Olympiad problems

2021-IMOC, A11

Tags: inequalities

Given $n \geq 2$ reals $x_1 , x_2 , \dots , x_n.$ Show that $$\prod_{1\leq i < j \leq n} (x_i - x_j)^2 \leq \prod_{i=0}^{n-1} \left(\sum_{j=1}^{n} x_j^{2i}\right)$$ and find all the $(x_1 , x_2 , \dots , x_n)$ where the equality holds.

2024 VJIMC, 3

Tags: combinatorics , inequalities , graph , degree

Let $n$ be a positive integer and let $G$ be a simple undirected graph on $n$ vertices. Let $d_i$ be the degree of its $i$-th vertex, $i = 1, \dots , n$. Denote $\Delta=\max d_i$. Prove that if \[\sum_{i=1}^n d_i^2>n\Delta(n-\Delta),\] then $G$ contains a triangle.

2019 China National Olympiad, 6

Tags: algebra , geometry , trigonometry , inequalities

The point $P_1, P_2,\cdots ,P_{2018} $ is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that $$S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2$$takes the maximum value.

2006 Pre-Preparation Course Examination, 3

Tags: abstract algebra , superior algebra , greatest common divisor , inequalities , number theory

a) If $K$ is a finite extension of the field $F$ and $K=F(\alpha,\beta)$ show that $[K: F]\leq [F(\alpha): F][F(\beta): F]$ b) If $gcd([F(\alpha): F],[F(\beta): F])=1$ then does the above inequality always become equality? c) By giving an example show that if $gcd([F(\alpha): F],[F(\beta): F])\neq 1$ then equality might happen.

2009 Abels Math Contest (Norwegian MO) Final, 4a

Tags: algebra , inequalities

Show that $\left(\frac{2010}{2009}\right)^{2009}> 2$.

1960 Poland - Second Round, 1

Tags: inequalities , sum , algebra

Prove that if the real numbers $ a $ and $ b $ are not both equal to zero, then for every natural $ n $ $$ a^{2n} + a^{2n-1}b + a^{2n-2} b^2 + \ldots + ab^{2n-1} + b^{2n} > 0. $$

2009 Greece National Olympiad, 3

Tags: calculus , inequalities

Let $ x,y,z$ be nonnegative real numbers such that $ x \plus{} y \plus{} z \equal{} 2$. Prove that $ x^{2}y^{2} \plus{} y^{2}z^{2} \plus{} z^{2}x^{2} \plus{} xyz\leq 1$. When does the equality occur?

1998 Singapore MO Open, 4

Tags: algebra , inequalities

Let $n$ be a fixed positive integer. Find all the positive integers $m$ such that $$\frac{m^2+4m}{a_1}+\frac{m^2+8m}{a_1+a_2}+\frac{m^2+12m}{a_1+a_2+a_3}+...+\frac{m^2+4nm}{a_1+a_2+...+a_n}<2500 \left(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}\right)$$ for any positive numbers $a_1,a_2,...,a_n$. Justify your answer.

2006 Tuymaada Olympiad, 2

Tags: geometry , triangle inequality , trigonometry , euler , circumcircle , inequalities

Let $ABC$ be a triangle, $G$ it`s centroid, $H$ it`s orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$). It is known that $MG=R$, where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$. [i]Proposed by F. Bakharev[/i]

2023 Middle European Mathematical Olympiad, 1

Tags: inequalities

For each pair $(\alpha, \beta)$ of non-negative reals with $\alpha+\beta \geq 2$, determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(x)f(y) \leq f(xy)+\alpha x+\beta y$$ for all reals $x, y$.

