Found problems: 85335
1976 AMC 12/AHSME, 15
If $r$ is the remainder when each of the numbers $1059,~1417,$ and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals
$\textbf{(A) }1\qquad\textbf{(B) }15\qquad\textbf{(C) }179\qquad\textbf{(D) }d-15\qquad \textbf{(E) }d-1$
2002 IMO Shortlist, 7
Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?
2009 JBMO Shortlist, 3
Parallelogram ${ABCD}$with obtuse angle $\angle ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$, we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear. Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects the straight line ${BD}$ at point ${P}$. Prove that ${PC}$is the bisector of the angle $\angle BP{D}'$.
2021 Durer Math Competition Finals, 1
In Sixcountry there are $ 12$ months, but each month consists of $6$ weeks. The month are named the same way we do, from January to December, but in each month the weeks have different lengths. In the $k$-th month the weeks consist of $6^{k-1}$ days. What is the number of days of the spring (March, April, May together)?
2004 Romania Team Selection Test, 12
Let $n\geq 2$ be an integer and let $a_1,a_2,\ldots,a_n$ be real numbers. Prove that for any non-empty subset $S\subset \{1,2,3,\ldots, n\}$ we have
\[ \left( \sum_{i \in S} a_i \right)^2 \leq \sum_{1\leq i \leq j \leq n } (a_i + \cdots + a_j ) ^2 . \]
[i]Gabriel Dospinescu[/i]
2005 May Olympiad, 5
The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this:
[img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img]
The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .
2022 Saudi Arabia JBMO TST, 3
Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ at points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.
2013 India PRMO, 11
Three real numbers $x, y, z$ are such that $x^2 + 6y = -17, y^2 + 4z = 1$ and $z^2 + 2x = 2$. What is the value of $x^2 + y^2 + z^2$?
2022 Iberoamerican, 2
Let $S=\{13, 133, \cdots\}$ be the set of the positive integers of the form $133 \cdots 3$. Consider a horizontal row of $2022$ cells. Ana and Borja play a game: they alternatively write a digit on the leftmost empty cell, starting with Ana. When the row is filled, the digits are read from left to right to obtain a $2022$-digit number $N$. Borja wins if $N$ is divisible by a number in $S$, otherwise Ana wins. Find which player has a winning strategy and describe it.
1986 China Team Selection Test, 1
If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.
2022 Thailand TST, 2
Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$
2019 Greece Junior Math Olympiad, 3
Determine all positive integers equal to 13 times the sum of their digits.
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 $X_1|X_1+X_2+X_3=k$.
$\textbf{(A)}~\operatorname{Uniform}(0,k)$
$\textbf{(B)}~\operatorname{Uniform}\left(0,\frac k3\right)$
$\textbf{(C)}~\operatorname{Uniform}\left(0,\frac{2k}3\right)$
$\textbf{(D)}~\text{None of the above}$