Found problems: 6530
2018 Moldova Team Selection Test, 10
The positive real numbers $a,b, c,d$ satisfy the equality $ \frac {1}{a+1} + \frac {1}{b+1} + \frac {1}{c+1} + \frac{ 1}{d+1} = 3 $ . Prove the inequality $\sqrt [3]{abc} + \sqrt [3]{bcd} + \sqrt [3]{cda} + \sqrt [3]{dab} \le \frac {4}{3} $.
2008 China Team Selection Test, 3
Let $ 0 < x_{1}\leq\frac {x_{2}}{2}\leq\cdots\leq\frac {x_{n}}{n}, 0 < y_{n}\leq y_{n \minus{} 1}\leq\cdots\leq y_{1},$ Prove that $ (\sum_{k \equal{} 1}^{n}x_{k}y_{k})^2\leq(\sum_{k \equal{} 1}^{n}y_{k})(\sum_{k \equal{} 1}^{n}(x_{k}^2 \minus{} \frac {1}{4}x_{k}x_{k \minus{} 1})y_{k}).$ where $ x_{0} \equal{} 0.$
2010 Contests, 2a
Show that $\frac{x^2}{1 - x}+\frac{(1 - x)^2}{x} \ge 1$ for all real numbers $x$, where $0 < x < 1$
2005 Iran MO (3rd Round), 5
Suppose $a,b,c \in \mathbb R^+$and \[\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2\]
Prove that $ab+ac+bc\leq \frac32$
Russian TST 2018, P2
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have
$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$
Find the largest constant $K = K(n)$ such that
$$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$
holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.
1966 IMO Longlists, 32
The side lengths $a,$ $b,$ $c$ of a triangle $ABC$ form an arithmetical progression (such that $b-a=c-b$). The side lengths $a_{1},$ $b_{1},$ $c_{1}$ of a triangle $A_{1}B_{1}C_{1}$ also form an arithmetical progression (with $b_{1}-a_{1}=c_{1}-b_{1}$). [Hereby, $a=BC,$ $b=CA,$ $c=AB, $ $a_{1}=B_{1}C_{1},$ $b_{1}=C_{1}A_{1},$ $c_{1}=A_{1}B_{1}.$] Moreover, we know that $\measuredangle CAB=\measuredangle C_{1}A_{1}B_{1}.$
Show that triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.
2001 Cuba MO, 6
The roots of the equation $ax^2 - 4bx + 4c = 0$ with $ a > 0$ belong to interval $[2, 3]$. Prove that:
a) $a \le b \le c < a + b.$
b) $\frac{a}{a+c} + \frac{b}{b+a} > \frac{c}{b+c} .$
2017 AMC 10, 3
Real numbers $x$, $y$, and $z$ satisfy the inequalities
$$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$
Which of the following numbers is nessecarily positive?
$\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\
\textbf{(E) } y+z$
2024 Bundeswettbewerb Mathematik, 2
Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties:
(1) No number occurs more than once in the sequence.
(2) The sum of two different elements of the sequence is never a power of two.
(3) For all positive integers $n$, we have $a_n<r \cdot n$.
2022 Canadian Junior Mathematical Olympiad, 3
If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$
2019 China National Olympiad, 1
Let $a,b,c,d,e\geq -1$ and $a+b+c+d+e=5.$ Find the maximum and minimum value of $S=(a+b)(b+c)(c+d)(d+e)(e+a).$
2012 Mediterranean Mathematics Olympiad, 2
In an acute $\triangle ABC$, prove that
\begin{align*}\frac{1}{3}\left(\frac{\tan^2A}{\tan B\tan C}+\frac{\tan^2 B}{\tan C\tan A}+\frac{\tan^2 C}{\tan A\tan B}\right) \\ +3\left(\frac{1}{\tan A+\tan B+\tan C}\right)^{\frac{2}{3}}\ge 2.\end{align*}
2003 Costa Rica - Final Round, 3
If $a>1$ and $b>2$ are positive integers, show that $a^{b}+1 \geq b(a+1)$, and determine when equality holds.
2022 Moscow Mathematical Olympiad, 1
$a,b,c$ are nonnegative and $a+b+c=2\sqrt{abc}$.
Prove $bc \geq b+c$
2013 Kosovo National Mathematical Olympiad, 1
Which number is bigger $\sqrt[2012]{2012!}$ or $\sqrt[2013]{2013!}$.
2011 Balkan MO Shortlist, A4
Let $x,y,z \in \mathbb{R}^+$ satisfying $xyz=3(x+y+z)$. Prove, that
\begin{align*} \sum \frac{1}{x^2(y+1)} \geq \frac{3}{4(x+y+z)} \end{align*}
1970 Canada National Olympiad, 5
A quadrilateral has one vertex on each side of a square of side-length 1. Show that the lengths $a$, $b$, $c$ and $d$ of the sides of the quadrilateral satisfy the inequalities \[ 2\le a^2+b^2+c^2+d^2\le 4. \]
2022 Tuymaada Olympiad, 4
For every positive $a_1, a_2, \dots, a_6$, prove the inequality
\[ \sqrt[4]{\frac{a_1}{a_2 + a_3 + a_4}} + \sqrt[4]{\frac{a_2}{a_3 + a_4 + a_5}} + \dots + \sqrt[4]{\frac{a_6}{a_1 + a_2 + a_3}} \ge 2 \]
1960 Putnam, B7
Let $g(t)$ and $h(t)$ be real, continuous functions for $t\geq 0.$ Show that any function $v(t)$ satisfying the differential inequality
$$\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,$$
satisfies the further inequality $v(t)\geq u(t),$ where
$$\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.$$
From this, conclude that for sufficiently small $t>0,$ the solution of
$$\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c$$
may be written
$$v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),$$
where the maximum is over all continuous functions $w(t)$ defined over some $t$-interval $[0,t_0 ].$
2004 India National Olympiad, 4
$ABC$ is a triangle, with sides $a$, $b$, $c$ , circumradius $R$, and exradii $r_a$, $r_b$, $r_c$If $2R\leq r_a$, show that $a > b$, $a > c$, $2R > r_b$, and $2R > r_c$.
1973 Putnam, B4
(a) On $[0, 1]$, let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$. Also suppose that $f(0) = 0.$ Prove that
$$ \left( \int_{0}^{1} f(x)\; dx \right)^{2} \geq \int_{0}^{1} f(x)^{3}\; dx.$$
(b) Show an example in which equality occurs.
2011 Morocco National Olympiad, 1
Prove that
\[2010< \frac{2^{2}+1}{2^{2}-1}+\frac{3^{2}+1}{3^{2}-1}+...+\frac{2010^{2}+1}{2010^{2}-1}< 2010+\frac{1}{2}.\]
2003 China Team Selection Test, 3
Given $S$ be the finite lattice (with integer coordinate) set in the $xy$-plane. $A$ is the subset of $S$ with most elements such that the line connecting any two points in $A$ is not parallel to $x$-axis or $y$-axis. $B$ is the subset of integer with least elements such that for any $(x,y)\in S$, $x \in B$ or $y \in B$ holds. Prove that $|A| \geq |B|$.
1993 Poland - Second Round, 3
A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.
1998 VJIMC, Problem 4-M
A function $f:\mathbb R\to\mathbb R$ has the property that for every
$x,y\in\mathbb R$ there exists a real number $t$ (depending on $x$ and $y$) such
that $0<t<1$ and
$$f(tx+(1-t)y)=tf(x)+(1-t)f(y).$$
Does it imply that
$$f\left(\frac{x+y}2\right)=\frac{f(x)+f(y)}2$$
for every $x,y\in\mathbb R$?