Found problems: 6530
1996 Bundeswettbewerb Mathematik, 4
Let $p$ be an odd prime. Determine the positive integers $x$ and $y$ with $x\leq y$ for which the number $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is non-negative and as small as possible.
1995 Poland - First Round, 7
Nonnegative numbers $a, b, c, p, q, r$ satisfy the conditions:
$a + b + c = p + q + r = 1; ~~~~~~ p, q, r \leq \frac{1}{2}$.
Prove that $8abc \leq pa + qb + rc$ and determine when equality holds.
2005 Vietnam National Olympiad, 1
Let $x,y$ be real numbers satisfying the condition:
\[x-3\sqrt {x+1}=3\sqrt{y+2} -y\]
Find the greatest value and the smallest value of:
\[P=x+y\]
2014 ELMO Shortlist, 8
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that
\[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]
1998 Brazil Team Selection Test, Problem 5
Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that
$$k\le2\lfloor\sqrt n\rfloor.$$
2012 239 Open Mathematical Olympiad, 4
For positive real numbers $a$, $b$, and $c$ with $a+b+c=1$, prove that:
$$ (a-b)^2 + (b-c)^2 + (c-a)^2 \geq \frac{1-27abc}{2}. $$
2002 China Team Selection Test, 1
Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that
\begin{align*}
P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right),
\end{align*}
where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]
1982 Putnam, B6
Denote by $S(a,b,c)$ the area of a triangle whose lengthes of three sides are $a,b,c$
Prove that for any positive real numbers $a_{1},b_{1},c_{1}$ and $a_{2},b_{2},c_{2}$ which can serve as the lengthes of three sides of two triangles respectively ,we have
$ \sqrt{S(a_{1},b_{1},c_{1})}+\sqrt{S(a_{2},b_{2},c_{2})}\le\sqrt{S(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})}$
1962 Polish MO Finals, 5
Prove that if $ n $ is a natural number greater than $ 2 $, then $$\sqrt[n + 1]{n+1} < \sqrt[n]{n}.$$
2009 Ukraine National Mathematical Olympiad, 3
Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that
\[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]
1940 Putnam, B7
Which is greater
$$\sqrt{n}^{\sqrt{n+1}} \;\; \; \text{or}\;\;\; \sqrt{n+1}^{\sqrt{n}}$$
where $n>8?$
2024 VJIMC, 1
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function. Prove that
\[\left\vert f(1)-\int_0^1 f(x) dx\right\vert \le \frac{1}{2} \max_{x \in [0,1]} \vert f'(x)\vert.\]
2015 Macedonia National Olympiad, Problem 2
Let $a,b,c \in \mathbb{R}^{+}$ such that $abc=1$. Prove that:
$$a^2b + b^2c + c^2a \ge \sqrt{(a+b+c)(ab + bc +ca)}$$
2010 Kazakhstan National Olympiad, 3
Positive real $A$ is given. Find maximum value of $M$ for which inequality
$ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $
holds for all $x, y>0$
2005 China Northern MO, 6
Let $0 \leq \alpha , \beta , \gamma \leq \frac{\pi}{2}$, such that $\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma = 1$. Prove that
$2 \leq (1 + \cos ^{2} \alpha ) ^{2} \sin^{4} \alpha + (1 + \cos ^{2} \beta ) ^{2} \sin ^{4} \beta + (1 + \cos ^{2} \gamma ) ^{2} \sin ^{4} \gamma \leq (1 + \cos ^{2} \alpha )(1 + \cos ^{2} \beta)(1 + \cos ^{2} \gamma ).$
2023 All-Russian Olympiad Regional Round, 9.9
Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.
2006 Federal Math Competition of S&M, Problem 1
Let $x,y,z$ be positive numbers with the sum $1$. Prove that
$$\frac x{y^2+z}+\frac y{z^2+x}+\frac z{x^2+y}\ge\frac94.$$
2011 Romania National Olympiad, 1
Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that
$$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$
Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $
2015 JBMO Shortlist, A5
The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that$$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$
1998 South africa National Olympiad, 2
Find the maximum value of \[ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} \] where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$.
2008 Germany Team Selection Test, 1
Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions:
[b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$
[b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$
such that for each sequence element we have the inequality $ a_n \leq Q.$
2008 Sharygin Geometry Olympiad, 18
(A.Abdullayev, 9--11) Prove that the triangle having sides $ a$, $ b$, $ c$ and area $ S$ satisfies the inequality
\[ a^2\plus{}b^2\plus{}c^2\minus{}\frac12(|a\minus{}b|\plus{}|b\minus{}c|\plus{}|c\minus{}a|)^2\geq 4\sqrt3 S.\]
Russian TST 2014, P1
Let $x,y,z$ be positive real numbers. Prove that \[\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.\]
1977 Vietnam National Olympiad, 5
The real numbers $a_0, a_1, ... , a_{n+1}$ satisfy $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_k + a_{k+1}| \le 1$ for $k = 1, 2, ... , n$. Show that $|a_k| \le \frac{ k(n + 1 - k)}{2}$ for all $k$.
1986 AMC 8, 14
If $ 200 \le a \le 400$ and $ 600 \le b \le 1200$, then the largest value of the quotient $ \frac{b}{a}$ is
\[ \textbf{(A)}\ \frac{3}{2} \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 300 \qquad
\textbf{(E)}\ 600 \qquad
\]