Olympiad problems

2005 Croatia National Olympiad, 2

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.

1968 German National Olympiad, 5

Tags: algebra , inequalities , trigonometry

Prove that for all real numbers $x$ of the interval $0 < x <\pi$ the inequality $$\sin x +\frac12 \sin 2x +\frac13 \sin 3x > 0$$ holds.

2008 China National Olympiad, 2

Tags: inequalities , induction , combinatorics

Given an integer $n\ge3$, prove that the set $X=\{1,2,3,\ldots,n^2-n\}$ can be divided into two non-intersecting subsets such that neither of them contains $n$ elements $a_1,a_2,\ldots,a_n$ with $a_1<a_2<\ldots<a_n$ and $a_k\le\frac{a_{k-1}+a_{k+1}}2$ for all $k=2,\ldots,n-1$.

2012 Cuba MO, 4

Tags: inequalities , algebra

Let $x, y, z$ be positive reals. Prove that $$\frac{xz}{x^2 + xy + y^2 + 6z^2} + \frac{zx}{z^2 + zy + y^2 + 6x^2} + \frac{xy}{x^2 + xz + z^2 + 6y^2} \le \frac13$$

2015 JBMO TST - Turkey, 7

Tags: inequalities , number theory

For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$ [i]Proposed by Melih Üçer[/i]

1984 IMO Longlists, 65

Tags: analytic geometry , inequalities , tetrahedron , 3d geometry , sphere , geometry

A tetrahedron is inscribed in a sphere of radius $1$ such that the center of the sphere is inside the tetrahedron. Prove that the sum of lengths of all edges of the tetrahedron is greater than 6.

2014 China Western Mathematical Olympiad, 7

Tags: inequalities

In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$. Prove that\[|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.\]

1968 Swedish Mathematical Competition, 1

Tags: inequalities , algebra , min , max

Find the maximum and minimum values of $x^2 + 2y^2 + 3z^2$ for real $x, y, z$ satisfying $x^2 + y^2 + z^2 = 1$.

2000 Greece JBMO TST, 4

Tags: geometry , geometric inequality , inequalities

Let $a,b,c$ be sidelengths with $a\ge b\ge c$ and $s\ge a+1$ where $s$ be the semiperimeter of the triangle. Prove that $$ \frac{s-c}{\sqrt{a}}+\frac{s-b}{\sqrt{c}}+\frac{s-a}{\sqrt{b}}\ge \frac{s-b}{\sqrt{a}}+\frac{s-c}{\sqrt{b}}+\frac{s-a}{\sqrt{c}}$$

2012 ELMO Shortlist, 2

Tags: inequalities

Let $a,b,c$ be three positive real numbers such that $ a \le b \le c$ and $a+b+c=1$. Prove that \[\frac{a+c}{\sqrt{a^2+c^2}}+\frac{b+c}{\sqrt{b^2+c^2}}+\frac{a+b}{\sqrt{a^2+b^2}} \le \frac{3\sqrt{6}(b+c)^2}{\sqrt{(a^2+b^2)(b^2+c^2)(c^2+a^2)}}.\] [i]Owen Goff.[/i]

2006 China Team Selection Test, 1

Tags: inequalities , induction , algebra , limit , ratio

Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy: (a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$. (b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$. Find the general term of $\{ a_{n}\}$.

1985 Canada National Olympiad, 5

Tags: triangle inequality , induction , inequalities

Let $1 < x_1 < 2$ and, for $n = 1$, 2, $\dots$, define $x_{n + 1} = 1 + x_n - \frac{1}{2} x_n^2$. Prove that, for $n \ge 3$, $|x_n - \sqrt{2}| < 2^{-n}$.

