Olympiad problems

2025 Malaysian IMO Team Selection Test, 7

Tags: inequalities , algebra

Given a real polynomial $P(x)=a_{2024}x^{2024}+\cdots+a_1x+a_0$ with degree $2024$, such that for all positive reals $b_1, b_2,\cdots, b_{2025}$ with product $1$, then; $$P(b_1)+P(b_2)+\cdots +P(b_{2025})\ge 0$$ Suppose there exist positive reals $c_1, c_2, \cdots, c_{2025}$ with product $1$, such that; $$P(c_1)+P(c_2)+ \cdots +P(c_{2025})=0$$ Is it possible that the values $c_1, c_2, \cdots, c_{2025}$ are all distinct? [i]Proposed by Ivan Chan Kai Chin[/i]

2000 All-Russian Olympiad, 5

Tags: trigonometry , inequalities

Prove the inequality \[ \sin^n (2x) + \left( \sin^n x - \cos^n x \right)^2 \le 1. \]

2023 Moldova Team Selection Test, 9

Tags: inequalities

Let $ n $ $(n\geq2)$ be an integer. Find the greatest possible value of the expression $$E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}$$ if the positive real numbers $a_1,a_2,\ldots,a_n$ satisfy $a_1+a_2+\ldots+a_n=\frac{n}{2}.$ What are the values of $a_1,a_2,\ldots,a_n$ when the greatest value is achieved?

2006 Moldova Team Selection Test, 3

Tags: inequalities

Positive real numbers $a,b,c$ satisfy the relation $abc=1$. Prove the inequality: $\frac{a+3}{(a+1)^{2}}+\frac{b+3}{(b+1)^{2}}+\frac{c+3}{(c+1)^{2}}\geq3$.

2004 Gheorghe Vranceanu, 2

Tags: equation , monotone , inequalities

Solve in $ \mathbb{R}^2 $ the following equation. $$ 9^{\sqrt x} +9^{\sqrt{y}} +9^{1/\sqrt{xy}} =\frac{81}{\sqrt{x} +\sqrt{y} +1/\sqrt{xy}} $$ [i]O. Trofin[/i]

2019 NMTC Junior, 3

Tags: inequalities , algebra

Find the number of permutations $x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8$ of the integers $-3, -2, -1, 0,1,2,3,4$ that satisfy the chain of inequalities $$x_1x_2\le x_2x_3\le x_3x_4\le x_4x_5\le x_5x_6\le x_6x_7\le x_7x_8.$$

2003 CHKMO, 3

Tags: inequalities

Let $a\geq b\geq c\geq 0$ are real numbers such that $a+b+c=3$. Prove that $ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}$ and find cases of equality.

2001 IMC, 6

Tags: inequalities , derivative

For each positive integer $n$, let $f_{n}(\vartheta)=\sin(\vartheta)\cdot \sin(2\vartheta) \cdot \sin(4\vartheta)\cdots \sin(2^{n}\vartheta)$. For each real $\vartheta$ and all $n$, prove that \[|f_{n}(\vartheta)| \leq \frac{2}{\sqrt{3}}|f_{n}(\frac{\pi}{3})| \]

2014 South East Mathematical Olympiad, 4

Tags: inequalities

Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\]

1989 APMO, 1

Tags: inequalities , n-variable inequality

Let $x_1$, $x_2$, $\cdots$, $x_n$ be positive real numbers, and let \[ S = x_1 + x_2 + \cdots + x_n. \] Prove that \[ (1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!} \]

2019 Korea National Olympiad, 1

Tags: inequalities

The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition. $a_1=1, a_{n+1}=2019a_{n}+1$ Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.) Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{2020-k}-1)^2$

2000 Moldova National Olympiad, Problem 1

Tags: inequalities

Positive numbers $a$ and $b$ satisfy $a^{1999}+b^{2000}{\ge}a^{2000}+b^{2001}$.Prove that $a^{2000}+b^{2000}{\leq}2$. _____________________________________ Azerbaijan Land of the Fire :lol:

