Olympiad problems

2014 Contests, 3

Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation \[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \] holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.

2022-2023 OMMC, 21

Tags: inequalities

Define the minimum real $C$ where for any reals $0 = a_0 < a_{1} < \dots < a_{1000}$ then $$\min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor.$

1991 Baltic Way, 5

Tags: inequalities , search

For any positive numbers $a, b, c$ prove the inequalities \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.\]

1995 Baltic Way, 6

Tags: search , inequalities

Prove that for positive $a,b,c,d$ \[\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\ge 4\]

1993 Chile National Olympiad, 5

Tags: inequalities , algebra

Let $a,b,c$ three positive numbers less than $ 1$. Prove that cannot occur simultaneously these three inequalities: $$a (1- b)>\frac14$$ $$b (1-c)>\frac14 $$ $$c (1-a)>\frac14$$

2014 Bosnia and Herzegovina Junior BMO TST, 3

Tags: algebra , inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality: $\frac{1}{\sqrt{(a+2b)(b+2a)}}+\frac{1}{\sqrt{(b+2c)(c+2b)}}+\frac{1}{\sqrt{(c+2a)(a+2c)}} \geq 3$

2008 China Team Selection Test, 2

Tags: inequalities

For a given integer $ n\geq 2,$ determine the necessary and sufficient conditions that real numbers $ a_{1},a_{2},\cdots, a_{n},$ not all zero satisfy such that there exist integers $ 0<x_{1}<x_{2}<\cdots<x_{n},$ satisfying $ a_{1}x_{1}\plus{}a_{2}x_{2}\plus{}\cdots\plus{}a_{n}x_{n}\geq 0.$

2014 Peru IMO TST, 15

Tags: inequalities , sequence , algebra

Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]

1965 AMC 12/AHSME, 23

Tags: inequalities , function

If we write $ |x^2 \minus{} 4| < N$ for all $ x$ such that $ |x \minus{} 2| < 0.01$, the smallest value we can use for $ N$ is: $ \textbf{(A)}\ .0301 \qquad \textbf{(B)}\ .0349 \qquad \textbf{(C)}\ .0399 \qquad \textbf{(D)}\ .0401 \qquad \textbf{(E)}\ .0499 \qquad$

2021 Science ON all problems, 4

Tags: inequality , inequalities , algebra

Consider positive real numbers $x,y,z$. Prove the inequality $$\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).$$ [i] (Vlad Robu \& Sergiu Novac)[/i]

1990 Tournament Of Towns, (251) 5

Tags: combinatorics , number theory , inequalities

Find the number of pairs $(m, n)$ of positive integers, both of which are $\le 1000$, such that $\frac{m}{n+1}< \sqrt2 < \frac{m+1}{n}$ (recalling that $ \sqrt2 = 1.414213..$.). (D. Fomin, Leningrad)

2002 Czech-Polish-Slovak Match, 1

Tags: inequalities , ceiling function , floor function , algebra

Let $a, b$ be distinct real numbers and $k,m$ be positive integers $k + m = n \ge 3, k \le 2m, m \le 2k$. Consider sequences $x_1,\dots , x_n$ with the following properties: (i) $k$ terms $x_i$, including $x_1$, are equal to $a$; (ii) $m$ terms $x_i$, including $x_n$, are equal to $b$; (iii) no three consecutive terms are equal. Find all possible values of $x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1$.

2013 Swedish Mathematical Competition, 5

Tags: combinatorics , inequalities

Let $n \geq 2$ be a positive integer. Show that there are exactly $2^{n-3}n(n-1)$ $n$-tuples of integers $(a_1,a_2,\dots,a_n)$, which satisfy the conditions: (i) $a_1=0$; (ii) for each $m$, $2 \leq m \leq n$, there is an index in $m$, $1 \leq i_m <m$, such that $\left|a_{i_m}-a_m\right|\leq 1$; (iii) the $n$-tuple $(a_1,a_2,\dots,a_n)$ contains exactly $n-1$ different numbers.

