Found problems: 6530
2001 Tuymaada Olympiad, 1
Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored $n$-th has $x_{n}$ points ($n=1, \dots, 10$), prove that $x_{1}+2x_{2}+\dots+10x_{10}\geq 165$.
[i]Proposed by D. Teryoshin[/i]
2019 Middle European Mathematical Olympiad, 1
Determine the smallest and the greatest possible values of the expression
$$\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right)$$ provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$.
[i]Proposed by Walther Janous, Austria [/i]
2013 USAJMO, 6
Find all real numbers $x,y,z\geq 1$ satisfying \[\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
1998 Akdeniz University MO, 3
Let $x,y,z$ be non-negative numbers such that $x+y+z \leq 3$. Prove that
$$\frac{2}{1+x}+\frac{2}{1+y}+\frac{2}{1+z} \geq 3$$
2022 Balkan MO Shortlist, A2
Let $k > 1{}$ be a real number, $n\geqslant 3$ be an integer, and $x_1 \geqslant x_2\geqslant\cdots\geqslant x_n$ be positive real numbers. Prove that \[\frac{x_1+kx_2}{x_2+x_3}+\frac{x_2+kx_3}{x_3+x_4}+\cdots+\frac{x_n+kx_1}{x_1+x_2}\geqslant\frac{n(k+1)}{2}.\][i]Ilija Jovcheski[/i]
2014 JBMO TST - Turkey, 4
Determine the smallest value of $(a+5)^2+(b-2)^2+(c-9)^2$ for all real numbers $a, b, c$ satisfying $a^2+b^2+c^2-ab-bc-ca=3$.
2010 Saint Petersburg Mathematical Olympiad, 6
For positive is true $$\frac{3}{abc} \geq a+b+c$$
Prove $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq a+b+c$$
2013 USAMTS Problems, 4
An infinite sequence of real numbers $a_1,a_2,a_3,\dots$ is called $\emph{spooky}$ if $a_1=1$ and for all integers $n>1$,
\[\begin{array}{c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c@{\;\,}c}
na_1&+&(n-1)a_2&+&(n-2)a_3&+&\dots&+&2a_{n-1}&+&a_n&<&0,\\
n^2a_1&+&(n-1)^2a_2&+&(n-2)^2a_3&+&\dots&+&2^2a_{n-1}&+&a_n&>&0.
\end{array}\]Given any spooky sequence $a_1,a_2,a_3,\dots$, prove that
\[2013^3a_1+2012^3a_2+2011^3a_3+\cdots+2^3a_{2012}+a_{2013}<12345.\]
2018 239 Open Mathematical Olympiad, 8-9.6
Petya wrote down 100 positive integers $n, n+1, \ldots, n+99$, and Vasya wrote down 99 positive integers $m, m-1, \ldots, m-98$. It turned out that for each of Petya's numbers, there is a number from Vasya that divides it. Prove that $m>n^3/10, 000, 000$.
[i]Proposed by Ilya Bogdanov[/i]
2016 South East Mathematical Olympiad, 1
The sequence $(a_n)$ is defined by $a_1=1,a_2=\frac{1}{2}$,$$n(n+1)
a_{n+1}a_{n}+na_{n}a_{n-1}=(n+1)^2a_{n+1}a_{n-1}(n\ge 2).$$
Prove that $$\frac{2}{n+1}<\sqrt[n]{a_n}<\frac{1}{\sqrt{n}}(n\ge 3).$$
1985 Vietnam Team Selection Test, 1
A convex polygon $ A_1,A_2,\cdots ,A_n$ is inscribed in a circle with center $ O$ and radius $ R$ so that $ O$ lies inside the polygon. Let the inradii of the triangles $ A_1A_2A_3, A_1A_3A_4, \cdots , A_1A_{n \minus{} 1}A_n$ be denoted by $ r_1,r_2,\cdots ,r_{n \minus{} 2}$. Prove that $ r_1 \plus{} r_2 \plus{} ... \plus{} r_{n \minus{} 2}\leq R(n\cos \frac {\pi}{n} \minus{} n \plus{} 2)$.
