Olympiad problems

2010 Tournament Of Towns, 3

Tags: inequalities

For each side of a given polygon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than $2$.

2017 Korea Winter Program Practice Test, 1

Tags: function , inequalities

Let $f : \mathbb{Z} \to \mathbb{R}$ be a function satisfying $f(x) + f(y) + f(z) \ge 0$ for all integers $x, y, z$ with $x + y + z = 0$. Prove that \[ f(-2017) + f(-2016) + \cdots + f(2016) + f(2017) \ge 0. \]

2010 China National Olympiad, 1

Tags: inequalities , floor function , algebra

Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that \[|T| \leq 1+ \frac{a_n-a_1}{2n+1}\] and for every $i \in \{1,2,\ldots , m\}$, there exists $t \in T$ and $s \in [-n,n]$, such that $a_i=t+s$.

2024 Poland - Second Round, 5

Tags: inequalities

The positive reals $a, b, c, x, y, z$ satisfy $$5a+4b+3c=5x+4y+3z.$$ Show that $$\frac{a^5}{x^4}+\frac{b^4}{y^3}+\frac{c^3}{z^2} \geq x+y+z.$$ [i]Proposed by Dominik Burek[/i]

2012 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , inequalities

For which real numbers $x$ and $\alpha$ inequality holds: $$\log _2 {x}+\log _x {2}+2\cos{\alpha} \leq 0$$

2018 International Zhautykov Olympiad, 5

Tags: inequalities , algebra , function , floor function

Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with $$f(x-f(y))=f(x)+a[y]$$ for all $x,y\in \mathbb{R}$

1978 IMO Longlists, 33

Tags: function , inequalities , algebra , logarithm

A sequence $(a_n)^{\infty}_0$ of real numbers is called [i]convex[/i] if $2a_n\le a_{n-1}+a_{n+1}$ for all positive integers $n$. Let $(b_n)^{\infty}_0$ be a sequence of positive numbers and assume that the sequence $(\alpha^nb_n)^{\infty}_0$ is convex for any choice of $\alpha > 0$. Prove that the sequence $(\log b_n)^{\infty}_0$ is convex.

2006 Korea - Final Round, 1

Tags: inequalities

Given three distinct real numbers $a_{1}, a_{2}, a_{3}$ , deﬁne $b_{j}= (1+\frac{a_{j}a_{i}}{a_{j}-a_{i}})(1+\frac{a_{j}a_{k}}{a_{j}-a_{k}})$, where $\{i, j, k\}= \{1, 2, 3\}$. Prove that $1+|a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}| \leq (1+|a_{1}|)(1+|a_{2}|)(1+|a_{3}|)$ and ﬁnd the cases of equality.

2012 ELMO Shortlist, 9

Tags: algebra , polynomial , function , induction , inequalities

Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that \[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\] and \[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\] [i]Calvin Deng.[/i]

2015 International Zhautykov Olympiad, 3

Tags: trigonometry , inequalities , geometry

The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of the triangles $ABC$, $BCD$, $CDE$, $DEA$, $EAB$ are $R_1$, $R_2$, $R_3$, $R_4$, $R_5$. Prove the inequality \[ R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2. \]

2009 China Western Mathematical Olympiad, 4

Tags: inequalities

The real numbers $a_{1},a_{2},\ldots ,a_{n}$ where $n\ge 3$ are such that $\sum_{i=1}^{n}a_{i}=0$ and $2a_{k}\le\ a_{k-1}+a_{k+1}$ for all $k=2,3,\ldots ,n-1$. Find the least $f(n)$ such that, for all $k\in\left\{1,2,\ldots ,n\right\}$, we have $|a_{k}|\le f(n)\max\left\{|a_{1}|,|a_{n}|\right\}$.

2023 China Second Round, 4

Tags: algebra , inequalities

Let $a=1+10^{-4}$. Consider some $2023\times 2023$ matrix with each entry a real in $[1,a]$. Let $x_i$ be the sum of the elements of the $i$-th row and $y_i$ be the sum of the elements of the $i$-th column for each integer $i\in [1,n]$. Find the maximum possible value of $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$ (the answer may be expressed in terms of $a$).