2017 Singapore Junior Math Olympiad, 1

Tags: geometry , rectangle , ratio , geometric inequalities , inequalities , square , sidelenghts

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.

2006 IMO Shortlist, 6

Tags: inequalities , function , algebra

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

2001 Estonia National Olympiad, 4

Tags: inequalities , algebra

If $x$ and $y$ are nonnegative real numbers with $x+y= 2$, show that $x^2y^2(x^2+y^2)\le 2$.

2024 Nepal TST, P2

Tags: function , natural number , inequalities

Let $f: \mathbb{N} \to \mathbb{N}$ be an arbitrary function. Prove that there exist two positive integers $x$ and $y$ which satisfy $f(x+y) \le f(2x+f(y))$. [i](Proposed by David Anghel, Romania)[/i]

2005 Abels Math Contest (Norwegian MO), 4b

Tags: inequalities , algebra

Let $a, b$ and $c$ be real numbers such that $ab + bc + ca> a + b + c> 0$. Show then that $a+b+c>3$

2011 Mongolia Team Selection Test, 1

Tags: inequalities

Let $t,k,m$ be positive integers and $t>\sqrt{km}$. Prove that $\dbinom{2m}{0}+\dbinom{2m}{1}+\cdots+\dbinom{2m}{m-t-1}<\dfrac{2^{2m}}{2k}$ (proposed by B. Amarsanaa, folklore)

1956 AMC 12/AHSME, 44

Tags: inequalities

If $ x < a < 0$ means that $ x$ and $ a$ are numbers such that $ x$ is less than $ a$ and $ a$ is less than zero, then: $ \textbf{(A)}\ x^2 < ax < 0 \qquad\textbf{(B)}\ x^2 > ax > a^2 \qquad\textbf{(C)}\ x^2 < a^2 < 0$ $ \textbf{(D)}\ x^2 > ax\text{ but }ax < 0 \qquad\textbf{(E)}\ x^2 > a^2\text{ but }a^2 < 0$

2020 Regional Olympiad of Mexico Center Zone, 2

Tags: algebra , inequalities

Let $a$, $b$ and $c$ be positive real numbers, prove that \[\frac{2a^2 b^2}{a^5+b^5}+\frac{2b^2 c^2}{b^5+c^5}+\frac{2c^2 a^2}{c^5+a^5}\le\frac{a+b}{2ab}+\frac{b+c}{2bc}+\frac{c+a}{2ca}\]

2010 Czech-Polish-Slovak Match, 2

Tags: inequalities

Let $x$, $y$, $z$ be positive real numbers satisfying $x+y+z\ge 6$. Find, with proof, the minimum value of \[ x^2+y^2+z^2+\frac{x}{y^2+z+1}+\frac{y}{z^2+x+1}+\frac{z}{x^2+y+1}. \]

2004 China National Olympiad, 2

Tags: inequalities , algebra , polynomial , n-variable inequality

For a given positive integer $n\ge 2$, suppose positive integers $a_i$ where $1\le i\le n$ satisfy $a_1<a_2<\ldots <a_n$ and $\sum_{i=1}^n \frac{1}{a_i}\le 1$. Prove that, for any real number $x$, the following inequality holds \[\left(\sum_{i=1}^n\frac{1}{a_i^2+x^2}\right)^2\le\frac{1}{2}\cdot\frac{1}{a_1(a_1-1)+x^2} \] [i]Li Shenghong[/i]

2018 Pan African, 5

Tags: inequalities , algebra

Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd. $$ Show that $$ \frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12 $$ and that $-12$ is the maximum.

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 \[ \left( \frac{x+y}{z+w} + \frac{z+w}{x+y} \right) ^{-1} + \frac{1}{2} \ge \left( \frac{x}{z} + \frac{z}{x} \right) ^{-1} + \left( \frac{y}{w} + \frac{w}{y} \right) ^{-1}\]