2010 Contests, 4

Tags: inequalities , number theory

Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]

2011 AMC 12/AHSME, 14

Tags: probability , inequalities , conic , parabola

Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. What is the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$? $ \textbf{(A)}\ \frac{11}{81} \qquad \textbf{(B)}\ \frac{13}{81} \qquad \textbf{(C)}\ \frac{5}{27} \qquad \textbf{(D)}\ \frac{17}{81} \qquad \textbf{(E)}\ \frac{19}{81} $

1992 Taiwan National Olympiad, 3

Tags: induction , algebra , inequalities

If $x_{1},x_{2},...,x_{n}(n>2)$ are positive real numbers with $x_{1}+x_{2}+...+x_{n}=1$. Prove that $x_{1}^{2}x_{2}+x_{2}^{2}x_{3}+...+x_{n}^{2}x_{1}\leq\frac{4}{27}$.

1959 Poland - Second Round, 3

Tags: trigonometry , inequalities

Prove that if $ 0 \leq \alpha < \frac{\pi}{2} $ and $ 0 \leq \beta < \frac{\pi}{2} $, then $$ tg \frac{\alpha + \beta}{2} \leq \frac{tg \alpha + tg \beta}{2}.$$

2021 Saudi Arabia BMO TST, 3

Tags: inequalities , algebra

Let $a$, $b$, and $c$ be positive real numbers. Prove that $$(a^5 - a^2 +3)(b^5 - b^2 +3)(c^5 - c^2 +3)\ge (a+b+c)^3$$

2000 Irish Math Olympiad, 2

Tags: cyclic quadrilateral , trigonometry , geometry , inequalities

In a cyclic quadrilateral $ ABCD, a,b,c,d$ are its side lengths, $ Q$ its area, and $ R$ its circumradius. Prove that: $ R^2\equal{}\frac{(ab\plus{}cd)(ac\plus{}bd)(ad\plus{}bc)}{16Q^2}$. Deduce that $ R \ge \frac{(abcd)^{\frac{3}{4}}}{Q\sqrt{2}}$ with equality if and only if $ ABCD$ is a square.

2019 Romania National Olympiad, 2

Tags: inequalities , algebra

If $a,b,c\in(0,\infty)$ such that $a+b+c=3$, then $$\frac{a}{3a+bc+12}+\frac{b}{3b+ca+12}+\frac{c}{3c+ab+12}\le \frac{3}{16}$$

2007 Iran Team Selection Test, 1

Tags: inequalities , algebra , polynomial

Find all polynomials of degree 3, such that for each $x,y\geq 0$: \[p(x+y)\geq p(x)+p(y)\]

2013 China National Olympiad, 1

Tags: inequalities , induction , combinatorics , floor function

Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition \[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\] Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.

2012 District Olympiad, 2

Tags: algebra , inequalities

If $ a,b,c>0, $ then $ \sum_{\text{cyc}} \frac{a}{2a+b+c}\le 3/4. $

2018 Turkey EGMO TST, 5

Tags: inequalities

Prove that $\dfrac {x^2+1}{(x+y)^2+4 (z+1)}+\dfrac {y^2+1}{(y+z)^2+4 (x+1)}+\dfrac {z^2+1}{(z+x)^2+4 (y+1)} \ge \dfrac{1}{2} $ for all positive reals $x,y,z$

2015 Harvard-MIT Mathematics Tournament, 5

Tags: algebra , maximization , inequalities

Let $a,b,c$ be positive real numbers such that $a+b+c=10$ and $ab+bc+ca=25$. Let $m=\min\{ab,bc,ca\}$. Find the largest possible value of $m$.

2010 IMC, 2

Tags: induction , inequalities

Let $a_0,a_1,\dots,a_n$ be positive real numbers such that $a_{k+1}-a_k \geq 1$ for all $k=0,1,\dots,n-1.$ Prove that \[1+\frac{1}{a_0} \left( 1+\frac1{a_1-a_0}\right)\cdots\left(1+\frac1{a_n-a_0}\right)\leq \left(1+\frac1{a_0}\right) \left(1+\frac1{a_1}\right)\cdots \left(1+\frac1{a_n}\right).\]