Olympiad problems

2002 Tournament Of Towns, 1

Tags: inequalities

Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.

1996 USAMO, 3

Tags: ratio , geometric transformation , trapezoid , geometry , angle bisector , inequalities , reflection

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

2020 Costa Rica - Final Round, 3

Tags: algebra , inequalities

Let $x, y, z \in R^+$. Prove that $$\frac{x}{x +\sqrt{(x + y)(x + z)}}+\frac{y}{y +\sqrt{(y + z)(y + x)}}+\frac{z}{z +\sqrt{(x + z)(z + y)}} \le 1$$

2001 National Olympiad First Round, 18

Tags: perpendicular bisector , inequalities , triangle inequality , geometry

A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

2008 India Regional Mathematical Olympiad, 3

Tags: inequality , positive integer , real number , inequalities

Prove that for every positive integer $n$ and a non-negative real number $a$, the following inequality holds: $$n(n+1)a+2n \geqslant 4\sqrt{a}(\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}).$$

2003 China Team Selection Test, 1

Tags: trigonometry , inequalities

Let $g(x)= \sum_{k=1}^{n} a_k \cos{kx}$, $a_1,a_2, \cdots, a_n, x \in R$. If $g(x) \geq -1$ holds for every $x \in R$, prove that $\sum_{k=1}^{n}a_k \leq n$.

2016 Iran MO (2nd Round), 1

Tags: inequalities , algebra

If $0<a\leq b\leq c$ prove that $$\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$

2005 Kyiv Mathematical Festival, 1

Tags: inequalities

Prove that there exists a positive integer $ n$ such that for every $ x\ge0$ the inequality $ (x\minus{}1)(x^{2005}\minus{}2005x^{n\plus{}1}\plus{}2005x^n\minus{}1)\ge0$ holds.

2000 Greece National Olympiad, 3

Tags: inequalities

Find the maximum value of $k$ such that \[\frac{xy}{\sqrt{(x^2 + y^2)(3x^2 + y^2)}}\leq \frac{1}{k}\] holds for all positive numbers $x$ and $y.$

2014 Korea Junior Math Olympiad, 6

Tags: maximum , inequalities , algebra

Let $p = 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}. $ For nonnegative reals $x, y,z$ satisfying $(x-1)^2 + (y-1)^2 + (z-1)^2 = 27,$ find the maximum value of $x^p + y^p + z^p.$

2012 Today's Calculation Of Integral, 843

Tags: special factorizations , function , integration , calculus computations , inequalities , calculus

Let $f(x)$ be a continuous function such that $\int_0^1 f(x)\ dx=1.$ Find $f(x)$ for which $\int_0^1 (x^2+x+1)f(x)^2dx$ is minimized.

2008 Hanoi Open Mathematics Competitions, 10

Tags: algebra , inequalities

Let $a,b,c \in [1, 3]$ and satisfy the following conditions: $ max \{a, b, c\}\ge 2$ and $ a + b + c = 5$ What is the smallest possible value of $a^2 + b^2 + c^2$?

2010 Polish MO Finals, 1

Tags: perimeter , geometry , inequalities , parallelogram

On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$. Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that \[p_1 > p_2 + 2\cdot \min\{BD, EC\}.\]

2015 USA TSTST, 4

Tags: inequalities , algebra

Let $x$, $y$, and $z$ be real numbers (not necessarily positive) such that $x^4+y^4+z^4+xyz=4$. Show that $x\le2$ and $\sqrt{2-x}\ge\frac{y+z}{2}$. [i]Proposed by Alyazeed Basyoni[/i]

2004 Moldova Team Selection Test, 2

Tags: 3d geometry , geometry , sphere , tetrahedron , inequalities

In the tetrahedron $ABCD$ the radius of its inscribed sphere is $r$ and the radiuses of the exinscribed spheres (each tangent with a face of the tetrahedron and with the planes of the other faces) are $r_A, r_B, r_C, r_D.$ Prove the inequality $$\frac{1}{\sqrt{r_A^2-r_Ar_B+r_B^2}}+\frac{1}{\sqrt{r_B^2-r_Br_C+r_C^2}}+\frac{1}{\sqrt{r_C^2-r_Cr_D+r_D^2}}+\frac{1}{\sqrt{r_D^2-r_Dr_A+r_A^2}}\leq\frac{2}{r}.$$

