Found problems: 85335
1992 Poland - First Round, 12
Prove that the polynomial $x^n+4$ can be expressed as a product of two polynomials (each with degree less than $n$) with integer coefficients, if and only if $n$ is divisible by $4$.
2013 Stanford Mathematics Tournament, 6
Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of 10 mph. If he completes the first three laps at a constant speed of only 9 mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?
2021 Grand Duchy of Lithuania, 3
Let $ABCD$ be a convex quadrilateral satisfying $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o$, $AD = BC$. Prove that there exists a right-angled triangle with side lengths $AC$, $BD$, $CD$.
2021 Bolivian Cono Sur TST, 3
Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$. Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$. Find
$$\frac{[ABKM]}{[ABCL]}$$
2008 Teodor Topan, 1
Solve in $ M_2(\mathbb{C})$ the equation $ X^2\equal{}\left(
\begin{array}{cc}
1 & 2 \\
3 & 6 \end{array}
\right)$
2017 NMTC Junior, 6
If $a,b,c,d$ are positive reals such that $a^2+b^2=c^2+d^2$ and $a^2+d^2-ad=b^2+c^2+bc$, find the value of $\frac{ab+cd}{ad+bc}$
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 $n?$
$\textbf{(A) } 4 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$