Olympiad problems

2019 Federal Competition For Advanced Students, P2, 4

Tags: inequalities

Let $a, b, c$ be the positive real numbers such that $a+b+c+2=abc .$ Prove that $$(a+1)(b+1)(c+1)\geq 27.$$

1995 Brazil National Olympiad, 3

Tags: greatest common divisor , number theory , prime divisor , inequalities

For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which \[ P\left(n\right)<P\left(n\plus{}1\right)<P\left(n\plus{}2\right).\]

KoMaL A Problems 2017/2018, A. 709

Tags: inequalities , algebra

Let $a>0$ be a real number. Find the minimal constant $C_a$ for which the inequality$$\displaystyle C_a\sum_{k=1}^n \frac1{x_k-x_{k-1}} >\sum_{k=1}^n \frac{k+a}{x_k}$$holds for any positive integer $n$ and any sequence $0=x_0<x_1<\cdots <x_n$ of real numbers.

2013 Moldova Team Selection Test, 1

Tags: inequalities

For any positive real numbers $x,y,z$, prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq \frac{z(x+y)}{y(y+z)} + \frac{x(z+y)}{z(x+z)} + \frac{y(x+z)}{x(x+y)}$

1997 Korea - Final Round, 2

Tags: trigonometry , perimeter , inequalities , geometry

The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that \[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\] and find the conditions for equality.

2003 AMC 12-AHSME, 7

Tags: perimeter , inequalities , congruent triangles , triangle inequality , geometry

How many non-congruent triangles with perimeter $ 7$ have integer side lengths? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

2000 Korea Junior Math Olympiad, 6

Tags: inequalities , algebra

$x, y, z$ are positive reals which their product is not smaller then their sum. Prove the inequality: $$\sqrt{2x^2+yz}+\sqrt{2y^2+zx}+\sqrt{2z^2+xy} \geq 9$$

2006 District Olympiad, 2

Tags: inequalities , trigonometry

In triangle $ABC$ we have $\angle ABC = 2 \angle ACB$. Prove that a) $AC^2 = AB^2 + AB \cdot BC$; b) $AB+BC < 2 \cdot AC$.

1983 AMC 12/AHSME, 21

Tags: function , inequalities , algebra , difference of squares , special factorizations

Find the smallest positive number from the numbers below $\text{(A)} \ 10-3\sqrt{11} \qquad \text{(B)} \ 3\sqrt{11}-10 \qquad \text{(C)} \ 18-5\sqrt{13} \qquad \text{(D)} \ 51-10\sqrt{26} \qquad \text{(E)} \ 10\sqrt{26}-51$

2021 Macedonian Mathematical Olympiad, Problem 1

Tags: sequence , recursive , inequality , factorial , am-gm , inequalities

Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$ For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.

2008 Saint Petersburg Mathematical Olympiad, 7

Tags: inequalities

In a sequence, $x_1=\frac{1}{2}$ and $x_{n+1}=1-x_1x_2x_3...x_n$ for $n\ge 1$. Prove that $0.99<x_{100}<0.991$. Fresh translation. This problem may be similar to one of the 9th grade problems.

2002 Turkey Team Selection Test, 3

Tags: inequalities

A positive integer $n$ and real numbers $a_1,\dots, a_n$ are given. Show that there exists integers $m$ and $k$ such that \[|\sum\limits_{i=1}^m a_i -\sum\limits_{i=m+1}^n a_i | \leq |a_k|.\]

2020 Jozsef Wildt International Math Competition, W19

Tags: inequalities

Prove the inequality $$\prod_{k=2}^n\left(1+\frac{k^{p-1}}{1^p+2^p+\ldots+k^p}\right)<e^{\frac{p-1}2}$$ [i]Proposed by Arkady Alt[/i]

2013 SDMO (Middle School), 4

Tags: inequalities

Let $a$, $b$, $c$, and $d$ be positive real numbers such that $a+b=c+d$ and $a^2+b^2>c^2+d^2$. Prove that $a^3+b^3>c^3+d^3$.

1998 Brazil Team Selection Test, Problem 5

Tags: inequalities , number theory , sequence

Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that $$k\le2\lfloor\sqrt n\rfloor.$$

2011 Romania Team Selection Test, 2

Tags: inequalities

Let $n$ be an integer number greater than $2$, let $x_{1},x_{2},\ldots ,x_{n}$ be $n$ positive real numbers such that \[\sum_{i=1}^{n}\frac{1}{x_{i}+1}=1\] and let $k$ be a real number greater than $1$. Show that: \[\sum_{i=1}^{n}\frac{1}{x_{i}^{k}+1}\ge\frac{n}{(n-1)^{k}+1}\] and determine the cases of equality.

2024 Junior Balkan MO, 1

Tags: inequalities , algebra

Let $a, b, c$ be positive real numbers such that $$a^2 + b^2 + c^2 = \frac{1}{4}.$$ Prove that $$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$ [i]Proposed by Petar Filipovski, Macedonia[/i]

2004 IMC, 4

Tags: 3d geometry , sphere , inequalities , geometry

Suppose $n\geq 4$ and let $S$ be a finite set of points in the space ($\mathbb{R}^3$), no four of which lie in a plane. Assume that the points in $S$ can be colored with red and blue such that any sphere which intersects $S$ in at least 4 points has the property that exactly half of the points in the intersection of $S$ and the sphere are blue. Prove that all the points of $S$ lie on a sphere.

1991 India National Olympiad, 4

Tags: inequalities

Let $a,b,c$ be real numbers with $0 < a< 1$, $0 < b < 1$, $0 < c < 1$, and $a+b + c = 2$. Prove that $\dfrac{a}{1-a} \cdot \dfrac{b}{1-b} \cdot \dfrac{c}{1-c} \geq 8$.

2006 China Team Selection Test, 1

Tags: inequalities , combinatorics

Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.

2005 Georgia Team Selection Test, 10

Tags: 3-variable inequality , algebra , inequalities

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

2011 South East Mathematical Olympiad, 2

Tags: vector , inequalities

Let $P_i$ $i=1,2,......n$ be $n$ points on the plane , $M$ is a point on segment $AB$ in the same plane , prove : $\sum_{i=1}^{n} |P_iM| \le \max( \sum_{i=1}^{n} |P_iA| , \sum_{i=1}^{n} |P_iB| )$. (Here $|AB|$ means the length of segment $AB$) .

2004 Kazakhstan National Olympiad, 7

Tags: inequalities

Prove that for any $a>0,b>0,c>0$ we have $8a^2 b^2 c^2 \geq (a^2 + ab + ac - bc)(b^2 + ba + bc - ac)(c^2 + ca + cb - ab)$.

2004 Switzerland - Final Round, 5

Tags: algebra , inequalities

Let $a$ and $b$ be fixed positive numbers. Find the smallest possible depending on $a$ and $b$ value of the sum $$\frac{x^2}{(ay + bz)(az + by)}+\frac{y^2}{(az + bx)(ax + bz)}+\frac{z^2}{(ax + by)(ay + bx)},$$ where $x, y, z$ are positive real numbers.

2007 Baltic Way, 17

Tags: inequalities , greatest common divisor , number theory

Let $x,y,z$ be positive integers such that $\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $d\le \sqrt[3]{xy+yz+zx}$.