Found problems: 6530
2015 Canadian Mathematical Olympiad Qualification, 5
Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Show that $$\left( \frac{x+1}{x} \right)^2 + \left( \frac{y+1}{y} \right)^2 \geq 18.$$
2002 Belarusian National Olympiad, 2
Given rational numbers $a_1,...,a_n$ such that $\sum_{i=1}^n \{ka_i\}<\frac{n}{2}$ for any positive integer $k$.
a) Prove that at least one of $a_1,...,a_n$ is integer.
b) Is the previous statement true, if the number $\frac{n}{2}$ is replaced by the greater number? (Here $\{x\}$ means a fractional part of $x$.)
(N. Selinger)
2024 Iran Team Selection Test, 3
For any real numbers $x , y ,z$ prove that :
$$(x+y+z)^2 + \sum_{cyc}{\frac{(x+y)(y+z)}{1+|x-z|}} \ge xy+yz+zx$$
[i]Proposed by Navid Safaei[/i]
2024 India Regional Mathematical Olympiad, 4
Let $a_1,a_2,a_3,a_4$ be real numbers such that $a_1^2 + a_2^2 + a_3^2 + a_4^2 = 1$. Show that there exist $i,j$ with $ 1 \leq i < j \leq 4$, such that $(a_i - a_j)^2 \leq \frac{1}{5}$.
2012 Tuymaada Olympiad, 3
A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$. Prove that the length of its radius is less than $3$.
[i]Proposed by K. Kokhas[/i]
2019 Regional Competition For Advanced Students, 1
Let $x,y$ be real numbers such that $(x+1)(y+2)=8.$ Prove that $$(xy-10)^2\ge 64.$$
2010 China Team Selection Test, 1
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$, we have $a_i+a_{i+1}\geq 0$;
(2) for $j=0,1,\cdots,n-1$, we have $a_{2j+1}\leq 0$;
(2) for any integer $p,q$, $0\leq p\leq q\leq n$, we have $\sum_{k=2p}^{2q}b_k>0$.
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$, and determine when the equality holds.
2016 MMATHS, 4
For real numbers $a, b, c$ with $a + b + c = 3$, prove that $$a^2(b - c)^2 + b^2(c - a)^2 + c^2(a - b)^2 \ge \frac9 2 abc(1 - abc)$$ and state when equality is reached.
2006 Mediterranean Mathematics Olympiad, 4
Let $0\le x_{i,j} \le 1$, where $i=1,2, \ldots m$ and $j=1,2, \ldots n$. Prove the inequality
\[ \prod_{j=1}^n\left(1-\prod_{i=1}^mx_{i,j} \right)+ \prod_{i=1}^m\left(1-\prod_{j=1}^n(1-x_{i,j}) \right) \ge 1 \]
2003 China Team Selection Test, 3
Suppose $A\subset \{(a_1,a_2,\dots,a_n)\mid a_i\in \mathbb{R},i=1,2\dots,n\}$. For any $\alpha=(a_1,a_2,\dots,a_n)\in A$ and $\beta=(b_1,b_2,\dots,b_n)\in A$, we define
\[ \gamma(\alpha,\beta)=(|a_1-b_1|,|a_2-b_2|,\dots,|a_n-b_n|), \] \[ D(A)=\{\gamma(\alpha,\beta)\mid\alpha,\beta\in A\}. \] Please show that $|D(A)|\geq |A|$.
2001 Junior Balkan Team Selection Tests - Romania, 3
Let $n\ge 2$ be a positive integer. Find the positive integers $x$
\[\sqrt{x+\sqrt{x+\ldots +\sqrt{x}}}<n \]
for any number of radicals.
JOM 2015, 3
Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.
1972 All Soviet Union Mathematical Olympiad, 172
Let the sum of positive numbers $x_1, x_2, ... , x_n$ be $1$. Let $s$ be the greatest of the numbers $$\left\{\frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, ..., \frac{x_n}{1+x_1+...+x_n}\right\}$$
What is the minimal possible $s$? What $x_i $correspond it?
