Found problems: 6530
2023 239 Open Mathematical Olympiad, 8
Let $r\geqslant 0$ be a real number and define $f(x)=1/(1+x^2)^r$. Prove that \[|f^{(k)}(x)|\leqslant\frac{2r\cdot(2r+1)\cdots(2r+k-1)}{(1+x^2)^{r+k/2}},\]for every natural number $k{}$. Here, $f^{(k)}(x)$ denotes the $k^{\text{th}}$ derivative of $f$.
2009 Polish MO Finals, 4
Let $ x_1,x_2,..,x_n$ be non-negative numbers whose sum is $ 1$ . Show that there exist numbers $ a_1,a_2,\ldots ,a_n$ chosen from amongst $ 0,1,2,3,4$ such that $ a_1,a_2,\ldots ,a_n$ are different from $ 2,2,\ldots ,2$ and $ 2\leq a_1x_1\plus{}a_2x_2\plus{}\ldots\plus{}a_nx_n\leq 2\plus{}\frac{2}{3^n\minus{}1}$.
2005 Georgia Team Selection Test, 10
Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that
\[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]
2011 China Girls Math Olympiad, 3
The positive reals $a,b,c,d$ satisfy $abcd=1$. Prove that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{9}{{a + b + c + d}} \geqslant \frac{{25}}{4}$.
1996 AMC 12/AHSME, 25
Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have?
$\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$
1983 IMO Longlists, 72
Prove that for all $x_1, x_2,\ldots , x_n \in \mathbb R$ the following inequality holds:
\[\sum_{n \geq i >j \geq 1} \cos^2(x_i - x_j ) \geq \frac{n(n-2)}{4}\]
2006 Poland - Second Round, 1
Positive integers $a,b,c,x,y,z$ satisfy:
$a^2+b^2=c^2$, $x^2+y^2=z^2$
and
$|x-a| \leq 1$ , $|y-b| \leq 1$.
Prove that sets $\{a,b\}$ and $\{x,y\}$ are equal.
PEN M Problems, 26
Let $p$ be an odd prime $p$ such that $2h \neq 1 \; \pmod{p}$ for all $h \in \mathbb{N}$ with $h< p-1$, and let $a$ be an even integer with $a \in] \tfrac{p}{2}, p [$. The sequence $\{a_n\}_{n \ge 0}$ is defined by $a_{0}=a$, $a_{n+1}=p -b_{n}$ \; $(n \ge 0)$, where $b_{n}$ is the greatest odd divisor of $a_n$. Show that the sequence $\{a_n\}_{n \ge 0}$ is periodic and find its minimal (positive) period.
1976 IMO Longlists, 12
Five points lie on the surface of a ball of unit radius. Find the maximum of the smallest distance between any two of them.
1999 Mediterranean Mathematics Olympiad, 3
Let $a,b,c\not= 0$ and $x,y,z\in\mathbb{R}^+$ such that $x+y+z=3$. Prove that \[\frac{3}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}\]
[color=#FF0000]Mod: before the edit, it was [/color]
\[\frac{3}{2}\left (\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right )\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}\]
1996 China Team Selection Test, 2
Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that
\[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\]
Define
\begin{align*}
A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\
B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\
W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2.
\end{align*}
Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.
1995 China National Olympiad, 3
Find the minimun value of $\sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10}|k(x+y-10i)(3x-6y-36j)(19x+95y-95k)|$ , where $x,y$ are integers.
2003 Baltic Way, 3
Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Prove that
$$\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).$$
STEMS 2021 Math Cat A, Q2
Suppose $f: \mathbb{R}^{+} \mapsto \mathbb{R}^{+}$ is a function such that $\frac{f(x)}{x}$ is increasing on $\mathbb{R}^{+}$. For $a,b,c>0$, prove that $$2\left (\frac{f(a)+f(b)}{a+b} + \frac{f(b)+f(c)}{b+c}+ \frac{f(c)+f(a)}{c+a} \right) \geq 3\left(\frac{f(a)+f(b)+f(c)}{a+b+c}\right) + \frac{f(a)}{a}+ \frac{f(b)}{b}+ \frac{f(c)}{c}$$
2023 Brazil Undergrad MO, 3
Prove that there exists a constant $C > 0$ such that, for any integers $m, n$ with $n \geq m > 1$ and any real number $x > 1$, $$\sum_{k=m}^{n}\sqrt[k]{x} \leq C\bigg(\frac{m^2 \cdot \sqrt[m-1]{x}}{\log{x}} + n\bigg)$$
2009 Indonesia MO, 2
For any real $ x$, let $ \lfloor x\rfloor$ be the largest integer that is not more than $ x$. Given a sequence of positive integers $ a_1,a_2,a_3,\ldots$ such that $ a_1>1$ and
\[ \left\lfloor\frac{a_1\plus{}1}{a_2}\right\rfloor\equal{}\left\lfloor\frac{a_2\plus{}1}{a_3}\right\rfloor\equal{}\left\lfloor\frac{a_3\plus{}1}{a_4}\right\rfloor\equal{}\cdots\]
Prove that
\[ \left\lfloor\frac{a_n\plus{}1}{a_{n\plus{}1}}\right\rfloor\leq1\]
holds for every positive integer $ n$.
