Olympiad problems

1998 Rioplatense Mathematical Olympiad, Level 3, 2

Given an integer $n > 2$, consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.

2015 China Second Round Olympiad, 1

Tags: inequalities

Let $a,b,c$ be nonnegative real numbers.Prove that$$\frac{(a-bc)^2+(b-ca)^2+(c-ab)^2}{(a-b)^2+(b-c)^2+(c-a)^2}\geq\frac{1}{2}.$$

2018 International Zhautykov Olympiad, 1

Tags: algebra , inequalities , triangle inequality , geometry

Let $\alpha,\beta,\gamma$ measures of angles of opposite to the sides of triangle with measures $a,b,c$ respectively.Prove that $$2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}$$

2005 Canada National Olympiad, 4

Tags: circumcircle , perimeter , inequalities , inradius , geometry

Let $ ABC$ be a triangle with circumradius $ R$, perimeter $ P$ and area $ K$. Determine the maximum value of: $ \frac{KP}{R^3}$.

2009 Today's Calculation Of Integral, 482

Tags: calculus , integration , limit , geometry , rectangle , inequalities , logarithm

Let $ n$ be natural number. Find the limit value of ${ \lim_{n\to\infty} \frac{1}{n}(\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}}+\cdots\cdots +\frac{n}{\sqrt{n^2+1}}).$

2007 Croatia Team Selection Test, 7

Tags: algorithm , inequalities , three variable inequality

Let $a,b,c>0$ such that $a+b+c=1$. Prove: \[\frac{a^{2}}b+\frac{b^{2}}c+\frac{c^{2}}a \ge 3(a^{2}+b^{2}+c^{2}) \]

2001 Moldova National Olympiad, Problem 8

Tags: inequalities , algebra

If $a_1,a_2,\ldots,a_n$ are positive real numbers, prove the inequality $$\dfrac1{\dfrac1{1+a_1}+\dfrac1{1+a_2}+\ldots+\dfrac1{1+a_n}}-\dfrac1{\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}}\ge\frac1n.$$

2016 Thailand TSTST, 2

Tags: inequalities , sequence

Find the number of sequences $a_1,a_2,\dots,a_{100}$ such that $\text{(i)}$ There exists $i\in\{1,2,\dots,100\}$ such that $a_i=3$, and $\text{(ii)}$ $|a_i-a_{i+1}|\leq 1$ for all $1\leq i<100$.

2013 ELMO Shortlist, 3

Tags: inequalities , induction , number theory

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

2004 USAMO, 6

Tags: geometry , trapezoid , inequalities , trigonometry

A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that \[ (AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2. \] Prove that $ABCD$ is an isosceles trapezoid.

2002 Moldova Team Selection Test, 1

Tags: inequalities

Positive numbers $\alpha ,\beta , x_1, x_2,\ldots, x_n$ ($n \geq 1$) satisfy $x_1+x_2+\cdots+x_n = 1$. Prove that \[\sum_{i=1}^{n} \frac{x_i^3}{\alpha x_i+\beta x_{i+1}} \geq \frac{1}{n(\alpha+\beta)}.\] [b]Note.[/b] $x_{n+1}=x_1$.

2022 Switzerland Team Selection Test, 5

Tags: inequalities , algebra

Let $a, b, c, \lambda$ be positive real numbers with $\lambda \geq 1/4$. Show that $$\frac{a}{\sqrt{b^2+\lambda bc+c^2}}+\frac{b}{\sqrt{c^2+\lambda ca+a^2}}+\frac{c}{\sqrt{a^2+\lambda ab+b^2}} \geq \frac{3}{\sqrt{\lambda +2}}.$$

2019 IFYM, Sozopol, 4

Tags: combinatorics , inequalities

On a competition called [i]"Mathematical duels"[/i] students were given $n$ problems and each student solved exactly 3 of them. For each two students there is at most one problem that is solved from both of them. Prove that, if $s\in \mathbb{N}$ is a number for which $s^2-s+1<2n$, then there are $s$ problems among the $n$, no three of which solved by one student.

