Olympiad problems

2003 India IMO Training Camp, 9

Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.

2000 Belarus Team Selection Test, 1.4

Tags: trigonometry , inequalities , geometric inequalities , sphere , geometry , 3d geometry

A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$. Prove that $\ell \le \sin \frac{2\pi}{5}$ .

1996 Irish Math Olympiad, 2

Tags: inequalities

Show that for every positive integer $ n$, $ 2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot ... \cdot (2^n)^{\frac{1}{2^n}}<4$.

KoMaL A Problems 2017/2018, A. 702

Tags: geometry , inequalities , optimization , geometric inequality

Fix a triangle $ABC$. We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$, $Y$ lies on segment $CA$, $Z$ lies on segment $AB$, and $XYZ$ is similar to $ABC$ (i.e., $\angle A=\angle X, \angle B=\angle Y, \angle C=\angle Z $). Of all the elegant triangles, which one has the smallest perimeter?

2024 Tuymaada Olympiad, 2

Tags: inequalities , combinatorics

We will call a [i]hedgehog[/i] a graph in which one vertex is connected to all the others and there are no other edges; the number of vertices of this graph will be called the size of the hedgehog. A graph $G$ is given on $n$ vertices (where $n > 1$). For each edge $e$, we denote by $s(e)$ the size of the maximum hedgehog in graph $G$, which contains this edge. Prove the inequality (summation is carried out over all edges of the graph $G$): \[\sum_e \frac{1}{s(e)} \leqslant \frac{n}{2}.\] [i]Proposed by D. Malec, C. Tompkins[/i]

2018 Iran Team Selection Test, 2

Tags: inequalities

Determine the least real number $k$ such that the inequality $$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$ holds for all real numbers $a,b,c$. [i]Proposed by Mohammad Jafari[/i]

1976 IMO Longlists, 34

Tags: inequalities

Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas \[a_{n+1} = a_n + b_n,\] \[ b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,\] and the initial values $a_0 = b_0 = 1$. Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegative integers $n$.

2020 Korea Junior Math Olympiad, 5

Tags: algebra , inequalities

Let $a, b, c, d, e$ be real numbers satisfying the following conditions. \[a \le b \le c \le d \le e, \quad a+e=1, \quad b+c+d=3, \quad a^2+b^2+c^2+d^2+e^2=14\]Determine the maximum possible value of $ae$.

1991 National High School Mathematics League, 15

Tags: inequalities

If $0<a<1,x^2+y=0$, prove that $\log_a(a^x+a^y)\leq\log_a2+\frac{1}{8}$.

1971 Polish MO Finals, 1

Tags: algebra , number theory , inequalities

Show that if $(a_n)$ is an infinite sequence of distinct positive integers, neither of which contains digit $0$ in the decimal expansion, then $$\sum_{n=1}^{\infty} \frac{1}{a_n}< 29.$$

2009 AMC 12/AHSME, 24

Tags: function , inequalities , logarithm

The [i]tower function of twos[/i] is defined recursively as follows: $ T(1) \equal{} 2$ and $ T(n \plus{} 1) \equal{} 2^{T(n)}$ for $ n\ge1$. Let $ A \equal{} (T(2009))^{T(2009)}$ and $ B \equal{} (T(2009))^A$. What is the largest integer $ k$ such that \[ \underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}} \]is defined? $ \textbf{(A)}\ 2009\qquad \textbf{(B)}\ 2010\qquad \textbf{(C)}\ 2011\qquad \textbf{(D)}\ 2012\qquad \textbf{(E)}\ 2013$

2000 Baltic Way, 19

Tags: inequalities , algebra

Let $t\ge\frac{1}{2}$ be a real number and $n$ a positive integer. Prove that \[t^{2n}\ge (t-1)^{2n}+(2t-1)^n\]

2003 District Olympiad, 3

Tags: trigonometry , inequalities , geometry

(a) If $\displaystyle ABC$ is a triangle and $\displaystyle M$ is a point from its plane, then prove that \[ \displaystyle AM \sin A \leq BM \sin B + CM \sin C . \] (b) Let $\displaystyle A_1,B_1,C_1$ be points on the sides $\displaystyle (BC),(CA),(AB)$ of the triangle $\displaystyle ABC$, such that the angles of $\triangle A_1 B_1 C_1$ are $\widehat{A_1} = \alpha, \widehat{B_1} = \beta, \widehat{C_1} = \gamma$. Prove that \[ \displaystyle \sum A A_1 \sin \alpha \leq \sum BC \sin \alpha . \] [i]Dan Ştefan Marinescu, Viorel Cornea[/i]

1964 Kurschak Competition, 3

Tags: algebra , inequalities

Show that for any positive reals $w, x, y, z$ we have $$\sqrt{\frac{w^2 + x^2 + y^2 + z^2}{4}}\ge \sqrt[3]{ \frac{wxy + wxz + wyz + xyz}{4}}$$

1997 Swedish Mathematical Competition, 1

Tags: geometric inequalities , geometry , inequalities

Let $AC$ be a diameter of a circle and $AB$ be tangent to the circle. The segment $BC$ intersects the circle again at $D$. Show that if $AC = 1$, $AB = a$, and $CD = b$, then $$\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}$$

1987 Tournament Of Towns, (147) 4

Tags: inequalities , radical , algebra

For any natural $n$ prove the inequality $$\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3$$

2007 Serbia National Math Olympiad, 3

Tags: modular arithmetic , quadratic , greatest common divisor , number theory , inequalities

Determine all pairs of natural numbers $(x; n)$ that satisfy the equation \[x^{3}+2x+1 = 2^{n}.\]

2006 China Girls Math Olympiad, 7

Tags: inequalities

Given that $x_{i}>0$, $i = 1, 2, \cdots, n$, $k \geq 1$. Show that: \[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]

2014 Iran Team Selection Test, 5

Tags: function , n-variable inequality , inequalities

$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that: $\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$

2009 Jozsef Wildt International Math Competition, W. 1

Tags: inequalities

Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\sqrt[3]{\left (\frac{1+a}{b+c}\right )^{\frac{1-a}{bc}}\left (\frac{1+b}{c+a}\right )^{\frac{1-b}{ca}}\left (\frac{1+c}{a+b}\right )^{\frac{1-c}{ab}}} \geq 64 $$

2007 Hanoi Open Mathematics Competitions, 2

Tags: number theory , inequalities

Which is largest positive integer n satisfying the following inequality: $n^{2007} > (2007)^n$.

2016 India Regional Mathematical Olympiad, 6

Tags: inequalities , algebra

Positive integers $a, b, c$ satisfy $\frac1a +\frac1b +\frac1c<1$. Prove that $\frac1a +\frac1b +\frac1c\le \frac{41}{42}$. Also prove that equality in fact holds in the second inequality.

2014 India National Olympiad, 3

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

Let $a,b$ be natural numbers with $ab>2$. Suppose that the sum of their greatest common divisor and least common multiple is divisble by $a+b$. Prove that the quotient is at most $\frac{a+b}{4}$. When is this quotient exactly equal to $\frac{a+b}{4}$

1931 Eotvos Mathematical Competition, 3

Tags: geometry , inequalities , geometric inequality

Let $A$ and $B$ be two given points, distance $1 $ apart. Determine a point $P$ on the line $AB$ such that $$\frac{1}{1 + AP}+\frac{1}{1 + BP}$$ is a maximum.

Russian TST 2015, P2

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers satisfying $a^2+b^2+c^2+d^2=1$. Prove that \[a^3+b^3+c^3+d^3+abcd\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)\leqslant 1.\]