Olympiad problems

2018 Brazil Team Selection Test, 4

Tags: algebra , combinatorics , subset , inequalities , boundary conditions

Given a set $S$ of positive real numbers, let $$\Sigma (S) = \Bigg\{ \sum_{x \in A} x : \emptyset \neq A \subset S \Bigg\}.$$ be the set of all the sums of elements of non-empty subsets of $S$. Find the least constant $L> 0$ with the following property: for every integer greater than $1$ and every set $S$ of $n$ positive real numbers, it is possible partition $\Sigma(S)$ into $n$ subsets $\Sigma_1,\ldots, \Sigma_n$ so that the ratio between the largest and smallest element of each $\Sigma_i$ is at most $L$.

2007 ITest, 9

Tags: inequalities

Suppose that $m$ and $n$ are positive integers such that $m<n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. How many pairs $(m,n)$ satisfy these conditions? $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$ $\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$ $\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }2007$

1989 China National Olympiad, 6

Tags: function , inequalities , algebra

Find all functions $f:(1,+\infty) \rightarrow (1,+\infty)$ that satisfy the following condition: for arbitrary $x,y>1$ and $u,v>0$, inequality $f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}$ holds.

2020 South East Mathematical Olympiad, 4

Tags: inequalities , algebra , n-variable inequality

Let $0\leq a_1\leq a_2\leq \cdots\leq a_{n-1}\leq a_n $ and $a_1+a_2+\cdots+a_n=1.$ Prove that: For any non-negative numbers $x_1,x_2,\cdots,x_n ; y_1, y_2,\cdots, y_n$ , have $$\left(\sum_{i=1}^n a_ix_i - \prod_{i=1}^n x_i^{a_i}\right) \left(\sum_{i=1}^n a_iy_i - \prod_{i=1}^n y_i^{a_i}\right) \leq a_n^2\left(n\sqrt{\sum_{i=1}^n x_i\sum_{i=1}^n y_i} - \sum_{i=1}^n\sqrt{x_i} \sum_{i=1}^n\sqrt{y_i}\right)^2.$$

2001 Czech-Polish-Slovak Match, 1

Tags: inequalities , n-variable inequality , holder , convexity

Let $n\ge2$ be a natural number, and $a_i$ be positive numbers, where $i=1,2,\cdots,n.$ Show that \[\left(a_1^3+1\right)\left(a_2^3+1\right)\cdots\left(a_n^3+1\right) \geq \left(a_1^2a_2+1\right)\left(a_2^2a_3+1\right)\cdots\left(a_n^2a_1+1\right)\]

2013 Silk Road, 3

Tags: function , inequalities , algebra

Find all non-decreasing functions $ f\,:\,\mathbb{N}\to\mathbb{N} $, such that $f(f(m)f(n)+m)=f(mf(n))+f(m)$

2007 India IMO Training Camp, 2

Tags: function , inequalities

Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that \[a^2+b^2+c^2\leq 2abc+1.\]

2022 VJIMC, 3

Tags: probability theory , inequalities

Let $x_1,\ldots,x_n$ be given real numbers with $0<m\le x_i\le M$ for each $i\in\{1,\ldots,n\}$. Let $X$ be the discrete random variable uniformly distributed on $\{x_1,\ldots,x_n\}$. The mean $\mu$ and the variance $\sigma^2$ of $X$ are defined as $$\mu(X)=\frac{x_1+\ldots+x_n}n\text{ and }\sigma^2(X)=\frac{(x_1-\mu(X))^2+\ldots+(x_n-\mu(X))^2}n.$$ By $X^2$ denote the discrete random variable uniformly distributed on $\{x_1^2,\ldots,x_n^2\}$. Prove that $$\sigma^2(X)\ge\left(\frac m{2M^2}\right)^2\sigma^2(X^2).$$

2005 National Olympiad First Round, 7

Tags: trigonometry , inequalities , rearrangement inequality

What is the greatest value of $\sin x \cos y + \sin y \cos z + \sin z \cos x$, where $x,y,z$ are real numbers? $ \textbf{(A)}\ \sqrt 2 \qquad\textbf{(B)}\ \dfrac 32 \qquad\textbf{(C)}\ \dfrac {\sqrt 3}2 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 3 $

2009 BAMO, 5

Tags: inequalities

Let $\triangle ABC$ be an acute triangle with angles $\alpha, \beta,$ and $\gamma$. Prove that $$\frac{\cos(\beta-\gamma)}{cos\alpha}+\frac{\cos(\gamma-\alpha)}{\cos \beta}+\frac{\cos(\alpha-\beta)}{\cos \gamma} \geq \frac{3}{2}$$

1998 Romania National Olympiad, 1

Tags: inequalities , algebra , perfect square , number theory

Let $n$ be a positive integer and $x_1,x_2,...,x_n$ be integer numbers such that $$x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2$$ . Show that : a) $x_1,x_2,...,x_n$ are non-negative integers b) the number $x_1+x_2+...+x_n+n+1$ is not a perfect square.

