Found problems: 6530
2021 China Girls Math Olympiad, 6
Given a finite set $S$, $P(S)$ denotes the set of all the subsets of $S$. For any $f:P(S)\rightarrow \mathbb{R}$ ,prove the following inequality:$$\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.$$
2010 China Team Selection Test, 1
Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds:
\[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\]
where $S=\sum_{i=1}^n x_i$.
2012 China Girls Math Olympiad, 1
Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that
$\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$
2014 Contests, 1
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
[i]Proposed by Gerhard Wöginger, Austria.[/i]
2022 OMpD, 2
We say that a sextuple of positive real numbers $(a_1, a_2, a_3, b_1, b_2, b_3)$ is $\textit{phika}$ if $a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = 1$.
(a) Prove that there exists a $\textit{phika}$ sextuple $(a_1, a_2, a_3, b_1, b_2, b_3)$ such that:
$$a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) > 1 - \frac{1}{2022^{2022}}$$
(b) Prove that for every $\textit{phika}$ sextuple $(a_1, a_2, a_3, b_1, b_2, b_3)$, we have:
$$a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) < 1$$
2009 Indonesia TST, 4
Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality
\[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3.
\]
2017 Taiwan TST Round 3, 5
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
\[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]
1999 Ukraine Team Selection Test, 4
If $n \in N$ and $0 < x <\frac{\pi}{2n}$, prove the inequality $\frac{\sin 2x}{\sin x}+\frac{\sin 3x}{\sin 2x} +...+\frac{\sin (n+1)x}{\sin nx} < 2\frac{\cos x}{\sin^2 x}$.
.
2020 Jozsef Wildt International Math Competition, W36
For all $x\in\left(0,\frac\pi4\right)$ prove
$$\frac{(\sin^2x)^{\sin^2x}+(\tan^2x)^{\tan^2x}}{(\sin^2x)^{\tan^2x}+(\tan^2x)^{\sin^2x}}<\frac{\sin x}{4\sin x-3x}$$
[i]Proposed by Pirkulyiev Rovsen[/i]
2008 Iran MO (3rd Round), 8
In an old script found in ruins of Perspolis is written:
[code]
This script has been finished in a year whose 13th power is
258145266804692077858261512663
You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.[/code]
Find the year the script is finished. Give a reason for your answer.
1970 Yugoslav Team Selection Test, Problem 1
Positive integers $a$ and $b$ have $n$ digits each in their decimal representation. Assume that $m$ is a positive integer such that $\frac n2<m<n$ and assume that each of the leftmost $m$ digits of $a$ is equal to the corresponding digit of $b$. Prove that
$$a^{\frac1n}-b^{\frac1n}<\frac1n.$$
2023 Stanford Mathematics Tournament, 8
If $x$ and $y$ are real numbers, compute the minimum possible value of
\[\frac{4xy(3x^2+10xy+6y^2)}{x^4+4y^4}.\]
2006 Lithuania Team Selection Test, 2
Solve in integers $x$ and $y$ the equation $x^3-y^3=2xy+8$.
2016 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.
2008 Romania Team Selection Test, 1
Let $ n \geq 3$ be an odd integer. Determine the maximum value of
\[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\]
where $ x_{i}$ are positive real numbers from the interval $ [0,1]$.
2022 Moldova Team Selection Test, 2
Real numbers $a, b, c, d$ satisfy $$a^2+b^2+c^2+d^2=4.$$
Find the greatest possible value of $$E(a,b,c,d)=a^4+b^4+c^4+d^4+4(a+b+c+d)^2 .$$
2001 Estonia National Olympiad, 4
If $x$ and $y$ are nonnegative real numbers with $x+y= 2$, show that $x^2y^2(x^2+y^2)\le 2$.
1970 Czech and Slovak Olympiad III A, 6
Determine all real $x$ such that \[\sqrt{\tan(x)-1}\,\Bigl(\log_{\tan(x)}\bigl(2+4\cos^2(x)-2\bigr)\Bigr)\ge0.\]
2005 Germany Team Selection Test, 1
Find the smallest positive integer $n$ with the following property:
For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that
\[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]
1993 Romania Team Selection Test, 1
Find max. numbers $A$ wich is true ineq.:
$\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{x^{2}+z^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}\geq A$
$x,y,z$ are positve reals numberes! :wink:
1998 Belarus Team Selection Test, 2
Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]
1996 IMO Shortlist, 3
Let $ a > 2$ be given, and starting $ a_0 \equal{} 1, a_1 \equal{} a$ define recursively:
\[ a_{n\plus{}1} \equal{} \left(\frac{a^2_n}{a^2_{n\minus{}1}} \minus{} 2 \right) \cdot a_n.\]
Show that for all integers $ k > 0,$ we have: $ \sum^k_{i \equal{} 0} \frac{1}{a_i} < \frac12 \cdot (2 \plus{} a \minus{} \sqrt{a^2\minus{}4}).$
2009 China Team Selection Test, 3
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$
2011 Abels Math Contest (Norwegian MO), 3a
The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$. Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ .
.
2023 SG Originals, Q2
Let $a, b, c, d$ be positive reals with $a - c = b - d > 0$. Show that
$$\frac{ab}{cd} \ge \left(\frac{\sqrt{a} +\sqrt{b}}{\sqrt{c}+\sqrt{d}}\right)^4$$