Found problems: 6530
2002 JBMO ShortLists, 3
Let $ a,b,c$ be positive real numbers such that $ abc\equal{}\frac{9}{4}$. Prove the inequality:
$ a^3 \plus{} b^3 \plus{} c^3 > a\sqrt {b \plus{} c} \plus{} b\sqrt {c \plus{} a} \plus{} c\sqrt {a \plus{} b}$
Jury's variant:
Prove the same, but with $ abc\equal{}2$
2010 Victor Vâlcovici, 2
Let $ f:[2,\infty )\rightarrow\mathbb{R} $ be a differentiable function satisfying $ f(2)=0 $ and
$$ \frac{df}{dx}=\frac{2}{x^2+f^4{x}} , $$
for any $ x\in [2,\infty ) . $ Show that there exists $ \lim_{x\to\infty } f(x) $ and is at most $ \ln 3. $
[i]Gabriel Daniilescu[/i]
2011 AMC 12/AHSME, 2
Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?
$ \textbf{(A)}\ 80 \qquad
\textbf{(B)}\ 82 \qquad
\textbf{(C)}\ 85 \qquad
\textbf{(D)}\ 90 \qquad
\textbf{(E)}\ 95 $
2023 South East Mathematical Olympiad, 4
Find the largest real number $c$, such that for any integer $s>1$, and positive integers $m, n$ coprime to $s$, we have$$ \sum_{j=1}^{s-1} \{ \frac{jm}{s} \}(1 - \{ \frac{jm}{s} \})\{ \frac{jn}{s} \}(1 - \{ \frac{jn}{s} \}) \ge cs$$
where $\{ x \} = x - \lfloor x \rfloor $.
2015 Czech-Polish-Slovak Junior Match, 5
Find the smallest real constant $p$ for which the inequality holds $\sqrt{ab}- \frac{2ab}{a + b} \le p \left( \frac{a + b}{2} -\sqrt{ab}\right)$ with any positive real numbers $a, b$.
2022 Junior Balkan Team Selection Tests - Moldova, 6
The non-negative numbers $x,y,z$ satisfy the relation $x + y+ z = 3$. Find the smallest possible numerical value and the largest possible numerical value for the expression
$$E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} .$$
2011 China National Olympiad, 2
Let $a_i,b_i,i=1,\cdots,n$ are nonnegitive numbers,and $n\ge 4$,such that $a_1+a_2+\cdots+a_n=b_1+b_2+\cdots+b_n>0$.
Find the maximum of $\frac{\sum_{i=1}^n a_i(a_i+b_i)}{\sum_{i=1}^n b_i(a_i+b_i)}$
2006 Romania Team Selection Test, 4
The real numbers $a_1,a_2,\dots,a_n$ are given such that $|a_i|\leq 1$
for all $i=1,2,\dots,n$ and
$a_1+a_2+\cdots+a_n=0$.
a) Prove that there exists $k\in\{1,2,\dots,n\}$ such that
\[ |a_1+2a_2+\cdots+ka_k|\leq\frac{2k+1}{4}. \]
b) Prove that for $n > 2$ the bound above is the best possible.
[i]Radu Gologan, Dan Schwarz[/i]
1990 Tournament Of Towns, (267) 1
Given $$a=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99}}}}, \,\,and\,\,\,
b=\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{...+\dfrac{...}{99+\dfrac{1}{100}}}}}$$
Prove that $$|a-b| <\frac{1}{99! 100!}$$
(G Galperin, Moscow)
2022 Kosovo National Mathematical Olympiad, 4
Assume that in the $\triangle ABC$ there exists a point $D$ on $BC$ and a line $l$ passing through $A$ such that $l$ is tangent to $(ADC)$ and $l$ bisects $BD.$
Prove that $a\sqrt{2}\geq b+c.$
2023 Middle European Mathematical Olympiad, 2
If $a, b, c, d>0$ and $abcd=1$, show that $$\frac{ab+1}{a+1}+\frac{bc+1}{b+1}+\frac{cd+1}{c+1}+\frac{da+1}{d+1} \geq 4. $$ When does equality hold?