1996 German National Olympiad, 2

Tags: inequalities , algebra

Let $a$ and $b$ be positive real numbers smaller than $1$. Prove that the following two statements are equivalent: (i) $a+b = 1$, (ii) Whenever $x,y$ are positive real numbers such that $x < 1, y < 1, ax+by < 1$, the following inequlity holds: $$\frac{1}{1-ax-by} \le \frac{a}{1-x} + \frac{b}{1-y}$$

2021 Thailand Mathematical Olympiad, 3

Tags: inequalities

Let $a$, $b$, and $c$ be positive real numbers satisfying $ab+bc+ca=abc$. Determine the minimum value of $$a^abc + b^bca + c^cab.$$

1982 IMO Longlists, 48

Tags: inequalities , induction , complex number , combinatorial geometry , algebra

Given a finite sequence of complex numbers $c_1, c_2, \ldots , c_n$, show that there exists an integer $k$ ($1 \leq k \leq n$) such that for every finite sequence $a_1, a_2, \ldots, a_n$ of real numbers with $1 \geq a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$, the following inequality holds: \[\left| \sum_{m=1}^n a_mc_m \right| \leq \left| \sum_{m=1}^k c_m \right|.\]

2007 Indonesia MO, 3

Tags: inequalities

Let $ a,b,c$ be positive real numbers which satisfy $ 5(a^2\plus{}b^2\plus{}c^2)<6(ab\plus{}bc\plus{}ca)$. Prove that these three inequalities hold: $ a\plus{}b>c$, $ b\plus{}c>a$, $ c\plus{}a>b$.

2013 Korea Junior Math Olympiad, 1

Tags: algebra , inequalities

Compare the magnitude of the following three numbers. $$ \sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}} $$

2006 Mathematics for Its Sake, 2

Tags: inequalities , algebra , logarithm

For three real numbers $ a,b,c>1, $ prove the inequality: $ \log_{a^2b} a +\log_{b^2c} b +\log_{c^2a} c\le 1. $

2016 Estonia Team Selection Test, 9

Tags: combinatorics , inequalities , permutation

Let $n$ be a positive integer such that there exists a positive integer that is less than $\sqrt{n}$ and does not divide $n$. Let $(a_1, . . . , a_n)$ be an arbitrary permutation of $1, . . . , n$. Let $a_{i1} < . . . < a_{ik}$ be its maximal increasing subsequence and let $a_{j1} > . . . > a_{jl}$ be its maximal decreasing subsequence. Prove that tuples $(a_{i1}, . . . , a_{ik})$ and $(a_{j1}, . . . , a_{jl} )$ altogether contain at least one number that does not divide $n$.

2012 Today's Calculation Of Integral, 791

Tags: calculus , integration , algebra , function , domain , analytic geometry , inequalities

Let $S$ be the domain in the coordinate plane determined by two inequalities: \[y\geq \frac 12x^2,\ \ \frac{x^2}{4}+4y^2\leq \frac 18.\] Denote by $V_1$ the volume of the solid by a rotation of $S$ about the $x$-axis and by $V_2$, by a rotation of $S$ about the $y$-axis. (1) Find the values of $V_1,\ V_2$. (2) Compare the size of the value of $\frac{V_2}{V_1}$ and 1.

2013 Iran MO (3rd Round), 2

Tags: inequalities

Real numbers $a_1 , a_2 , \dots, a_n$ add up to zero. Find the maximum of $a_1 x_1 + a_2 x_2 + \dots + a_n x_n$ in term of $a_i$'s, when $x_i$'s vary in real numbers such that $(x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_{n-1} - x_n)^2 \leq 1$. (15 points)

2010 Baltic Way, 2

Tags: trigonometry , inequalities

Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that \[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]

2013 USAMTS Problems, 3

Tags: geometry , induction , ratio , inequalities , limit , arithmetic sequence

An infinite sequence of positive real numbers $a_1,a_2,a_3,\dots$ is called [i]territorial[/i] if for all positive integers $i,j$ with $i<j$, we have $|a_i-a_j|\ge\tfrac1j$. Can we find a territorial sequence $a_1,a_2,a_3,\dots$ for which there exists a real number $c$ with $a_i<c$ for all $i$?

2016 Dutch IMO TST, 1

Tags: algebra , inequalities

Prove that for all positive reals $a, b,c$ we have: $a +\sqrt{ab}+ \sqrt[3]{abc}\le \frac43 (a + b + c)$

Estonia Open Senior - geometry, 2005.2.4

Tags: trigonometry , geometric inequality , inequalities , 3d geometry , angle , geometry

Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} > \sin \frac{\gamma}{2}$.