Found problems: 6530
2015 All-Russian Olympiad, 3
Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$.
Russian TST 2017, P3
Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have
\[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]
2017 District Olympiad, 1
Let $ f,g:[0,1]\longrightarrow{R} $ be two continuous functions such that $ f(x)g(x)\ge 4x^2, $ for all $ x\in [0,1] . $ Prove that
$$ \left| \int_0^1 f(x)dx \right| \ge 1\text{ or } \left| \int_0^1 g(x)dx \right| \ge 1. $$
2013 ELMO Shortlist, 9
Let $a, b, c$ be positive reals, and let $\sqrt[2013]{\frac{3}{a^{2013}+b^{2013}+c^{2013}}}=P$. Prove that
\[\prod_{\text{cyc}}\left(\frac{(2P+\frac{1}{2a+b})(2P+\frac{1}{a+2b})}{(2P+\frac{1}{a+b+c})^2}\right)\ge \prod_{\text{cyc}}\left(\frac{(P+\frac{1}{4a+b+c})(P+\frac{1}{3b+3c})}{(P+\frac{1}{3a+2b+c})(P+\frac{1}{3a+b+2c})}\right).\][i]Proposed by David Stoner[/i]
1978 IMO Longlists, 14
Let $p(x, y)$ and $q(x, y)$ be polynomials in two variables such that for $x \ge 0, y \ge 0$ the following conditions hold:
$(i) p(x, y)$ and $q(x, y)$ are increasing functions of $x$ for every fixed $y$.
$(ii) p(x, y)$ is an increasing and $q(x)$ is a decreasing function of $y$ for every fixed $x$.
$(iii) p(x, 0) = q(x, 0)$ for every $x$ and $p(0, 0) = 0$.
Show that the simultaneous equations $p(x, y) = a, q(x, y) = b$ have a unique solution in the set $x \ge 0, y \ge 0$ for all $a, b$ satisfying $0 \le b \le a$ but lack a solution in the same set if $a < b$.
PEN E Problems, 15
Show that there exist two consecutive squares such that there are at least $1000$ primes between them.
2017 Pan African, Problem 2
Let $x,y$, and $z$ be positive real numbers such that $xy+yz+zx=3xyz$. Prove that $$x^2y+y^2z+z^2x \geq 2(x+y+z)-3.$$
In which cases do we have equality?
2008 Bulgaria National Olympiad, 2
Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which
\[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\]
is satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.
2005 Austria Beginners' Competition, 2
Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .
1995 Taiwan National Olympiad, 1
Let $P(x)=a_{0}+a_{1}x+...+a_{n}x^{n}\in\mathbb{C}[x]$ , where $a_{n}=1$. The roots of $P(x)$ are $b_{1},b_{2},...,b_{n}$, where $|b_{1}|,|b_{2}|,...,|b_{j}|>1$ and $|b_{j+1}|,...,|b_{n}|\leq 1$. Prove that $\prod_{i=1}^{j}|b_{i}|\leq\sqrt{|a_{0}|^{2}+|a_{1}|^{2}+...+|a_{n}|^{2}}$.
2012 Uzbekistan National Olympiad, 4
Given $a,b$ and $c$ positive real numbers with $ab+bc+ca=1$. Then prove that
$\frac{a^3}{1+9b^2ac}+\frac{b^3}{1+9c^2ab}+\frac{c^3}{1+9a^2bc} \geq \frac{(a+b+c)^3}{18}$
2002 China Team Selection Test, 2
Circles $ \omega_{1}$ and $ \omega_{2}$ intersect at points $ A$ and $ B.$ Points $ C$ and $ D$ are on circles $ \omega_{1}$ and $ \omega_{2},$ respectively,
such that lines $ AC$ and $ AD$ are tangent to circles $ \omega_{2}$ and $ \omega_{1},$ respectively.
