Found problems: 6530
2012 Grigore Moisil Intercounty, 2
[b]a)[/b] Prove that
$$ k+\frac{1}{2}-\frac{1}{8k}<\sqrt{k^2+k}<k+\frac{1}{2}-\frac{1}{8k}+\frac{1}{16k^2} , $$
for any natural number $ k. $
[b]b)[/b] Prove that there exists four numbers $ \alpha,\beta,\gamma,\delta\in\{0,1,2,3,4,5,6,7,8,9\} $ such that
$$ \left\lfloor\sum_{k=1}^{2012} \sqrt{k(k+1)\left( k^2+k+1 \right)}\right\rfloor =\underbrace{\ldots\alpha \beta\gamma\delta}_{\text{decimal form}} $$
and $ \alpha +\delta =\gamma . $
2024-IMOC, A1
Given a positive integer $N$. Prove that
\[\sum_{m=1}^N \sum_{n=1}^N \frac{1}{mn^2+m^2n+2mn}<\frac{7}{4}.\]
[i]Proposed by tan-1[/i]
2005 VJIMC, Problem 2
Let $(a_{i,j})^n_{i,j=1}$ be a real matrix such that $a_{i,i}=0$ for $i=1,2,\ldots,n$. Prove that there exists a set $\mathcal J\subset\{1,2,\ldots,n\}$ of indices such that
$$\sum_{\begin{smallmatrix}i\in\mathcal J\\j\notin\mathcal J\end{smallmatrix}}a_{i,j}+\sum_{\begin{smallmatrix}i\notin\mathcal J\\j\in\mathcal J\end{smallmatrix}}a_{i,j}\ge\frac12\sum_{i,j=1}^na_{i,j}.$$
1988 India National Olympiad, 4
If $ a$ and $ b$ are positive and $ a \plus{} b \equal{} 1$, prove that
\[ \left(a\plus{}\frac{1}{a}\right)^2\plus{}\left(b\plus{}\frac{1}{b}\right)^2 \geq \frac{25}{2}\]
2016 China Second Round Olympiad, 1
Let $a_1, a_2, \ldots, a_{2016}$ be real numbers such that $9a_i\ge 11a^2_{i+1}$ $(i=,2,\cdots,2015)$.
Find the maximum value of $(a_1-a^2_2)(a_2-a^2_3)\cdots (a_{2015}-a^2_{2016})(a_{2016}-a^2_{1}).$
2014 Czech-Polish-Slovak Junior Match, 6
Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$
if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.
2011 Iran Team Selection Test, 10
Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality
\[ \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\ \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array}\]
holds.
2005 Unirea, 4
$a>0$ $f:[-a,a]\rightarrow R$ such that $f''$ exist and Riemann-integrable
suppose $f(a)=f(-a)$
$ f'(-a)=f'(a)=a^2$
Prove that $6a^3\leq \int_{-a}^{a}{f''(x)}^2dx$
Study equality case ?
Radu Miculescu
1966 IMO Longlists, 5
Prove the inequality
\[\tan \frac{\pi \sin x}{4\sin \alpha} + \tan \frac{\pi \cos x}{4\cos \alpha} >1\]
for any $x, \alpha$ with $0 \leq x \leq \frac{\pi }{2}$ and $\frac{\pi}{6} < \alpha < \frac{\pi}{3}.$
2009 Today's Calculation Of Integral, 490
For a positive real number $ a > 1$, prove the following inequality.
$ \frac {1}{a \minus{} 1}\left(1 \minus{} \frac {\ln a}{a\minus{}1}\right) < \int_0^1 \frac {x}{a^x}\ dx < \frac {1}{\ln a}\left\{1 \minus{} \frac {\ln (\ln a \plus{} 1)}{\ln a}\right\}$
2019 Polish Junior MO First Round, 3
The integers $a, b, c$ are not $0$ such that $\frac{a}{b + c^2}=\frac{a + c^2}{b}$. Prove that $a + b + c \le 0$.
2002 India Regional Mathematical Olympiad, 6
Prove that for any natural number $n > 1$, \[ \frac{1}{2} < \frac{1}{n^2+1} + \frac{2}{n^2 +2} + \ldots + \frac{n}{n^2 + n} < \frac{1}{2} + \frac{1}{2n}. \]
Russian TST 2016, P2
Let $x,y,z{}$ be positive real numbers. Prove that \[(xy+yz+zx)\left(\frac{1}{x^2+y^2}+\frac{1}{y^2+z^2}+\frac{1}{z^2+x^2}\right)>\frac{5}{2}.\]
2015 Chile TST Ibero, 4
Let $x, y \in \mathbb{R}^+$. Prove that:
\[
\left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2.
\]
1982 Swedish Mathematical Competition, 6
Show that
\[
(2a-1) \sin x + (1-a) \sin(1-a)x \geq 0
\]
for $0 \leq a \leq 1$ and $0 \leq x \leq \pi$.
1989 All Soviet Union Mathematical Olympiad, 507
Find the least possible value of $(x + y)(y + z)$ for positive reals satisfying $(x + y + z) xyz = 1$.
2010 Germany Team Selection Test, 2
Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too.
[i]Proposed by Mirsaleh Bahavarnia, Iran[/i]
2019 Singapore Junior Math Olympiad, 4
Let $a>b>0$. Prove that $\sqrt2 a^3+ \frac{3}{ab-b^2}\ge 10$
When does equality hold?
2018 JBMO Shortlist, A6
For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove:
$ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
1972 Miklós Schweitzer, 7
Let $ f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $ \mathbb{R}^3$ for which the inequality $ f(x_0,0,0) \leq \varepsilon^2$ holds for some $ 0\leq x_0 \leq 1$ and $ 0<\varepsilon<(1\minus{}x_0)^2$. Prove that $ f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $ (1\minus{}3\varepsilon^{1/4}).$
[i]P. Turan[/i]
2009 Poland - Second Round, 1
Let $a_1\ge a_2\ge \ldots \ge a_n>0$ be $n$ reals. Prove the inequality
\[a_1a_2\ldots a_{n-1}+(2a_2-a_1)(2a_3-a_2)\ldots (2a_n-a_{n-1})\ge 2a_2a_3\ldots a_n\]
2024 CMI B.Sc. Entrance Exam, 4
(a) For non negetive $a,b,c, r$ prove that
\[a^r(a-b)(a-c) + b^r(b-a)(b-c) + c^r (c-a)(c-b) \geq 0 \]
(b) Find an inequality for non negative $a,b,c$ with $a^4+b^4+c^4 + abc(a+b+c)$ on the greater side.
(c) Prove that if $abc = 1$ for non negative $a,b,c$, $a^4+b^4+c^4+a^3+b^3+c^3+a+b+c \geq \frac{a^2+b^2}{c}+\frac{b^2+c^2}{a}+\frac{c^2+a^2}{b}+3$
1970 Vietnam National Olympiad, 1
Prove that for an arbitrary triangle $ABC$ : $sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2} < \frac{1}{4}$.
2002 China Team Selection Test, 3
$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and:
\[ \sum_{i\equal{}1}^n x_i < \frac{1}{k\plus{}1} \left[ k \cdot \frac{n(n\plus{}1)(2n\plus{}1)}{6} \minus{} (k\plus{}1)^2 \cdot \frac{n(n\plus{}1)}{2} \right]\]
and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i \plus{} x_j \equal{} x_t \plus{} x_l$.
1993 India Regional Mathematical Olympiad, 6
If $a,b,c,d$ are four positive reals such that $abcd= 1$ , prove that $(1+a) (1+b) (1 +c ) (1 +d ) \geq 16.$