Found problems: 6530
1984 Polish MO Finals, 2
Let $n$ be a positive integer. For all $i, j \in \{1,2,...,n\}$ define $a_{j,i} = 1$ if $j = i$ and $a_{j,i} = 0$ otherwise. Also, for $i = n+1,...,2n$ and $j = 1,...,n$ define $a_{j,i} = -\frac{1}{n}$.
Prove that for any permutation $p$ of the set $\{1,2,...,2n\}$ the following inequality holds: $\sum_{j=1}^{n}\left|\sum_{k=1}^{n} a_{j,p}(k)\right| \ge \frac{n}{2}$
1975 Kurschak Competition, 3
Let $$x_0 = 5\,\, ,\, \,\,x_{n+1} = x_n +\frac{1}{x_n}.$$
Prove that $45 < x_{1000} < 45.1$.
2005 Federal Math Competition of S&M, Problem 1
If $x,y,z$ are positive numbers, prove that
$$\frac x{\sqrt{y+z}}+\frac y{\sqrt{z+x}}+\frac z{\sqrt{x+y}}\ge\sqrt{\frac32(x+y+z)}.$$
2005 Moldova National Olympiad, 11.2
Let $a$ and $b$ be two real numbers.
Find these numbers given that the graphs of $f:\mathbb{R} \to \mathbb{R} , f(x)=2x^4-a^2x^2+b-1$ and $g:\mathbb{R} \to \mathbb{R} ,g(x)=2ax^3-1$ have exactly two points of intersection.
1983 IMO Longlists, 49
Given positive integers $k,m, n$ with $km \leq n$ and non-negative real numbers $x_1, \ldots , x_k$, prove that
\[n \left( \prod_{i=1}^k x_i^m -1 \right) \leq m \sum_{i=1}^k (x_i^n-1).\]
2016 China Western Mathematical Olympiad, 1
Let $a,b,c,d$ be real numbers such that $abcd>0$. Prove that:There exists a permutation $x,y,z,w$ of $a,b,c,d$ such that $$2(xy+zw)^2>(x^2+y^2)(z^2+w^2)$$.
2021 Junior Balkan Team Selection Tests - Romania, P1
Let $a,b,c>0$ be real numbers with the property that $a+b+c=1$. Prove that \[\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.\]
2024 Belarus - Iran Friendly Competition, 1.2
Given $n \geq 2$ positive real numbers $x_1 \leq x_2 \leq \ldots \leq x_n$ satisfying the equalities
$$x_1+x_2+\ldots+x_n=4n$$
$$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}=n$$
Prove that $\frac{x_n}{x_1} \geq 7+4\sqrt{3}$
2006 Finnish National High School Mathematics Competition, 2
Show that the inequality \[3(1 + a^2 + a^4)\geq (1 + a + a^2)^2\]
holds for all real numbers $a.$
1996 Iran MO (2nd round), 4
Let $n$ blue points $A_i$ and $n$ red points $B_i \ (i = 1, 2, \ldots , n)$ be situated on a line. Prove that
\[\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.\]
2005 Taiwan National Olympiad, 1
Let $a,b,c$ be three positive real numbers such that $abc=1$. Prove that: \[ 1+\frac{3}{a+b+c}\ge{\frac{6}{ab+bc+ca}} . \]
2007 Mongolian Mathematical Olympiad, Problem 4
Let $ a,b,c>0$. Prove that
$ \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}$
2024 SG Originals, Q4
Alice and Bob play a game. Bob starts by picking a set $S$ consisting of $M$ vectors of length $n$ with entries either $0$ or $1$. Alice picks a sequence of numbers $y_1\le y_2\le\dots\le y_n$ from the interval $[0,1]$, and a choice of real numbers $x_1,x_2\dots,x_n\in \mathbb{R}$. Bob wins if he can pick a vector $(z_1,z_2,\dots,z_n)\in S$ such that $$\sum_{i=1}^n x_iy_i\le \sum_{i=1}^n x_iz_i,$$otherwise Alice wins. Determine the minimum value of $M$ so that Bob can guarantee a win.
[i]Proposed by DVDthe1st[/i]
1996 North Macedonia National Olympiad, 3
Prove that if $\alpha, \beta, \gamma$ are angles of a triangle, then $\frac{1}{\sin \alpha}+ \frac{1}{\sin \beta} \ge \frac{8}{ 3+2 \ cos\gamma}$
.
2011 Middle European Mathematical Olympiad, 2
Let $a, b, c$ be positive real numbers such that
\[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2.\]
Prove that
\[\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.\]
2001 Switzerland Team Selection Test, 2
If $a,b$, and $c$ are the sides of a triangle, prove the inequality $\sqrt{a+b-c}+\sqrt{c+a-b}+\sqrt{b+c-a } \le \sqrt{a}+\sqrt{b}+\sqrt{c}$.
When does equality occur?
2012 Portugal MO, 1
A five-digit positive integer $abcde_{10}$ ($a\neq 0$) is said to be a [i]range[/i] if its digits satisfy the inequalities $a<b>c<d>e$. For example, $37452$ is a range. How many ranges are there?
1968 Swedish Mathematical Competition, 5
Let $a, b$ be non-zero integers. Let $m(a, b)$ be the smallest value of $\cos ax + \cos bx$ (for real $x$).
Show that for some $r$, $m(a, b) \le r < 0$ for all $a, b$.
2010 ELMO Shortlist, 2
Let $a,b,c$ be positive reals. Prove that
\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]
[i]Calvin Deng.[/i]
PEN G Problems, 2
Prove that for any positive integers $ a$ and $ b$
\[ \left\vert a\sqrt{2}\minus{}b\right\vert >\frac{1}{2(a\plus{}b)}.\]
2015 Indonesia MO Shortlist, A5
Let $a,b,c$ be positive real numbers. Prove that
$\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$
2010 Iran MO (3rd Round), 5
$x,y,z$ are positive real numbers such that $xy+yz+zx=1$. prove that:
$3-\sqrt{3}+\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge(x+y+z)^2$
(20 points)
the exam time was 6 hours.
2003 China Team Selection Test, 2
In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.
1970 All Soviet Union Mathematical Olympiad, 130
The product of three positive numbers equals to one, their sum is strictly greater than the sum of the inverse numbers. Prove that one and only one of them is greater than one.
1993 French Mathematical Olympiad, Problem 5
(a) Let there be two given points $A,B$ in the plane.
i. Find the triangles $MAB$ with the given area and the minimal perimeter.
ii. Find the triangles $MAB$ with a given perimeter and the maximal area.
(b) In a tetrahedron of volume $V$, let $a,b,c,d$ be the lengths of its four edges, no three of which are coplanar, and let $L=a+b+c+d$. Determine the maximum value of $\frac V{L^3}$.