Found problems: 85335
1984 Iran MO (2nd round), 3
Let $f : \mathbb R \to \mathbb R$ be a function such that
\[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\]
Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$
2008 Sharygin Geometry Olympiad, 20
(A.Zaslavsky, 10--11) a) Some polygon has the following property:
if a line passes through two points which bisect its perimeter then this line bisects the area of the polygon. Is it true that the polygon is central symmetric?
b) Is it true that any figure with the property from part a) is central symmetric?
2020 Germany Team Selection Test, 2
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
2020 APMO, 3
Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.
2023 CCA Math Bonanza, TB1
$\text{Find }\left(\sum_{k=1}^{2023}{(k^{42432})}\right)\text{ mod 2023}$
[i]Tiebreaker #1[/i]
2019 BMT Spring, 1
Find the maximum integral value of $k$ such that $0 \le k \le 2019$ and $|e^{2\pi i \frac{k}{2019}} - 1|$ is maximal.
1996 May Olympiad, 1
Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .
2016 Harvard-MIT Mathematics Tournament, 9
The incircle of a triangle $ABC$ is tangent to $BC$ at $D$.
Let $H$ and $\Gamma$ denote the orthocenter and circumcircle of $\triangle ABC$.
The \emph{$B$-mixtilinear incircle}, centered at $O_B$,
is tangent to lines $BA$ and $BC$ and internally tangent to $\Gamma$.
The \emph{$C$-mixtilinear incircle}, centered at $O_C$, is defined similarly.
Suppose that $\overline{DH} \perp \overline{O_BO_C}$, $AB = \sqrt3$ and $AC = 2$. Find $BC$.
2017 Princeton University Math Competition, 14
Eric rolls a ten-sided die (with sides labeled $1$ through $10$) repeatedly until it lands on $3, 5$, or $7$. Conditional on all of Eric’s rolls being odd, the expected number of rolls can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2009 Stanford Mathematics Tournament, 2
The pattern in the fi gure below continues inward in finitely. The base of the biggest triangle is 1. All triangles are equilateral. Find the shaded area.
[asy]
defaultpen(linewidth(0.8));
pen blu = rgb(0,112,191);
real r=sqrt(3);
fill((8,0)--(0,8r)--(-8,0)--cycle, blu);
fill(origin--(4,4r)--(-4,4r)--cycle, white);
fill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);
fill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]
1964 Dutch Mathematical Olympiad, 1
Given a triangle $ABC$, $\angle C= 60^o$. Construct a point $P$ on the side $AC$ and a point $Q$ on side $BC$ such that $ABQP$ is a trapezoid whose diagonals make an angle of $60^o$ with each other.
2017 Iran MO (3rd round), 1
Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then
$$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$
is not an integer.
2019 LIMIT Category C, Problem 11
Let $X_1,X_2,X_3$ be $\exp(1)$. Find the conditional distribution of $X_1|X_1+X_2+X_3=k$.
$\textbf{(A)}~\operatorname{Uniform}(0,k)$
$\textbf{(B)}~\operatorname{Uniform}\left(0,\frac k3\right)$
$\textbf{(C)}~\operatorname{Uniform}\left(0,\frac{2k}3\right)$
$\textbf{(D)}~\text{None of the above}$
1995 Baltic Way, 11
In how many ways can the set of integers $\{1,2,\ldots ,1995\}$ be partitioned into three non-empty sets so that none of these sets contains any pair of consecutive integers?
1986 IMO Longlists, 56
Let $A_1A_2A_3A_4A_5A_6$ be a hexagon inscribed into a circle with center $O$. Consider the circular arc with endpoints $A_1,A_6$ not containing $A_2$. For any point $M$ of that arc denote by $h_i$ the distance from $M$ to the line $A_iA_{i+1} \ (1 \leq i \leq 5)$. Construct $M$ such that the sum $h_1 + \cdots + h_5$ is maximal.
1996 Turkey MO (2nd round), 2
Prove that $\prod\limits_{k=0}^{n-1}{({{2}^{n}}-{{2}^{k}})}$ is divisible by $n!$ for all positive integers $n$.
2016 BMT Spring, 5
Find
$$\frac{\tan 1^o}{1 + \tan 1^o }+\frac{\tan 2^o}{1 + \tan 2^o } + ... + \frac{\tan 89^o}{1 + \tan 89^o}$$
2020 USMCA, 22
Kelvin the Frog places $40$ rooks on a uniformly random subset of $40$ squares of a $20 \times 20$ chessboard. Then, Alex the Kat chooses two of the $40$ rooks uniformly at random. What is the probability that Alex's two rooks attack each other? Two rooks attack each other if they are on the same row or column, and no piece stands between them.
2014 NIMO Summer Contest, 6
Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
[i]Proposed by Lewis Chen[/i]
1999 All-Russian Olympiad, 2
There are several cities in a country. Some pairs of the cities are connected by a two-way airline of one of the $N$ companies, so that each company serves exactly one airline from each city, and one can travel between any two cities, possibly with transfers. During a financial crisis, $N-1$ airlines have been canceled, all from different companies. Prove that it is still possible to travel between any two cities.
1991 Arnold's Trivium, 68
Find
\[\inf\iint_{x^2+y^2\le1}\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial u}{\partial y}\right)^2dxdy\]
1969 IMO Longlists, 34
$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$
2020 Caucasus Mathematical Olympiad, 1
Determine if there exists a finite set $A$ of positive integers satisfying the following condition: for each $a\in{A}$ at least one of two numbers $2a$ and
$\frac{a}{3}$ belongs to $A$.
2009 IMO Shortlist, 6
Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$.
[i]Proposed by Okan Tekman, Turkey[/i]
2017 Ukraine Team Selection Test, 12
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]