Olympiad problems

1940 Moscow Mathematical Olympiad, 069

Tags: algebra , inequalities

Let $a_1, ...,, a_n$ be positive numbers. Prove the inequality: $$\frac{a_1}{a_2}+\frac{a_2}{a_3}+\frac{a_3}{a_4}+ ... +\frac{a_{n-1}}{a_n}+ \frac{a_n}{a_1} \ge n$$

2013 Czech-Polish-Slovak Junior Match, 5

Tags: inequalities , algebra

Let $a, b, c$ be positive real numbers for which $ab + ac + bc \ge a + b + c$. Prove that $a + b + c \ge 3$.

2012 Indonesia TST, 3

Tags: inequalities

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that \[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\] and \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\] Prove that \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]

2015 IMC, 6

Tags: series , inequalities

Prove that $$\sum\limits_{n = 1}^{\infty}\frac{1}{\sqrt{n}\left(n+1\right)} < 2.$$ Proposed by Ivan Krijan, University of Zagreb

2015 Iran Team Selection Test, 1

Tags: inequalities

$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$. Prove that : $abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$

2002 AMC 10, 4

Tags: inequalities

For how many positive integers $ m$ does there exist at least one positive integer $ n$ such that $ m\cdot n \le m \plus{} n$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}$ infinitely many

1992 Iran MO (2nd round), 2

Tags: induction , inequalities

In the sequence $\{a_n\}_{n=0}^{\infty}$ we have $a_0=1$, $a_1=2$ and \[a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1\] Prove that \[52 < a_{1371} < 65\]

2010 Contests, 3

Tags: inequalities

Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$. (Dan Schwarz)

2022 Kosovo National Mathematical Olympiad, 4

Tags: inequalities , geometric inequality

Assume that in the $\triangle ABC$ there exists a point $D$ on $BC$ and a line $l$ passing through $A$ such that $l$ is tangent to $(ADC)$ and $l$ bisects $BD.$ Prove that $a\sqrt{2}\geq b+c.$

2005 Cuba MO, 6

Tags: algebra , inequalities

All positive differences $a_i -a_j$ of five different positive integers $a_1$, $a_2$, $a_3$, $a_4$ and $a_5$ are all different. Let $A$ be the set formed by the largest elements of each group of $5$ elements that meet said condition. Determine the minimum element of $A$.

1994 All-Russian Olympiad Regional Round, 11.1

Tags: inequality , trigonometry , trigonometric inequality , inequalities

Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.

2008 China Team Selection Test, 1

Tags: inequalities , circumcircle , trigonometry , ratio , cyclic quadrilateral , geometry

Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.

2007 Grigore Moisil Intercounty, 1

Tags: inequalities , geometric inequality , geometry

For a point $ P $ situated in the plane determined by a triangle $ ABC, $ prove the following inequality: $$ BC\cdot PB\cdot PC+AC\cdot PC\cdot PA +AB\cdot PA\cdot PB\ge AB\cdot BC\cdot CA $$

2008 Turkey MO (2nd round), 3

Tags: inequalities

Let a.b.c be positive reals such that their sum is 1. Prove that $ \frac{a^{2}b^{2}}{c^{3}(a^{2}\minus{}ab\plus{}b^{2})}\plus{}\frac{b^{2}c^{2}}{a^{3}(b^{2}\minus{}bc\plus{}c^{2})}\plus{}\frac{a^{2}c^{2}}{b^{3}(a^{2}\minus{}ac\plus{}c^{2})}\geq \frac{3}{ab\plus{}bc\plus{}ac}$

V Soros Olympiad 1998 - 99 (Russia), 10.8

Tags: inequalities , algebra

It is known that for all $x$ such that $|x| < 1$, the following inequality holds $$ax^2+bx+c\le \frac{1}{\sqrt{1-x^2}}$$Find the greatest value of $a + 2c$.