1968 All Soviet Union Mathematical Olympiad, 113
The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $$a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.$$ Prove that $$(a_1+a_2+...+a_n)/n \ge -1/2$$
2016 Turkmenistan Regional Math Olympiad, Problem 2
If $a,b,c$ are triangle sides then prove that $(\sum_{cyc}\sqrt{\frac{a}{-a+b+c}} \geq 3$
2017 Saint Petersburg Mathematical Olympiad, 6
Given three real numbers $a,b,c\in [0,1)$ such that $a^2+b^2+c^2=1$. Find the smallest possible value of
$$\frac{a}{\sqrt{1-a^2}}+\frac{b}{\sqrt{1-b^2}}+\frac{c}{\sqrt{1-c^2}}.$$
2010 Ukraine Team Selection Test, 4
For the nonnegative numbers $a, b, c$ prove the inequality:
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge \frac52$$
2015 Romania Team Selection Test, 2
Let $ABC$ be a triangle, and let $r$ denote its inradius. Let $R_A$ denote the radius of the circle internally tangent at $A$ to the circle $ABC$ and tangent to the line $BC$; the radii $R_B$ and $R_C$ are defined similarly. Show that $\frac{1}{R_A} + \frac{1}{R_B} + \frac{1}{R_C}\leq\frac{2}{r}$.
2011 ISI B.Math Entrance Exam, 3
For $n\in\mathbb{N}$ prove that
\[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdots\frac{2n-1}{2n}\leq\frac{1}{\sqrt{2n+1}}.\]
2003 Indonesia MO, 6
The hall in a castle is a regular hexagon where its sides' length is 6 meters. The floor of the hall is to be tiled with equilateral triangular tiles where its sides' length is 50 centimeters. Each tile is divided into three congruent triangles by their altitudes up to its orthocenter (see below). Each of these small triangles are colored such that each tile has different colors and no two tiles have identical colorings. How many colors at least are required?
A tile's pattern is:
[asy]
draw((0,0.000)--(2,0.000));
draw((2,0.000)--(1,1.732));
draw((1,1.732)--(0,0.000));
draw((1,0.577)--(0,0.000));
draw((1,0.577)--(2,0.000));
draw((1,0.577)--(1,1.732));
[/asy]
1971 Swedish Mathematical Competition, 1
Show that
\[
\left(1 + a + a^2\right)^2 < 3\left(1 + a^2 + a^4\right)
\]
for real $a \neq 1$.
2015 CCA Math Bonanza, L3.4
Compute the greatest constant $K$ such that for all positive real numbers $a,b,c,d$ measuring the sides of a cyclic quadrilateral, we have
\[
\left(\frac{1}{a+b+c-d}+\frac{1}{a+b-c+d}+\frac{1}{a-b+c+d}+\frac{1}{-a+b+c+d}\right)(a+b+c+d)\geq K.
\]
[i]2015 CCA Math Bonanza Lightning Round #3.4[/i]
2013 ELMO Shortlist, 4
Let $n$ be a positive integer. The numbers $\{1, 2, ..., n^2\}$ are placed in an $n \times n$ grid, each exactly once. The grid is said to be [i]Muirhead-able[/i] if the sum of the entries in each column is the same, but for every $1 \le i,k \le n-1$, the sum of the first $k$ entries in column $i$ is at least the sum of the first $k$ entries in column $i+1$. For which $n$ can one construct a Muirhead-able array such that the entries in each column are decreasing?
[i]Proposed by Evan Chen[/i]
2008 Cuba MO, 7
For non negative reals $a,b$ we know that $a^2+a+b^2\ge a^4+a^3+b^4$. Prove that $$\frac{1-a^4}{a^2}\ge \frac{b^2-1}{b}$$
2019 Romania National Olympiad, 1
If $a,b,c>0$ then
$$\frac{1}{abc}+1\ge3\left(\frac{1}{a^2+b^2+c^2}+\frac{1}{a+b+c}\right)$$
1999 Polish MO Finals, 2
Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}$.
2004 Greece JBMO TST, 4
Let $a,b$ be positive real numbers such that $b^3+b\le a-a^3$. Prove that:
i) $b<a<1$
ii) $a^2+b^2<1$