Found problems: 85335
1960 IMO Shortlist, 3
In a given right triangle $ABC$, the hypotenuse $BC$, of length $a$, is divided into $n$ equal parts ($n$ and odd integer). Let $\alpha$ be the acute angel subtending, from $A$, that segment which contains the mdipoint of the hypotenuse. Let $h$ be the length of the altitude to the hypotenuse fo the triangle. Prove that: \[ \tan{\alpha}=\dfrac{4nh}{(n^2-1)a}. \]
2003 Gheorghe Vranceanu, 1
Prove that any permutation group of an order equal to a power of $ 2 $ contains a commutative subgroup whose order is the square of the exponent of the order of the group.
2016 Thailand TSTST, 2
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2019 Germany Team Selection Test, 2
Does there exist a subset $M$ of positive integers such that for all positive rational numbers $r<1$ there exists exactly one finite subset of $M$ like $S$ such that sum of reciprocals of elements in $S$ equals $r$.
2009 Harvard-MIT Mathematics Tournament, 4
If $\tan x + \tan y = 4$ and $\cot x + \cot y = 5$, compute $\tan(x + y)$.
2007 IMS, 2
Does there exist two unfair dices such that probability of their sum being $j$ be a number in $\left(\frac2{33},\frac4{33}\right)$ for each $2\leq j\leq 12$?
1999 Kazakhstan National Olympiad, 4
Seven dwarfs live in one house and each has its own hat. One morning one day, two dwarfs inadvertently exchanged hats. At any time, any three gnomes can sit down at the round table and exchange hats clockwise. Is it possible that by evening all the gnomes will be with their hats.
1981 AMC 12/AHSME, 3
For $x \neq 0$, $\frac{1}{x}+ \frac{1}{2x}+\frac{1}{3x}$ equals
$\text{(A)}\ \frac{1}{2x} \qquad \text{(B)}\ \frac{1}{6} \qquad \text{(C)}\ \frac{5}{6x} \qquad \text{(D)}\ \frac{11}{6x} \qquad \text{(E)}\ \frac{1}{6x^3}$
2010 Contests, 2
Find all prime numbers $p, q, r$ such that
\[15p+7pq+qr=pqr.\]
2021 Germany Team Selection Test, 1
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
2010 Contests, 1
Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and
\[\gcd(P(0), P(1), P(2), \ldots ) = 1.\]
Show there are infinitely many $n$ such that
\[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]
2001 Bosnia and Herzegovina Team Selection Test, 5
Let $n$ be a positive integer, $n \geq 1$ and $x_1,x_2,...,x_n$ positive real numbers such that $x_1+x_2+...+x_n=1$. Does the following inequality hold $$\sum_{i=1}^{n} {\frac{x_i}{1-x_1\cdot...\cdot x_{i-1} \cdot x_{i+1} \cdot ... x_n}} \leq \frac{1}{1-\left(\frac{1}{n}\right)^{n-1}} $$
2024 Caucasus Mathematical Olympiad, 6
The integers from $1$ to $320000$ are placed in the cells of a $8 \times 40000$ board. Prove that it is possible to permute the rows of the table so that the numbers in each column will not be sorted from the top to the bottom in increasing order.
2022 Rioplatense Mathematical Olympiad, 1
Prove that there exists infinitely many positive integers $n$ for which the equation$$x^2+y^{11}-z^{2022!}=n$$has no solution $(x,y,z)$ over the integers.
Estonia Open Senior - geometry, 1995.1.3
We call a tetrahedron a "trirectangular " if it has a vertex (we call this is called a "right-angled" vertex) in which the planes of the three sides of the tetrahedron intersect at right angles.
Prove the "three-dimensional Pythagorean theorem":
The square of the area of the opposite face of the "right-angled" vertex of the ""trirectangular " tetrahedron is equal to the sum of the squares of the areas of three other sides of the tetrahedron .
2023 CMIMC Geometry, 4
A rhombus $\mathcal R$ has short diagonal of length $1$ and long diagonal of length $2023$. Let $\mathcal R'$ be the rotation of $\mathcal R$ by $90^\circ$ about its center. If $\mathcal U$ is the set of all points contained in either $\mathcal R$ or $\mathcal R'$ (or both; this is known as the [i]union[/i] of $\mathcal R$ and $\mathcal R'$) and $\mathcal I$ is the set of all points contained in both $\mathcal R$ and $\mathcal R'$ (this is known as the [i]intersection[/i] of $\mathcal R$ and $\mathcal R'$, compute the ratio of the area of $\mathcal I$ to the area of $\mathcal U$.
[i]Proposed by Connor Gordon[/i]
2014 Contests, 2
$2014$ triangles have non-overlapping interiors contained in a circle of radius $1$. What is the largest possible value of the sum of their areas?
2005 Irish Math Olympiad, 3
Prove that the sum of the lengths of the medians of a triangle is at least three quarters of its perimeter.
2011 Belarus Team Selection Test, 1
Is it possible to arrange the numbers $1,2,...,2011$ over the circle in some order so that among any $25$ successive numbers at least $8$ numbers are multiplies of $5$ or $7$ (or both $5$ and $7$) ?
I. Gorodnin
2016 Azerbaijan JBMO TST, 2
Let $ABCD$ be a quadrilateral ,circumscribed about a circle. Let $M$ be a point on the side $AB$. Let $I_{1}$,$I_{2}$ and $I_{3}$ be the incentres of triangles $AMD$, $CMD$ and $BMC$ respectively. Prove that $I_{1}I_{2}I_{3}M$ is circumscribed.
2017 MMATHS, 3
Let $f : R \to R$, and let $P$ be a nonzero polynomial with degree no more than $2015$. For any nonnegative integer $n$, $f^{(n)}(x)$ denotes the function defined as $f$ composed with itself $n$ times. For example, $f^{(0)}(x) = x$, $f^{(1)}(x) = f(x)$, $f^{(2)}(x) = f(f(x))$, etc. Show that there always exists a real number $q$ such that $$f^{((2017^{2017})!)(q)} \ne (q + 2017)(qP(q) - 1).$$
2023 MMATHS, 12
Let $ABC$ be a triangle with incenter $I.$ The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA,$ and $AB$ at points $D, E,$ and $F,$ respectively. Let $D'$ be the reflection of $D$ over $I.$ Let $P$ be a point on $\omega$ such that $\angle{ADP}=90^\circ.$ $\mathcal{H}$ is a hyperbola passing through $D', E, F, I,$ and $P.$ Given that $\angle{BAD}=45^\circ$ and $\angle{CAD}=30^\circ,$ the acute angle between the asymptotes of $\mathcal{H}$ can be expressed as $\left(\tfrac{m}{n}\right)^\circ,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2017 Germany Team Selection Test, 3
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2008 Greece Team Selection Test, 4
Given is the equation $x^2+y^2-axy+2=0$ where $a$ is a positive integral parameter.
$i.$Show that,for $a\neq 4$ there exist no pairs $(x,y)$ of positive integers satisfying the equation.
$ii.$ Show that,for $a=4$ there exist infinite pairs $(x,y)$ of positive integers satisfying the equation,and determine those pairs.
2022 Harvard-MIT Mathematics Tournament, 5
Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number n when read from left to right. Compute the expected value of $n$.