Olympiad problems

2018 Caucasus Mathematical Olympiad, 8

Tags: inequalities

Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality $$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$

2007 Bulgarian Autumn Math Competition, Problem 12.3

Tags: symmetric inequality , three variable inequality , algebra , inequalities

Find all real numbers $r$, such that the inequality \[r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9\] holds for any real $a,b,c>0$.

2022 New Zealand MO, 2

Tags: algebra , inequalities

Find all triples $(a, b, c) $ of real numbers such that $a^2 + b^2 + c^2 = 1$ and $a(2b - 2a - c) \ge \frac12$.

2009 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , system , inequalities , minimum

Find minimal value of $a \in \mathbb{R}$ such that system $$\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}=a-1$$ $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}=a+1$$ has solution in set of real numbers

2020 Jozsef Wildt International Math Competition, W9

Tags: inequalities , geometry , triangle , inradius

In any triangle $ABC$ prove that the following relationship holds: $$\begin{vmatrix}(b+c)^2&a^2&a^2\\b^2&(c+a)^2&b^2\\c^2&c^2&(a+b)^2\end{vmatrix}\ge93312r^6$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

1991 AIME Problems, 15

Tags: ratio , trigonometry , rotation , triangle inequality , inequalities

For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$.

2013 Brazil Team Selection Test, 2

Tags: function , floor function , modular arithmetic , quadratic , algebra , inequalities

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

1997 IMO Shortlist, 26

Tags: minimization , sequence , fibonacci , algebra , inequalities

For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i\equal{}0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 \equal{} 1,$ $ a_i \leq a_{i\plus{}1} \plus{} a_{i\plus{}2}$ for $ i \equal{} 0, \ldots, n \minus{} 2.$

1971 IMO Longlists, 54

Tags: inequalities , combinatorics

A set $M$ is formed of $\binom{2n}{n}$ men, $n=1,2,\ldots$. Prove that we can choose a subset $P$ of the set $M$ consisting of $n+1$ men such that one of the following conditions is satisfied: $(1)$ every member of the set $P$ knows every other member of the set $P$; $(2)$ no member of the set $P$ knows any other member of the set $P$.

2008 ISI B.Stat Entrance Exam, 1

Tags: geometry , trigonometry , function , inequalities , perimeter

Of all triangles with given perimeter, find the triangle with the maximum area. Justify your answer

1998 APMO, 3

Tags: algebra , inequalities , trigonometry

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr). \]

2014 Contests, 3

Tags: number theory , inequalities

For all integers $n\ge 2$ with the following property: [list] [*] for each pair of positive divisors $k,~\ell <n$, at least one of the numbers $2k-\ell$ and $2\ell-k$ is a (not necessarily positive) divisor of $n$ as well.[/list]

2004 IMO Shortlist, 8

Tags: inequalities , graph theory , extremal combinatorics

For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \[g(G)^3\le c\cdot f(G)^4\] for every graph $G$. [i]Proposed by Marcin Kuczma, Poland [/i]

2024 Tuymaada Olympiad, 8

Tags: graph theory , graph , inequalities

A graph $G$ has $n$ vertices ($n>1$). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$. Prove that the inequality (the summation is over all edges of $G$): \[\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.\]

2020 Mediterranean Mathematics Olympiad, 3

Tags: algebra , inequalities

Prove that all postive real numbers $a,b,c$ with $a+b+c=4$ satisfy the inequality $$\frac{ab}{\sqrt[4]{3c^2+16}}+ \frac{bc}{\sqrt[4]{3a^2+16}}+ \frac{ca}{\sqrt[4]{3b^2+16}} \le\frac43 \sqrt[4]{12}$$

2017 Regional Competition For Advanced Students, 1

Tags: inequalities

Let $x_1, x_2, \dots, x_n$ be non-negative real numbers such that $$x_1^2+x_2^2 + \dots x_9^2 \ge 25.$$ Prove that one can choose three of these numbers such that their sum is at least $5$. [i]Proposed by Karl Czakler[/i]

2016 Germany Team Selection Test, 2

Tags: number theory , integer , n-variable inequality , inequalities

The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]

1966 IMO Shortlist, 32

Tags: geometry , triangle inequality , similar triangles , inequalities

The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). [Hereby, $a=BC,$ $b=CA,$ $c=AB, $ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$ Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

2012 Tuymaada Olympiad, 3

Tags: geometry , trigonometry , inequalities

A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$. Prove that the length of its radius is less than $3$. [i]Proposed by K. Kokhas[/i]

2003 China Team Selection Test, 3

Tags: number theory , analytic geometry , graph theory , inequalities

Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.

1994 French Mathematical Olympiad, Problem 2

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

Let be given a semi-sphere $\Sigma$ whose base-circle lies on plane $p$. A variable plane $Q$, parallel to a fixed plane non-perpendicular to $P$, cuts $\Sigma$ at a circle $C$. We denote by $C'$ the orthogonal projection of $C$ onto $P$. Find the position of $Q$ for which the cylinder with bases $C$ and $C'$ has the maximum volume.

1996 Polish MO Finals, 3

Tags: inequalities

$a_i, x_i$ are positive reals such that $a_1 + a_2 + ... + a_n = x_1 + x_2 + ... + x_n = 1$. Show that \[ 2 \sum_{i<j} x_ix_j \leq \frac{n-2}{n-1} + \sum \frac{a_ix_i ^2}{1-a_i} \] When do we have equality?

2014 Belarusian National Olympiad, 5

Tags: algebra , inequalities

Prove that $\frac{1}{x+y+1}-\frac{1}{(x+1)(y+1)}<\frac{1}{11}$ for all positive $x$ and $y$.

2001 India IMO Training Camp, 3

Tags: algebra , polynomial , induction , inequalities

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

2002 Abels Math Contest (Norwegian MO), 2ab

Tags: algebra , inequalities

a) Let $x$ be a positive real number. Show that $x + 1 / x\ge 2$. b) Let $n\ge 2$ be a positive integer and let $x _1,y_1,x_2,y_2,...,x_n,y_n$ be positive real numbers such that $x _1+x _2+...+x _n \ge x _1y_1+x _2y_2+...+x _ny_n$. Show that $x _1+x _2+...+x _n \le \frac{x _1}{y_1}+\frac{x _2}{y_2}+...+\frac{x _n}{y_n}$