Olympiad problems

2016 Ukraine Team Selection Test, 6

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

2012 NIMO Problems, 2

Tags: inequalities

Compute the number of positive integers $n$ satisfying the inequalities \[ 2^{n-1} < 5^{n-3} < 3^n. \][i]Proposed by Isabella Grabski[/i]

2022 Belarusian National Olympiad, 9.2

Tags: algebra , factorial , inequalities

Prove the inequality $$\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\ldots+\frac{1}{2022!}>\frac{1^2}{2!}+\frac{2^2}{3!}+\frac{3^2}{4!}+\ldots+\frac{2022^2}{2023!}$$.

2013 ISI Entrance Examination, 3

Tags: real analysis , inequalities , function

Let $f:\mathbb R\to\mathbb R$ satisfy \[|f(x+y)-f(x-y)-y|\leq y^2\] For all $(x,y)\in\mathbb R^2.$ Show that $f(x)=\frac x2+c$ where $c$ is a constant.

2014 South East Mathematical Olympiad, 4

Tags: inequalities

Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\]

2011 Iran MO (3rd Round), 1

Tags: inequalities , polynomial , algebra

We define the recursive polynomial $T_n(x)$ as follows: $T_0(x)=1$ $T_1(x)=x$ $T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$ $\forall n \in \mathbb N$. [b]a)[/b] find $T_2(x),T_3(x),T_4(x)$ and $T_5(x)$. [b]b)[/b] find all the roots of the polynomial $T_n(x)$ $\forall n \in \mathbb N$. [i]Proposed by Morteza Saghafian[/i]

2018 Romania Team Selection Tests, 1

Tags: n-variable inequality , inequalities

Find the least number $ c$ satisfyng the condition $\sum_{i=1}^n {x_i}^2\leq cn$ and all real numbers $x_1,x_2,...,x_n$ are greater than or equal to $-1$ such that $\sum_{i=1}^n {x_i}^3=0$

1997 Korea - Final Round, 2

Tags: geometry , perimeter , trigonometry , inequalities

The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that \[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\] and find the conditions for equality.

2009 India IMO Training Camp, 7

Tags: geometry , inequalities

Let $ P$ be any point in the interior of a $ \triangle ABC$.Prove That $ \frac{PA}{a}\plus{}\frac{PB}{b}\plus{}\frac{PC}{c}\ge \sqrt{3}$.

2023 All-Russian Olympiad, 8

Tags: algebra , inequalities

Given is a real number $a \in (0,1)$ and positive reals $x_0, x_1, \ldots, x_n$ such that $\sum x_i=n+a$ and $\sum \frac{1}{x_i}=n+\frac{1}{a}$. Find the minimal value of $\sum x_i^2$.

2010 Romanian Master of Mathematics, 2

Tags: algebra , cauchy inequality , absolute value , inequalities , function

For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]

2001 Austrian-Polish Competition, 10

Tags: inequalities

The sequence $a_{1},a_{2},\cdots,a_{2010}$ has the following properties: (1) each sum of the 20 successive values of the sequence is nonnegative, (2) $|a_{i}a_{i+1}| \leq 1$ for $i=1,2,\cdots,2009$. Determine the maximal value of the expression $\sum_{i=1}^{2010}a_{i}$.

2006 MOP Homework, 4

Tags: system of equations , algebra , inequalities , function , polynomial

Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.

2009 Jozsef Wildt International Math Competition, W. 13

Tags: inequalities

If $a_k >0$ [ $k=$1, 2, $\cdots$, $n$], then prove the following inequality $$\left (\sum \limits_{k=1}^n a_k^5 \right )^4 \geq \frac{1}{n} \left (\frac{2}{n-1} \right )^5 \left (\sum \limits_{1\leq i<j\leq n} a_i^2a_j^2 \right )^5$$

2005 MOP Homework, 7

Tags: inequalities

Let $n$ be a positive integer with $n>1$, and let $a_1$, $a_2$, ..., $a_n$ be positive integers such that $a_1<a_2<...<a_n$ and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n} \le 1$. Prove that $(\frac{1}{a_1^2+x^2}+\frac{1}{a_2^2+x^2}+...+\frac{1}{a_n^2+x^2})^2 \le \frac{1}{2} \cdot \frac{1}{a_1(a_1-1)+x^2}$ for all real numbers $x$.

2003 Germany Team Selection Test, 3

Tags: least common multiple , ceiling function , number theory , floor function , inequalities

Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$

2022 Junior Macedonian Mathematical Olympiad, P2

Tags: algebra , rational function , inequalities , function

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=3$. Prove the inequality $$\frac{a^3}{a^2+1}+\frac{b^3}{b^2+1}+\frac{c^3}{c^2+1} \geq \frac{3}{2}.$$ [i]Proposed by Anastasija Trajanova[/i]

2008 Hungary-Israel Binational, 1

Tags: geometry , triangle inequality , inequalities , perpendicular bisector

Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.

2002 Olympic Revenge, 3

Tags: four variables , euclidean distance , inequalities

Show that if $x,y,z,w$ are positive reals, then \[ \frac{3}{2}\sqrt{(x^2+y^2)(w^2+z^2)} + \sqrt{(x^2+w^2)(y^2+z^2) - 3xyzw} \geq (x+z)(y+w) \]

2010 Contests, 3

Tags: n-variable inequality , inequalities

Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$ [i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]

1990 Baltic Way, 4

Tags: inequalities

Prove that, for any real numbers $a_1, a_2, \dots , a_n$, \[ \sum_{i,j=1}^n \frac{a_ia_j}{i+j-1}\ge 0.\]

2009 Sharygin Geometry Olympiad, 2

Tags: inequalities , convex quadrilateral , circumradius

Given a convex quadrilateral $ABCD$. Let $R_a, R_b, R_c$ and $R_d$ be the circumradii of triangles $DAB, ABC, BCD, CDA$. Prove that inequality $R_a < R_b < R_c < R_d$ is equivalent to $180^o - \angle CDB < \angle CAB < \angle CDB$ . (O.Musin)

2010 AMC 12/AHSME, 24

Tags: algebra , vieta , inequalities , polynomial

The set of real numbers $ x$ for which \[ \frac{1}{x\minus{}2009}\plus{}\frac{1}{x\minus{}2010}\plus{}\frac{1}{x\minus{}2011}\ge 1\] is the union of intervals of the form $ a<x\le b$. What is the sum of the lengths of these intervals? $ \textbf{(A)}\ \frac{1003}{335} \qquad \textbf{(B)}\ \frac{1004}{335} \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac{403}{134} \qquad \textbf{(E)}\ \frac{202}{67}$

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: sum , absolute value , algebra , inequalities

Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$. Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.

2014 PUMaC Individual Finals A, 2

Tags: algebra , function , hmmt , inequalities , polynomial

Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that \[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]