2010 Cuba MO, 4

Tags: algebra , inequalities

Prove that for all positive real numbers $x, y$ holds the inequality $$x^4 + y^3 + x^2 + y + 1 > \frac92 xy.$$

2010 Balkan MO Shortlist, A3

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers. Prove that \[(\frac{a}{a+b})^{5}+(\frac{b}{b+c})^{5}+(\frac{c}{c+d})^{5}+(\frac{d}{d+a})^{5}\ge \frac{1}{8}\]

1982 Tournament Of Towns, (017) 3

Tags: sequence , inequalities , combinatorics , algebra

a) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$, one can find $100$ members for which $a_k > k$. b) Prove that in an infinite sequence ${a_k}$ of integers, pairwise distinct and each member greater than $1$ there are infinitely many such numbers $a_k$ such that $a_k > k$. (A Andjans, Riga) PS. (a) for juniors (b) for seniors

2014 ELMO Shortlist, 1

Tags: inequalities , trigonometry , calculus , topology , three variable inequality

In a non-obtuse triangle $ABC$, prove that \[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]

2018 China Team Selection Test, 3

Tags: inequalities , algebra

Prove that there exists a constant $C>0$ such that $$H(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i}$$ holds for arbitrary positive integer $m$ and any $m$ positive integer $a_1,a_2,\cdots,a_m$, where $$H(n)=\sum_{k=1}^{n}\frac{1}{k}.$$

2002 Tournament Of Towns, 6

Tags: induction , inequalities , number theory

In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

1998 VJIMC, Problem 4-M

Tags: inequalities , integration , calculus , riemann sum

Prove the inequality $$\frac{n\pi}4-\frac1{\sqrt{8n}}\le\frac12+\sum_{k=1}^{n-1}\sqrt{1-\frac{k^2}{n^2}}\le\frac{n\pi}4$$for every integer $n\ge2$.

2008 Serbia National Math Olympiad, 6

Tags: trigonometry , inequalities , cyclic quadrilateral , geometry

In a convex pentagon $ ABCDE$, let $ \angle EAB \equal{} \angle ABC \equal{} 120^{\circ}$, $ \angle ADB \equal{} 30^{\circ}$ and $ \angle CDE \equal{} 60^{\circ}$. Let $ AB \equal{} 1$. Prove that the area of the pentagon is less than $ \sqrt {3}$.

2004 Romania National Olympiad, 2

Tags: function , algebra , polynomial , calculus , integration , inequalities , logarithm

Let $P(n)$ be the number of functions $f: \mathbb{R} \to \mathbb{R}$, $f(x)=a x^2 + b x + c$, with $a,b,c \in \{1,2,\ldots,n\}$ and that have the property that $f(x)=0$ has only integer solutions. Prove that $n<P(n)<n^2$, for all $n \geq 4$. [i]Laurentiu Panaitopol[/i]

1996 IMO Shortlist, 3

Tags: inequalities , sequence , recurrence , algebra

Let $ a > 2$ be given, and starting $ a_0 \equal{} 1, a_1 \equal{} a$ define recursively: \[ a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n.\] Show that for all integers $ k > 0,$ we have: $ \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).$

2011 Serbia JBMO TST, 4

Tags: inequalities

If a, b, c are positive real numbers with $ a+b+c=1 $. Find the minimum value of $ \sqrt{a}+\sqrt{b}+\sqrt{c}+\frac{1}{\sqrt{abc}} $

2012 Polish MO Finals, 6

Tags: algebra , inequality , inequalities

Show that for any positive real numbers $a, b, c$ true is inequality: $\left(\frac{a - b}{c}\right)^2 + \left(\frac{b - c}{a}\right)^2 + \left(\frac{c - a}{b}\right)^2 \ge 2\sqrt{2}\left(\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} \right)$.