Olympiad problems

2005 Miklós Schweitzer, 6

$SU_2(\mathbb{C})=\left\{\begin{pmatrix} z & w \\ -\bar{w} & \bar{z} \end{pmatrix} : z,w\in\mathbb{C} , z\bar{z}+w\bar{w}=1\right\}$ A and B are 2 elements of the above matrix group and have eigenvalues $e^{i\theta_1}$ , $e^{-i\theta_1}$ and $e^{i\theta_2}$ , $e^{-i\theta_2}$respectively, where $0\leq\theta_i\leq\pi$ . Prove that if AB has eigenvalue $e^{i\theta_3}$ , then $\theta_3$ satisfies the inequality $|\theta_1-\theta_2|\leq\theta_3\leq \min\{\theta_1+\theta_2 , 2\pi-(\theta_1+\theta_2)\}$

2024 Mozambique National Olympiad, P2

Tags: algebra , ident , identity , inequalities

Prove that if $a+b+c=0$ then $a^3+b^3+c^3=3abc$

1990 Poland - Second Round, 5

Tags: algebra , inequalities

There are $ n $ natural numbers ($ n\geq 2 $) whose sum is equal to their product. Prove that this common value does not exceed $2n$.

2013 All-Russian Olympiad, 2

Tags: inequalities

Let $a,b,c,d$ be positive real numbers such that $ 2(a+b+c+d)\ge abcd $. Prove that \[ a^2+b^2+c^2+d^2 \ge abcd .\]

2007 Mongolian Mathematical Olympiad, Problem 4

Tags: algebra , inequalities

Let $ a,b,c>0$. Prove that $ \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}$

2020 DMO Stage 1, 3.

Tags: inequality , inequalities

[b]Q.[/b] Prove that: $$\sum_{\text{cyc}}\tan (\tan A) - 2 \sum_{\text{cyc}} \tan \left(\cot \frac{A}{2}\right) \geqslant -3 \tan (\sqrt 3)$$where $A, B$ and $C$ are the angles of an acute-angled $\triangle ABC$. [i]Proposed by SA2018[/i]

2008 China Team Selection Test, 5

Tags: function , inequalities , cauchy inequality , algebra

For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]

2022 Puerto Rico Team Selection Test, 6

Tags: algebra , inequalities

Let $f$ be a function defined on $[0, 2022]$, such that $f(0) = f(2022) = 2022$, and $$|f(x) - f(y)| \le 2|x -y|,$$ for all $x, y$ in $[0, 2022]$. Prove that for each $x, y$ in $[0, 2022]$, the distance between $f(x)$ and $f(y)$ does not exceed $2022$.

2011 Balkan MO Shortlist, A3

Tags: inequalities

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

1997 Putnam, 4

Tags: inequalities , floor function

Let $a_{m,n}$ denote the coefficient of $x^n$ in the expansion $(1+x+x^2)^n$. Prove the inequality for all integers $k\ge 0$ : \[ 0\le \sum_{\ell=0}^{\left\lfloor{\frac{2k}{3}}\right\rfloor} (-1)^{\ell} a_{k-\ell,\ell}\le 1 \]

2014 Singapore Senior Math Olympiad, 7

Tags: trigonometry , inequalities

Find the largest number among the following numbers: $ \textbf{(A) }\tan47^{\circ}+\cos47^{\circ}\qquad\textbf{(B) }\cot 47^{\circ}+\sqrt{2}\sin 47^{\circ}\qquad\textbf{(C) }\sqrt{2}\cos47^{\circ}+\sin47^{\circ}\qquad\textbf{(D) }\tan47^{\circ}+\cot47^{\circ}\qquad\textbf{(E) }\cos47^{\circ}+\sqrt{2}\sin47^{\circ} $

2012 Thailand Mathematical Olympiad, 9

Tags: polynomial , inequalities , real root , algebra

Let $n$ be a positive integer and let $P(x) = x^n + a_{n-1}x^{n-1} +... + a_1x + 1$ be a polynomial with positive real coefficients. Under the assumption that the roots of $P$ are all real, show that $P(x) \ge (x + 1)^n$ for all $x > 0$.

2022 Indonesia TST, A

Tags: algebra , fe , inequalities , function

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying \[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.

