Olympiad problems

1993 Vietnam Team Selection Test, 3

Tags: inequalities

Let's consider the real numbers $x_1, x_2, x_3, x_4$ satisfying the condition \[ \dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1 \] Find the maximal and the minimal values of expression: \[ A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2 \]

2016 China Team Selection Test, 2

Tags: geometry , geometric inequality , inequalities

Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.

1973 Putnam, B4

Tags: integral , inequalities , integration

(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that $$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$ (b) Show an example in which equality occurs.

2018 China Western Mathematical Olympiad, 2

Tags: inequalities , fractional part , n-variable inequality , algebra

Let $n \geq 2$ be an integer. Positive reals $x_1, x_2, \cdots, x_n$ satisfy $x_1x_2 \cdots x_n = 1$. Show: $$\{x_1\} + \{x_2\} + \cdots + \{x_n\} < \frac{2n-1}{2}$$ Where $\{x\}$ denotes the fractional part of $x$.

MathLinks Contest 2nd, 3.1

Tags: functional , inequalities , algebra

Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$

2010 CentroAmerican, 5

Tags: algebra , triangle inequality , polynomial , inequalities

If $p$, $q$ and $r$ are nonzero rational numbers such that $\sqrt[3]{pq^2}+\sqrt[3]{qr^2}+\sqrt[3]{rp^2}$ is a nonzero rational number, prove that $\frac{1}{\sqrt[3]{pq^2}}+\frac{1}{\sqrt[3]{qr^2}}+\frac{1}{\sqrt[3]{rp^2}}$ is also a rational number.

2003 China National Olympiad, 3

Tags: inequalities

Suppose $a,b,c,d$ are positive reals such that $ab+cd=1$ and $x_i,y_i$ are real numbers such that $x_i^2+y_i^2=1$ for $i=1,2,3,4$. Prove that \[(ax_1+bx_2+cx_3+dx_4)^2+(ay_4+by_3+cy_2+dy_1)^2\le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).\] [i]Li Shenghong[/i]

2018 VJIMC, 2

Tags: integer part , floor function , inequalities , algebra

Let $n$ be a positive integer and let $a_1\le a_2 \le \dots \le a_n$ be real numbers such that \[a_1+2a_2+\dots+na_n=0.\] Prove that \[a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0\] for every real number $x$. (Here $[t]$ denotes the integer satisfying $[t] \le t<[t]+1$.)

1996 APMO, 2

Tags: inequalities

Let $m$ and $n$ be positive integers such that $n \leq m$. Prove that \[ 2^n n! \leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n \]

2010 South africa National Olympiad, 4

Tags: inequalities

Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that \[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]

1984 IMO Longlists, 36

Tags: inequalities , number theory

The set $\{1, 2, \cdots, 49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a, b, c$ such that $a + b = c$.

2017 Swedish Mathematical Competition, 6

Tags: algebra , inequalities

Let $a,b,c,x,y,z$ be real numbers such that $x+y+z=0$, $a+b+c\geq 0$, $ab+bc+ca \ge 0$. Prove that $$ ax^2+by^2+cz^2\ge 0 $$

2014 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , quadratic , complex number , dilation , geometric transformation , geometry , inequalities

Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.

1999 All-Russian Olympiad, 6

Tags: floor function , inequalities , number theory , algebra , induction

Prove that for all natural numbers $n$, \[ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. \] Here, $\{x\}$ denotes the fractional part of $x$.

2005 Regional Competition For Advanced Students, 1

Tags: inequalities

Show for all integers $ n \ge 2005$ the following chaine of inequalities: $ (n\plus{}830)^{2005}<n(n\plus{}1)\dots(n\plus{}2004)<(n\plus{}1002)^{2005}$

2019 China Girls Math Olympiad, 6

Tags: algebra , inequalities

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

2016 Iran MO (3rd Round), 1

Tags: sequence , algebra , inequalities

The sequence $(a_n)$ is defined as: $$a_1=1007$$ $$a_{i+1}\geq a_i+1$$ Prove the inequality: $$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$

2010 Vietnam Team Selection Test, 1

Tags: inequalities

Let $a,b,c$ be positive integers which satisfy the condition: $16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. Prove that \[\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq \frac{8}{9}\]

1962 Swedish Mathematical Competition, 4

Tags: trigonometry , geometry , number theory , inequalities

Which of the following statements are true? (A) $X$ implies $Y$, or $Y$ implies $X$, where $X$ is the statement, the lines $L_1, L_2, L_3$ lie in a plane, and $Y$ is the statement, each pair of the lines $L_1, L_2, L_3$ intersect. (B) Every sufficiently large integer $n$ satisfies $n = a^4 + b^4$ for some integers a, b. (C) There are real numbers $a_1, a_2,... , a_n$ such that $a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0$ for all real $x$.

1998 Romania Team Selection Test, 2

Tags: 3d geometry , geometry , prism , inequalities , trigonometry

A parallelepiped has surface area 216 and volume 216. Show that it is a cube.

2004 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities

Let $ABC$ be a triangle with side lengths $a,b,c$, such that $a$ is the longest side. Prove that $\angle BAC = 90^\circ$ if and only if \[ (\sqrt { a+b } + \sqrt { a-b} )(\sqrt {a+c } + \sqrt { a-c } ) = (a+b+c) \sqrt 2. \]

2002 Turkey Team Selection Test, 3

Tags: inequalities

A positive integer $n$ and real numbers $a_1,\dots, a_n$ are given. Show that there exists integers $m$ and $k$ such that \[|\sum\limits_{i=1}^m a_i -\sum\limits_{i=m+1}^n a_i | \leq |a_k|.\]

2022 Taiwan TST Round 2, A

Tags: inequality , algebra , inequalities

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

2012 Bosnia Herzegovina Team Selection Test, 2

Tags: trigonometry , inequalities

Prove for all positive real numbers $a,b,c$, such that $a^2+b^2+c^2=1$: \[\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.\]

2011 239 Open Mathematical Olympiad, 7

Tags: inequalities

Prove for positive reals $a,b,c$ that $(ab+bc+ca+1)(a+b)(b+c)(c+a) \ge 2abc(a+b+c+1)^2$