Olympiad problems

2005 Thailand Mathematical Olympiad, 20

Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?

2014 Turkey MO (2nd round), 6

Tags: combinatorics , inequalities

$5$ airway companies operate in a country consisting of $36$ cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities $A, B$ and $B, C$ we say that the triple $A, B, C$ is [i]properly-connected[/i]. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.

2010 China Team Selection Test, 2

Tags: inequalities

Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds: $\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$.

V Soros Olympiad 1998 - 99 (Russia), 11.2

Tags: inequalities , algebra

Find the greatest value of $C$ for which, for any $x, y, z,u$, and such that for $0\le x\le y \le z\le u$, holds the inequality $$(x + y +z + u)^2 \ge Cyz .$$

2012 Macedonia National Olympiad, 2

Tags: inequalities

If $~$ $a,\, b,\, c,\, d$ $~$ are positive real numbers such that $~$ $abcd=1$ $~$ then prove that the following inequality holds \[ \frac{1}{bc+cd+da-1} + \frac{1}{ab+cd+da-1} + \frac{1}{ab+bc+da-1} + \frac{1}{ab+bc+cd-1}\; \le\; 2\, . \] When does inequality hold?

2024 Poland - Second Round, 5

Tags: inequalities

The positive reals $a, b, c, x, y, z$ satisfy $$5a+4b+3c=5x+4y+3z.$$ Show that $$\frac{a^5}{x^4}+\frac{b^4}{y^3}+\frac{c^3}{z^2} \geq x+y+z.$$ [i]Proposed by Dominik Burek[/i]

2013 IMC, 4

Tags: function , inequalities

Let $\displaystyle{n \geqslant 3}$ and let $\displaystyle{{x_1},{x_2},...,{x_n}}$ be nonnegative real numbers. Define $\displaystyle{A = \sum\limits_{i = 1}^n {{x_i}} ,B = \sum\limits_{i = 1}^n {x_i^2} ,C = \sum\limits_{i = 1}^n {x_i^3} }$. Prove that: \[\displaystyle{\left( {n + 1} \right){A^2}B + \left( {n - 2} \right){B^2} \geqslant {A^4} + \left( {2n - 2} \right)AC}.\] [i]Proposed by Géza Kós, Eötvös University, Budapest.[/i]

2005 China Team Selection Test, 1

Tags: algebra , inequalities

Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.

2008 Thailand Mathematical Olympiad, 6

Tags: functional inequality , algebra , inequalities , functional

Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$. Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right| < 1$ for all real numbers $x$.

2015 Iran MO (3rd round), 6

Tags: inequalities , algebra

$a_1,a_2,\dots ,a_n>0$ are positive real numbers such that $\sum_{i=1}^{n} \frac{1}{a_i}=n$ prove that: $\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)$

2013 Czech-Polish-Slovak Match, 2

Tags: perimeter , inequalities , geometry , combinatorics

Triangular grid divides an equilateral triangle with sides of length $n$ into $n^2$ triangular cells as shown in figure for $n=12$. Some cells are infected. A cell that is not yet infected, ia infected when it shares adjacent sides with at least two already infected cells. Specify for $n=12$, the least number of infected cells at the start in which it is possible that over time they will infected all the cells of the original triangle. [asy] unitsize(0.25cm); path p=polygon(3); for(int m=0; m<=11;++m){ for(int n=0 ; n<= 11-m; ++n){ draw(shift((n+0.5*m)*sqrt(3),1.5*m)*p); } } [/asy]

2012 Germany Team Selection Test, 3

Tags: inequalities

Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that: $$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$

2009 Indonesia TST, 1

Tags: logarithm , combinatorics , inequalities , induction

2008 persons take part in a programming contest. In one round, the 2008 programmers are divided into two groups. Find the minimum number of groups such that every two programmers ever be in the same group.

2004 Austrian-Polish Competition, 2

Tags: geometry , trigonometry , inequalities , angle bisector

In a triangle $ABC$ let $D$ be the intersection of the angle bisector of $\gamma$, angle at $C$, with the side $AB.$ And let $F$ be the area of the triangle $ABC.$ Prove the following inequality: \[2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.\]

1991 Greece National Olympiad, 1

Tags: inequalities , algebra

Let $a, b$ be two reals such that $a+b<2ab$. Prove that $a+b>2$

1985 Canada National Olympiad, 4

Tags: prime factorization , logarithm , inequalities , number theory , floor function

Prove that $2^{n - 1}$ divides $n!$ if and only if $n = 2^{k - 1}$ for some positive integer $k$.

2009 Miklós Schweitzer, 9

Tags: inequalities , invariant , integration , real analysis , function , calculus

Let $ P\subseteq \mathbb{R}^m$ be a non-empty compact convex set and $ f: P\rightarrow \mathbb{R}_{ \plus{} }$ be a concave function. Prove, that for every $ \xi\in \mathbb{R}^m$ \[ \int_{P}\langle \xi,x \rangle f(x)dx\leq \left[\frac {m \plus{} 1}{m \plus{} 2}\sup_{x\in P}{\langle\xi,x\rangle} \plus{} \frac {1}{m \plus{} 2}\inf_{x\in P}{\langle\xi,x\rangle}\right] \cdot\int_{P}f(x)dx.\]

2011 All-Russian Olympiad Regional Round, 9.4

Tags: three variable inequality , calculus , inequalities

$x$, $y$ and $z$ are positive real numbers. Prove the inequality \[\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\] (Authors: A. Khrabrov, B. Trushin)

2024 ISI Entrance UGB, P2

Tags: algebra , polynomial , inequalities

Suppose $n\ge 2$. Consider the polynomial \[Q_n(x) = 1-x^n - (1-x)^n .\] Show that the equation $Q_n(x) = 0$ has only two real roots, namely $0$ and $1$.

2006 Moldova Team Selection Test, 3

Tags: geometry , circumcircle , function , inequalities , abstract algebra , trigonometry

Let $a,b,c$ be sides of a triangle and $p$ its semiperimeter. Show that $a\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p$

2017 District Olympiad, 3

Tags: inequalities

Let $ a $ be a positive real number. Prove that $$ a^{\sin x}\cdot (a+1)^{\cos x}\ge a,\quad\forall x\in \left[ 0,\frac{\pi }{2} \right] . $$

2011 Brazil Team Selection Test, 3

Tags: algebra , polynomial , inequalities

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

2018 Czech-Polish-Slovak Match, 4

Tags: geometric inequalities , geometry , inequalities

Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i]

2006 Junior Balkan Team Selection Tests - Romania, 3

Tags: algebra , inequalities , trigonometry

Find all real numbers $ a$ and $ b$ such that \[ 2(a^2 \plus{} 1)(b^2 \plus{} 1) \equal{} (a \plus{} 1)(b \plus{} 1)(ab \plus{} 1). \] [i]Valentin Vornicu[/i]

2013 Romania Team Selection Test, 3

Tags: induction , function , inequalities , algebra

Determine all injective functions defined on the set of positive integers into itself satisfying the following condition: If $S$ is a finite set of positive integers such that $\sum\limits_{s\in S}\frac{1}{s}$ is an integer, then $\sum\limits_{s\in S}\frac{1}{f\left( s\right) }$ is also an integer.