Olympiad problems

2017 Balkan MO Shortlist, C3

In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality: $A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$

2009 China Team Selection Test, 3

Tags: induction , inequalities

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$

2011 China Team Selection Test, 1

Tags: inequalities

Let $n\geq 3$ be an integer. Find the largest real number $M$ such that for any positive real numbers $x_1,x_2,\cdots,x_n$, there exists an arrangement $y_1,y_2,\cdots,y_n$ of real numbers satisfying \[\sum_{i=1}^n \frac{y_i^2}{y_{i+1}^2-y_{i+1}y_{i+2}+y_{i+2}^2}\geq M,\] where $y_{n+1}=y_1,y_{n+2}=y_2$.

2009 Princeton University Math Competition, 5

Tags: trigonometry , inequalities , calculus , derivative

Find the maximal positive integer $n$, so that for any real number $x$ we have $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$.

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 $a_i=t+s$.

2012 Dutch IMO TST, 2

Tags: inequalities

Let $a, b, c$ and $d$ be positive real numbers. Prove that $$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$

2011 Morocco National Olympiad, 3

Tags: inequalities

Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.

II Soros Olympiad 1995 - 96 (Russia), 10.1

Tags: trigonometry , algebra , inequalities

Find the largest and smallest value of the function $$y=\sqrt{7+5\cos x}-\cos x.$$

2012 ELMO Shortlist, 4

Tags: induction , floor function , ceiling function , inequalities , pigeonhole principle , combinatorics

A tournament on $2k$ vertices contains no $7$-cycles. Show that its vertices can be partitioned into two sets, each with size $k$, such that the edges between vertices of the same set do not determine any $3$-cycles. [i]Calvin Deng.[/i]

2010 Moldova Team Selection Test, 2

Tags: trigonometry , inequalities

Let $ x_1, x_2, \ldots, x_n$ be positive real numbers with sum $ 1$. Find the integer part of: $ E\equal{}x_1\plus{}\dfrac{x_2}{\sqrt{1\minus{}x_1^2}}\plus{}\dfrac{x_3}{\sqrt{1\minus{}(x_1\plus{}x_2)^2}}\plus{}\cdots\plus{}\dfrac{x_n}{\sqrt{1\minus{}(x_1\plus{}x_2\plus{}\cdots\plus{}x_{n\minus{}1})^2}}$

2013 Irish Math Olympiad, 10

Tags: inequalities

Let $a,b,c $ be real numbers and let $x=a+b+c,y=a^2+b^2+c^2,z=a^3+b^3+c^3$ and $S=2x^3-9xy+9z .$ $(a)$ Prove that $S$ is unchanged when $a,b,c$ are replaced by $a+t,b+t,c+t $ , respectively , for any real number $t$. $(b)$ Prove that $ (3y-x^2)^3\ge S^2 .$

2022 Philippine MO, 7

Tags: inequalities , algebra

Let $a, b,$ and $c$ be positive real numbers such that $ab + bc + ca = 3$. Show that \[ \dfrac{bc}{1 + a^4} + \dfrac{ca}{1 + b^4} + \dfrac{ab}{1 + c^4} \geq \dfrac{3}{2}. \]

1991 China Team Selection Test, 3

Tags: inequalities , geometry , 3d geometry , tetrahedron , combinatorics , graph theory

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

2005 Germany Team Selection Test, 3

Tags: inradius , trigonometry , inequalities , function , triangle inequality , geometry

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

2002 Kazakhstan National Olympiad, 2

Tags: inequalities , trigonometry , algebra

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

2018 IFYM, Sozopol, 4

Tags: algebra , inequalities

The real numbers $a$, $b$, $c$ are such that $a+b+c+ab+bc+ca+abc \geq 7$. Prove that $\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2} \geq 6$

1993 Moldova Team Selection Test, 7

Tags: inequalities

If $x_1 + x_2 + \cdots + x_n = \sum_{i=1}^{n} x_i = \frac{1}{2}$ and $x_i > 0$ ; then prove that: $ \frac{1-x_1}{1+x_1} \cdot \frac{1-x_2}{1+x_2} \cdots \frac{1-x_n}{1+x_n} = \prod_{i=1}^{n} \frac{1-x_i}{1+x_i} \geq \frac{1}{3}$

2007 Balkan MO Shortlist, C2

Tags: function , inequalities , domain , combinatorics , algebra

Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find \[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]

2000 India National Olympiad, 6

Tags: geometry , perimeter , inequalities , triangle inequality , combinatorics

For any natural numbers $n$, ( $n \geq 3$), let $f(n)$ denote the number of congruent integer-sided triangles with perimeter $n$. Show that (i) $f(1999) > f (1996)$; (ii) $f(2000) = f(1997)$.

2018 Cyprus IMO TST, 3

Tags: inequalities

Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.

2023 Romania Team Selection Test, P5

Tags: geometry , inequalities

Let $ABCDEF$ be a convex hexagon. The diagonals $AC$ and $BD$ cross at $P,$ the diagonals $AE{}$ and $DF$ cross at $Q,$ and the line $PQ$ crosses the sides $BC$ and $EF$ at $X$ and $Y,{}$ respectively. Prove that the length of the segment $XY$ does not exceed the sum of the lengths of one of the diagonals through $P{}$ and one of the diagonals through $Q{}$. [i]The Problem Selection Committee[/i]

1996 IMO Shortlist, 3

Tags: inequalities , algebra , sequence , recurrence

Let $ a > 2$ be given, and starting $ a_0 \equal{} 1, a_1 \equal{} a$ define recursively: \[ a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n.\] Show that for all integers $ k > 0,$ we have: $ \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).$

2008 ITest, 23

Tags: inequalities

Find the number of positive integers $n$ that are solutions to the simultaneous system of inequalities \begin{align*}4n-18&<2008,\\7n+17&>2008.\end{align*}

2007 Bulgaria Team Selection Test, 4

Tags: inequalities , combinatorics

Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.

2014 Contests, 3

Tags: number theory , combinatorics , algebra , inequalities

$N$ in natural. There are natural numbers from $N^3$ to $N^3+N$ on the board. $a$ numbers was colored in red, $b$ numbers was colored in blue. Sum of red numbers in divisible by sum of blue numbers. Prove, that $b|a$