Found problems: 6530
2009 Hanoi Open Mathematics Competitions, 7
Let $a,b,c,d$ be positive integers such that $a+b+c+d=99$. Find the maximum and minimum of product $abcd$
2010 ISI B.Stat Entrance Exam, 1
Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that
\[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]
2005 Romania Team Selection Test, 2
Let $ABC$ be a triangle, and let $D$, $E$, $F$ be 3 points on the sides $BC$, $CA$ and $AB$ respectively, such that the inradii of the triangles $AEF$, $BDF$ and $CDE$ are equal with half of the inradius of the triangle $ABC$. Prove that $D$, $E$, $F$ are the midpoints of the sides of the triangle $ABC$.
2008 Irish Math Olympiad, 5
Suppose that $ x, y$ and $ z$ are positive real numbers such that $ xyz \ge 1$.
(a) Prove that
$ 27 \le (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$,
with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.
(b) Prove that
$ (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2$ $ \le 3(x \plus{} y \plus{} z)^2$,
with equality if and only if $ x \equal{} y \equal{} z \equal{} 1$.
2018 Mathematical Talent Reward Programme, SAQ: P 1
Let $x, y, z$ are real numbers such that $x<y<z .$ Prove that
$$
(x-y)^{3}+(y-z)^{3}+(z-x)^{3}>0
$$
2018 Kyiv Mathematical Festival, 3
For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge2\sqrt{2xy}.$
Today's calculation of integrals, 884
Prove that :
\[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]
2013 China National Olympiad, 3
Find all positive real numbers $t$ with the following property: there exists an infinite set $X$ of real numbers such that the inequality \[ \max\{|x-(a-d)|,|y-a|,|z-(a+d)|\}>td\] holds for all (not necessarily distinct) $x,y,z\in X$, all real numbers $a$ and all positive real numbers $d$.
1996 All-Russian Olympiad, 7
Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$?
[i]E. Malinnikova[/i]
2011 Balkan MO Shortlist, C2
Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.
1999 Romania National Olympiad, 2b
Let $a, b, c$ be positive real numbers such that $ab +be + ba \le 3abc$. Prove that
$$a^3+b^3+c^3 \ge a+b+c.$$
1961 Putnam, B4
Let $x_1 , x_2 ,\ldots, x_n$ be real numbers in $[0,1].$ Determine the maximum value of the sum of the $\frac{n(n-1)}{2}$ terms:
$$\sum_{i<j}|x_i -x_j |.$$
2016 Romania National Olympiad, 2
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions:
$$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$
for all real numbers $ x,y,t $ with $ t\in [0,1] . $
Prove that:
[b]a)[/b] $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $
[b]b)[/b] for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds.
$$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$
2020 European Mathematical Cup, 4
Let \(a,b,c\) be positive real numbers such that \(ab+bc+ac = a+b+c\). Prove the following inequality:
\[\sqrt{a+\frac{b}{c}} + \sqrt{b+\frac{c}{a}} + \sqrt{c+\frac{a}{b}} \leq \sqrt{2} \cdot \min \left\{ \frac{a}{b}+\frac{b}{c}+\frac{c}{a},\ \frac{b}{a}+\frac{c}{b}+\frac{a}{c} \right\}.\] \\ \\ [i]Proposed by Dorlir Ahmeti.[/i]
2020 Moldova Team Selection Test, 10
Let $n$ be a positive integer. Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Find the greatest possible value of $$E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}$$
2012 Stars of Mathematics, 3
For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality
$$ \left | \dfrac {a} {b-c} \right | + \left | \dfrac {b} {c-a} \right | + \left | \dfrac {c} {a-b} \right | \geq 2$$
and determine all cases of equality.
Prove that if we also impose $a,b,c$ positive, then all equality cases disappear, but the value $2$ remains the best constant possible.
([i]Dan Schwarz[/i])
2018 Moldova Team Selection Test, 6
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Show that
$$\frac{a}{1+b^2}+\frac{b}{1+c^2}+\frac{c}{1+a^2}\geq \frac{3}{2}.$$
2007 USAMO, 6
Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that
\[8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; , \]
with equality if and only if triangle $ABC$ is equilateral.
2017 Switzerland - Final Round, 10
Let $x, y, z$ be nonnegative real numbers with $xy + yz + zx = 1$. Show that:
$$\frac{4}{x + y + z} \le (x + y)(\sqrt3 z + 1).$$
VMEO III 2006 Shortlist, N9
Assume the $m$ is a given integer greater than $ 1$. Find the largest number $C$ such that for all $n \in N$ we have
$$\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}$$
2005 Today's Calculation Of Integral, 82
Let $0<a<b$.Prove the following inequaliy.
\[\frac{1}{b-a}\int_a^b \left(\ln \frac{b}{x}\right)^2 dx<2\]
2012 India Regional Mathematical Olympiad, 8
Let $x, y, z$ be positive real numbers such that $2(xy + yz + zx) = xyz$.
Prove that $\frac{1}{(x-2)(y-2)(z-2)} + \frac{8}{(x+2)(y+2)(z+2)} \le \frac{1}{32}$
2017 Saudi Arabia IMO TST, 2
Denote by $\{x\}$ the fractional part of a real number $x$, that is $\{x\} = x - \rfloor x \lfloor $ where $\rfloor x \lfloor $ is the maximum integer not greater than$ x$ . Prove that
a) For every integer $n$, we have $\{n\sqrt{17}\}> \frac{1}{2\sqrt{17} n}$
b) The value $\frac{1}{2\sqrt{17} }$ is the largest constant $c$ such that the inequality $\{n\sqrt{17}\}> c n $ holds for all positive integers $n$
2010 ELMO Shortlist, 2
Let $a,b,c$ be positive reals. Prove that
\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]
[i]Calvin Deng.[/i]
1970 Swedish Mathematical Competition, 6
Show that $\frac{(n - m)!}{m!} \le \left(\frac{n}{2} + \frac{1}{2}\right)^{n-2m}$ for positive integers $m, n$ with $2m \le n$.