1962 AMC 12/AHSME, 29
Which of the following sets of $ x$-values satisfy the inequality $ 2x^2 \plus{} x < 6?$
$ \textbf{(A)}\ \minus{} 2 < x < \frac{3}{2} \qquad \textbf{(B)}\ x > \frac32 \text{ or }x < \minus{} 2 \qquad \textbf{(C)}\ x < \frac32 \qquad \textbf{(D)}\ \frac32 < x < 2 \qquad \textbf{(E)}\ x < \minus{} 2$
1997 Iran MO (3rd Round), 2
Show that for any arbitrary triangle $ABC$, we have
\[\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.\]
2008 Bosnia And Herzegovina - Regional Olympiad, 2
For arbitrary reals $ x$, $ y$ and $ z$ prove the following inequality:
$ x^{2} \plus{} y^{2} \plus{} z^{2} \minus{} xy \minus{} yz \minus{} zx \geq \max \{\frac {3(x \minus{} y)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} \}$
1989 Tournament Of Towns, (203) 1
The positive numbers $a, b$ and $c$ satisfy $a \ge b \ge c$ and $a + b + c \le 1$ . Prove that $a^2 + 3b^2 + 5c^2 \le 1$ .
(F . L . Nazarov)
2007 Kyiv Mathematical Festival, 4
Let $a,b,c>0$ and $abc\ge1.$ Prove that
a) $\left(a+\frac{1}{a+1}\right)\left(b+\frac{1}{b+1}\right) \left(c+\frac{1}{c+1}\right)\ge\frac{27}{8}.$
b)$27(a^{3}+a^{2}+a+1)(b^{3}+b^{2}+b+1)(c^{3}+c^{2}+c+1)\ge$
$\ge 64(a^{2}+a+1)(b^{2}+b+1)(c^{2}+c+1).$
[hide="Generalization"]$n^{3}(a^{n}+\ldots+a+1)(b^{n}+\ldots+b+1)(c^{n}+\ldots+c+1)\ge$
$\ge (n+1)^{3}(a^{n-1}+\ldots+a+1)(b^{n-1}+\ldots+b+1)(c^{n-1}+\ldots+c+1),\ n\ge1.$ [/hide]
2009 China Team Selection Test, 3
Prove that for any odd prime number $ p,$ the number of positive integer $ n$ satisfying $ p|n! \plus{} 1$ is less than or equal to $ cp^\frac{2}{3}.$ where $ c$ is a constant independent of $ p.$
2002 All-Russian Olympiad, 1
For positive real numbers $a, b, c$ such that $a+b+c=3$, show that:
\[\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab+bc+ca.\]
2007 Baltic Way, 1
For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that
\[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]
2021 HMNT, 10
Real numbers $x, y, z$ satisfy $$x + xy + xyz = 1, y + yz + xyz = 2, z + xz + xyz = 4.$$
The largest possible value of $xyz$ is $\frac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, $d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d$.
2005 Moldova Team Selection Test, 1
Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that
$\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$,
where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.
2005 China Second Round Olympiad, 2
Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]
2007 District Olympiad, 2
Let $f : \left[ 0, 1 \right] \to \mathbb R$ be a continuous function and $g : \left[ 0, 1 \right] \to \left( 0, \infty \right)$.
Prove that if $f$ is increasing, then
\[\int_{0}^{t}f(x) g(x) \, dx \cdot \int_{0}^{1}g(x) \, dx \leq \int_{0}^{t}g(x) \, dx \cdot \int_{0}^{1}f(x) g(x) \, dx .\]
2002 Pan African, 6
If $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ and $a_1+a_2+\cdots+a_n=1$, then prove:
\[a_1^2+3a_2^2+5a_3^2+ \cdots +(2n-1)a_n^2 \leq 1\]
2013 Bogdan Stan, 1
Let be three real numbers $ u,v,t $ under the condition $ u+v+t=0. $ Prove that for any positive real number $ a\neq 1 $ the following inequality is true with equality only and only if $ u=v=t=0: $
$$ a^u/a^v+a^v/a^t+a^{v+t}\ge a^u+a^v+1 $$
[i]Ion Tecu[/i]