Found problems: 85335
2012 Czech-Polish-Slovak Junior Match, 4
Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.
2018 Saint Petersburg Mathematical Olympiad, 3
Point $T$ lies on the bisector of $\angle B$ of acuteangled $\triangle ABC$. Circle $S$ with diameter $BT$ intersects $AB$ and $BC$ at points $P$ and $Q$. Circle, that goes through point $A$ and tangent to $S$ at $P$ intersects line $AC$ at $X$. Circle, that goes through point $C$ and tangent to $S$ at $Q$ intersects line $AC$ at $Y$. Prove, that $TX=TY$
2007 AMC 12/AHSME, 22
For each positive integer $ n,$ let $ S(n)$ denote the sum of the digits of $ n.$ For how many values of $ n$ is $ n \plus{} S(n) \plus{} S(S(n)) \equal{} 2007?$
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2014 Paraguay Mathematical Olympiad, 5
Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$.
2017 Costa Rica - Final Round, A1
Let $P (x)$ be a polynomial of degree $2n$, such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$. Determine $P (2n + 1)$.
2001 India IMO Training Camp, 1
Let $x$ , $y$ , $z>0$. Prove that if $xyz\geq xy+yz+zx$, then $xyz \geq 3(x+
y+z)$.
2013 ELMO Shortlist, 5
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
1991 Cono Sur Olympiad, 3
It is known that the number of real solutions of the following system if finite. Prove that this system has an even number of solutions:
$(y^2+6)(x-1)=y(x^2+1)$
$(x^2+6)(y-1)=x(y^2+1)$
2007 Junior Balkan Team Selection Tests - Romania, 3
Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.
2002 Croatia National Olympiad, Problem 3
Let $f(x)=x^{2002}-x^{2001}+1$. Prove that for every positive integer $m$, the numbers $m,f(m),f(f(m)),\ldots$ are pairwise coprime.
2009 Harvard-MIT Mathematics Tournament, 3
A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines shown below, so that the folded flaps just touch at the corners as shown by the dotted lines. Find the area of the resulting trapezoid.
[asy]
size(150);
defaultpen(linewidth(0.8));
draw(origin--(8,0)--(8,5)--(0,5)--cycle,linewidth(1));
draw(origin--(8/3,5)^^(16/3,5)--(8,0),linetype("4 4"));
draw(origin--(4,3)--(8,0)^^(8/3,5)--(4,3)--(16/3,5),linetype("0 4"));
label("$5$",(0,5/2),W);
label("$8$",(4,0),S);
[/asy]
2016 Middle European Mathematical Olympiad, 1
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$
1973 Polish MO Finals, 2
Let $p_n$ denote the probability that, in $n$ tosses, a fair coin shows the head up $100$ consecutive times. Prove that the sequence $(p_n)$ converges and determine its limit.
1989 Greece Junior Math Olympiad, 4
Simplify
i) $1+\frac{2a+\dfrac{2}{a}}{a+\dfrac{1}{a}}$
ii) $\frac{3b+\dfrac{3}{b}+\dfrac{3}{b^2}}{b+\dfrac{1}{b}+\dfrac{1}{b^2}}$
iii) $\frac{\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{ab}\right)a^6b^2-a^6-a^5b}{a^4b}$
LMT Guts Rounds, 29
Let $S$ be the set of integers that represent the number of intersections of some four distinct lines in the plane. List the elements of $S$ in ascending order.
2008 Indonesia MO, 2
Prove that for $ x,y\in\mathbb{R^ \plus{} }$,
$ \frac {1}{(1 \plus{} \sqrt {x})^{2}} \plus{} \frac {1}{(1 \plus{} \sqrt {y})^{2}} \ge \frac {2}{x \plus{} y \plus{} 2}$
1993 AMC 8, 19
$(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) = $
$\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142$
2016 Serbia Additional Team Selection Test, 1
Let $P_0(x)=x^3-4x$. Sequence of polynomials is defined as following:\\
$P_{n+1}=P_n(1+x)P_n(1-x)-1$.\\
Prove that $x^{2016}|P_{2016}(x)$.
1994 Czech And Slovak Olympiad IIIA, 5
In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?
2011 Middle European Mathematical Olympiad, 7
Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.
2011 AIME Problems, 13
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integers, and $r+s+t<1000$. Find $r+s+t$.
2023 Korea Junior Math Olympiad, 5
For a positive integer $n(\geq 5)$, there are $n$ white stones and $n$ black stones (total $2n$ stones) lined up in a row. The first $n$ stones from the left are white, and the next $n$ stones are black. $$\underbrace{\Circle \Circle \cdots \Circle}_n \underbrace{\CIRCLE \CIRCLE \cdots \CIRCLE}_n $$
You can swap the stones by repeating the following operation.
[b](Operation)[/b] Choose a positive integer $k (\leq 2n - 5)$, and swap $k$-th stone and $(k+5)$-th stone from the left.
Find all positive integers $n$ such that we can make first $n$ stones to be black and the next $n$ stones to be white in finite number of operations.
2004 National High School Mathematics League, 10
$p$ is a give odd prime, if $\sqrt{k^2-pk}$ is a positive integer, then the value of positive integer $k$ is________.
2007 Princeton University Math Competition, 2
A black witch's hat is in the classic shape of a cone on top of a circular brim. The cone has a slant height of $18$ inches and a base radius of $3$ inches. The brim has a radius of $5$ inches. What is the total surface area of the hat?
2004 India IMO Training Camp, 1
Prove that in any triangle $ABC$,
\[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]