2006 District Olympiad, 4

Tags: inequalities , algebra

For each positive integer $n\geq 2$ we denote with $p(n)$ the largest prime number less than or equal to $n$, and with $q(n)$ the smallest prime number larger than $n$. Prove that \[ \sum^n_{k=2} \frac 1{p(k)q(k)} < \frac 12. \]

2019 LIMIT Category B, Problem 12

Tags: inequalities

The system of inequalities $$a-b^2\ge\frac14$$$$b-c^2\ge\frac14$$$$c-d^2\ge\frac14$$$$d-a^2\ge\frac14$$where $a,b,c,d$ are real numbers has $\textbf{(A)}~\text{no solutions}$ $\textbf{(B)}~\text{exactly one solution}$ $\textbf{(C)}~\text{exactly two solutions}$ $\textbf{(D)}~\text{infinitely many solutions}$

2019 China National Olympiad, 6

Tags: inequalities , algebra , geometry , trigonometry

The point $P_1, P_2,\cdots ,P_{2018} $ is placed inside or on the boundary of a given regular pentagon. Find all placement methods are made so that $$S=\sum_{1\leq i<j\leq 2018}|P_iP_j| ^2$$takes the maximum value.

2008 SEEMOUS, Problem 4

Tags: calculus , integration , inequalities

Let $n$ be a positive integer and $f:[0,1]\to\mathbb R$ be a continuous function such that $$\int^1_0x^kf(x)dx=1$$for every $k\in\{0,1,\ldots,n-1\}$. Prove that $$\int^1_0f(x)^2dx\ge n^2.$$

2000 India Regional Mathematical Olympiad, 3

Tags: inequalities , integration , calculus , induction

Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$, and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3. \]

2018 Spain Mathematical Olympiad, 3

Tags: geometry , inequalities , geometric inequality

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and let $M$ be a point on $AB$. The circumcircle of $AMO$ intersects $AC$ a second time on $K$ and the circumcircle of $BOM$ intersects $BC$ a second time on $N$. Prove that $\left[MNK\right] \geq \frac{\left[ABC\right]}{4}$ and determine the equality case.

2004 South East Mathematical Olympiad, 3

Tags: polynomial , inequalities , algebra , logarithm

(1) Determine if there exists an infinite sequence $\{a_n\}$ with positive integer terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$. (2) Determine if there exists an infinite sequence $\{a_n\}$ with positive irrational terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.

1962 Swedish Mathematical Competition, 4

Tags: geometry , number theory , trigonometry , inequalities

Which of the following statements are true? (A) $X$ implies $Y$, or $Y$ implies $X$, where $X$ is the statement, the lines $L_1, L_2, L_3$ lie in a plane, and $Y$ is the statement, each pair of the lines $L_1, L_2, L_3$ intersect. (B) Every sufficiently large integer $n$ satisfies $n = a^4 + b^4$ for some integers a, b. (C) There are real numbers $a_1, a_2,... , a_n$ such that $a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0$ for all real $x$.

2019 China Western Mathematical Olympiad, 4

Tags: inequalities

Let $n$ be a given integer such that $n\ge 2$. Find the smallest real number $\lambda$ with the following property: for any real numbers $x_1,x_2,\ldots ,x_n\in [0,1]$ , there exists integers $\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n\in\{0,1\}$ such that the inequality $$\left\vert \sum^j_{k=i} (\varepsilon_k-x_k)\right\vert\le \lambda$$holds for all pairs of integers $(i,j)$ where $1\le i\le j\le n$.

2016 Middle European Mathematical Olympiad, 1

Tags: inequalities

Let $n \ge 2$ be an integer, and let $x_1, x_2, \ldots, x_n$ be reals for which: (a) $x_j > -1$ for $j = 1, 2, \ldots, n$ and (b) $x_1 + x_2 + \ldots + x_n = n.$ Prove that $$ \sum_{j = 1}^{n} \frac{1}{1 + x_j} \ge \sum_{j = 1}^{n} \frac{x_j}{1 + x_j^2} $$ and determine when does the equality occur.