2011 Junior Balkan MO, 4

Tags: rectangle , inequalities , trigonometry , vector , triangle inequality , geometry

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

2012 IMC, 5

Tags: inequalities , abstract algebra , induction , group theory

Let $c \ge 1$ be a real number. Let $G$ be an Abelian group and let $A \subset G$ be a finite set satisfying $|A+A| \le c|A|$, where $X+Y:= \{x+y| x \in X, y \in Y\}$ and $|Z|$ denotes the cardinality of $Z$. Prove that \[|\underbrace{A+A+\dots+A}_k| \le c^k |A|\] for every positive integer $k$. [i]Proposed by Przemyslaw Mazur, Jagiellonian University.[/i]

2015 Dutch Mathematical Olympiad, 5

Tags: algebra , inequalities

Given are (not necessarily positive) real numbers $a, b$, and $c$ for which $|a - b| \ge |c| , |b - c| \ge |a|$ and $|c - a| \ge |b|$ . Prove that one of the numbers $a, b$, and $c$ is the sum of the other two.

2011 AMC 12/AHSME, 19

Tags: floor function , function , inequalities , algebra , logarithm

At a competition with $N$ players, the number of players given elite status is equal to \[2^{1+\lfloor\log_2{(N-1)}\rfloor} - N. \] Suppose that $19$ players are given elite status. What is the sum of the two smallest possible values of $N$? $ \textbf{(A)}\ 38\qquad \textbf{(B)}\ 90 \qquad \textbf{(C)}\ 154 \qquad \textbf{(D)}\ 406 \qquad \textbf{(E)}\ 1024$

2009 Indonesia TST, 2

Tags: parameterization , inequalities , algebra , logarithm

Find the value of real parameter $ a$ such that $ 2$ is the smallest integer solution of \[ \frac{x\plus{}\log_2 (2^x\minus{}3a)}{1\plus{}\log_2 a} >2.\]

2017 Iran Team Selection Test, 1

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

2010 Germany Team Selection Test, 2

Tags: inequalities

Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\] [i]Proposed by Dzianis Pirshtuk, Belarus[/i]

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 $$\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3$$

2020 Jozsef Wildt International Math Competition, W52

Tags: calculus , derivative , inequalities

If $f\in C^{(3)}([0,1])$ such that $f(0)=f(1)=f'(0)=0$ and $|f'''(x)|\le1,(\forall)x\in[0,1]$, show that: a) $$|f(x)|\le\frac{x(1-x)}{\sqrt3}\cdot\left(\int^x_0\frac{f(t)}{t(1-t)}dt\right)^{1/2},(\forall)x\in[0,1]$$ b) $$|f'(x)|\le\frac{1-2x}{\sqrt3}\cdot\left(\int^x_0\frac{|f(t)|}{t(1-t)}dt\right)^{1/2},(\forall)x\in\left[0,\frac12\right]$$ c) $$\int^1_0(1-x)^2\cdot\frac{|f(x)|}xdx\ge9\int^1_0\left(\frac{f(x)}x\right)^2dx$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

1992 China Team Selection Test, 2

Tags: inequalities , algebra

Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

2008 Germany Team Selection Test, 2

Tags: inequalities , circumcircle , trigonometry , geometry

For three points $ X,Y,Z$ let $ R_{XYZ}$ be the circumcircle radius of the triangle $ XYZ.$ If $ ABC$ is a triangle with incircle centre $ I$ then we have: \[ \frac{1}{R_{ABI}} \plus{} \frac{1}{R_{BCI}} \plus{} \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} \plus{} \frac{1}{\bar{BI}} \plus{} \frac{1}{\bar{CI}}.\]

2009 Romania National Olympiad, 3

Tags: inequalities , permutation , arrangement

Let be a natural number $ n, $ a permutation $ \sigma $ of order $ n, $ and $ n $ nonnegative real numbers $ a_1,a_2,\ldots , a_n. $ Prove the following inequality. $$ \left( a_1^2+a_{\sigma (1)} \right)\left( a_2^2+a_{\sigma (2)} \right)\cdots \left( a_n^2+a_{\sigma (n)} \right)\ge \left( a_1^2+a_1 \right)\left( a_2^2+a_{2} \right)\cdots \left( a_n^2+a_n \right) $$