Olympiad problems

2016 Saudi Arabia GMO TST, 1

Let $S = x + y +z$ where $x, y, z$ are three nonzero real numbers satisfying the following system of inequalities: $$xyz > 1$$ $$x + y + z >\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$ Prove that $S$ can take on any real values when $x, y, z$ vary

1972 IMO Longlists, 40

Tags: calculus , inequalities , trigonometry , function

Prove the inequalities \[\frac{u}{v}\le \frac{\sin u}{\sin v}\le \frac{\pi}{2}\times\frac{u}{v},\text{ for }0 \le u < v \le \frac{\pi}{2}\]

2010 India IMO Training Camp, 4

Tags: inequalities , function

Let $a,b,c$ be positive real numbers such that $ab+bc+ca\le 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\le \sqrt{2} (\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a})\]

1987 IMO Longlists, 26

Tags: inequalities , algebra

Prove that if $x, y, z$ are real numbers such that $x^2+y^2+z^2 = 2$, then \[x + y + z \leq xyz + 2.\]

2000 Junior Balkan MO, 4

Tags: combinatorics , inequalities

At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is knowns that there were no draws. Find $n$. [i]Serbia[/i]

1999 Romania Team Selection Test, 9

Tags: reflection , geometry , trigonometry , inequalities , parallelogram , algebra , geometric transformation

Let $O,A,B,C$ be variable points in the plane such that $OA=4$, $OB=2\sqrt3$ and $OC=\sqrt {22}$. Find the maximum value of the area $ABC$. [i]Mihai Baluna[/i]

1997 Romania Team Selection Test, 1

Tags: polynomial , algebra , inequalities

Let $P(X),Q(X)$ be monic irreducible polynomials with rational coefficients. suppose that $P(X)$ and $Q(X)$ have roots $\alpha$ and $\beta$ respectively, such that $\alpha + \beta $ is rational. Prove that $P(X)^2-Q(X)^2$ has a rational root. [i]Bogdan Enescu[/i]

2023 Indonesia TST, A

Tags: algebra , inequalities

Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that \[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]

2022 Moscow Mathematical Olympiad, 1

Tags: algebra , inequalities

$a,b,c$ are nonnegative and $a+b+c=2\sqrt{abc}$. Prove $bc \geq b+c$

2010 Contests, 2

Tags: inequalities , combinatorics

Let $n$ be a positive integer. Find the number of sequences $x_{1},x_{2},\ldots x_{2n-1},x_{2n}$, where $x_{i}\in\{-1,1\}$ for each $i$, satisfying the following condition: for any integer $k$ and $m$ such that $1\le k\le m\le n$ then the following inequality holds \[\left|\sum_{i=2k-1}^{2m}x_{i}\right|\le\ 2\]

2009 China Girls Math Olympiad, 3

Tags: inequalities , analytic geometry

Let $ n$ be a given positive integer. In the coordinate set, consider the set of points $ \{P_{1},P_{2},...,P_{4n\plus{}1}\}\equal{}\{(x,y)|x,y\in \mathbb{Z}, xy\equal{}0, |x|\le n, |y|\le n\}.$ Determine the minimum of $ (P_{1}P_{2})^{2} \plus{} (P_{2}P_{3})^{2} \plus{}...\plus{} (P_{4n}P_{4n\plus{}1})^{2} \plus{} (P_{4n\plus{}1}P_{1})^{2}.$

1994 North Macedonia National Olympiad, 3

Tags: sum , product , max , algebra , inequalities

a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $ b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $

2022 Cyprus JBMO TST, 3

Tags: inequalities , algebra

If $x,y$ are real numbers with $x+y\geqslant 0$, determine the minimum value of the expression \[K=x^5+y^5-x^4y-xy^4+x^2+4x+7\] For which values of $x,y$ does $K$ take its minimum value?

2005 Colombia Team Selection Test, 4

Tags: inequalities , induction

1. Prove the following inequality for positive reals $a_1,a_2...,a_n$ and $b_1,b_2...,b_n$: $(\sum a_i)(\sum b_i)\geq (\sum a_i+b_i)(\sum\frac{a_ib_i}{a_i+b_i})$

1986 IMO Longlists, 30

Tags: 3d geometry , icosahedron , octahedron , dodecahedron , tetrahedron , geometry , inequalities

Prove that a convex polyhedron all of whose faces are equilateral triangles has at most $30$ edges.

2018 Saudi Arabia IMO TST, 3

Tags: fibonacci number , inequalities , max , algebra

Consider the function $f (x) = (x - F_1)(x - F_2) ...(x -F_{3030})$ with $(F_n)$ is the Fibonacci sequence, which defined as $F_1 = 1, F_2 = 2$, $F_{n+2 }=F_{n+1} + F_n$, $n \ge 1$. Suppose that on the range $(F_1, F_{3030})$, the function $|f (x)|$ takes on the maximum value at $x = x_0$. Prove that $x_0 > 2^{2018}$.

2006 Moldova National Olympiad, 10.5

Tags: derivative , function , algebra , inequalities , calculus , logarithm

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

2013 BMT Spring, 2

Tags: geometry , inequalities , algebra

A point $P$ is given on the curve $x^4+y^4=1$. Find the maximum distance from the point $P$ to the origin.

2008 Indonesia MO, 2

Tags: function , inequalities

Prove that for $ x,y\in\mathbb{R^ \plus{} }$, $ \frac {1}{(1 \plus{} \sqrt {x})^{2}} \plus{} \frac {1}{(1 \plus{} \sqrt {y})^{2}} \ge \frac {2}{x \plus{} y \plus{} 2}$

1977 AMC 12/AHSME, 5

Tags: parabola , triangle inequality , geometry , conic , inequalities , perpendicular bisector

The set of all points $P$ such that the sum of the (undirected) distances from $P$ to two fixed points $A$ and $B$ equals the distance between $A$ and $B$ is $\textbf{(A) }\text{the line segment from }A\text{ to }B\qquad$ $\textbf{(B) }\text{the line passing through }A\text{ and }B\qquad$ $\textbf{(C) }\text{the perpendicular bisector of the line segment from }A\text{ to }B\qquad$ $\textbf{(D) }\text{an elllipse having positive area}\qquad$ $\textbf{(E) }\text{a parabola}$

2004 Croatia Team Selection Test, 2

Tags: inequalities

Prove that if $a,b,c$ are positive numbers with $abc=1$, then \[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]

2004 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities

Find all positive reals $a,b,c$ which fulfill the following relation \[ 4(ab+bc+ca)-1 \geq a^2+b^2+c^2 \geq 3(a^3+b^3+c^3) . \] created by Panaitopol Laurentiu.

2013 China Girls Math Olympiad, 3

Tags: pigeonhole principle , combinatorics , graph theory , symmetry , inequalities

In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.

2009 Czech and Slovak Olympiad III A, 3

Tags: inequalities

Find the least value of $x>0$ such that for all positive real numbers $a,b,c,d$ satisfying $abcd=1$, the inequality $a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ is true.

2014 Contests, 3

Tags: extremal combinatorics , inequalities , combinatorics , graph theory , induction

Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable. [i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]