Found problems: 85335
2022 Purple Comet Problems, 7
The value of
$$\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)$$
can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $2m - n.$
Kvant 2023, M2772
7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?
India EGMO 2025 TST, 5
Let acute scalene $\Delta ABC$ have circumcircle $\omega$. Let $M$ be the midpoint of $BC$, define $X$ as the other intersection of $AM$ with $\omega$. Let $E,F$ be the feet of altitudes from $B,C$ to $AC, AB$ respectively. Let $Q$ be the second intersection of the circumcircle of $\Delta AEF$ and $\omega$. Let $Y\neq X$ be a point such that $MX=MY$ and $QMXY$ is cyclic. Finally, let $S$ be a point on $BC$ such that $\angle BAS=\angle MAC.$ Prove that the quadrilaterals $BFYS$ and $CEYS$ are cyclic.
Proposed by Kanav Talwar and Malay Mahajan
2024 Kyiv City MO Round 1, Problem 2
Let $BL, AD$ be the bisector and the altitude correspondingly of an acute triangle ABC. They intersect at point $T$. It turned out that the altitude $LK$ of $\triangle ALB$ is divided in half by the line $AD$. Prove that $KT \perp BL$.
[i]Proposed by Mariia Rozhkova[/i]
2017 Online Math Open Problems, 5
There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac mn$ for relatively prime positive integers $m,n$. Find $100m+n$
[i]Proposed by Yannick Yao[/i]
2016 Online Math Open Problems, 10
Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
[i]Proposed by Yannick Yao[/i]
2011 Romania Team Selection Test, 2
Prove that the set $S=\{\lfloor n\pi\rfloor \mid n=0,1,2,3,\ldots\}$ contains arithmetic progressions of any finite length, but no infinite arithmetic progressions.
[i]Vasile Pop[/i]
2020 Jozsef Wildt International Math Competition, W45
Let $a_1,a_2,a_3,a_4$ be strictly positive numbers. Then is the following inequality true:
$$4\left(a_1a_2^n+a_2a_3^n+a_3a_4^n+a_4a_1^n\right)^n\le\left(a_1^n+a_2^n+a_3^n+a_4^n\right)^{n+1}$$
for each $n\in\mathbb N$?
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
2004 National Olympiad First Round, 29
Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral?
$
\textbf{(A)}\ 5\sqrt 3
\qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3}
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \sqrt{34}
$
2019 Caucasus Mathematical Olympiad, 4
Vova has a square grid $72\times 72$. Unfortunately, $n$ cells are stained with coffee. Determine if Vova always can cut out a clean square $3\times 3$ without its central cell, if
a) $n=699$;
b) $n=750$.
2007 Harvard-MIT Mathematics Tournament, 6
Triangle $ABC$ has $\angle A=90^\circ$, side $BC=25$, $AB>AC$, and area $150$. Circle $\omega$ is inscribed in $ABC$, with $M$ its point of tangency on $AC$. Line $BM$ meets $\omega$ a second time at point $L$. Find the length of segment $BL$.
2018 Sharygin Geometry Olympiad, 5
The vertex $C$ of equilateral triangles $ABC$ and $CDE$ lies on the segment $AE$, and the vertices $B$ and $D$ lie on the same side with respect to this segment. The circumcircles of these triangles centered at $O_1$ and $O_2$ meet for the second time at point $F$. The lines $O_1O_2$ and $AD$ meet at point $K$. Prove that $AK = BF$.
2015 South East Mathematical Olympiad, 6
Given a positive integer $n\geq 2$. Let $A=\{ (a,b)\mid a,b\in \{ 1,2,…,n\} \}$ be the set of points in Cartesian coordinate plane. How many ways to colour points in $A$, each by one of three fixed colour, such that, for any $a,b\in \{ 1,2,…,n-1\}$, if $(a,b)$ and $(a+1,b)$ have same colour, then $(a,b+1)$ and $(a+1,b+1)$ also have same colour.
2023 Euler Olympiad, Round 1, 3
Leonard has a hand clock with only hour and minute hands. Determine the number of minutes in a day where the angle between the clock hands is not more than 1 degree. Both clock hands move continuously and at a constant speed.
[i]Proposed by Giorgi Arabidze, Georgia [/i]
1978 AMC 12/AHSME, 17
If $k$ is a positive number and $f$ is a function such that, for every positive number $x$, \[\left[f(x^2+1)\right]^{\sqrt{x}}=k;\] then, for every positive number $y$, \[\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}\] is equal to
$\textbf{(A) }\sqrt{k}\qquad\textbf{(B) }2k\qquad\textbf{(C) }k\sqrt{k}\qquad\textbf{(D) }k^2\qquad \textbf{(E) }y\sqrt{k}$
2019 China Northern MO, 1
Find all positive intengers $x,y$, satisfying:
$$3^x+x^4=y!+2019.$$
2001 USAMO, 5
Let $S$ be a set of integers (not necessarily positive) such that
(a) there exist $a,b \in S$ with $\gcd(a,b)=\gcd(a-2,b-2)=1$;
(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2-y$ also belongs to $S$.
Prove that $S$ is the set of all integers.
JBMO Geometry Collection, 2019
Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.
2020 AMC 12/AHSME, 10
There is a unique positive integer $n$ such that \[\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.\]
What is the sum of the digits of $n?$
$\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$
2016 Middle European Mathematical Olympiad, 3
A $8 \times 8$ board is given, with sides directed north-south and east-west.
It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one [i]house[/i]. A house occupies only one cell.
A house is [i] in the shade[/i] if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade.
Find the maximal number of houses that can be placed on the board such that no house is in the shade.
2012 Hanoi Open Mathematics Competitions, 2
[b]Q2.[/b] Let be given a parallegogram $ABCD$ with the area of $12 \ \text{cm}^2$. The line through $A$ and the midpoint $M$ of $BC$ mects $BD$ at $N.$ Compute the area of the quadrilateral $MNDC.$
\[(A) \; 4 \text{cm}^2; \qquad (B) \; 5 \text{cm}^2; \qquad (C ) \; 6 \text{cm}^2; \qquad (D) \; 7 \text{cm}^2; \qquad (E) \; \text{None of the above.}\]
2015 IMO Shortlist, G7
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
2005 USAMO, 5
Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a [i]balancing line[/i] if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.
Prove that there exist at least two balancing lines.
2020 Junior Balkan Team Selection Tests - Moldova, 12
Find all numbers $n \in \mathbb{N}^*$ for which there exists a finite set of natural numbers $A=(a_1, a_2,...a_n)$ so that for any $k$ $(1\leq k \leq n)$ the number $a_k$ is the number of all multiples of $k$ in set $A$.
2015 HMNT, 8
Find $ \textbf{any} $ quadruple of positive integers $(a,b,c,d)$ satisfying $a^3+b^4+c^5=d^{11}$ and $abc<10^5$.