Found problems: 6530
1998 IMC, 2
$S$ ist the set of all cubic polynomials $f$ with $|f(\pm 1)| \leq 1$ and $|f(\pm \frac{1}{2})| \leq 1$. Find $\sup_{f \in S} \max_{-1 \leq x \leq 1} |f''(x)|$ and all members of $f$ which give equality.
2013 IMC, 1
Let $\displaystyle{A}$ and $\displaystyle{B}$ be real symmetric matrixes with all eigenvalues strictly greater than $\displaystyle{1}$. Let $\displaystyle{\lambda }$ be a real eigenvalue of matrix $\displaystyle{{\rm A}{\rm B}}$. Prove that $\displaystyle{\left| \lambda \right| > 1}$.
[i]Proposed by Pavel Kozhevnikov, MIPT, Moscow.[/i]
1997 Taiwan National Olympiad, 8
Let $O$ be the circumcenter and $R$ be the circumradius of an acute triangle $ABC$. Let $AO$ meet the circumcircle of $OBC$ again at $D$, $BO$ meet the circumcircle of $OCA$ again at $E$, and $CO$ meet the circumcircle of $OAB$ again at $F$. Show that $OD.OE.OF\geq 8R^{3}$.
1997 Baltic Way, 14
In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.
1979 IMO Shortlist, 13
Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$
2000 Romania National Olympiad, 1
For the real numbers $a, b, c, d$, the following inequalities hold:
$$a + b + c \le 3d, \,\,\, b + c + d \le 3a, \,\,\,c + d + a \le 3b, \,\,\,d + a + b\le 3c.$$
Compare the numbers $a, b, c, d$.
2009 Costa Rica - Final Round, 1
Let $ x$ and $ y$ positive real numbers such that $ (1\plus{}x)(1\plus{}y)\equal{}2$. Show that $ xy\plus{}\frac{1}{xy}\geq\ 6$
2009 Tuymaada Olympiad, 4
The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50.
[i]Proposed by A. Khabrov[/i]
2014 Contests, 3
Positive real numbers $a, b, c$ satisfy $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.$ Prove the inequality \[\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.\]
1963 Putnam, B5
Let $(a_n )$ be a sequence of real numbers satisfying the inequalities
$$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$
and such that the series
$$\sum_{n=0}^{\infty} a_n $$
converges. Prove that
$$\lim_{n\to \infty} n a_n = 0.$$
2014 Contests, 4
Let $a,b,c$ be real numbers such that $a+b+c = 4$ and $a,b,c > 1$. Prove that:
\[\frac 1{a-1} + \frac 1{b-1} + \frac 1{c-1} \ge \frac 8{a+b} + \frac 8{b+c} + \frac 8{c+a}\]
I Soros Olympiad 1994-95 (Rus + Ukr), 10.7
Without using a calculator, prove that $$2^{1995} >5^{854},$$
2014 Romania Team Selection Test, 3
Determine the smallest real constant $c$ such that
\[\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2\]
for all positive integers $n$ and all positive real numbers $x_1,\cdots ,x_n$.
2004 Iran MO (3rd Round), 7
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$.
Find the smallest k that:
$S(F) \leq k.P(F)^2$
2019 Kyiv Mathematical Festival, 2
Let $a,b,c>0$ and $abc\ge1.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$
2013 JBMO TST - Turkey, 5
Let $a, b, c ,d$ be real numbers greater than $1$ and $x, y$ be real numbers such that
\[ a^x+b^y = (a^2+b^2)^x \quad \text{and} \quad c^x+d^y = 2^y(cd)^{y/2} \]
Prove that $x<y$.
2015 Federal Competition For Advanced Students, P2, 4
Let $x,y,z$ be positive real numbers with $x+y+z \ge 3$. Prove that
$\frac{1}{x+y+z^2} + \frac{1}{y+z+x^2} + \frac{1}{z+x+y^2} \le 1$
When does equality hold?
(Karl Czakler)
1999 Polish MO Finals, 2
Prove that for any $ 2n$ real numbers $ a_{1}$, $ a_{2}$, ..., $ a_{n}$, $ b_{1}$, $ b_{2}$, ..., $ b_{n}$, we have $ \sum_{i < j}{\left|a_{i}\minus{}a_{j}\right|}\plus{}\sum_{i < j}{\left|b_{i}\minus{}b_{j}\right|}\leq\sum_{i,j\in\left[1,n\right]}{\left|a_{i}\minus{}b_{j}\right|}$.
1995 Israel Mathematical Olympiad, 5
Let $n$ be an odd positive integer and let $x_1,x_2,...,x_n$ be n distinct real numbers that satisfy $|x_i -x_j| \le 1$ for $1 \le i < j \le n$. Prove that
$$\sum_{i<j} |x_i -x_j| \le \left[\frac{n}{2} \right] \left(\left[\frac{n}{2} \right]-1 \right)$$
2009 All-Russian Olympiad, 5
Prove that \[ \log_ab\plus{}\log_bc\plus{}\log_ca\le \log_ba\plus{}\log_cb\plus{}\log_ac\] for all $ 1<a\le b\le c$.
2011 Turkey Junior National Olympiad, 1
Show that
\[1 \leq \frac{(x+y)(x^3+y^3)}{(x^2+y^2)^2} \leq \frac98\]
holds for all positive real numbers $x,y$.
1978 Swedish Mathematical Competition, 1
Let $a,b,c,d$ be real numbers such that $a>b>c>d\geq 0$ and $a + d = b + c$. Show that
\[
x^a + x^d \geq x^b + x^c
\]
for $x>0$.
2015 Azerbaijan JBMO TST, 1
With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]
1970 IMO Longlists, 19
Let $1<n\in\mathbb{N}$ and $1\le a\in\mathbb{R}$ and there are $n$ number of $x_i, i\in\mathbb{N}, 1\le i\le n$ such that $x_1=1$ and $\frac{x_{i}}{x_{i-1}}=a+\alpha _ i$ for $2\le i\le n$, where $\alpha _i\le \frac{1}{i(i+1)}$. Prove that $\sqrt[n-1]{x_n}< a+\frac{1}{n-1}$.
2018 Switzerland - Final Round, 9
Let $n$ be a positive integer and let $G$ be the set of points $(x, y)$ in the plane such that $x$ and $y$ are integers with $1 \leq x, y \leq n$. A subset of $G$ is called [i]parallelogram-free[/i] if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of $G$ can have?