Olympiad problems

1996 Polish MO Finals, 3

From the set of all permutations $f$ of $\{1, 2, ... , n\}$ that satisfy the condition: $f(i) \geq i-1$ $i=1,...,n$ one is chosen uniformly at random. Let $p_n$ be the probability that the chosen permutation $f$ satisfies $f(i) \leq i+1$ $i=1,...,n$ Find all natural numbers $n$ such that $p_n > \frac{1}{3}$.

2013 ELMO Problems, 2

Tags: function , calculus , derivative , 3-variable inequality , logarithm , inequalities

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

2003 China Team Selection Test, 3

Tags: polynomial , inequalities , algebra

The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.

2014 Cono Sur Olympiad, 1

Tags: inequalities , number theory , least common multiple , invariant , greatest common divisor , combinatorics

Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$. Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$

2025 AIME, 12

Tags: inequalities

The set of points in $3$-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $$x-yz<y-zx<z-xy$$forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b},$ where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b.$

2023 China Second Round, 5

Tags: trigonometry , algebra , inequalities , equation

Find the sum of the smallest 20 positive real solutions of the equation $\sin x=\cos 2x .$

2011 IFYM, Sozopol, 3

Tags: algebra , maximum value , inequalities

If $x$ and $y$ are real numbers, determine the greatest possible value of the expression $\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.

2013 Uzbekistan National Olympiad, 2

Tags: trigonometry , algebra , inequality , inequalities

Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$, find (a)max$S$ and (b) min$S$.

1993 Hungary-Israel Binational, 3

Tags: trigonometry , symmetry , function , vector , inequalities

Distinct points $A, B , C, D, E$ are given in this order on a semicircle with radius $1$. Prove that \[AB^{2}+BC^{2}+CD^{2}+DE^{2}+AB \cdot BC \cdot CD+BC \cdot CD \cdot DE < 4.\]

2010 Contests, 1

Tags: incenter , perimeter , trigonometry , inradius , inequalities , geometry

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

2011 AMC 12/AHSME, 19

Tags: analytic geometry , calculus , integration , inequalities

A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$? $ \textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25} $

2020 Moldova Team Selection Test, 2

Tags: inequality , inequalities

Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \geq 3+\sqrt{3}$

2002 Bulgaria National Olympiad, 6

Tags: inequalities

Find the smallest number $k$, such that $ \frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively. [i]Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov[/i]

1988 IMO Longlists, 64

Tags: inequalities , number theory

Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$

2008 AMC 12/AHSME, 14

Tags: inequalities , geometry , invariant , geometric transformation , analytic geometry

What is the area of the region defined by the inequality $ |3x\minus{}18|\plus{}|2y\plus{}7|\le 3$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ \frac{9}{2} \qquad \textbf{(E)}\ 5$

1998 Belarus Team Selection Test, 2

Tags: ring theory , algebra , inequalities

Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]

2012 Romania Team Selection Test, 3

Tags: inequalities , combinatorics

Let $A$ and $B$ be finite sets of real numbers and let $x$ be an element of $A+B$. Prove that \[|A\cap (x-B)|\leq \frac{|A-B|^2}{|A+B|}\] where $A+B=\{a+b: a\in A, b\in B\}$, $x-B=\{x-b: b\in B\}$ and $A-B=\{a-b: a\in A, b\in B\}$.

2010 Lithuania National Olympiad, 1

Tags: inequalities

$a,b$ are real numbers such that: \[ a^3+b^3=8-6ab. \] Find the maximal and minimal value of $a+b$.

1990 Baltic Way, 18

Tags: inequalities

Numbers $1, 2,\dots , 101$ are written in the cells of a $101\times 101$ square board so that each number is repeated $101$ times. Prove that there exists either a column or a row containing at least $11$ different numbers.

1989 IMO Longlists, 12

Tags: polynomial , algebra , inequalities

Let $ P(x)$ be a polynomial such that the following inequalities are satisfied: \[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\] and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\] Prove that for every positive natural number $ n,$ $ P(n)$ is positive.

2005 Iran MO (3rd Round), 3

Tags: function , inequalities , algebra

Find all $\alpha>0$ and $\beta>0$ that for each $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)\in\mathbb {R^+}^n$ that:\[(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i\]

2005 France Team Selection Test, 4

Tags: floor function , ceiling function , induction , inequalities , number theory , logarithm

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

2017 Korea National Olympiad, problem 7

Tags: algebra , functional equation , function , inequalities

Find all real numbers $c$ such that there exists a function $f: \mathbb{R}_{ \ge 0} \rightarrow \mathbb{R}$ which satisfies the following. For all nonnegative reals $x, y$, $f(x+y^2) \ge cf(x)+y$. Here $\mathbb{R}_{\ge 0}$ is the set of all nonnegative reals.

2007 ITest, 31

Tags: calculus , integration , geometry , inequalities

Let $x$ be the length of one side of a triangle and let $y$ be the height to that side. If $x+y=418$, find the maximum possible $\textit{integral value}$ of the area of the triangle.

2007 Princeton University Math Competition, 6

Tags: inequalities

If $a, b, c, d$ are reals with $a \ge b \ge c \ge d \ge 0$ and $b(b-a)+c(c-b)+d(d-c) \le 2 - \frac{a^2}{2}$, find the minimum value of the expression \begin{align*}\frac{1}{b+2006c-2006d}+\frac{1}{a+2006b-2006c-d} + \frac{1}{2007a-2006b-c+d} + \frac{1}{a-b+c+2006d}.\end{align*}