Found problems: 592
2016 JBMO Shortlist, 1
Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that
$\frac{ab + 4}{a + 2}+\frac{bc + 4}{b + 2}+\frac{ca + 4}{c + 2}\ge 6$.
1983 Czech and Slovak Olympiad III A, 2
Given a triangle $ABC$, prove that for every inner point $P$ of the side $AB$ the inequality $$PC\cdot AB<PA\cdot BC+PB\cdot AC$$ holds.
2024 Korea National Olympiad, 5
Find the smallest real number $M$ such that
$$\sum_{k = 1}^{99}\frac{a_{k+1}}{a_k+a_{k+1}+a_{k+2}} < M$$
for all positive real numbers $a_1, a_2, \dots, a_{99}$. ($a_{100} = a_1, a_{101} = a_2$)
2004 USAMO, 5
Let $a, b, c > 0$. Prove that $(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \geq (a + b + c)^3$.
2020 India National Olympiad, 4
Let $n \geqslant 2$ be an integer and let $1<a_1 \le a_2 \le \dots \le a_n$ be $n$ real numbers such that $a_1+a_2+\dots+a_n=2n$. Prove that$$a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.$$
[i]Proposed by Kapil Pause[/i]
2005 Korea Junior Math Olympiad, 7
If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}}
$$
2009 Korea Junior Math Olympiad, 6
If positive reals $a,b,c,d$ satisfy $abcd = 1.$ Prove the following inequality $$1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.$$
2024 Belarusian National Olympiad, 11.7
Positive real numbers $a_1,a_2,\ldots, a_n$ satisfy the equation $$2a_1+a_2+\ldots+a_{n-1}=a_n+\frac{n^2-3n+2}{2}$$
For every positive integer $n \geq 3$ find the smallest possible value of the sum $$\frac{(a_1+1)^2}{a_2}+\ldots+\frac{(a_{n-1}+1)^2}{a_n}$$
[i]M. Zorka[/i]
2021 Serbia JBMO TSTs, 1
Prove that for positive real numbers $a, b, c$ the following inequality holds:
\begin{align*}
\frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2}
\end{align*}
When does equality occur?
2006 Singapore Team Selection Test, 2
Let n be an integer greater than 1 and let $x_1, x_2, . . . , x_n$ be real numbers such that
$|x_1| + |x_2| + ... + |x_n| = 1$ and $x_1 + x_2 + ... + x_n = 0$
Prove that
$\left| \frac{x_1}{1}+\frac{x_2}{2}+\cdots+\frac{x_n}{n} \right| \leq \frac{1}{2} \left(1-\frac{1}{n}\right)$
1985 IMO Shortlist, 18
Let $x_1, x_2, \cdots , x_n$ be positive numbers. Prove that
\[\frac{x_1^2}{x_1^2+x_2x_3} + \frac{x_2^2}{x_2^2+x_3x_4} + \cdots +\frac{x_{n-1}^2}{x_{n-1}^2+x_nx_1} +\frac{x_n^2}{x_n^2+x_1x_2} \leq n-1\]
1975 IMO Shortlist, 14
Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that
\[45 < x_{1000} < 45. 1.\]
2025 JBMO TST - Turkey, 6
Find the minimum value of
\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]
where $x,y,z>1$ are reals.
2002 Rioplatense Mathematical Olympiad, Level 3, 2
Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5 $.
1987 Czech and Slovak Olympiad III A, 4
Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$
2025 Taiwan Mathematics Olympiad, 2
Let $a, b, c, d$ be four positive reals such that $abc+abd+acd+bcd = 1$. Determine all possible values for
$$(ab + cd)(ac + bd)(ad + bc).$$
[i]Proposed by usjl and YaWNeeT[/i]
2013 Vietnam Team Selection Test, 4
Find the greatest positive integer $k$ such that the following inequality holds for all $a,b,c\in\mathbb{R}^+$ satisfying $abc=1$ \[ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{k}{a+b+c+1}\geqslant 3+\frac{k}{4} \]
2021 Alibaba Global Math Competition, 6
Let $M(t)$ be measurable and locally bounded function, that is,
\[M(t) \le C_{a,b}, \quad \forall 0 \le a \le t \le b<\infty\]
with some constant $C_{a,b}$, from $[0,\infty)$ to $[0,\infty)$ such that
\[M(t) \le 1+\int_0^t M(t-s)(1+t)^{-1}s^{-1/2} ds, \quad \forall t \ge 0.\]
Show that
\[M(t) \le 10+2\sqrt{5}, \quad \forall t \ge 0.\]
1972 IMO Shortlist, 3
The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that
\[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]
2016 Nigerian Senior MO Round 2, Problem 6
Given that $a, b, c, d \in \mathbb{R}$, prove that $(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)$.
1993 IMO Shortlist, 8
Let $c_1, \ldots, c_n \in \mathbb{R}$ with $n \geq 2$ such that \[ 0 \leq \sum^n_{i=1} c_i \leq n. \] Show that we can find integers $k_1, \ldots, k_n$ such that \[ \sum^n_{i=1} k_i = 0 \] and \[ 1-n \leq c_i + n \cdot k_i \leq n \] for every $i = 1, \ldots, n.$
[hide="Another formulation:"]
Let $x_1, \ldots, x_n,$ with $n \geq 2$ be real numbers such that \[ |x_1 + \ldots + x_n| \leq n. \] Show that there exist integers $k_1, \ldots, k_n$ such that \[ |k_1 + \ldots + k_n| = 0. \] and \[ |x_i + 2 \cdot n \cdot k_i| \leq 2 \cdot n -1 \] for every $i = 1, \ldots, n.$ In order to prove this, denote $c_i = \frac{1+x_i}{2}$ for $i = 1, \ldots, n,$ etc.
[/hide]
1989 IMO Longlists, 55
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
2021 CHKMO, 4
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that
\[\dfrac{1}{a^3+2b^2+2b+4}+\dfrac{1}{b^3+2c^2+2c+4}+\dfrac{1}{c^3+2a^2+2a+4}\leq \dfrac13.\]
1967 IMO Shortlist, 6
Prove the following inequality:
\[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1}
x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in
\mathbb{N}.$
2021 Switzerland - Final Round, 4
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]