2021-IMOC, A10
For any positive reals $x$, $y$, $z$ with $xyz + xy + yz + zx = 4$, prove that
$$\sqrt{\frac{xy+x+y}{z}}+\sqrt{\frac{yz+y+z}{x}}+\sqrt{\frac{zx+z+x}{y}}\geq 3\sqrt{\frac{3(x+2)(y+2)(z+2)}{(2x + 1)(2y + 1)(2z + 1).
}}$$
1999 Balkan MO, 3
Let $ABC$ be an acute-angled triangle of area 1. Show that the triangle whose vertices are the feet of the perpendiculars from the centroid $G$ to
$AB$, $BC$, $CA$ has area between $\frac 4{27}$ and $\frac 14$.
2007 China Team Selection Test, 3
Find the smallest constant $ k$ such that
$ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$
for all positive $ x$, $ y$, $ z$.
2025 Iran MO (2nd Round), 5
Find all functions $f:\mathbb{R}^+ \to \mathbb{R}$ such that for all $x,y,z>0$
$$
3(x^3+y^3+z^3)\geq f(x+y+z)\cdot f(xy+yz+xz) \geq (x+y+z)(xy+yz+xz).
$$
2011 Albania National Olympiad, 3
In a convex quadrilateral $ABCD$ ,$\angle ABC$ and $\angle BCD$ are $\geq 120^o$. Prove that $|AC|$ + $|BD| \geq |AB|+|BC|+|CD|$. (With $|XY|$ we understand the length of the segment $XY$).
2021 239 Open Mathematical Olympiad, 5
Let $a,b,c$ be some complex numbers. Prove that
$$|\dfrac{a^2}{ab+ac-bc}| + |\dfrac{b^2}{ba+bc-ac}| + |\dfrac{c^2}{ca+cb-ab}| \ge \dfrac{3}{2}$$
if the denominators are not 0
2022 Canadian Mathematical Olympiad Qualification, 8
Let $\{m, n, k\}$ be positive integers. $\{k\}$ coins are placed in the squares of an $m \times n$ grid. A square may contain any number of coins, including zero. Label the $\{k\}$ coins $C_1, C_2, · · · C_k$. Let $r_i$ be the number of coins in the same row as $C_i$, including $C_i$ itself. Let $s_i$ be the number of coins in the same column as $C_i$, including $C_i$ itself. Prove that
\[\sum_{i=1}^k \frac{1}{r_i+s_i} \leq \frac{m+n}{4}\]
2018 Macedonia National Olympiad, Problem 4
Let $t_{k} = a_{1}^k + a_{2}^k +...+a_{n}^k$, where $a_{1}$, $a_{2}$, ... $a_{n}$ are positive real numbers and $k \in \mathbb{N}$. Prove that $$\frac{t_{5}^2 t_1^{6}}{15} - \frac{t_{4}^4 t_{2}^2 t_{1}^2}{6} + \frac{t_{2}^3 t_{4}^5}{10} \geq 0 $$
[i]Proposed by Daniel Velinov[/i]
2013 Middle European Mathematical Olympiad, 2
Let $ x, y, z, w $ be nonzero real numbers such that $ x+y \ne 0$, $ z+w \ne 0 $, and $ xy+zw \ge 0 $. Prove that
\[ \left( \frac{x+y}{z+w} + \frac{z+w}{x+y} \right) ^{-1} + \frac{1}{2} \ge \left( \frac{x}{z} + \frac{z}{x} \right) ^{-1} + \left( \frac{y}{w} + \frac{w}{y} \right) ^{-1}\]