(n-1)x_2}+...+\frac{x_n}{1+(n-1)x_n}\le 1$$
2023 China Second Round, 4
Let $a=1+10^{-4}$. Consider some $2023\times 2023$ matrix with each entry a real in $[1,a]$. Let $x_i$ be the sum of the elements of the $i$-th row and $y_i$ be the sum of the elements of the $i$-th column for each integer $i\in [1,n]$. Find the maximum possible value of $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$ (the answer may be expressed in terms of $a$).
2009 CentroAmerican, 6
Find all prime numbers $ p$ and $ q$ such that $ p^3 \minus{} q^5 \equal{} (p \plus{} q)^2$.
2024 Tuymaada Olympiad, 2
We will call a [i]hedgehog[/i] a graph in which one vertex is connected to all the others and there are no other edges; the number of vertices of this graph will be called the size of the hedgehog. A graph $G$ is given on $n$ vertices (where $n > 1$). For each edge $e$, we denote by $s(e)$ the size of the maximum hedgehog in graph $G$, which contains this edge. Prove the inequality (summation is carried out over all edges of the graph $G$):
\[\sum_e \frac{1}{s(e)} \leqslant \frac{n}{2}.\]
[i]Proposed by D. Malec, C. Tompkins[/i]