Found problems: 6530
2004 Switzerland - Final Round, 5
Let $a$ and $b$ be fixed positive numbers. Find the smallest possible depending on $a$ and $b$ value of the sum
$$\frac{x^2}{(ay + bz)(az + by)}+\frac{y^2}{(az + bx)(ax + bz)}+\frac{z^2}{(ax + by)(ay + bx)},$$
where $x, y, z$ are positive real numbers.
2022 China Team Selection Test, 5
Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions:
(1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ such that $z_j \in \Gamma$.
(2) For any open arc $\gamma$ of length $\pi/3$ on $C$, there are at most $120$ of $j ~(1 \le j \le 240)$ such that $z_j \in \gamma$.
Find the maximum of $|z_1+z_2+\ldots+z_{240}|$.
2013 Spain Mathematical Olympiad, 1
Let $a,b,n$ positive integers with $a>b$ and $ab-1=n^2$. Prove that $a-b \geq \sqrt{4n-3}$ and study the cases where the equality holds.
2004 Junior Balkan Team Selection Tests - Romania, 2
The real numbers $a_1,a_2,\ldots,a_{100}$ satisfy the relationship
\[ a_1^2+ a_2^2 + \cdots +a_{100}^2 + ( a_1+a_2 + \cdots + a_{100})^2 = 101. \]
Prove that $|a_k|\leq 10$, for all $k=1,2,\ldots,100$.
2022 Kosovo & Albania Mathematical Olympiad, 0
Let $a>0$. If the inequality $22<ax<222$ holds for precisely $10$ positive integers $x$, find how many positive integers satisfy the inequality $222<ax<2022$?
[i]Note: The first 8 problems of the competition are questions which the contestants are expected to solve quickly and only write the answer of. This problem turned out to be a lot more difficult than anticipated for an answer-only question.[/i]
Russian TST 2016, P3
Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2\geqslant 3$. Prove that \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geqslant\frac{3}{2}.\]
2001 Baltic Way, 12
Let $a_1, a_2,\ldots , a_n$ be positive real numbers such that $\sum_{i=1}^na_i^3=3$ and $\sum_{i=1}^na_i^5=5$. Prove that $\sum_{i=1}^na_i>\frac{3}{2}$.
2018 China National Olympiad, 1
Let $n$ be a positive integer. Let $A_n$ denote the set of primes $p$ such that there exists positive integers $a,b$ satisfying
$$\frac{a+b}{p} \text{ and } \frac{a^n + b^n}{p^2}$$
are both integers that are relatively prime to $p$. If $A_n$ is finite, let $f(n)$ denote $|A_n|$.
a) Prove that $A_n$ is finite if and only if $n \not = 2$.
b) Let $m,k$ be odd positive integers and let $d$ be their gcd. Show that
$$f(d) \leq f(k) + f(m) - f(km) \leq 2 f(d).$$
2000 Belarus Team Selection Test, 5.1
Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$).
If $BC = a, AM = m_a, AL = l_a$, prove the inequalities:
(a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$
(b) $2l_a \le a\cot \frac{a}{2} $.
1997 VJIMC, Problem 3
Let $u\in C^2(\overline D)$, $u=0$ on $\partial D$ where $D$ is the open unit ball in $\mathbb R^3$. Prove that the following inequality holds for all $\varepsilon>0$:
$$\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.$$(We recall that $\nabla u$ and $\Delta u$ are the gradient and Laplacian, respectively.)
2024 Middle European Mathematical Olympiad, 1
Consider two infinite sequences $a_0,a_1,a_2,\dots$ and $b_0,b_1,b_2,\dots$ of real numbers such that $a_0=0$, $b_0=0$ and
\[a_{k+1}=b_k, \quad b_{k+1}=\frac{a_kb_k+a_k+1}{b_k+1}\]
for each integer $k \ge 0$. Prove that $a_{2024}+b_{2024} \ge 88$.
