Found problems: 6530
2012 Korea National Olympiad, 3
Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of
\[ \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) \]
2018 Saudi Arabia IMO TST, 3
Consider the function $f (x) = (x - F_1)(x - F_2) ...(x -F_{3030})$ with $(F_n)$ is the Fibonacci sequence, which defined as $F_1 = 1, F_2 = 2$, $F_{n+2 }=F_{n+1} + F_n$, $n \ge 1$. Suppose that on the range $(F_1, F_{3030})$, the function $|f (x)|$ takes on the maximum value at $x = x_0$. Prove that $x_0 > 2^{2018}$.
2017 South East Mathematical Olympiad, 3
Let $a_1,a_2,\cdots,a_{n+1}>0$. Prove that$$\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})$$
1978 All Soviet Union Mathematical Olympiad, 257
Prove that there exists such an infinite sequence $\{x_i\}$, that for all $m$ and all $k$ ($m\ne k$) holds the inequality $$|x_m-x_k|>1/|m-k|$$
1985 Balkan MO, 2
Let $a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ be real numbers such that
$\sin{a}+\sin{b}+\sin{c}+\sin{d}=1$ and $\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}$.
Prove that $a,b,c,d \in [0, \frac{\pi}{6}]$
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 $strong$.
2022 Romania Team Selection Test, 3
Let $n\geq 2$ be an integer. Let $a_{ij}, \ i,j=1,2,\ldots,n$ be $n^2$ positive real numbers satisfying the following conditions:
[list=1]
[*]For all $i=1,\ldots,n$ we have $a_{ii}=1$ and,
[*]For all $j=2,\ldots,n$ the numbers $a_{ij}, \ i=1,\ldots, j-1$ form a permutation of $1/a_{ji}, \ i=1,\ldots, j-1.$
[/list]
Given that $S_i=a_{i1}+\cdots+a_{in}$, determine the maximum value of the sum $1/S_1+\cdots+1/S_n.$
2022 Austrian MO Regional Competition, 1
Let $a$ and $b$ be positive real numbers with $a^2 + b^2 =\frac12$. Prove that
$$\frac{1}{1 - a}+\frac{1}{1-b}\ge 4.$$
When does equality hold?
[i](Walther Janous)[/i]
2014 Contests, 3
Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]
2009 Romania National Olympiad, 1
Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and
$$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$
2008 IMC, 4
We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$.
For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?
1993 Kurschak Competition, 1
Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.
2010 China National Olympiad, 3
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.
2010 China Team Selection Test, 2
Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set
$A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$
$B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$
Prove that $2|A|\geq |B|$.
PEN N Problems, 9
Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that
a) for any $ m > n$, $ m \minus{} n$ is a factor of $ q_{m} \minus{} q_{n}$,
b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$.
Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} \equal{} Q(n)$ for all $ n$.
1980 Vietnam National Olympiad, 3
Let be given an integer $n\ge 2$ and a positive real number $p$. Find the maximum of
\[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\]
where $x_i$ are non-negative real numbers with sum $p$.
2008 China Western Mathematical Olympiad, 2
Given $ x,y,z\in (0,1)$ satisfying that
$ \sqrt{\frac{1 \minus{} x}{yz}} \plus{} \sqrt{\frac{1 \minus{} y}{xz}} \plus{} \sqrt{\frac{1 \minus{} z}{xy}} \equal{} 2$.
Find the maximum value of $ xyz$.
2011 Balkan MO, 2
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
2003 China Western Mathematical Olympiad, 4
$ 1650$ students are arranged in $ 22$ rows and $ 75$ columns. It is known that in any two columns, the number of pairs of students in the same row and of the same sex is not greater than $ 11$. Prove that the number of boys is not greater than $ 928$.
1992 Vietnam Team Selection Test, 3
Let $ABC$ a triangle be given with $BC = a$, $CA = b$, $AB = c$ ($a \neq b \neq c \neq a$). In plane ($ABC$) take the points $A'$, $B'$, $C'$ such that:
[b]I.[/b] The pairs of points $A$ and $A'$, $B$ and $B'$, $C$ and $C'$ either all lie in one side either all lie in different sides under the lines $BC$, $CA$, $AB$ respectively;
[b]II.[/b] Triangles $A'BC$, $B'CA$, $C'AB$ are similar isosceles triangles.
Find the value of angle $A'BC$ as function of $a, b, c$ such that lengths $AA', BB', CC'$ are not sides of an triangle. (The word "triangle" must be understood in its ordinary meaning: its vertices are not collinear.)
1994 Hong Kong TST, 1
Suppose, $x, y, z \in \mathbb{R}_+$ such that $xy+yz+zx=1$. Prove that, \[x(1-y^2)(1-z^2)+y(1-z^2)(1-x^2)+z(1-x^2)(1-y^2)\leq \frac{4\sqrt{3}}{9}\]
1983 Polish MO Finals, 5
On the plane are given unit vectors $\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}$. Show that one can choose numbers $c_1,c_2,c_3 \in \{-1,1\}$ such that the length of the vector $c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3}$ is at least $2$.
2021 ISI Entrance Examination, 7
Let $a, b, c$ be three real numbers which are roots of a cubic polynomial, and satisfy $a+b+c=6$ and $ab+bc+ca=9$. Suppose $a<b<c$. Show that $$0<a<1<b<3<c<4.$$
2021 China Second Round Olympiad, Problem 15
Positive real numbers $x, y, z$ satisfy $\sqrt x + \sqrt y + \sqrt z = 1$. Prove that $$\frac{x^4+y^2z^2}{x^{\frac 52}(y+z)} + \frac{y^4+z^2x^2}{y^{\frac 52}(z+x)} + \frac{z^4+y^2x^2}{z^{\frac 52}(y+x)} \geq 1.$$
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 15)[/i]
2006 JBMO ShortLists, 2
Let $ x,y,z$ be positive real numbers such that $ x\plus{}2y\plus{}3z\equal{}\frac{11}{12}$. Prove the inequality $ 6(3xy\plus{}4xz\plus{}2yz)\plus{}6x\plus{}3y\plus{}4z\plus{}72xyz\le \frac{107}{18}$.