Olympiad problems

2024 Kyiv City MO Round 2, Problem 2

Tags: algebra , inequalities

For any positive real numbers $a, b, c, d$, prove the following inequality: $$(a^2+b^2)(b^2+c^2)(c^2+d^2)(d^2+a^2) \geq 64abcd|(a-b)(b-c)(c-d)(d-a)|$$ [i]Proposed by Anton Trygub[/i]

2017 Thailand TSTST, 3

Tags: inequalities

Let $a, b, c \in\mathbb{R}^+$. Prove that $$\sum_{cyc}ab\left(\frac{1}{2a+c}+\frac{1}{2b+c}\right)<\sum_{cyc}\frac{a^3+b^3}{c^2+ab}.$$

2021 IMO Shortlist, A4

Tags: n-variable inequality , convexity , concavity , inequalities , algebra

Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$

2014 Contests, 1

Tags: inequality , algebra , inequalities

Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true: $\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.

1998 Canada National Olympiad, 3

Tags: inequalities

Let $ n$ be a natural number such that $ n \geq 2$. Show that \[ \frac {1}{n \plus{} 1} \left( 1 \plus{} \frac {1}{3} \plus{} \cdot \cdot \cdot \plus{} \frac {1}{2n \minus{} 1} \right) > \frac {1}{n} \left( \frac {1}{2} \plus{} \frac {1}{4} \plus{} \cdot \cdot \cdot \plus{} \frac {1}{2n} \right). \]

2006 AMC 12/AHSME, 22

Tags: floor function , inequalities , function , algebra

Suppose $ a, b,$ and $ c$ are positive integers with $ a \plus{} b \plus{} c \equal{} 2006$, and $ a!b!c! \equal{} m\cdot10^n$, where $ m$ and $ n$ are integers and $ m$ is not divisible by 10. What is the smallest possible value of $ n$? $ \textbf{(A) } 489 \qquad \textbf{(B) } 492 \qquad \textbf{(C) } 495 \qquad \textbf{(D) } 498 \qquad \textbf{(E) } 501$

2014 IFYM, Sozopol, 6

Tags: inequalities

$x_1,...,x_n$ are non-negative reals and $n \geq 3$. Prove that at least one of the following inequalities is true: \[ \sum_{i=1} ^n \frac{x_i}{x_{i+1}+x_{i+2}} \geq \frac{n}{2}, \] \[ \sum_{i=1} ^n \frac{x_i}{x_{i-1}+x_{i-2}} \geq \frac{n}{2} . \]

1997 Vietnam Team Selection Test, 3

Tags: polynomial , inequalities , induction , number theory , algebra

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

2010 Belarus Team Selection Test, 1.3

Tags: algebra , inequalities

Given $a, b,c \ge 0, a + b + c = 1$, prove that $(a^2 + b^2 + c^2)^2 + 6abc \ge ab + bc + ac$ (I. Voronovich)

2006 Romania Team Selection Test, 2

Tags: geometry , inequalities

Let $ABC$ be a triangle with $\angle B = 30^{\circ }$. We consider the closed disks of radius $\frac{AC}3$, centered in $A$, $B$, $C$. Does there exist an equilateral triangle with one vertex in each of the 3 disks? [i]Radu Gologan, Dan Schwarz[/i]

2010 Contests, 1

Tags: perimeter , trigonometry , inequalities , incenter , inradius , geometry

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

2010 All-Russian Olympiad, 2

Tags: combinatorics , inequalities

There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.

2018 Greece Team Selection Test, 1

Tags: inequalities

If $x, y, z$ are positive real numbers such that $x + y + z = 9xyz.$ Prove that: $$\frac {x} {\sqrt {x^2+2yz+2}}+\frac {y} {\sqrt {y^2+2zx+2}}+\frac {z} {\sqrt {z^2+2xy+2}}\ge 1.$$ Consider when equality applies.