2017 China Girls Math Olympiad, 3
Given $a_i\ge 0,x_i\in\mathbb{R},(i=1,2,\ldots,n)$. Prove that
$$((1-\sum_{i=1}^n a_i\cos x_i)^2+(1-\sum_{i=1}^n a_i\sin x_i)^2)^2\ge 4(1-\sum_{i=1}^n a_i)^3$$
1982 Czech and Slovak Olympiad III A, 2
Given real numbers $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$. Let $M$ denote the maximum of their absolute values. Prove that it is valid $$
| x_1x_4-x_1x_5 +x_2x_5 -x_2x_6+x_3x_6-x_3x_4| \le 4M^2$$
2021 Indonesia TST, A
A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying
$$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$
then the following inequality holds:
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$
(a) Prove that $M=20-\frac{1}{20}$ is not $strong$.
(b) Prove that $M=20-\frac{1}{21}$ is $strong$.
1999 Polish MO Finals, 1
Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD > BC$. Let $E$ be a point on the side $AC$ such that $\frac{AE}{EC} = \frac{BD}{AD-BC}$. Show that $AD > BE$.
2001 VJIMC, Problem 1
Let $A$ be a set of positive integers such that for any $x,y\in A$,
$$x>y\implies x-y\ge\frac{xy}{25}.$$Find the maximal possible number of elements of the set $A$.
2011 Today's Calculation Of Integral, 747
Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$
2009 China Team Selection Test, 5
Let $ m > 1$ be an integer, $ n$ is an odd number satisfying $ 3\le n < 2m,$ number $ a_{i,j} (i,j\in N, 1\le i\le m, 1\le j\le n)$ satisfies $ (1)$ for any $ 1\le j\le n, a_{1,j},a_{2,j},\cdots,a_{m,j}$ is a permutation of $ 1,2,3,\cdots,m; (2)$ for any $ 1 < i\le m, 1\le j\le n \minus{} 1, |a_{i,j} \minus{} a_{i,{j \plus{} 1}}|\le 1$ holds. Find the minimal value of $ M$, where $ M \equal{} max_{1 < i < m}\sum_{j \equal{} 1}^n{a_{i,j}}.$
2010 ELMO Shortlist, 1
For a positive integer $n$, let $\mu(n) = 0$ if $n$ is not squarefree and $(-1)^k$ if $n$ is a product of $k$ primes, and let $\sigma(n)$ be the sum of the divisors of $n$. Prove that for all $n$ we have
\[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \]
and determine when equality holds.
[i]Wenyu Cao.[/i]
2014 Putnam, 4
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E[X]=1,$ $E[X^2]=2,$ and $E[X^3]=5.$ (Here $E[Y]$ denotes the expectation of the random variable $Y.$) Determine the smallest possible value of the probability of the event $X=0.$
2004 China Team Selection Test, 3
Let $k \geq 2, 1 < n_1 < n_2 < \ldots < n_k$ are positive integers, $a,b \in \mathbb{Z}^+$ satisfy \[ \prod^k_{i=1} \left( 1 - \frac{1}{n_i} \right) \leq \frac{a}{b} < \prod^{k-1}_{i=1} \left( 1 - \frac{1}{n_i} \right) \]
Prove that: \[ \prod^k_{i=1} n_i \geq (4 \cdot a)^{2^k - 1}. \]
1992 IMO Longlists, 62
Let $c_1, \cdots, c_n \ (n \geq 2)$ be real numbers such that $0 \leq \sum c_i \leq n$. Prove that there exist integers $k_1, \cdots , k_n$ such that $\sum k_i=0$ and $1-n \leq c_i + nk_i \leq n$ for every $i = 1, \cdots , n.$
2013 ELMO Shortlist, 2
Prove that for all positive reals $a,b,c$,
\[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \][i]Proposed by David Stoner[/i]
2020 Thailand Mathematical Olympiad, 8
For all positive real numbers $a,b,c$ with $a+b+c=3$, prove the inequality
$$\frac{a^6}{c^2+2b^3} + \frac{b^6}{a^2+2c^3} + \frac{c^6}{b^2+2a^3} \geq 1.$$
1963 Kurschak Competition, 2
$A$ is an acute angle. Show that
$$\left(1 +\frac{1}{sen A}\right)\left(1 +\frac{1}{cos A}\right)> 5$$