2017 Mathematical Talent Reward Programme, SAQ: P 2
Let $a$, $b$, $c$ be positive reals such that $a+b+c=3$. Show that $$\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} \leq \frac{6}{\sqrt(a+b)(b+c)(c+a)}$$
2011 Morocco National Olympiad, 1
Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.
1968 All Soviet Union Mathematical Olympiad, 113
The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $$a_1=0, |a_2|=|a_1+1|, ..., |a_n|=|a_{n-1}+1|.$$ Prove that $$(a_1+a_2+...+a_n)/n \ge -1/2$$
1990 Dutch Mathematical Olympiad, 1
Prove that for every integer $ n>1, 1 \cdot 3 \cdot 5 \cdot ... \cdot (2n\minus{}1)<n^n.$
2010 Estonia Team Selection Test, 3
Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?
2012 ELMO Shortlist, 6
Consider a directed graph $G$ with $n$ vertices, where $1$-cycles and $2$-cycles are permitted. For any set $S$ of vertices, let $N^{+}(S)$ denote the out-neighborhood of $S$ (i.e. set of successors of $S$), and define $(N^{+})^k(S)=N^{+}((N^{+})^{k-1}(S))$ for $k\ge2$.
For fixed $n$, let $f(n)$ denote the maximum possible number of distinct sets of vertices in $\{(N^{+})^k(X)\}_{k=1}^{\infty}$, where $X$ is some subset of $V(G)$. Show that there exists $n>2012$ such that $f(n)<1.0001^n$.
[i]Linus Hamilton.[/i]
2013 Mexico National Olympiad, 5
A pair of integers is special if it is of the form $(n, n-1)$ or $(n-1, n)$ for some positive integer $n$. Let $n$ and $m$ be positive integers such that pair $(n, m)$ is not special. Show $(n, m)$ can be expressed as a sum of two or more different special pairs if and only if $n$ and $m$ satisfy the inequality $ n+m\geq (n-m)^2 $.
Note: The sum of two pairs is defined as $ (a, b)+(c, d) = (a+c, b+d) $.
2010 Today's Calculation Of Integral, 556
Prove the following inequality.
\[ \sqrt[3]{\int_0^{\frac {\pi}{4}} \frac {x}{\cos ^ 2 x\cos ^ 2 (\tan x)\cos ^ 2(\tan (\tan x))\cos ^ 2(\tan (\tan (\tan x)))}dx}<\frac{4}{\pi}\]
Last Edited.
Sorry, I have changed the problem.
kunny
2014 District Olympiad, 2
Let real numbers $a,b,c$ such that $\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|.$
Prove that $a=b+c$ or $b=c+a$ or $c=a+b.$
2007 District Olympiad, 4
[b]a)[/b] Let $ \bold {u,v,w,} $ be three coplanar vectors of absolute value $ 1. $ Show that there exist $ \varepsilon_1 ,\varepsilon_2, \varepsilon_3\in \{ \pm 1\} $ such that
$$ \big| \varepsilon_1\bold u +\varepsilon_2\bold v +\varepsilon_3\bold w \big|\le 1. $$
[b]b)[/b] Give an example of three vectors such that the inequality above does not work for any sclaras from $ \{ \pm 1\} . $
1999 Balkan MO, 4
Let $\{a_n\}_{n\geq 0}$ be a non-decreasing, unbounded sequence of non-negative integers with $a_0=0$. Let the number of members of the sequence not exceeding $n$ be $b_n$. Prove that \[ (a_0 + a_1 + \cdots + a_m)( b_0 + b_1 + \cdots + b_n ) \geq (m + 1)(n + 1). \]
2009 AIME Problems, 13
The terms of the sequence $ (a_i)$ defined by $ a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}}$ for $ n \ge 1$ are positive integers. Find the minimum possible value of $ a_1 \plus{} a_2$.