Found problems: 85335
2010 Iran MO (3rd Round), 6
In a triangle $ABC$, $\angle C=45$. $AD$ is the altitude of the triangle. $X$ is on $AD$ such that $\angle XBC=90-\angle B$ ($X$ is in the triangle). $AD$ and $CX$ cut the circumcircle of $ABC$ in $M$ and $N$ respectively. if tangent to circumcircle of $ABC$ at $M$ cuts $AN$ at $P$, prove that $P$,$B$ and $O$ are collinear.(25 points)
the exam time was 4 hours and 30 minutes.
2021 China Second Round Olympiad, Problem 14
Define the set $P=\{a_1, a_2, a_3, \cdots, a_n\}$ and its arithmetic mean $$C_p = \frac{a_1+a_2+\cdots+a_m}m.$$ If we divide $S = \{1, 2, 3, \cdots, n\}$ into two disjoint subsets $A, B$, compute the greatest possible value of $|C_A-C_B|$. For how many $(A, B)$ is equality attained?
[i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 14)[/i]
2016 BMT Spring, 10
Evaluate $$\sum^{\infty}_{k=0} \left( \frac{-1}{8}\right)^k {2k \choose k}$$
Today's calculation of integrals, 848
Evaluate $\int_0^{\frac {\pi}{4}} \frac {\sin \theta -2\ln \frac{1-\sin \theta}{\cos \theta}}{(1+\cos 2\theta)\sqrt{\ln \frac{1+\sin \theta}{\cos \theta}}}d\theta .$
2008 Saint Petersburg Mathematical Olympiad, 1
Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point.
EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet.
Note: fresh translation
2013 India Regional Mathematical Olympiad, 6
Suppose that the vertices of a regular polygon of $20$ sides are coloured with three colours - red, blue and green - such that there are exactly three red vertices. Prove that there are three vertices $A,B,C$ of the polygon having the same colour such that triangle $ABC$ is isosceles.
2005 Junior Balkan Team Selection Tests - Romania, 13
The positive integers from 1 to $n^2$ are placed arbitrarily on the $n^2$ squares of a $n\times n$ chessboard. Two squares are called [i]adjacent[/i] if they have a common side. Show that two opposite corner squares can be joined by a path of $2n-1$ adjacent squares so that the sum of the numbers placed on them is at least $\left\lfloor \frac{n^3} 2 \right\rfloor + n^2 - n + 1$.
[i]Radu Gologan[/i]
1997 Tournament Of Towns, (541) 2
$D$ and $E$ are points on the sides $BC$ and $AC$ of a triangle $ABC$ such that $AD$ and $BE$ are angle bisectors of the triangle $ABC$. If $DE$ bisects $\angle ADC$, find $\angle A$.
(SI Tokarev)
2012 All-Russian Olympiad, 3
A plane is coloured into black and white squares in a chessboard pattern. Then, all the white squares are coloured red and blue such that any two initially white squares that share a corner are different colours. (One is red and the other is blue.) Let $\ell$ be a line not parallel to the sides of any squares. For every line segment $I$ that is parallel to $\ell$, we can count the difference between the length of its red and its blue areas. Prove that for every such line $\ell$ there exists a number $C$ that exceeds all those differences that we can calculate.
1984 AMC 12/AHSME, 14
The product of all real roots of the equation $x^{\log_{10} x} = 10$ is
A. 1
B. -1
C. 10
D. $10^{-1}$
E. None of these
2020 LIMIT Category 1, 8
Kunal and Arnab play a game as follows. Initially there are $2$ piles of coins with $x$ and $y$ coins respectively. The game starts with Kunal. In each turn a player chooses one pile and removes as many coins as he wants from that pile. The game goes on and the last one to remove a coin loses. Determine all possible values of $(x,y)$ which ensure Kunal's victory against Arnab given both os them play optimally.
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[i]You are required to find an exhaustive set of solutions[/i]
2013 China Girls Math Olympiad, 7
As shown in the figure, $\odot O_1$ and $\odot O_2$ touches each other externally at a point $T$, quadrilateral $ABCD$ is inscribed in $\odot O_1$, and the lines $DA$, $CB$ are tangent to $\odot O_2$ at points $E$ and $F$ respectively. Line $BN$ bisects $\angle ABF$ and meets segment $EF$ at $N$. Line $FT$ meets the arc $\widehat{AT}$ (not passing through the point $B$) at another point $M$ different from $A$. Prove that $M$ is the circumcenter of $\triangle BCN$.
2006 IMO Shortlist, 5
If $a,b,c$ are the sides of a triangle, prove that
\[\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3 \]
[i]Proposed by Hojoo Lee, Korea[/i]
2025 Thailand Mathematical Olympiad, 9
Let $p$ be an odd prime and $S = \{1,2,3,\dots, p\}$
Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that $$B\cdot U(U(a)) - a \: \text{ is a multiple of} \: p \: \text{for all} \: a \in S$$
Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.
1997 Iran MO (2nd round), 1
Let $x_1,x_2,x_3,x_4$ be positive reals such that $x_1x_2x_3x_4=1$. Prove that:
\[ \sum_{i=1}^{4}{x_i^3}\geq\max\{ \sum_{i=1}^{4}{x_i},\sum_{i=1}^{4}{\frac{1}{x_i}} \}. \]
1912 Eotvos Mathematical Competition, 1
How many positive integers of $n$ digits exist such that each digit is $1, 2$, or $3$? How many of these contain all three of the digits $1, 2$, and $3$ at least once?
2014 USAMO, 5
Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.
2014 NIMO Problems, 3
In land of Nyemo, the unit of currency is called a [i]quack[/i]. The citizens use coins that are worth $1$, $5$, $25$, and $125$ quacks. How many ways can someone pay off $125$ quacks using these coins?
[i]Proposed by Aaron Lin[/i]
2025 Harvard-MIT Mathematics Tournament, 25
Let $ABCD$ be a trapezoid such that $AB \parallel CD, AD=13, BC=15, AB=20,$ and $CD=34.$ Point $X$ lies inside the trapezoid such that $\angle{XAB}=2\angle{XBA}$ and $\angle{XDC}=2\angle{XCD}.$ Compute $XD-XA.$
1966 IMO Longlists, 2
Given $n$ positive numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ such that $a_{1}\cdot a_{2}\cdot ...\cdot a_{n}=1.$ Prove \[ \left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.\]
2011 Paraguay Mathematical Olympiad, 2
In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AC$ and $BC$ respectively. The distance from the midpoint of $BD$ to the midpoint of $AE$ is $4.5$. What is the length of side $AB$?
2017 Brazil National Olympiad, 6.
[b]6.[/b] Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.
PEN G Problems, 3
Prove that there exist positive integers $ m$ and $ n$ such that
\[ \left\vert\frac{m^{2}}{n^{3}}\minus{}\sqrt{2001}\right\vert <\frac{1}{10^{8}}.\]
2011 Laurențiu Duican, 2
$ \lim_{n\to\infty } \int_{\pi }^{2\pi } \frac{|\sin (nx) +\cos (nx)|}{ x} dx ? $
[i]Gabriela Boeriu[/i]
2011 Romania National Olympiad, 4
Let $ f,F:\mathbb{R}\longrightarrow\mathbb{R} $ be two functions such that $ f $ is nondecreasing, $ F $ admits finite lateral derivates in every point of its domain,
$$ \lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} , $$
for all real numbers $ y, $ and $ F(0)=0. $
Prove that $ F(x)=\int_0^x f(t)dt, $ for all real numbers $ x. $