Found problems: 6530
2008 Iran MO (3rd Round), 8
In an old script found in ruins of Perspolis is written:
[code]
This script has been finished in a year whose 13th power is
258145266804692077858261512663
You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.[/code]
Find the year the script is finished. Give a reason for your answer.
I Soros Olympiad 1994-95 (Rus + Ukr), 11.1
Without using a calculator, prove that
$$2^{1995} > 5^{856}$$
2006 CHKMO, 2
Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.
2017 Taiwan TST Round 3, 1
There are $m$ real numbers $x_i \geq 0$ ($i=1,2,...,m$), $n \geq 2$, $\sum_{i=1}^{m} x_i=S$. Prove that\\
\[
\sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S-x_i}} \geq 2,
\]
The equation holds if and only if there are exactly two of $x_i$ are equal(not equal to $0$), and the rest are equal to $0$.
1995 All-Russian Olympiad Regional Round, 10.5
Consider all quadratic functions $f(x) = ax^2 +bx+c$ with $a < b$ and $f(x) \ge 0$ for all $x$. What is the smallest possible value of the expression $\frac{a+b+c}{b-a}$?
1991 Iran MO (2nd round), 2
Triangle $ABC$ is inscribed in circle $C.$ The bisectors of the angles $A,B$ and $C$ meet the circle $C$ again at the points $A', B', C'$. Let $I$ be the incenter of $ABC,$ prove that
\[\frac{IA'}{IA} + \frac{IB'}{IB}+\frac{IC'}{IC} \geq 3\]\[, IA'+IB'+IC' \geq IA+IB+IC\]
2021-IMOC, A7
For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that
$$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$
1978 Putnam, A5
Let $0 < x_i < \pi$ for $i=1,2,\ldots, n$ and set
$$x= \frac{ x_1 +x_2 + \ldots+ x_n }{n}.$$
Prove that
$$ \prod_{i=1}^{n} \frac{ \sin x_i }{x_i } \leq \left( \frac{ \sin x}{x}\right)^{n}.$$
1998 China National Olympiad, 2
Let $D$ be a point inside acute triangle $ABC$ satisfying the condition
\[DA\cdot DB\cdot AB+DB\cdot DC\cdot BC+DC\cdot DA\cdot CA=AB\cdot BC\cdot CA.\]
Determine (with proof) the geometric position of point $D$.
2015 AIME Problems, 8
Let $a$ and $b$ be positive integers satisfying $\frac{ab+1}{a+b}<\frac{3}{2}$. The maximum possible value of $\frac{a^3b^3+1}{a^3+b^3}$ is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2007 Moldova National Olympiad, 11.5
Real numbers $a_{1},a_{2},\dots,a_{n}$ satisfy $a_{i}\geq\frac{1}{i}$, for all $i=\overline{1,n}$. Prove the inequality:
\[\left(a_{1}+1\right)\left(a_{2}+\frac{1}{2}\right)\cdot\dots\cdot\left(a_{n}+\frac{1}{n}\right)\geq\frac{2^{n}}{(n+1)!}(1+a_{1}+2a_{2}+\dots+na_{n}).\]
1980 All Soviet Union Mathematical Olympiad, 299
Let the edges of rectangular parallelepiped be $x,y$ and $z$ ($x<y<z$). Let
$$p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2}$$ be its perimeter, surface area and diagonal length, respectively. Prove that $$x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )$$
2018 Nepal National Olympiad, 2b
[b]Problem Section #2
b) Find the maximal value of $(x^3+1)(y^3+1)$, where $x,y \in \mathbb{R}$, $x+y=1$.
2012 AMC 10, 21
Four distinct points are arranged in a plane so that the segments connecting them has lengths $a,a,a,a,2a,$ and $b$. What is the ratio of $b$ to $a$?
