Found problems: 85335
2004 Belarusian National Olympiad, 8
Tom Sawyer must whitewash a circular fence consisting of $N$ planks. He whitewashes the fence going clockwise and following the rule: He whitewashes the first plank, skips two planks, whitewashes one, skips three, and so on. Some planks may be whitewashed several times. Tom believes that all planks will be whitewashed sooner or later, but aunt Polly is sure that some planks will remain unwhitewashed forever. Prove that Tom is right if $N$ is a power of two, otherwise
aunt Polly is right.
1994 Taiwan National Olympiad, 4
Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers.
[i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.
1987 IMO Longlists, 34
(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$
(b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$
[i]Proposed by Hungary.[/i]
1990 Greece National Olympiad, 2
Let $ACBD$ be a asquare and $K,L,M,N$ be points of $AB,BC,CD,DA$ respectively. If $O$ is the center of the square , prove that the expression $$ \overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK}$$
is independent of positions of $K,L,M,N$, (i.e. is constant )
2020 CHMMC Winter (2020-21), 5
Thanos establishes $5$ settlements on a remote planet, randomly choosing one of them to stay in, and then he randomly builds a system of roads between these settlements such that each settlement has exactly one outgoing (unidirectional) road to another settlement. Afterwards, the Avengers randomly choose one of the $5$ settlements to teleport to. Then, they (the Avengers) must use the system of roads to travel from one settlement to another. The probability that the Avengers can find Thanos can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$.
2003 Junior Balkan Team Selection Tests - Romania, 4
Let $E$ be the midpoint of the side $CD$ of a square $ABCD$. Consider the point $M$ inside the square such that $\angle MAB = \angle MBC = \angle BME = x$. Find the angle $x$.
2007 District Olympiad, 4
The points of a circle are colored in green and yellow, such that every equilateral triangle inscribed in the circle has exactly 2 vertices colored in yellow. Prove that there exist a square inscribed in the circle which has at least 3 vertices colored in yellow.
2023 Romania National Olympiad, 3
Determine all natural numbers $m$ and $n$ such that
\[
n \cdot (n + 1) = 3^m + s(n) + 1182,
\]
where $s(n)$ represents the sum of the digits of the natural number $n$.
2014 Turkey Junior National Olympiad, 2
Determine the minimum possible amount of distinct prime divisors of $19^{4n}+4$, for a positive integer $n$.
2013 National Olympiad First Round, 14
Let $d(n)$ be the number of positive integers that divide the integer $n$. For all positive integral divisors $k$ of $64800$, what is the sum of numbers $d(k)$?
$
\textbf{(A)}\ 1440
\qquad\textbf{(B)}\ 1650
\qquad\textbf{(C)}\ 1890
\qquad\textbf{(D)}\ 2010
\qquad\textbf{(E)}\ \text{None of above}
$
2019-2020 Winter SDPC, 7
Let $a,b$ be positive integers. Find, with proof, the maximum possible value of $a\lceil b\lambda \rceil - b \lfloor a \lambda \rfloor$ for irrational $\lambda$.
2009 AMC 12/AHSME, 16
Trapezoid $ ABCD$ has $ AD\parallel{}BC$, $ BD \equal{} 1$, $ \angle DBA \equal{} 23^{\circ}$, and $ \angle BDC \equal{} 46^{\circ}$. The ratio $ BC: AD$ is $ 9: 5$. What is $ CD$?
$ \textbf{(A)}\ \frac {7}{9}\qquad \textbf{(B)}\ \frac {4}{5}\qquad \textbf{(C)}\ \frac {13}{15} \qquad \textbf{(D)}\ \frac {8}{9}\qquad \textbf{(E)}\ \frac {14}{15}$
Ukrainian TYM Qualifying - geometry, VII.12
Let $a, b$, and $c$ be the lengths of the sides of an arbitrary triangle, and let $\alpha,\beta$, and $\gamma$ be the radian measures of its corresponding angles. Prove that $$ \frac{\pi}{3}\le \frac{\alpha a +\beta b + \gamma c}{a+b+c} < \frac{\pi}{2}.$$ Suggest spatial analogues of this inequality.
