Found problems: 85335
2020 CMIMC Geometry, 9
In triangle $ABC$, points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$. Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$. Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that $PB = 5$, compute the length of $BC$.
2020 Romanian Masters In Mathematics, 3
Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$.
Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.
2022 Argentina National Olympiad Level 2, 1
Find all real numbers $x$ such that exactly one of the four numbers $x-\sqrt 2$, $x-\dfrac{1}{x}$, $x+\dfrac{1}{x}$ and $x^2+2\sqrt{2}$ is [b]not[/b] an integer.
2000 Tuymaada Olympiad, 5
Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?
2002 National Olympiad First Round, 17
Let $ABCD$ be a trapezoid and a tangential quadrilateral such that $AD || BC$ and $|AB|=|CD|$. The incircle touches $[CD]$ at $N$. $[AN]$ and $[BN]$ meet the incircle again at $K$ and $L$, respectively. What is $\dfrac {|AN|}{|AK|} + \dfrac {|BN|}{|BL|}$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ 16
$
2019 SIMO, Q1
[i]George the grasshopper[/i] lives of the real line, starting at $0$ . He is given the following sequence of numbers: $2, 3, 4, 8, 9, ... ,$ which are all the numbers of the form $2^k$ or $3^l$, $k, l \in \mathbb{N}$, arranged in increasing order. Starting from $2$, for each number $x$ in the sequence in order, he (currently at $a$) must choose to jump to either $a+x$ or $a-x$. Show that [i]George the grasshopper[/i] can jump in a way that he reaches every integer on the real line.
2023 Math Prize for Girls Problems, 19
Let $\displaystyle{N = \prod_{k=1}^{1000} (4^k - 1)}$. Determine the largest positive integer $n$ such that $5^n$ divides evenly into $N$.
2012 Online Math Open Problems, 5
Congruent circles $\Gamma_1$ and $\Gamma_2$ have radius $2012,$ and the center of $\Gamma_1$ lies on $\Gamma_2.$ Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. The line through $A$ perpendicular to $AB$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$, respectively. Find the length of $CD$.
[i]Author: Ray Li[/i]
2020 OMpD, 3
Determine all integers $n$ such that both of the numbers:
$$|n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8|$$
are both prime numbers.
2006 Kyiv Mathematical Festival, 2
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then
a)$800000000$; b)$880000000$ by such replacements?
1993 Baltic Way, 16
Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$, so that $|AB|=|BC|=|CD|=14\text{cm}$. Another line intersects the circles at $E,F$, respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$. Find the radius $r$.
2023 Middle European Mathematical Olympiad, 8
Let $A, B \in \mathbb{N}$. Consider a sequence $x_1, x_2, \ldots$ such that for all $n\geq 2$, $$x_{n+1}=A \cdot \gcd(x_n, x_{n-1})+B. $$ Show that the sequence attains only finitely many distinct values.
2003 Romania National Olympiad, 1
Let $ m,n$ be positive integers. Prove that the number $ 5^n\plus{}5^m$ can be represented as sum of two perfect squares if and only if $ n\minus{}m$ is even.
[i]Vasile Zidaru[/i]
2015 USAMTS Problems, 3
For all positive integers $n$, show that:
$$ \dfrac1n \sum^n _{k=1} \dfrac{k \cdot k! \cdot {n\choose k}}{n^k} = 1$$
2000 Brazil Team Selection Test, Problem 3
Let $BB',CC'$ be altitudes of $\triangle ABC$ and assume $AB$ ≠ $AC$.Let $M$ be the midpoint of $BC$ and $H$ be orhocenter of $\triangle ABC$ and $D$ be the intersection of $BC$ and $B'C'$.Show that $DH$ is perpendicular to $AM$.
2017 South East Mathematical Olympiad, 8
Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and
$$(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.$$Determine the largest possible number of elements in $A$.
2018 Sharygin Geometry Olympiad, 14
Let $ABC$ be a right-angled triangle with $\angle C = 90^{\circ}$, $K$, $L$, $M$ be the midpoints of sides $AB$, $BC$, $CA$ respectively, and $N$ be a point of side $AB$. The line $CN$ meets $KM$ and $KL$ at points $P$ and $Q$ respectively. Points $S$, $T$ lying on $AC$ and $BC$ respectively are such that $APQS$ and $BPQT$ are cyclic quadrilaterals. Prove that
a) if $CN$ is a bisector, then $CN$, $ML$ and $ST$ concur;
b) if $CN$ is an altitude, then $ST$ bisects $ML$.
2018 CHMMC (Fall), 3
Let $p$ be the third-smallest prime number greater than $5$ such that:
$\bullet$ $2p + 1$ is prime, and
$\bullet$ $5^p \not\equiv 1$ (mod $2p + 1$).
Find $p$.
1952 AMC 12/AHSME, 26
If $ \left(r \plus{} \frac {1}{r}\right)^2 \equal{} 3$, then $ r^3 \plus{} \frac {1}{r^3}$ equals
$ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$
2012 Iran MO (3rd Round), 4
Prove that from an $n\times n$ grid, one can find $\Omega (n^{\frac{5}{3}})$ points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of $\frac{5}{3}$!
2004 Nicolae Coculescu, 3
Solve in $ \mathcal{M}_2(\mathbb{R}) $ the equation $ X^3+X+2I=0. $
[i]Florian Dumitrel[/i]
1991 Tournament Of Towns, (318) 5
Let $M$ be a centre of gravity (the intersection point of the medians) of a triangle $ABC$. Under rotation by $120$ degrees about the point $M$, the point $B$ is taken to the point $P$; under rotation by $240$ degrees about $M$, the point $C$ is taken to the point $Q$. Prove that either $APQ$ is an equilateral triangle, or the points $A, P, Q$ coincide.
(Bykovsky, Khabarovsksk)
2022 CCA Math Bonanza, I1
Asteroids A and B have circular orbits around the same star. Asteroid A is located 400 km away from the star and takes 8000 hours to complete one full revolution. Asteroid B is located 100 km away and the speed of Asteroid B is twice the speed of Asteroid A. Find how long it takes for Asteroid B to complete one full revolution in hours.
[i]2022 CCA Math Bonanza Individual Round #1[/i]
2021 JBMO Shortlist, C2
Let $n$ be a positive integer. We are given a $3n \times 3n$ board whose unit squares are colored in black and white in such way that starting with the top left square, every third diagonal is colored in black and the rest of the board is in white. In one move, one can take a $2 \times 2$ square and change the color of all its squares in such way that white squares become orange, orange ones become black and black ones become white. Find all $n$ for which, using a finite number of moves, we can make all the squares which were initially black white, and all squares which were initially white black.
Proposed by [i]Boris Stanković and Marko Dimitrić, Bosnia and Herzegovina[/i]
2011 ELMO Shortlist, 4
Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal.
[i]David Yang.[/i]