Let $ I_{1}$ and $ I_{2}$ be the incenters of triangles $ ABC$ and $ ABD,$ respectively. Segments $ I_{1}I_{2}$ and $ AB$ intersect at $ E$.
Prove that: $ \frac {1}{AE} \equal{} \frac {1}{AC} \plus{} \frac {1}{AD}$
Fractal Edition 1, P3
Let \( a \), \( b \), and \( c \) be three positive real numbers that satisfy \( ab + bc + ca = 1 \). Show that:
\[
\frac{a}{a^2+1} + \frac{b}{b^2+1} + \frac{c}{c^2+1} \le \frac{1}{4abc}
\]
2014 Peru MO (ONEM), 1
Find all triples ( $\alpha, \beta,\theta$) of acute angles such that the following inequalities are fulfilled at the same time
$$(\sin \alpha + \cos \beta + 1)^2 \ge 2(\sin \alpha + 1)(\cos \beta + 1)$$
$$(\sin \beta + \cos \theta + 1)^2 \ge 2(\sin \beta + 1)(\cos \theta + 1)$$
$$(\sin \theta + \cos \alpha + 1)^2 \ge 2(\sin \theta + 1)(\cos \alpha + 1).$$
VI Soros Olympiad 1999 - 2000 (Russia), 9.3
The quadratic trinomial $x^2 + bx + c$ has two roots belonging to the interval $(2, 3)$. Prove that $5b+2c+12 < 0$.
2003 China Team Selection Test, 3
The $ n$ roots of a complex coefficient polynomial $ f(z) \equal{} z^n \plus{} a_1z^{n \minus{} 1} \plus{} \cdots \plus{} a_{n \minus{} 1}z \plus{} a_n$ are $ z_1, z_2, \cdots, z_n$. If $ \sum_{k \equal{} 1}^n |a_k|^2 \leq 1$, then prove that $ \sum_{k \equal{} 1}^n |z_k|^2 \leq n$.
2016 Iran Team Selection Test, 2
Let $a,b,c,d$ be positive real numbers such that $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2$. Prove that $$\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8$$
2002 Moldova National Olympiad, 1
Consider the real numbers $ a\ne 0,b,c$ such that the function $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ satisfies $ |f(x)|\le 1$ for all $ x\in [0,1]$. Find the greatest possible value of $ |a| \plus{} |b| \plus{} |c|$.
1971 IMO Longlists, 42
Show that for nonnegative real numbers $a,b$ and integers $n\ge 2$,
\[\frac{a^n+b^n}{2}\ge\left(\frac{a+b}{2}\right)^n\]
When does equality hold?
2017 Kazakhstan National Olympiad, 2
For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds
$$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$
2024 Baltic Way, 5
Find all positive real numbers $\lambda$ such that every sequence $a_1, a_2, \ldots$ of positive real numbers satisfying
\[
a_{n+1}=\lambda\cdot\frac{a_1+a_2+\ldots+a_n}{n}
\]
for all $n\geq 2024^{2024}$ is bounded.
[i]Remark:[/i] A sequence $a_1,a_2,\ldots$ of positive real numbers is \emph{bounded} if there exists a real number $M$ such that $a_i<M$ for all $i=1,2,\ldots$
2023 Taiwan TST Round 3, 4
Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and
\[a+b+c-3\sqrt[3]{abc}=1.\]
[i]Proposed by usjl[/i]
2016 Taiwan TST Round 3, 2
Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions:
$(i)$ $a_0=a_n=1$;
$(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$;
$(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
$(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$.
Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.
1965 IMO, 1
Determine all values of $x$ in the interval $0 \leq x \leq 2\pi$ which satisfy the inequality \[ 2 \cos{x} \leq \sqrt{1+\sin{2x}}-\sqrt{1-\sin{2x}} \leq \sqrt{2}. \]
2014 India Regional Mathematical Olympiad, 4
let $ABC$ be a right angled triangle with inradius $1$
find the minimum area of triangle $ABC$