Olympiad problems

1993 All-Russian Olympiad, 1

Tags: prime number , area of a triangle , heron's formula , perimeter , number theory , inequalities , geometry

The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.

1953 Moscow Mathematical Olympiad, 238

Tags: inequalities , algebra , radical , nested radical

Prove that if in the following fraction we have $n$ radicals in the numerator and $n - 1$ in the denominator, then $$\frac{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}{2-\sqrt{2+\sqrt{2+...+\sqrt{2}}}}>\frac14$$

2007 Italy TST, 3

Tags: geometric transformation , geometry , prime number , number theory , reflection , inequalities

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

2002 Greece Junior Math Olympiad, 4

Tags: function , induction , inequalities

Prove that $1\cdot2\cdot3\cdots 2002<\left(\frac{2003}{2}\right)^{2002}.$

1999 Kazakhstan National Olympiad, 8

Tags: permutation , max , n-variable inequality , inequalities , algebra

Let $ {{a} _ {1}}, {{a} _ {2}}, \ldots, {{a} _ {n}} $ be permutation of numbers $ 1,2, \ldots, n $, where $ n \geq 2 $. Find the maximum value of the sum $$ S (n) = | {{a} _ {1}} - {{a} _ {2}} | + | {{a} _ {2}} - {{a} _ {3}} | + \cdots + | {{a} _ {n-1}} - {{a} _ {n}} |. $$

2015 South East Mathematical Olympiad, 2

Tags: algebra , inequalities

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 $n$, $a_{2^n}>2^{\frac{n^2}{4}}$.

2001 National High School Mathematics League, 2

Tags: inequalities

If $x_i\geq0(i=1,2,\cdots,n)$, and $$\sum_{i=1}^n x_i^2 + 2\sum_{1 \leq k < j \leq n} \sqrt{\frac{k}{j}}x_kx_j = 1.$$ Find the maximum and minumum value of $\sum_{i=1}^n x_i$.

2024 Balkan MO, 3

Tags: number theory , divisibility , inequality , inequalities

Let $a$ and $b$ be distinct positive integers such that $3^a + 2$ is divisible by $3^b + 2$. Prove that $a > b^2$. [i]Proposed by Tynyshbek Anuarbekov, Kazakhstan[/i]

2019 Junior Balkan MO, 2

Tags: algebra , inequalities

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

2021 Kyiv City MO Round 1, 11.4

Tags: inequalities , algebra

For positive real numbers $a, b, c$ with sum $\frac{3}{2}$, find the smallest possible value of the following expression: $$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}$$ [i]Proposed by Serhii Torba[/i]

1998 Korea Junior Math Olympiad, 6

Tags: inequalities

For positive reals $a \geq b \geq c \geq 0$ prove the following inequality: $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a+b}{a+c}+\frac{b+c}{b+a}+\frac{c+a}{c+b}$$

2011 Canadian Students Math Olympiad, 3

Tags: inequalities

Find the largest $C \in \mathbb{R}$ such that \[\frac{x+z}{(x-z)^2} +\frac{x+w}{(x-w)^2} +\frac{y+z}{(y-z)^2}+\frac{y+w}{(y-w)^2} + \sum_{cyc} \frac{1}{x} \ge \frac{C}{x+y+z+w}\] where $x,y,z,w \in \mathbb{R^+}$. [i]Author: Hunter Spink[/i]

2016 China Western Mathematical Olympiad, 8

Tags: number theory , inequalities , algebra

For any given integers $m,n$ such that $2\leq m<n$ and $(m,n)=1$. Determine the smallest positive integer $k$ satisfying the following condition: for any $m$-element subset $I$ of $\{1,2,\cdots,n\}$ if $\sum_{i\in I}i> k$, then there exists a sequence of $n$ real numbers $a_1\leq a_2 \leq \cdots \leq a_n$ such that $$\frac1m\sum_{i\in I} a_i>\frac1n\sum_{i=1}^na_i$$

1994 Tuymaada Olympiad, 5

Tags: trigonometry , inequalities , trigonometric inequality , minimum

Find the smallest natural number $n$ for which $sin \Big(\frac{1}{n+1934}\Big)<\frac{1}{1994}$ .

2011 N.N. Mihăileanu Individual, 3

Tags: inequalities

Let a,b,c>0 with ab+bc+ca=1. Prove that: $\frac{b^3c}{a^2+b^2}+\frac{c^3a}{b^2+c^2}+\frac{a^3b}{c^2+a^2}\ge\frac{1}{2}.$

2003 Irish Math Olympiad, 1

Tags: geometry , inequalities , perimeter , triangle inequality

If $a,b,c$ are the sides of a triangle whose perimeter is equal to 2 then prove that: a) $abc+\frac{28}{27}\geq ab+bc+ac$; b) $abc+1<ab+bc+ac$ See also [url]http://www.mathlinks.ro/Forum/viewtopic.php?t=47939&view=next[/url] (problem 1) :)

1992 Romania Team Selection Test, 9

Tags: inequalities

Let $x, y$ be real numbers such that $1\le x^2-xy+y^2\le2$. Show that: a) $\dfrac{2}{9}\le x^4+y^4\le 8$; b) $x^{2n}+y^{2n}\ge\dfrac{2}{3^n}$, for all $n\ge3$. [i]Laurențiu Panaitopol[/i] and [i]Ioan Tomescu[/i]

2021 Canadian Junior Mathematical Olympiad, 4

Tags: n-variable inequality , inequalities , algebra

Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$. Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$

2021 Israel TST, 3

Tags: inequalities

What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*} holds for any $8$ real numbers $a,b,c,d,x,y,z,t$? Edit: Fixed a mistake! Thanks @below.

2012 European Mathematical Cup, 3

Tags: inequalities

Prove that the following inequality holds for all positive real numbers $a$, $b$, $c$, $d$, $e$ and $f$ \[\sqrt[3]{\frac{abc}{a+b+d}}+\sqrt[3]{\frac{def}{c+e+f}} < \sqrt[3]{(a+b+d)(c+e+f)} \text{.}\] [i]Proposed by Dimitar Trenevski.[/i]

2010 Germany Team Selection Test, 3

Tags: function , inequalities , algebra , functional inequality

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

1966 IMO Shortlist, 5

Tags: trigonometry , geometric inequality , inequalities

Prove the inequality \[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\] for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$

2009 Irish Math Olympiad, 5

Tags: inequalities

Hello. Suppose $a$, $b$, $c$ are real numbers such that $a+b+c = 0$ and $a^{2}+b^{2}+c^{2} = 1$. Prove that $a^{2}b^{2}c^{2}\leq \frac{1}{54}$ and determine the cases of equality.

2004 District Olympiad, 2

Tags: algebra , inequalities

The real numbers $a, b, c, d$ satisfy $a > b > c > d$ and $$a + b + c + d = 2004 \,\,\, and \,\,\, a^2 - b^2 + c^2 - d^2 = 2004.$$ Answer, with proof, to the following questions: a) What is the smallest possible value of $a$? b) What is the number of possible values of $a$?

1987 IMO Longlists, 44

Tags: trigonometry , modular arithmetic , inequalities

Let $\theta_1,\theta_2,\cdots,\theta_n$ be $n$ real numbers such that $\sin \theta_1+\sin \theta_2+\cdots+\sin \theta_n=0$. Prove that \[|\sin \theta_1+2 \sin \theta_2+\cdots +n \sin \theta_n| \leq \left[ \frac{n^2}{4} \right]\]