2007 Pre-Preparation Course Examination, 4

Tags: inequalities

Prove that \[\sum_{i=-2007}^{2007}\frac{\sqrt{|i+1|}}{(\sqrt2)^{|i|}}>\sum_{i=-2007}^{2007}\frac{\sqrt{|i|}}{(\sqrt2)^{|i|}}\]

1989 APMO, 1

Tags: inequalities , n-variable inequality

Let $x_1$, $x_2$, $\cdots$, $x_n$ be positive real numbers, and let \[ S = x_1 + x_2 + \cdots + x_n. \] Prove that \[ (1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!} \]

PEN D Problems, 15

Tags: congruence , inequalities , modular arithmetic

Let $n_{1}, \cdots, n_{k}$ and $a$ be positive integers which satify the following conditions:[list][*] for any $i \neq j$, $(n_{i}, n_{j})=1$, [*] for any $i$, $a^{n_{i}} \equiv 1 \pmod{n_i}$, [*] for any $i$, $n_{i}$ does not divide $a-1$. [/list] Show that there exist at least $2^{k+1}-2$ integers $x>1$ with $a^{x} \equiv 1 \pmod{x}$.

2001 USA Team Selection Test, 1

Tags: inequalities

Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$. Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$.

2008 Greece Junior Math Olympiad, 2

Tags: inequalities

If $x,y,z$ are positive real numbers with $x^2+y^2+z^2=3$, prove that $\frac32<\frac{1+y^2}{x+2}+\frac{1+z^2}{y+2}+\frac{1+x^2}{z+2}<3$

2011 Junior Balkan Team Selection Tests - Romania, 3

Tags: max , inequalities , algebra

a) Find the largest possible value of the number $x_1x_2 + x_2x_3 + ... + x_{n-1}x_n$, if $x_1, x_2, ... , x_n$ ($n \ge 2$) are non-negative integers and their sum is $2011$. b) Find the numbers $x_1, x_2, ... , x_n$ for which the maximum value determined at a) is obtained

2004 Poland - First Round, 3

Tags: inequalities , geometry , vector , derivative , calculus , trigonometry

3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and: $\frac{EF}{FD}=\frac{AD}{DB}$ Prove that $CF \bot AE$

2018 Saint Petersburg Mathematical Olympiad, 6

Tags: algebra , inequalities

Let $a,b,c,d>0$ . Prove that $a^4+b^4+c^4+d^4 \geq 4abcd+4(a-b)^2 \sqrt{abcd}$

1954 Polish MO Finals, 4

Tags: inequalities , algebra

Find the values of $ x $ that satisfy the inequality $$ \sqrt{x} - \sqrt{x- a} > 2,$$ where $ a $ is a gicen poistive number.

2009 Vietnam Team Selection Test, 1

Tags: function , quadratic , logarithm , inequalities , calculus

Let $ a,b,c$ be positive numbers.Find $ k$ such that: $ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$

2013 Mexico National Olympiad, 5

Tags: inequalities , number theory

A pair of integers is special if it is of the form $(n, n-1)$ or $(n-1, n)$ for some positive integer $n$. Let $n$ and $m$ be positive integers such that pair $(n, m)$ is not special. Show $(n, m)$ can be expressed as a sum of two or more different special pairs if and only if $n$ and $m$ satisfy the inequality $ n+m\geq (n-m)^2 $. Note: The sum of two pairs is defined as $ (a, b)+(c, d) = (a+c, b+d) $.

2004 IMO, 4

Tags: triangle inequality , n-variable inequality , inequalities

Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

1999 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: inequalities , number theory

Let $p_1, p_2, ..., p_k$ be $k$ different primes. We consider all positive integers that use only these primes (not necessarily all) in their prime factorization, and arrange those numbers in increasing order, forming an infinite sequence: $a_1 < a_2 < ... < a_n < ...$ Prove that, for every number $c$, there exists $n$ such that $a_{n+1} -a_n > c$.

2017 Baltic Way, 2

Tags: inequalities , algebra , system of equations

Does there exist a finite set of real numbers such that their sum equals $2$, the sum of their squares equals $3$, the sum of their cubes equals $4$, ..., and the sum of their ninth powers equals $10$?

2013 Middle European Mathematical Olympiad, 2

Tags: inequalities

Let $ x, y, z, w $ be nonzero real numbers such that $ x+y \ne 0$, $ z+w \ne 0 $, and $ xy+zw \ge 0 $. Prove that \[ \left( \frac{x+y}{z+w} + \frac{z+w}{x+y} \right) ^{-1} + \frac{1}{2} \ge \left( \frac{x}{z} + \frac{z}{x} \right) ^{-1} + \left( \frac{y}{w} + \frac{w}{y} \right) ^{-1}\]