Olympiad problems

2008 Iran Team Selection Test, 6

Tags: inequalities , combinatorics

Prove that in a tournament with 799 teams, there exist 14 teams, that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.

1978 Bulgaria National Olympiad, Problem 6

Tags: pyramid , geometry , inequalities , 3d geometry , geometric inequalities

The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$. If $AS$ is the longest edge of the pyramid prove that $BS>CS$. [i]Jordan Tabov[/i]

2019 Romania National Olympiad, 1

Tags: divisor , inequalities , number theory

Consider $A$, the set of natural numbers with exactly $2019$ natural divisors , and for each $n \in A$, denote $$S_n=\frac{1}{d_1+\sqrt{n}}+\frac{1}{d_2+\sqrt{n}}+...+\frac{1}{d_{2019}+\sqrt{n}}$$ where $d_1,d_2, .., d_{2019}$ are the natural divisors of $n$. Determine the maximum value of $S_n$ when $n$ goes through the set $ A$.

2013 AMC 12/AHSME, 24

Tags: trig identities , inequalities , triangle inequality , law of sines , trigonometry , probability , geometry

Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? $ \textbf{(A)} \ \frac{553}{715} \qquad \textbf{(B)} \ \frac{443}{572} \qquad \textbf{(C)} \ \frac{111}{143} \qquad \textbf{(D)} \ \frac{81}{104} \qquad \textbf{(E)} \ \frac{223}{286}$

2013 Puerto Rico Team Selection Test, 7

Tags: inequalities

Show that if $\sqrt{x}-\sqrt{y}=10$, then $x-2y\leq200$.

2015 Saudi Arabia GMO TST, 1

Tags: algebra , inequalities

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Prove that $$2 \left( \frac{ab}{a + b} +\frac{bc}{b + c} +\frac{ca}{c+ a}\right)+ 1 \ge 6(ab + bc + ca)$$ Trần Nam Dũng

2022 Czech-Polish-Slovak Junior Match, 4

Tags: number theory , inequalities

Let $a$ and $b$ be positive integers with the property that $\frac{a}{b} > \sqrt2$. Prove that $$\frac{a}{b} - \frac{1}{2ab} > \sqrt2$$

2010 Contests, 2

Tags: inequalities

Show that \[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \] for all positive real numbers $a, \: b, \: c.$

2005 Croatia National Olympiad, 3

Tags: logarithm , inequalities

If $a, b, c$ are real numbers greater than $1$, prove that for any real number $r$ \[(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}. \]

2013 Mediterranean Mathematics Olympiad, 3

Tags: inequalities

Let $x,y,z$ be positive reals for which: $\sum (xy)^{2}=6xyz$ Prove that: $\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}$.

2020 Thailand TST, 5

Tags: inequalities , algebra

Let $x, y, z$ be nonnegative real numbers such that $x + y + z = 3$. Prove that $$\frac{x}{4-y}+\frac{y}{4-z}+\frac{z}{4-x}+\frac{1}{16}(1-x)^2(1-y)^2(1-z)^2\leq 1,$$ and determine all such triples $(x, y, z)$ where the equality holds.

2005 MOP Homework, 5

Tags: inequalities , algebra

Let $a_1$, $a_2$, ..., $a_{2004}$ be non-negative real numbers such that $a_1+...+ a_{2004} \le 25$. Prove that among them there exist at least two numbers $a_i$ and $a_j$ ($i \neq j$) such that $|\sqrt{a_i}-\sqrt{a_j}| \le \frac{5}{2003}$.

2012 Online Math Open Problems, 20

Tags: strong induction , pigeonhole principle , induction , algorithm , inequalities

The numbers $1, 2, \ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$, erases them, and writes the number $2x+2y$ on the board. This continues until only one number $N$ remains. Find the remainder when the maximum possible value of $N$ is divided by 1000. [i]Victor Wang.[/i]

V Soros Olympiad 1998 - 99 (Russia), 11.3

Tags: algebra , trigonometry , inequalities

For what a from the interval $[0,\pi]$ do there exist $a$ and $b$ that are not simultaneously equal to zero, for which the inequality $$a \cos x + b \cos 2x \le 0$$ is satisfied for all $x$ belonging to the segment $[a, \pi]$?

1967 AMC 12/AHSME, 7

Tags: inequalities

If $\frac{a}{b}<-\frac{c}{d}$ where $a$, $b$, $c$, and $d$ are real numbers and $bd \not= 0$, then: $ \text{(A)}\ a \; \text{must be negative} \qquad \text{(B)}\ a \; \text{must be positive} \qquad$ $\text{(C)}\ a \; \text{must not be zero} \qquad \text{(D)}\ a \; \text{can be negative or zero, but not positive } \\ \text{(E)}\ a \; \text{can be positive, negative, or zero}$

2018 District Olympiad, 2

Tags: inequalities , algebra

Let $a,b,c \in [1, \infty)$. Prove that: $$\frac{a\sqrt{b}}{a+b}+\frac{b\sqrt{c}}{b+c}+\frac{c\sqrt{b}}{c+a}+\frac32 \le a+b+c$$

2018 Kyiv Mathematical Festival, 4

Tags: inequalities

For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge\frac{8y\sqrt{xy}}{3\sqrt{3}}.$

2022 Thailand TSTST, 2

Tags: inequalities

Let $a,b,c>0$ satisfy $a\geq b\geq c$. Prove that $$\frac{4}{a^2(b+c)}+\frac{4}{b^2(c+a)}+\frac{4}{c^2(a+b)} \leq \left(\sum_{cyc} \frac{a^2+1} {b^2} \right)\left(\sum_{cyc} \frac{b^3}{a^2(a^3+2b^3)}\right).$$

2009 Middle European Mathematical Olympiad, 5

Tags: inequalities , search

Let $ x$, $ y$, $ z$ be real numbers satisfying $ x^2\plus{}y^2\plus{}z^2\plus{}9\equal{}4(x\plus{}y\plus{}z)$. Prove that \[ x^4\plus{}y^4\plus{}z^4\plus{}16(x^2\plus{}y^2\plus{}z^2) \ge 8(x^3\plus{}y^3\plus{}z^3)\plus{}27\] and determine when equality holds.

2012 Romania National Olympiad, 3

Tags: real analysis , cauchy inequality , integration , function , inequalities

[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by \[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\] Determine the following two sets: [list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$ [b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]

2013 Saint Petersburg Mathematical Olympiad, 2

Tags: inequalities

if $a^2+b^2+c^2+d^2=1$ prove that \[ (1-a)(1-b)\ge cd. \] A. Khrabrov

2021 Thailand TSTST, 1

Tags: inequalities , algebra

Let $a,b,c$ be distinct positive real numbers such that $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\leq 1$. Prove that $$2\left(\sqrt{\frac{a+b}{ac}}+\sqrt{\frac{b+c}{ba}}+\sqrt{\frac{c+a}{cb}}\right)<\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)}.$$

1980 Bulgaria National Olympiad, Problem 4

Tags: inequalities , inequality

Let $a $, $b $, and $c $ be non-negative reals. Prove that $a^3+b^3+c^3+6abc\ge \frac{(a+b+c)^3}{4} $.

PEN R Problems, 7

Tags: limit , integration , 3d geometry , triangle inequality , geometry , calculus , inequalities

Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]

2003 China Team Selection Test, 2

Tags: geometry , linear algebra , ratio , inequalities , matrix , analytic geometry

In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.