Found problems: 6530
2011 Postal Coaching, 5
The seats in the Parliament of some country are arranged in a rectangle of $10$ rows of $10$ seats each. All the $100$ $MP$s have different salaries. Each of them asks all his neighbours (sitting next to, in front of, or behind him, i.e. $4$ members at most) how much they earn. They feel a lot of envy towards each other: an $MP$ is content with his salary only if he has at most one neighbour who earns more than himself. What is the maximum possible number of $MP$s who are satisfied with their salaries?
2006 AMC 10, 10
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
$ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$
2017 Thailand TSTST, 5
Let $a, b, c \in \mathbb{R}^+$ such that $a + b + c = 3$. Prove that $$\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.$$
2003 India IMO Training Camp, 9
Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.
2006 China Team Selection Test, 2
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
2020 Taiwan TST Round 1, 1
Let $a$, $b$, $c$, $d$ be real numbers satisfying
\begin{align*}
(a + c)(b + d) = \sqrt{2}(ac - 2bd - 1).
\end{align*}
Show that
\begin{align*}
(ab - 1)^2 + (bc - 1)^2 + (cd - 1)^2 + (da - 1)^2 + (ac - 1)^2 + (2bd + 1)^2 \ge 4.
\end{align*}
2015 IFYM, Sozopol, 4
For all real numbers $a,b,c>0$ such that $abc=1$, prove that
$\frac{a}{1+b^3}+\frac{b}{1+c^3}+\frac{c}{1+a^3}\geq \frac{3}{2}$.
2011 Postal Coaching, 5
Let $a, b$ and $c$ be positive real numbers. Prove that
\[\frac{\sqrt{a^2+bc}}{b+c}+\frac{\sqrt{b^2+ca}}{c+a}+\frac{\sqrt{c^2+ab}}{a+b}\ge\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\]
2006 China Team Selection Test, 3
Given $n$ real numbers $a_1$, $a_2$ $\ldots$ $a_n$. ($n\geq 1$). Prove that there exists real numbers $b_1$, $b_2$ $\ldots$ $b_n$ satisfying:
(a) For any $1 \leq i \leq n$, $a_i - b_i$ is a positive integer.
(b)$\sum_{1 \leq i < j \leq n} (b_i - b_j)^2 \leq \frac{n^2-1}{12}$
1990 Greece National Olympiad, 1
Let $a,b$, be two real numbers. If for any $x>0$ holds that $|a-b|<x$, then prove that $a=b$.
2006 Romania Team Selection Test, 4
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]
2021-IMOC, A10
For any positive reals $x$, $y$, $z$ with $xyz + xy + yz + zx = 4$, prove that
$$\sqrt{\frac{xy+x+y}{z}}+\sqrt{\frac{yz+y+z}{x}}+\sqrt{\frac{zx+z+x}{y}}\geq 3\sqrt{\frac{3(x+2)(y+2)(z+2)}{(2x + 1)(2y + 1)(2z + 1).
}}$$
2021 China Team Selection Test, 1
Given positive integers $m$ and $n$. Let $a_{i,j} ( 1 \le i \le m, 1 \le j \le n)$ be non-negative real numbers, such that
$$ a_{i,1} \ge a_{i,2} \ge \cdots \ge a_{i,n} \text{ and } a_{1,j} \ge a_{2,j} \ge \cdots \ge a_{m,j} $$
holds for all $1 \le i \le m$ and $1 \le j \le n$. Denote
$$ X_{i,j}=a_{1,j}+\cdots+a_{i-1,j}+a_{i,j}+a_{i,j-1}+\cdots+a_{i,1},$$
$$ Y_{i,j}=a_{m,j}+\cdots+a_{i+1,j}+a_{i,j}+a_{i,j+1}+\cdots+a_{i,n}.$$
Prove that
$$ \prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.$$
2001 Moldova National Olympiad, Problem 5
For each integer $n\ge2$ prove the inequality
$$\log_23+\log_34+\ldots+\log_n(n+1)<n+\ln n-0.9.$$
2012 Romania Team Selection Test, 3
Let $A$ and $B$ be finite sets of real numbers and let $x$ be an element of $A+B$. Prove that \[|A\cap (x-B)|\leq \frac{|A-B|^2}{|A+B|}\] where $A+B=\{a+b: a\in A, b\in B\}$, $x-B=\{x-b: b\in B\}$ and $A-B=\{a-b: a\in A, b\in B\}$.
1990 Polish MO Finals, 2
Let $x_1, x_2, . . . , x_n$ be positive numbers. Prove that
\[ \sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1 \]
Where $x_{n+1}=x_1$ and $x_{n+2}=x_2$.
2014 Middle European Mathematical Olympiad, 1
Determine the lowest possible value of the expression
\[ \frac{1}{a+x} + \frac{1}{a+y} + \frac{1}{b+x} + \frac{1}{b+y} \]
where $a,b,x,$ and $y$ are positive real numbers satisfying the inequalities
\[ \frac{1}{a+x} \ge \frac{1}{2} \] \[\frac{1}{a+y} \ge \frac{1}{2} \] \[ \frac{1}{b+x} \ge \frac{1}{2} \] \[ \frac{1}{b+y} \ge 1. \]
1970 All Soviet Union Mathematical Olympiad, 130
The product of three positive numbers equals to one, their sum is strictly greater than the sum of the inverse numbers. Prove that one and only one of them is greater than one.
2012 Serbia Team Selection Test, 1
Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?
2011 NIMO Problems, 4
Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality
\[
1 < a < b+2 < 10.
\]
[i]Proposed by Lewis Chen
[/i]
2005 Romania Team Selection Test, 2
Let $n\geq 2$ be an integer. Find the smallest real value $\rho (n)$ such that for any $x_i>0$, $i=1,2,\ldots,n$ with $x_1 x_2 \cdots x_n = 1$, the inequality
\[ \sum_{i=1}^n \frac 1{x_i} \leq \sum_{i=1}^n x_i^r \] is true for all $r\geq \rho (n)$.
2008 IberoAmerican Olympiad For University Students, 5
Find all positive integers $n$ such that there are positive integers $a_1,\cdots,a_n, b_1,\cdots,b_n$ that satisfy
\[(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)-(a_1b_1+\cdots+a_nb_n)^2=n\]
2007 District Olympiad, 1
Let be three real numbers $ a,b,c, $ all in the interval $ (0,\infty ) $ or all in the interval $ (0,1). $ Prove the following inequality:
$$ \sum_{\text{cyc}}\log_a bc\ge 4\cdot\sum_{\text{cyc}} \log_{ab} c . $$
2013 Czech-Polish-Slovak Match, 2
Prove that for every real number $x>0$ and each integer $n>0$ we have
\[x^n+\frac1{x^n}-2 \ge n^2\left(x+\frac1x-2\right)\]
1987 Romania Team Selection Test, 4
Let $ P(X) \equal{} a_{n}X^{n} \plus{} a_{n \minus{} 1}X^{n \minus{} 1} \plus{} \ldots \plus{} a_{1}X \plus{} a_{0}$ be a real polynomial of degree $ n$. Suppose $ n$ is an even number and:
a) $ a_{0} > 0$, $ a_{n} > 0$;
b) $ a_{1}^{2} \plus{} a_{2}^{2} \plus{} \ldots \plus{} a_{n \minus{} 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n \minus{} 1}$.
Prove that $ P(x)\geq 0$ for all real values $ x$.
[i]Laurentiu Panaitopol[/i]