2014 Kyiv Mathematical Festival, 4a
a) Prove that for every positive integer $y$ the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=x(x+1)$ holds for infinitely many positive integers $x.$
b) Prove that there exists positive integer $y$ such that the equality ${\rm lcm}(x,y+1)\cdot {\rm lcm}(x+1,y)=y(y+1)$ holds for at least 2014 positive integers $x.$
1996 Miklós Schweitzer, 10
Let $Y_1 , ..., Y_n$ be exchangeable random variables, ie for all permutations $\pi$ , the distribution of $(Y_{\pi (1)}, \dots, Y_{\pi (n)} )$ is equal to the distribution of $(Y_1 , ..., Y_n)$. Let $S_0 = 0$ and
$$S_j = \sum_{i = 1}^j Y_i \qquad j = 1,\dots,n$$
Denote $S_{(0)} , ..., S_{(n)}$ by the ordered statistics formed by the random variables $S_0 , ..., S_n$. Show that the distribution of $S_{(j)}$ is equal to the distribution of $\max_{0 \le i \le j} S_i + \min_ {0 \le i \le n-j} (S_{j + i} -S_j)$.
2019 Balkan MO Shortlist, G2
Let be a triangle $\triangle ABC$ with $m(\angle ABC) = 75^{\circ}$ and $m(\angle ACB) = 45^{\circ}$. The angle bisector of $\angle CAB$ intersects $CB$ at point $D$. We consider the point $E \in (AB)$, such that $DE = DC$. Let $P$ be the intersection of lines $AD$ and $CE$. Prove that $P$ is the midpoint of segment $AD$.
MOAA Gunga Bowls, 2023.20
Big Bad Brandon is assigning groups of his Big Bad Burglars to attack 7 different towers. Each Burglar can only belong to one attack group and Brandon takes over a tower if the number of Burglars attacking the tower strictly exceeds the number of knights guarding it. He knows there the total number of knights guarding the towers is 99 but does not know the exact number of knights guarding each tower. What is the minimum number of Burglars that Brandon needs to guarantee he can take over at least 4 of the 7 towers?
[i]Proposed by Eric Wang[/i]
1985 Greece National Olympiad, 1
Find all arcs $\theta$ such that $\frac{1}{\sin ^2 \theta}, \frac{1}{\cos ^2 \theta} $ are integer numbers and roots of equation $$x^2-ax+a=0.$$
2009 IMO, 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP \equal{} OQ.$
[i]Proposed by Sergei Berlov, Russia [/i]
2016 Thailand TSTST, 2
Let $\omega$ be a circle touching two parallel lines $\ell_1, \ell_2$, $\omega_1$ a circle touching $\ell_1$ at $A$ and $\omega$ externally at $C$, and $\omega_2$ a circle touching $\ell_2$ at $B$, $\omega$ externally at $D$, and $\omega_1$ externally at $E$. Prove that $AD, BC$ intersect at the circumcenter of $\vartriangle CDE$.
2012 Bogdan Stan, 3
Find the real numbers $ x,y,z $ that satisfy the following:
$ \text{(i)} -2\le x\le y\le z $
$ \text{(ii)} x+y+z=2/3 $
$ \text{(iii)} \frac{1}{x^2} +\frac{1}{y^2} +\frac{1}{z^2} =\frac{1}{x} +\frac{1}{y} +\frac{1}{z} +\frac{3}{8} $
[i]Cristinel Mortici[/i]
2005 ISI B.Stat Entrance Exam, 5
Consider an acute angled triangle $PQR$ such that $C,I$ and $O$ are the circumcentre, incentre and orthocentre respectively. Suppose $\angle QCR, \angle QIR$ and $\angle QOR$, measured in degrees, are $\alpha, \beta$ and $\gamma$ respectively. Show that \[\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}>\frac{1}{45}\]
1985 AMC 12/AHSME, 23
If \[x \equal{} \frac { \minus{} 1 \plus{} i\sqrt3}{2}\qquad\text{and}\qquad y \equal{} \frac { \minus{} 1 \minus{} i\sqrt3}{2},\] where $ i^2 \equal{} \minus{} 1$, then which of the following is [i]not[/i] correct?
$ \textbf{(A)}\ x^5 \plus{} y^5 \equal{} \minus{} 1 \qquad \textbf{(B)}\ x^7 \plus{} y^7 \equal{} \minus{} 1 \qquad \textbf{(C)}\ x^9 \plus{} y^9 \equal{} \minus{} 1$
$ \textbf{(D)}\ x^{11} \plus{} y^{11} \equal{} \minus{} 1 \qquad \textbf{(E)}\ x^{13} \plus{} y^{13} \equal{} \minus{} 1$
2000 Korea Junior Math Olympiad, 7
$ABC$ is a triangle that $2\angle B < \angle A <90^{\circ}$, and $P$ is a point on $AB$ satisfying $\angle A=2\angle APC$. If $BC=a$, $AC=b$, $BP=1$, express $AP$ as a function of $a, b$.
2014 IMO Shortlist, N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]