Olympiad problems

2005 China Team Selection Test, 1

Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

2002 All-Russian Olympiad, 4

Tags: floor function , pigeonhole principle , inequalities , number theory

From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other.

1978 IMO Longlists, 8

Tags: inequalities , geometry

For two given triangles $A_1A_2A_3$ and $B_1B_2B_3$ with areas $\Delta_A$ and $\Delta_B$, respectively, $A_iA_k \ge B_iB_k, i, k = 1, 2, 3$. Prove that $\Delta_A \ge \Delta_B$ if the triangle $A_1A_2A_3$ is not obtuse-angled.

1949-56 Chisinau City MO, 50

Tags: trigonometry , inequalities , algebra

Prove the inequality: $ctg \frac{a}{2}> 1 + ctg a$ for $0 <a <\frac{\pi}{2}$

2014 Contests, 1

Tags: binomial coefficient , inequalities , algebra , polynomial , binomial theorem

Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that \[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]

2015 Saudi Arabia JBMO TST, 4

Tags: inequalities , algebra

Let $a,b$ and $c$ be positive numbers with $a^2+b^2+c^2=3$. Prove that $a+b+c\ge 3\sqrt[5]{abc}$.

2012 Chile National Olympiad, 2

Tags: number theory , sum , inequalities

Let $a_1,a_2,...,a_n$ be all positive integers with $2012$ digits or less, none of which is a $9$. Prove that $$ \frac{1}{a_1}+\frac{1}{a_2}+ ... +\frac{1}{a_{n}}\le 80.$$

2005 IMO, 3

Tags: inequalities , algebra

Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that \[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \] [i]Hojoo Lee, Korea[/i]

I Soros Olympiad 1994-95 (Rus + Ukr), 11.1

Tags: algebra , inequalities

Prove that for real $x\ge 1$, holds the inequality $$\frac{2^x +3^x }{3^x +4^x} \le \frac57$$

2019 IFYM, Sozopol, 5

Tags: algebra , inequalities

Let $a>0$ and $12a+5b+2c>0$. Prove that it is impossible for the equation $ax^2+bx+c=0$ to have two real roots in the interval $(2,3)$.

India EGMO 2022 TST, 1

Tags: inequalities , weighted am-gm

Let $n\ge 3$ be an integer, and suppose $x_1,x_2,\cdots ,x_n$ are positive real numbers such that $x_1+x_2+\cdots +x_n=1.$ Prove that $$x_1^{1-x_2}+x_2^{1-x_3}\cdots+x_{n-1}^{1-x_n}+x_n^{1-x_1}<2.$$ [i] ~Sutanay Bhattacharya[/i]

2018 Regional Olympiad of Mexico West, 2

Tags: algebra , inequalities

Let $a,b,c,d, e$ be real numbers such that they simultaneously satisfy the following equations $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ Determine the smallest and largest value that $a$ can take.

1982 All Soviet Union Mathematical Olympiad, 346

Tags: inequalities , algebra , floor function

Prove that the following inequality holds for all real $a$ and natural $n$: $$|a| \cdot |a-1|\cdot |a-2|\cdot ...\cdot |a-n| \ge \frac{n!F(a)}{2n}$$ $F(a)$ is the distance from $a$ to the closest integer.

2006 Romania Team Selection Test, 4

Tags: induction , inequalities

The real numbers $a_1,a_2,\dots,a_n$ are given such that $|a_i|\leq 1$ for all $i=1,2,\dots,n$ and $a_1+a_2+\cdots+a_n=0$. a) Prove that there exists $k\in\{1,2,\dots,n\}$ such that \[ |a_1+2a_2+\cdots+ka_k|\leq\frac{2k+1}{4}. \] b) Prove that for $n > 2$ the bound above is the best possible. [i]Radu Gologan, Dan Schwarz[/i]

1998 Singapore Senior Math Olympiad, 3

Tags: inequalities , radical , algebra

Prove that $\sqrt1+ \sqrt2+\sqrt3+...+ \sqrt{n^2-1}+\sqrt{n^2} \ge \frac{2n^3+n}{3}$ for any positive integer $n$.

1972 Polish MO Finals, 3

Tags: inequalities , algebra , polynomial

Prove that there is a polynomial $P(x)$ with integer coefficients such that for all $x$ in the interval $\left[ \frac{1}{10} , \frac{9}{10}\right]$ we have $$\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.$$

2004 Brazil Team Selection Test, Problem 1

Tags: inequality , inequalities , algebra

Let $x,y,z$ be positive numbers such that $x^2+y^2+z^2=1$. Prove that $$\frac x{1-x^2}+\frac y{1-y^2}+\frac z{1-z^2}\ge\frac{3\sqrt3}2$$

2012 Kazakhstan National Olympiad, 3

Tags: kanat satylkhanov , inequalities

Let $ a,b,c,d>0$ for which the following conditions:: $a)$ $(a-c)(b-d)=-4$ $b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$ Find the minimum of expression $a+c$

1978 Poland - Second Round, 1

Tags: inequalities , algebra

Prove that for positive real numbers $x$ and $y$ smaller than or equal to $1/2$, \[\frac{(x+y)^2}{xy} \geq \frac{(2-xy)^2}{(1-x)(1-y)}.\]

2012 Tuymaada Olympiad, 2

Tags: polynomial , quadratic , inequalities , sum of roots , algebra

Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$. [i]Proposed by A. Golovanov, M. Ivanov, K. Kokhas[/i]

1989 IberoAmerican, 3

Tags: inequalities

Let $a,b$ and $c$ be the side lengths of a triangle. Prove that: \[\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}<\frac{1}{16}\]

2007 Gheorghe Vranceanu, 2

Tags: inequalities , combinatorics , geometry

In the Euclidean plane, let be a point $ O $ and a finite set $ \mathcal{M} $ of points having at least two points. Prove that there exists a proper subset of $ \mathcal{M}, $ namely $ \mathcal{M}_0, $ such that the following inequality is true: $$ \sum_{P\in \mathcal{M}_0} OP\ge \frac{1}{4}\sum_{Q\in\mathcal{M}} OQ $$

2019 Jozsef Wildt International Math Competition, W. 46

Tags: inequalities , trigonometry

Let $x$, $y$, $z > 0$ such that $x^2 + y^2 + z^2 = 3$. Then $$x^3\tan^{-1}\frac{1}{x}+y^3\tan^{-1}\frac{1}{y}+z^3\tan^{-1}\frac{1}{z}<\frac{\pi \sqrt{3}}{2}$$

2011 National Olympiad First Round, 1

Tags: inequalities

Which one is true for a quadrilateral $ABCD$ such that perpendicular bisectors of $[AB]$ and $[CD]$ meet on the diagonal $[AC]$? $\textbf{(A)}\ |BA| + |AD| \leq |BC| + |CD| \\ \textbf{(B)}\ |BD| \leq |AC| \\ \textbf{(C)}\ |AC| \leq |BD| \\ \textbf{(D)}\ |AD| + |DC| \leq |AB| + |BC| \\ \textbf{(E)}\ \text{None}$

2002 Switzerland Team Selection Test, 6

Tags: sequence , inequalities , algebra

A sequence $x_1,x_2,x_3,...$ has the following properties: (a) $1 = x_1 < x_2 < x_3 < ...$ (b) $x_{n+1} \le 2n$ for all $n \in N$. Prove that for each positive integer $k$ there exist indices $i$ and $j$ such that $k =x_i -x_j$.