2005 India IMO Training Camp, 3

Tags: function , trigonometry , inequalities , algebra

For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$

2005 Moldova Team Selection Test, 2

Tags: function , inequalities

Let $ a$, $ b$, $ c$ be positive reals such that $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$. Prove that $ \sum\frac1{4 \minus{} ab}\leq1$, where the $ \sum$ sign stands for cyclic summation. [i]Alternative formulation:[/i] For any positive reals $ a$, $ b$, $ c$ satisfying $ a^4 \plus{} b^4 \plus{} c^4 \equal{} 3$, prove the inequality $ \frac{1}{4\minus{}bc}\plus{}\frac{1}{4\minus{}ca}\plus{}\frac{1}{4\minus{}ab}\leq 1$.

2010 Contests, 2

Tags: inequalities

Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]

1962 All Russian Mathematical Olympiad, 023

Tags: inequalities , geometry , maximum , area

What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality $0\le a\le 1\le b\le 2\le c\le 3$ ?

1990 Czech and Slovak Olympiad III A, 2

Tags: inequalities , algebra , triangle inequality , parameter , positive , quadratic

Determine all values $\alpha\in\mathbb R$ with the following property: if positive numbers $(x,y,z)$ satisfy the inequality \[x^2+y^2+z^2\le\alpha(xy+yz+zx),\] then $x,y,z$ are sides of a triangle.

2018-IMOC, A6

Tags: inequalities

Given $ a, b, c > 0$. Prove that: $ (1\plus{}a\plus{}b\plus{}c)(1\plus{}ab\plus{}bc\plus{}ca) \ge 4\sqrt{2}\sqrt{(a\plus{}bc)(b\plus{}ca)(c\plus{}ab)}$ :)

1995 India National Olympiad, 5

Tags: inequalities

Let $n \geq 2$. Let $a_1 , a_2 , a_3 , \ldots a_n$ be $n$ real numbers all less than $1$ and such that $|a_k - a_{k+1} | < 1$ for $1 \leq k \leq n-1$. Show that \[ \dfrac{a_1}{a_2} + \dfrac{a_2}{a_3} + \dfrac{a_3}{a_4} + \ldots + \dfrac{a_{n-1}}{a_n} + \dfrac{a_n}{a_1} < 2 n - 1 . \]

2012 Putnam, 4

Tags: limit , inequalities , induction , calculus , integration , logarithm

Suppose that $a_0=1$ and that $a_{n+1}=a_n+e^{-a_n}$ for $n=0,1,2,\dots.$ Does $a_n-\log n$ have a finite limit as $n\to\infty?$ (Here $\log n=\log_en=\ln n.$)

2011 Harvard-MIT Mathematics Tournament, 1

Tags: polynomial , inequalities , algebra

Let $a,b,c$ be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them: $ax^2+bx+c, bx^2+cx+a,$ and $cx^2+ax+b $.

2007 Balkan MO Shortlist, C2

Tags: function , inequalities , algebra , domain , combinatorics

Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find \[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]

2010 China Team Selection Test, 2

Tags: inequalities

Find all positive real numbers $\lambda$ such that for all integers $n\geq 2$ and all positive real numbers $a_1,a_2,\cdots,a_n$ with $a_1+a_2+\cdots+a_n=n$, the following inequality holds: $\sum_{i=1}^n\frac{1}{a_i}-\lambda\prod_{i=1}^{n}\frac{1}{a_i}\leq n-\lambda$.

1971 Polish MO Finals, 5

Tags: algebra , inequalities , number theory

Find the largest integer $A$ such that, for any permutation of the natural numbers not exceeding $100$, the sum of some ten successive numbers is at least $A$.

2000 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities

Let be a natural number $ n\ge 2, n $ real numbers $ b_1,b_2,\ldots ,b_n , $ and $ n-1 $ positive real numbers $ a_1,a_2,\ldots ,a_{n-1} $ such that $ a_1+a_2+\cdots +a_{n-1} =1. $ Prove the inequality $$ b_1^2+\frac{b_2^2}{a_1} +\frac{b_3^2}{a_2} +\cdots +\frac{b_n^2}{a_{n-1}} \ge 2b_1\left( b_2+b_3+\cdots +b_n \right) , $$ and specify when equality is attained. [i]Dumitru Acu[/i]

2009 USAMO, 2

Tags: pigeonhole principle , induction , ceiling function , symmetry , inequalities , combinatorics , oron wang orz

Let $n$ be a positive integer. Determine the size of the largest subset of $\{ -n, -n+1, \dots, n-1, n\}$ which does not contain three elements $a$, $b$, $c$ (not necessarily distinct) satisfying $a+b+c=0$.