2011 National Olympiad First Round, 1

Tags: inequalities

Which one is true for a quadrilateral $ABCD$ such that perpendicular bisectors of $[AB]$ and $[CD]$ meet on the diagonal $[AC]$? $\textbf{(A)}\ |BA| + |AD| \leq |BC| + |CD| \\ \textbf{(B)}\ |BD| \leq |AC| \\ \textbf{(C)}\ |AC| \leq |BD| \\ \textbf{(D)}\ |AD| + |DC| \leq |AB| + |BC| \\ \textbf{(E)}\ \text{None}$

2008 Irish Math Olympiad, 2

Tags: inequalities

For positive real numbers $ a$, $ b$, $ c$ and $ d$ such that $ a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2 \equal{} 1$ prove that $ a^2b^2cd \plus{} \plus{}ab^2c^2d \plus{} abc^2d^2 \plus{} a^2bcd^2 \plus{} a^2bc^2d \plus{} ab^2cd^2 \le 3/32,$ and determine the cases of equality.

JBMO Geometry Collection, 1997

Tags: inequalities

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$, $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $AI+BI+CI>BC+KL$. [i]Greece[/i]

2012 ELMO Shortlist, 6

Tags: vieta , quadratic , number theory , floor function , algebra , inequalities , polynomial

Prove that if $a$ and $b$ are positive integers and $ab>1$, then \[\left\lfloor\frac{(a-b)^2-1}{ab}\right\rfloor=\left\lfloor\frac{(a-b)^2-1}{ab-1}\right\rfloor.\]Here $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. [i]Calvin Deng.[/i]

2013 Today's Calculation Of Integral, 870

Tags: conic , hyperbola , ellipse , geometry , calculus , inequalities , integration

Consider the ellipse $E: 3x^2+y^2=3$ and the hyperbola $H: xy=\frac 34.$ (1) Find all points of intersection of $E$ and $H$. (2) Find the area of the region expressed by the system of inequality \[\left\{ \begin{array}{ll} 3x^2+y^2\leq 3 &\quad \\ xy\geq \frac 34 , &\quad \end{array} \right.\]

2023 Turkey Team Selection Test, 6

Tags: minimum value , inequalities

Let $a,b,c,d$ be positive real numbers. What is the minimum value of $$ \frac{(a^2+b^2+2c^2+3d^2)(2a^2+3b^2+6c^2+6d^2)}{(a+b)^2(c+d)^2}$$

2023 Macedonian Team Selection Test, Problem 2

Tags: geometric inequality , inequalities , geometry , reflection , geometric transformation

Let $ABC$ be an acute triangle such that $AB<AC$ and $AB<BC$. Let $P$ be a point on the segment $BC$ such that $\angle APB = \angle BAC$. The tangent to the circumcircle of triangle $ABC$ at $A$ meets the circumcircle of triangle $APB$ at $Q \neq A$. Let $Q'$ be the reflection of $Q$ with respect to the midpoint of $AB$. The line $PQ$ meets the segment $AQ'$ at $S$. Prove that $$\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.$$ [i]Authored by Nikola Velov[/i]

1989 Iran MO (2nd round), 1

Tags: limit , function , floor function , inequalities

[b](a)[/b] Let $n$ be a positive integer, prove that \[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\] [b](b)[/b] Find a positive integer $n$ for which \[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]

2007 All-Russian Olympiad, 5

Tags: vector , geometry , inequalities

Given a set of $n>2$ planar vectors. A vector from this set is called [i]long[/i], if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero. [i]N. Agakhanov[/i]

2005 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: circumcircle , geometry , inequalities

Let $P$ be a point inside triangle $ABC$ and let $R$ denote the circumradius of triangle $ABC$. Prove that \[ \frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.\]