Olympiad problems

1972 USAMO, 2

A given tetrahedron $ ABCD$ is isoceles, that is, $ AB\equal{}CD$, $ AC\equal{}BD$, $ AD\equal{}BC$. Show that the faces of the tetrahedron are acute-angled triangles.

1984 IMO Longlists, 67

Tags: inequalities , circumcircle , geometry

With the medians of an acute-angled triangle another triangle is constructed. If $R$ and $R_m$ are the radii of the circles circumscribed about the first and the second triangle, respectively, prove that \[R_m>\frac{5}{6}R\]

2002 VJIMC, Problem 3

Tags: calculus , integration , inequalities

Let $E$ be the set of all continuous functions $u:[0,1]\to\mathbb R$ satisfying $$u^2(t)\le1+4\int^t_0s|u(s)|\text ds,\qquad\forall t\in[0,1].$$Let $\varphi:E\to\mathbb R$ be defined by $$\varphi(u)=\int^1_0\left(u^2(x)-u(x)\right)\text dx.$$Prove that $\varphi$ has a maximum value and find it.

2011 Hanoi Open Mathematics Competitions, 4

Tags: inequalities , polynomial , algebra

Prove that $1 + x + x^2 + x^3 + ...+ x^{2011} \ge 0$ for every $x \ge - 1$ .

1996 Romania National Olympiad, 4

Tags: algebra , geometric inequality , geometric inequalities , 3d geometry , inequalities , geometry

a) Let $AB CD$ be a regular tetrahedron. On the sides $AB$, $AC$ and $AD$, the points $M$, $N$ and $P$, are considered. Determine the volume of the tetrahedron $AMNP$ in terms of $x, y, z$, where $x=AM$, $y=AN$, $z=AP$. b) Show that for any real numbers $x, y, z, t, u, v \in (0, 1)$ : $$xyz + uv(1- x) + (1- y)(1- v)t + (1- z)(1- w)(1- t) < 1.$$

2010 Contests, 2

Tags: n-variable inequality , inequalities

Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality \[ \frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}\] does holds.

1969 AMC 12/AHSME, 14

Tags: quadratic , inequalities

The complete set of $x$-values satisfying the inequality $\dfrac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that: $\textbf{(A) }x>2\text{ or }x<-2\text{ or }-1<x<1\qquad\, \textbf{(B) }x>2\text{ or }x<-2$ $\textbf{(C) }x>1\text{ or }x<-2\qquad\qquad\qquad\qquad\,\,\,\,\,\,\, \textbf{(D) }x>1\text{ or }x<-2\qquad$ $\textbf{(E) }x\text{ is any real number except }1\text{ or }-1$

2012 ELMO Shortlist, 1

Tags: inequalities , trigonometry

Let $x_1,x_2,x_3,y_1,y_2,y_3$ be nonzero real numbers satisfying $x_1+x_2+x_3=0, y_1+y_2+y_3=0$. Prove that \[\frac{x_1x_2+y_1y_2}{\sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)}}+\frac{x_2x_3+y_2y_3}{\sqrt{(x_2^2+y_2^2)(x_3^2+y_3^2)}}+\frac{x_3x_1+y_3y_1}{\sqrt{(x_3^2+y_3^2)(x_1^2+y_1^2)}} \ge -\frac32.\] [i]Ray Li, Max Schindler.[/i]

2014 VJIMC, Problem 4

Tags: inequalities , integration , calculus

Let $0<a<b$ and let $f:[a,b]\to\mathbb R$ be a continuous function with $\int^b_af(t)dt=0$. Show that $$\int^b_a\int^b_af(x)f(y)\ln(x+y)dxdy\le0.$$

2024 Thailand Mathematical Olympiad, 8

Tags: vector geometry , geometry , inequalities

Let $ABCDEF$ be a convex hexagon and denote $U$,$V$,$W$,$X$,$Y$ and $Z$ be the midpoint of $AB$,$BC$,$CD$,$DE$,$EF$ and $FA$ respectively. Prove that the length of $UX$,$VY$,$WZ$ can be the length of each sides of some triangle.

2013 Tuymaada Olympiad, 3

Tags: inequalities

For every positive real numbers $a$ and $b$ prove the inequality \[\displaystyle \sqrt{ab} \leq \dfrac{1}{3} \sqrt{\dfrac{a^2+b^2}{2}}+\dfrac{2}{3} \dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}.\] [i]A. Khabrov[/i]

1997 Romania Team Selection Test, 1

Tags: inequalities , geometry , trigonometry

We are given in the plane a line $\ell$ and three circles with centres $A,B,C$ such that they are all tangent to $\ell$ and pairwise externally tangent to each other. Prove that the triangle $ABC$ has an obtuse angle and find all possible values of this this angle. [i]Mircea Becheanu[/i]

2008 Bulgarian Autumn Math Competition, Problem 12.1

Tags: algebra , interval , parameterization , inequalities

Determine the values of the real parameter $a$, such that the solutions of the system of inequalities $\begin{cases} \log_{\frac{1}{3}}{(3^{x}-6a)}+\frac{2}{\log_{a}{3}}<x-3\\ \log_{\frac{1}{3}}{(3^{x}-18)}>x-5\\ \end{cases}$ form an interval of length $\frac{1}{3}$.

2013 Hanoi Open Mathematics Competitions, 12

Tags: quadratic , root , inequalities , algebra

If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$, prove that the equation $f(x) = 2x^2 - 1$ has two real roots.

1985 Traian Lălescu, 2.2

Tags: inequalities , calculus , young s inequality , integral inequality , bijective , integral , real analysis

Let $ a,b,c\in\mathbb{R}_+^*, $ and $ f:[0,a]\longrightarrow [0,b] $ bijective and non-decreasing. Prove that: $$ \frac{1}{b}\int_0^a f^2 (x)dx +\frac{1}{a}\int_0^b \left( f^{-1} (x)\right)^2dx\le ab. $$

2008 Germany Team Selection Test, 3

Tags: inequalities , polynomial , algebra

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

2021 Iran Team Selection Test, 4

Tags: number theory , function , algebra , inequalities

Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$f(ac)+f(bc)-f(c)f(ab) \ge 1$$ Proposed by [i]Mojtaba Zare[/i]

1999 National High School Mathematics League, 13

Tags: inequalities

If $x^2\cos\theta-x(1-x)+(1-x)^2\sin\theta>0$ for all $x\in[0,1]$, find the range value of $\theta$.

1988 Romania Team Selection Test, 11

Tags: inequalities , trigonometry

2003 India IMO Training Camp, 9

Tags: inequalities , induction , combinatorics , number theory

Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.

2003 USA Team Selection Test, 5

Tags: function , trigonometry , inequalities

Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$. Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} \\ Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} \\ Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$.