$ \textbf{(A)}\ \sqrt{3}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \sqrt{5}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \pi $
2013 Singapore Senior Math Olympiad, 3
Let $b_1,b_2,... $ be a sequence of positive real numbers such that for each $ n\ge 1$, $$b_{n+1}^2 \ge \frac{b_1^2}{1^3}+\frac{b_2^2}{2^3}+...+\frac{b_n^2}{n^3}$$
Show that there is a positive integer $M$ such that $$\sum_{n=1}^M \frac{b_{n+1}}{b_1+b_2+...+b_n} > \frac{2013}{1013}$$
2015 Bosnia And Herzegovina - Regional Olympiad, 1
Let $a$, $b$, $c$ and $d$ be real numbers such that $a+b+c+d=8$. Prove the inequality:
$$\frac{a}{\sqrt[3]{8+b-d}}+\frac{b}{\sqrt[3]{8+c-a}}+\frac{c}{\sqrt[3]{8+d-b}}+\frac{d}{\sqrt[3]{8+a-c}} \geq 4$$
2001 Abels Math Contest (Norwegian MO), 3b
The diagonals $AC$ and $BD$ in the convex quadrilateral $ABCD$ intersect in $S$. Let $F_1$ and $F_2$ be the areas of $\vartriangle ABS$ and $\vartriangle CSD$. and let $F$ be the area of the quadrilateral $ABCD$. Show that $\sqrt{ F_1 }+\sqrt{ F_2}\le \sqrt{ F}$
2009 IMAC Arhimede, 1
Prove for the sidelengths $a,b,c$ of a triangle $ABC$ the inequality $\frac{a^3}{b+c-a}+\frac{b^3}{c+a-b}+\frac{c^3}{a+b-c}\ge a^2+b^2+c^2$
2002 China Team Selection Test, 2
$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB \equal{} c$, $ AC \equal{} b$, $ BC \equal{} a$ and $ AC_1 \equal{} 2t AB$, $ BA_1 \equal{} 2rBC$, $ CB_1 \equal{} 2 \mu AC$. Prove that:
\[ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 \minus{} 2t} \right)^2 \plus{} \frac {b^2}{c^2} \cdot \left( \frac {r}{1 \minus{} 2r} \right)^2 \plus{} \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 \minus{} 2\mu} \right)^2 \plus{} 16tr \mu \geq 1
\]
2009 Jozsef Wildt International Math Competition, W. 29
Prove that for all triangle $\triangle ABC$ holds the following inequality $$\sum \limits_{cyc} \left (1-\sqrt{\sqrt{3}\tan \frac{A}{2}}+\sqrt{3}\tan \frac{A}{2}\right )\left (1-\sqrt{\sqrt{3}\tan \frac{B}{2}}+\sqrt{3}\tan \frac{B}{2}\right )\geq 3$$
2005 Putnam, A4
Let $H$ be an $n\times n$ matrix all of whose entries are $\pm1$ and whose rows are mutually orthogonal. Suppose $H$ has an $a\times b$ submatrix whose entries are all $1.$ Show that $ab\le n.$
2008 Federal Competition For Advanced Students, Part 2, 1
Prove the inequality
\[ \sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3}
\]
holds for all positive real numbers $ a$, $ b$ and $ c$ with $ a \plus{} b \plus{} c \equal{} 1$.
1997 Greece National Olympiad, 4
A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.
2014 Turkey Team Selection Test, 3
Prove that for all all non-negative real numbers $a,b,c$ with $a^2+b^2+c^2=1$
\[\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.\]
2011 Ukraine Team Selection Test, 1
Given a right $ n $ -angle $ {{A} _ {1}} {{A} _ {2}} \ldots {{A} _ {n}} $, $n \ge 4 $, and a point $ M $ inside it. Prove the inequality $$\sin (\angle {{A} _ {1}} M {{A} _ {2}}) + \sin (\angle {{A} _ {2}} M {{A} _ {3}} ) + \ldots + \sin (\angle {{A} _ {n}} M {{A} _ {1}})> \sin \frac{2 \pi}{n} + (n-2) sin \frac{\pi}{n}$$