1998 Yugoslav Team Selection Test, Problem 2

Tags: inequalities , geometric inequalities , geometry , geometric inequality

In a convex quadrilateral $ABCD$, the diagonal $AC$ intersects the diagonal $BD$ at its midpoint $S$. The radii of incircles of triangles $ABS,BCS,CDS,DAS$ are $r_1,r_2,r_3,r_4$, respectively. Prove that $$|r_1-r_2+r_3-r_4|\le\frac18|AB-BC+CD-DA|.$$

2007 ISI B.Math Entrance Exam, 7

Tags: function , trigonometry , inequalities

Let $ 0\leq \theta\leq \frac{\pi}{2}$ . Prove that $\sin \theta \geq \frac{2\theta}{\pi}$.

2003 China Girls Math Olympiad, 4

Tags: inequalities

(1) Prove that there exist five nonnegative real numbers $ a, b, c, d$ and $ e$ with their sum equal to 1 such that for any arrangement of these numbers around a circle, there are always two neighboring numbers with their product not less than $ \frac{1}{9}.$ (2) Prove that for any five nonnegative real numbers with their sum equal to 1 , it is always possible to arrange them around a circle such that there are two neighboring numbers with their product not greater than $ \frac{1}{9}.$

1999 APMO, 2

Tags: inequalities , logarithm , induction

Let $a_1, a_2, \dots$ be a sequence of real numbers satisfying $a_{i+j} \leq a_i+a_j$ for all $i,j=1,2,\dots$. Prove that \[ a_1 + \frac{a_2}{2} + \frac{a_3}{3} + \cdots + \frac{a_n}{n} \geq a_n \] for each positive integer $n$.

2024 South Africa National Olympiad, 2

Tags: inequalities

Determine which of the following is larger: \[ \sqrt{2+\sqrt[3]{5}}\qquad \text{or}\qquad \sqrt[3]{5+\sqrt{2}}.\] Fully explain your reasoning.

2018 AIME Problems, 15

Tags: inequalities

Find the number of functions $f$ from $\{0,1,2,3,4,5,6\}$ to the integers such that $f(0)=0, f(6)=12$, and \[|x-y| \le |f(x)-f(y)| \le 3 |x-y| \]for all $x$ and $y$ in $\{0,1,2,3,4,5,6\}$.

2009 China Team Selection Test, 3

Tags: inequalities , induction

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$

1992 IMO Longlists, 11

Tags: function , relatively prime , number theoretic functions , number theory , inequalities

Let $\phi(n,m), m \neq 1$, be the number of positive integers less than or equal to $n$ that are coprime with $m.$ Clearly, $\phi(m,m) = \phi(m)$, where $\phi(m)$ is Euler’s phi function. Find all integers $m$ that satisfy the following inequality: \[\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}\] for every positive integer $n.$

1990 Bulgaria National Olympiad, Problem 6

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

The base $ABC$ of a tetrahedron $MABC$ is an equilateral triangle, and the lateral edges $MA,MB,MC$ are sides of a triangle of the area $S$. If $R$ is the circumradius and $V$ the volume of the tetrahedron, prove that $RS\ge2V$. When does equality hold?

1999 Romania Team Selection Test, 11

Tags: polynomial , absolute value , inequalities , algebra

Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials. [i]Laurentiu Panaitopol[/i]

1988 Romania Team Selection Test, 4

Tags: induction , inequalities

Prove that for all positive integers $0<a_1<a_2<\cdots <a_n$ the following inequality holds: \[ (a_1+a_2+\cdots + a_n)^2 \leq a_1^3+a_2^3 + \cdots + a_n^3 . \] [i]Viorel Vajaitu[/i]

2023 Abelkonkurransen Finale, 4a

Tags: geometry , inequalities

Assuming $a,b,c$ are the side-lengths of a triangle, show that \begin{align*} \frac{a^2+b^2-c^2}{ab} + \frac{b^2+c^2-a^2}{bc} + \frac{c^2+a^2-b^2}{ca} > 2. \end{align*} Also show that the inequality does not necessarily hold if you replace $2$ (on the right-hand side) by a bigger by a bigger number.