Found problems: 6530
2005 IMO Shortlist, 5
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
[i]Hojoo Lee, Korea[/i]
1975 Chisinau City MO, 104
Prove that $x^2+y^2 \ge 2\sqrt2 (x-y)$ if $xy = 1$
2024 Mathematical Talent Reward Programme, 2
Find positive reals $a,b,c$ such that: $$\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} = 2$$
2016 China Girls Math Olympiad, 5
Define a sequence $\{a_n\}$ by\[S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).\]
Where $S_n$ the sum of first $n$ terms of sequence $\{a_n\}$.
For any positive integer $n$ ,prove that\[a_{n}\ge \frac{4}{\sqrt{9n+7}}.\]
2011 District Olympiad, 1
Find the real numbers $x$ and $y$ such that
$$(x^2 -x +1)(3y^2-2y + 3) -2=0.$$
2021 Tuymaada Olympiad, 8
An acute triangle $ABC$ is given, $AC \not= BC$. The altitudes drawn from $A$ and $B$ meet at $H$ and intersect the external bisector of the angle $C$ at $Y$ and $X$ respectively. The external bisector of the angle $AHB$ meets the segments $AX$ and $BY$ at $P$ and $Q$ respectively. If $PX = QY$, prove that $AP + BQ \ge 2CH$.
IV Soros Olympiad 1997 - 98 (Russia), 9.2
Find all values of the parameter $a$ for which there exist exactly two integer values of $x$ that satisfy the inequality $$x^2+5\sqrt2 x+a<0.$$
2002 Romania National Olympiad, 4
Find all functions $f: \mathbb{N}\to\mathbb{N}$ which satisfy the inequality:
\[f(3x+2y)=f(x)f(y)\]
for all non-negative integers $x,y$.
1978 Putnam, B6
Let $p$ and $n$ be positive integers. Suppose that the numbers $c_{hk}$ ($h=1,2,\ldots,n$ ; $k=1,2,\ldots,ph$) satisfy $0 \leq c_{hk} \leq 1.$ Prove that
$$ \left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,$$
where each summation is over all admissible ordered pairs $(h,k).$
2000 IMO, 2
Let $ a, b, c$ be positive real numbers so that $ abc \equal{} 1$. Prove that
\[ \left( a \minus{} 1 \plus{} \frac 1b \right) \left( b \minus{} 1 \plus{} \frac 1c \right) \left( c \minus{} 1 \plus{} \frac 1a \right) \leq 1.
\]
2010 BAMO, 3
Suppose $a,b,c$ are real numbers such that $a+b \ge 0, b+c \ge 0$, and $c+a \ge 0$.
Prove that $a+b+c \ge \frac{|a|+|b|+|c|}{3}$ .
(Note: $|x|$ is called the absolute value of $x$ and is defined as follows.
If $x \ge 0$ then $|x|= x$, and if $x < 0$ then $|x| = -x$. For example, $|6|= 6, |0| = 0$ and $|-6| = 6$.)
2003 Baltic Way, 3
Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Prove that
$$\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).$$
2003 China Team Selection Test, 3
Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.
2012 APMO, 5
Let $ n $ be an integer greater than or equal to $ 2 $. Prove that if the real numbers $ a_1 , a_2 , \cdots , a_n $ satisfy $ a_1 ^2 + a_2 ^2 + \cdots + a_n ^ 2 = n $, then
\[\sum_{1 \le i < j \le n} \frac{1}{n- a_i a_j} \le \frac{n}{2} \]
must hold.
2002 Singapore Team Selection Test, 1
Let $x_1, x_2, x_3$ be positive real numbers. Prove that
$$\frac{(x_1^2+x_2^2+x_3^2)^3}{(x_1^3+x_2^3+x_3^3)^2}\le 3$$
2007 District Olympiad, 3
Find all continuous functions $f : \mathbb R \to \mathbb R$ such that:
(a) $\lim_{x \to \infty}f(x)$ exists;
(b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.
2023 ISL, A5
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
2014 Saudi Arabia BMO TST, 4
Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. ([i]A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex[/i] )
2004 Austria Beginners' Competition, 2
For what pairs of integers $(x,y)$ does the inequality $x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)}$ hold?
1999 USAMO, 2
Let $ABCD$ be a cyclic quadrilateral. Prove that \[ |AB - CD| + |AD - BC| \geq 2|AC - BD|. \]
1986 China Team Selection Test, 3
Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that:
i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$
ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.
2011 Kosovo National Mathematical Olympiad, 3
If $a,b,c$ are real positive numbers prove that the inequality holds:
\[ \frac{\sqrt{a^3+b^3}}{a^2+b^2}+\frac{\sqrt{b^3+c^3}}{b^2+c^2}+\frac{\sqrt{c^3+a^3}}{c^2+a^2} \ge \frac{6(ab+bc+ac)}{(a+b+c)\sqrt{(a+b)(b+c)(c+a)}} \]
2017 Turkey EGMO TST, 3
For all positive real numbers $x,y,z$ satisfying the inequality $$\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,$$ prove that
$$\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.$$
1988 Nordic, 2
Let $a, b,$ and $c$ be non-zero real numbers and let $a \ge b \ge c$. Prove the inequality
$\frac{a^3 - c^3}{3} \ge abc (\frac{a- b}{c}+ \frac{b- c}{a})$ . When does equality hold?
1928 Eotvos Mathematical Competition, 1
Prove that, among the positive numbers
$$a,2a, ...,(n - 1)a.$$
there is one that differs from an integer by at most $1/n$.