Found problems: 6530
2015 Balkan MO Shortlist, A1
If ${a, b}$ and $c$ are positive real numbers, prove that
\begin{align*}
a ^ 3b ^ 6 + b ^ 3c ^ 6 + c ^ 3a ^ 6 + 3a ^ 3b ^ 3c ^ 3 &\ge{ abc \left (a ^ 3b ^ 3 + b ^ 3c ^ 3 + c ^ 3a ^ 3 \right) + a ^ 2b ^ 2c ^ 2 \left (a ^ 3 + b ^ 3 + c ^ 3 \right)}.
\end{align*}
[i](Montenegro).[/i]
1978 All Soviet Union Mathematical Olympiad, 264
Given $0 < a \le x_1\le x_2\le ... \le x_n \le b$. Prove that $$(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2$$
2007 Iran MO (3rd Round), 2
$ a,b,c$ are three different positive real numbers. Prove that:\[ \left|\frac{a\plus{}b}{a\minus{}b}\plus{}\frac{b\plus{}c}{b\minus{}c}\plus{}\frac{c\plus{}a}{c\minus{}a}\right|>1\]
2010 Contests, 4
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
1996 Tournament Of Towns, (488) 1
Prove that if $a, b$ and $c$ are positive numbers such that
$$a^2 + b^2 - ab = c^2,$$
then $(a - c)(b - c) < 0.$
(A Egorov)
1992 Swedish Mathematical Competition, 3
Solve:
$$\begin{cases} 2x_1 - 5x_2 + 3x_3 \ge 0 \\
2x_2 - 5x_3 + 3x4 \ge 0 \\
...\\
2x_{23} - 5x_{24} + 3x_{25} \ge 0\\
2x_{24} - 5x_{25} + 3x_1 \ge 0\\
2x_{25} - 5x_1 + 3x_2 \ge 0 \end{cases}$$
2012 India IMO Training Camp, 2
Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent:
$(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$
$(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$
1998 India Regional Mathematical Olympiad, 3
Prove that for every natural number $n > 1$ \[ \frac{1}{n+1} \left( 1 + \frac{1}{3} +\frac{1}{5} + \ldots + \frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2n} \right) . \]
2010 Moldova Team Selection Test, 2
Prove that for any real number $ x$ the following inequality is true:
$ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$
1998 Mediterranean Mathematics Olympiad, 1
A square $ABCD$ is inscribed in a circle. If $M$ is a point on the shorter arc $AB$, prove that
\[MC \cdot MD > 3\sqrt{3} \cdot MA \cdot MB.\]
1974 Czech and Slovak Olympiad III A, 1
Let $\left(a_k\right)_{k=1}^\infty$ be a sequence of positive numbers such that \[a_{k-1}a_{k+1}\ge a_k^2\] for all $k>1.$ For $n\ge1$ denote \[b_n=\left(a_1a_2\cdots a_n\right)^{1/n}.\] Show that also the inequality \[b_{n-1}b_{n+1}\ge b_n^2\] holds for every $n>1.$
2021 Iran Team Selection Test, 4
Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds :
$$f(ac)+f(bc)-f(c)f(ab) \ge 1$$
Proposed by [i]Mojtaba Zare[/i]
2013 Tuymaada Olympiad, 7
Points $A_1$, $A_2$, $A_3$, $A_4$ are the vertices of a regular tetrahedron of edge length $1$. The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$, $A_2$, $A_3$.
Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$.
[i] A. Kupavsky [/i]
2009 Turkey Team Selection Test, 2
In a triangle $ ABC$ incircle touches the sides $ AB$, $ AC$ and $ BC$ at $ C_1$, $ B_1$ and $ A_1$ respectively. Prove that $ \sqrt {\frac {AB_1}{AB}} \plus{} \sqrt {\frac {BC_1}{BC}} \plus{} \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}}$ is true.
2002 Balkan MO, 4
Determine all functions $f: \mathbb N\to \mathbb N$ such that for every positive integer $n$ we have: \[ 2n+2001\leq f(f(n))+f(n)\leq 2n+2002. \]
2004 France Team Selection Test, 2
Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively.
Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.
1997 Korea National Olympiad, 3
Let $ABCDEF$ be a convex hexagon such that $AB=BC,CD=DE, EF=FA.$
Prove that $\frac{BC}{BE}+\frac{DE}{DA}+\frac{FA}{FC}\ge\frac{3}{2}$ and find when equality holds.
2024 Mozambique National Olympiad, P2
Prove that if $a+b+c=0$ then $a^3+b^3+c^3=3abc$
2000 Baltic Way, 16
Prove that for all positive real numbers $a,b,c$ we have
\[\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2} \]
2009 Mathcenter Contest, 1
Let $m,n$ be natural numbers. Prove that $$m^{m^{m^m}}+n^{n^{n^n}}\geq m^{n^{n^n}}+ n^{m^{m^m}}$$
[i](nooonuii)[/i]
2006 India IMO Training Camp, 1
Let $ABC$ be a triangle with inradius $r$, circumradius $R$, and with sides $a=BC,b=CA,c=AB$. Prove that
\[\frac{R}{2r} \ge \left(\frac{64a^2b^2c^2}{(4a^2-(b-c)^2)(4b^2-(c-a)^2)(4c^2-(a-b)^2)}\right)^2.\]
2007 Romania Team Selection Test, 3
Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true.
[i]Dan Schwarz[/i]
2007 All-Russian Olympiad, 2
Given polynomial $P(x) = a_{0}x^{n}+a_{1}x^{n-1}+\dots+a_{n-1}x+a_{n}$. Put $m=\min \{ a_{0}, a_{0}+a_{1}, \dots, a_{0}+a_{1}+\dots+a_{n}\}$. Prove that $P(x) \ge mx^{n}$ for $x \ge 1$.
[i]A. Khrabrov [/i]
2020 Silk Road, 1
Given a strictly increasing infinite sequence of natural numbers $ a_1, $ $ a_2, $ $ a_3, $ $ \ldots $. It is known that $ a_n \leq n + 2020 $ and the number $ n ^ 3 a_n - 1 $ is divisible by $ a_ {n + 1} $ for all natural numbers $ n $. Prove that $ a_n = n $ for all natural numbers $ n $.
2006 Iran Team Selection Test, 4
Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that
\[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]