Olympiad problems

2015 Indonesia MO Shortlist, A3

Tags: inequality , inequalities

Let $a,b,c$ positive reals such that $a^2+b^2+c^2=1$. Prove that $$\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3$$

2021 Science ON all problems, 1

Tags: algebra , complex number , inequality

Consider the complex numbers $x,y,z$ such that $|x|=|y|=|z|=1$. Define the number $$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$ $\textbf{(a)}$ Prove that $a$ is a real number. $\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$. [i] (Stefan Bălăucă & Vlad Robu)[/i]

Russian TST 2016, P3

Tags: inequality , geometry

Prove that for any points $A,B,C,D$ in the plane, the following inequality holds \[\frac{AB}{DA+DB}+\frac{BC}{DB+DC}\geqslant\frac{AC}{DA+DC}.\]

1972 IMO Shortlist, 3

Tags: inequality , optimization , maximization

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 \[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]

BIMO 2022, 1

Tags: algebra , inequalities , inequality

Let $a, b, c,$ be nonnegative reals with $ a+b+c=3 $, find the largest positive real $ k $ so that for all $a,b,c,$ we have $$ a^2+b^2+c^2+k(abc-1)\ge 3 $$

1994 All-Russian Olympiad Regional Round, 11.1

Tags: inequality , trigonometry , trigonometric inequality , inequalities

Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.

1969 IMO Shortlist, 56

Tags: decimal representation , inequality , calculus

Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

2021 Azerbaijan Junior NMO, 3

Tags: inequality

$a,b,c $ are positive real numbers . Prove that $\sqrt[7]{\frac{a}{b+c}+\frac{b}{c+a}} +\sqrt[7]{\frac{b}{c+a}+\frac{c}{b+a}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}}\geq 3$

2023 Junior Balkan Team Selection Tests - Romania, P2

Tags: inequality , shortlist , algebra

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

2020 Tuymaada Olympiad, 2

Tags: inequality , inequalities , n-variable inequality

Given positive real numbers $a_1, a_2, \dots, a_n$. Let \[ m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). \] Prove the inequality \[ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. \]

1967 IMO Shortlist, 2

Tags: factorial , inequality

Prove that \[\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\frac{2}{n}},\] and let $n \geq 1$ be an integer. Prove that this inequality is only possible in the case $n = 1.$

1983 IMO Longlists, 4

Tags: number theory , sum of divisors , inequality , divisor

Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

2014 Contests, 1

Tags: algebra , inequality , inequalities

Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.

2001 China Team Selection Test, 2

Tags: trigonometry , inequality , sequence , inequalities , analysis

Let ${a_n}$ be a non-increasing sequence of positive numbers. Prove that if for $n \ge 2001$, $na_{n} \le 1$, then for any positive integer $m \ge 2001$ and $x \in \mathbb{R}$, the following inequality holds: $\left | \sum_{k=2001}^{m} a_{k} \sin kx \right | \le 1 + \pi$

2021 Switzerland - Final Round, 4

Tags: algebra , inequality , 4-variable inequality , inequalities

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]

2022 Greece National Olympiad, 3

Tags: inequality , inequalities

The positive real numbers $a,b,c,d$ satisfy the equality $$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$ Find the maximum possible value of $a$.

2024 Greece Junior Math Olympiad, 1

Tags: algebra , inequality

a) Prove that for all real numbers $k,l,m$ holds : $$(k+l+m)^2 \ge 3 (kl+lm+mk)$$ When does equality holds? b) If $x,y,z$ are positive real numbers and $a,b$ real numbers such that $$a(x+y+z)=b(xy+yz+zx)=xyz,$$ prove that $a \ge 3b^2$. When does equality holds?