2013 China Second Round Olympiad, 3

Tags: inequalities

The integers $n>1$ is given . The positive integer $a_1,a_2,\cdots,a_n$ satisfing condition : (1) $a_1<a_2<\cdots<a_n$; (2) $\frac{a^2_1+a^2_2}{2},\frac{a^2_2+a^2_3}{2},\cdots,\frac{a^2_{n-1}+a^2_n}{2}$ are all perfect squares . Prove that :$a_n\ge 2n^2-1.$

2018 Romania National Olympiad, 2

Tags: inequalities

Let $a,b,c \geq 0$ and $a+b+c=3.$ Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \geq \frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+a}$$

2009 Baltic Way, 2

Tags: algebra , polynomial , inequalities

Let $ a_1,a_{2},\ldots ,a_{100}$ be nonnegative integers satisfying the inequality \[a_1\cdot (a_1-1)\cdot\ldots\cdot (a_1-20)+a_2\cdot (a_2-1)\cdot\ldots\cdot (a_2-20)+\\ \ldots+a_{100}\cdot (a_{100}-1)\cdot\ldots\cdot (a_{100}-20)\le 100\cdot 99\cdot 98\cdot\ldots\cdot 79.\] Prove that $a_1+a_2+\ldots+a_{100}\le 9900$.

2015 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , maximum value

Let $a,b,c>0$ such that $a \geq bc^2$ , $b \geq ca^2$ and $c \geq ab^2$ . Find the maximum value that the expression : $$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive.

2007 IMO Shortlist, 3

Tags: inequalities , function , algebra , calculus

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

2015 South East Mathematical Olympiad, 2

Tags: inequalities , algebra

Given a sequence $\{ a_n\}_{n\in \mathbb{Z}^+}$ defined by $a_1=1$ and $a_{2k}=a_{2k-1}+a_k,a_{2k+1}=a_{2k}$ for all positive integer $k$. Prove that, for any positive integer $n$, $a_{2^n}>2^{\frac{n^2}{4}}$.

2002 IMO Shortlist, 2

Tags: geometry , homothety , inequalities , geometric inequality , triangle

Let $ABC$ be a triangle for which there exists an interior point $F$ such that $\angle AFB=\angle BFC=\angle CFA$. Let the lines $BF$ and $CF$ meet the sides $AC$ and $AB$ at $D$ and $E$ respectively. Prove that \[ AB+AC\geq4DE. \]

2001 Romania National Olympiad, 1

Tags: inequalities

Determine all real numbers $a$ and $b$ such that $a+b\in\mathbb{Z}$ and $a^2+b^2=2$.

2007 Indonesia TST, 1

Tags: inequalities

Let $ a,b,c$ be real numbers. Prove that $ (ab\plus{}bc\plus{}ca\minus{}1)^2 \le (a^2\plus{}1)(b^2\plus{}1)(c^2\plus{}1)$.

1979 IMO Longlists, 59

Tags: inequalities

Determine the maximum value of $x^2 y^2 z^2 w$ for $\{x,y,z,w\}\in\mathbb{R}^{+} \cup\{0\}$ and $2x+xy+z+yzw=1$.

1993 AMC 12/AHSME, 29

Tags: geometry , 3d geometry , prism , analytic geometry , inequalities , trigonometry , triangle inequality

Which of the following sets could NOT be the lengths of the external diagonals of a right rectangular prism [a "box"]? (An [i]external diagonal[/i] is a diagonal of one of the rectangular faces of the box.) $ \textbf{(A)}\ \{4, 5, 6\} \qquad\textbf{(B)}\ \{4, 5, 7\} \qquad\textbf{(C)}\ \{4, 6, 7\} \qquad\textbf{(D)}\ \{5, 6, 7\} \qquad\textbf{(E)}\ \{5, 7, 8\} $

2008 Bulgaria National Olympiad, 2

Tags: induction , inequalities

Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which \[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\] is satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.

2024 New Zealand MO, 5

Tags: inequalities

Determine the least real number $L$ such that $$\dfrac{1}{a}+\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}\leqslant L$$ for all quadruples $(a,b,c,d)$ of integers satisfying $1<a<b<c<d$.