Found problems: 85335
1987 ITAMO, 3
Show how to construct (by a ruler and a compass) a right-angled triangle, given its inradius and circumradius.
2021-IMOC, C8
Find all positive integers $m,n$ such that the $m \times n$ grid can be tiled with figures formed by deleting one of the corners of a $2 \times 3$ grid.
[i]usjl, ST[/i]
2004 Peru MO (ONEM), 3
Let $x,y,z$ be positive real numbers, less than $\pi$, such that:
$$\cos x + \cos y + \cos z = 0$$
$$\cos 2x + \cos 2 y + \cos 2z = 0$$
$$\cos 3x + \cos 3y + \cos 3z = 0$$
Find all the values that $\sin x + \sin y + \sin z$ can take.
2014 IMC, 4
Let $n>6$ be a perfect number, and let $n=p_1^{e_1}\cdot\cdot\cdot p_k^{e_k}$ be its prime factorisation with $1<p_1<\dots <p_k$. Prove that $e_1$ is an even number.
A number $n$ is [i]perfect[/i] if $s(n)=2n$, where $s(n)$ is the sum of the divisors of $n$.
(Proposed by Javier Rodrigo, Universidad Pontificia Comillas)
1946 Putnam, A5
Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid
$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1.$$
2022 Czech and Slovak Olympiad III A, 5
Find all integers $n$ such that $2^n + n^2$ is a square of an integer.
[i](Tomas Jurik )[/i]
1957 Moscow Mathematical Olympiad, 348
A snail crawls over a table at a constant speed. Every $15$ minutes it turns by $90^o$, and in between these turns it crawls along a straight line. Prove that it can return to the starting point only in an integer number of hours.
2018 Turkey Team Selection Test, 2
Find all $f:\mathbb{R}\to\mathbb{R}$ surjective functions such that
$$f(xf(y)+y^2)=f((x+y)^2)-xf(x) $$ for all real numbers $x,y$.
2016 Harvard-MIT Mathematics Tournament, 1
Dodecagon $QWARTZSPHINX$ has all side lengths equal to $2$, is not self-intersecting (in particular, the twelve vertices are all distinct), and moreover each interior angle is either $90^{\circ}$ or $270^{\circ}$. What are all possible values of the area of $\triangle SIX$?
2015 ASDAN Math Tournament, 5
Four men are each given a unique number from $1$ to $4$, and four women are each given a unique number from $1$ to $4$. How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to each other? Note that two configurations are considered to be the same if one can be rotated to obtain the other one.
2022 Thailand Mathematical Olympiad, 2
Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that
$$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$
for every positive integers $m,n$. Determine the minimum possible value of
$$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$
across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.
1985 Traian Lălescu, 1.3
Let $ H $ be the orthocenter of $ ABC $ and $ A',B',C', $ the symmetric points of $ A,B,C $ with respect to $ H. $ The intersection of the segments $ BC,CA, AB $ with the circles of diameter $ A'H,B'H, $ respectively, $ C'H, $ consists of $ 6 $ points. Prove that these are concyclic.
1995 Poland - First Round, 11
In a skiing jump competition $65$ contestants take part. They jump with the previously established order. Each of them jumps once. We assume that the obtained results are different and the orders of the contestants after the competition are equally likely. In each moment of the competition by a leader we call a person who is scored the best at this moment. Denote by $p$ the probability that during the whole competition there was exactly one change of the leader. Prove that $p > 1/16$.
1995 AMC 8, 20
Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number?
$\text{(A)}\ \dfrac{1}{3} \qquad \text{(B)}\ \dfrac{5}{12} \qquad \text{(C)}\ \dfrac{4}{9} \qquad \text{(D)}\ \dfrac{17}{36} \qquad \text{(E)}\ \dfrac{1}{2}$
2017 Vietnamese Southern Summer School contest, Problem 4
In a summer school, there are $n>4$ students. It is known that, among these students,
i. If two ones are friends, then they don't have any common friends.
ii If two ones are not friends, then they have exactly two common friends.
1. Prove that $8n-7$ must be a perfect square.
2. Determine the smallest possible value of $n$.
2004 Singapore Team Selection Test, 1
Let $D$ be a point in the interior of $\bigtriangleup ABC$ such that $AB = ab$, $AC = ac$, $AD = ad$, $BC = bc$, $BD = bd$ and $CD = cd$. Prove that $\angle ABD + \angle ACD = \frac{\pi}{3}$.
2017 ASDAN Math Tournament, 8
Let $\triangle ABC$ be a right triangle with right angle $\angle ACB$. Square $DEFG$ is contained inside triangle $ABC$ such that $D$ lies on $AB$, $E$ lies on $BC$, $F$ lies on $AC$, $AD=AF$, and $GA=GD=GF$. Suppose that $CE=2$. If $M$ is the area of triangle $ABC$ and $N$ is the area of square $DEFG$, compute $M-N$.
2003 Purple Comet Problems, 21
Let $a_n = \sqrt{1 + (1 - \tfrac{1}{n})^2} + \sqrt{1 + (1 + \tfrac{1}{n})^2}, n \ge 1$. Evaluate $\tfrac{1}{a_1} + \tfrac{1}{a_2} + \ldots + \tfrac{1}{a_{20}}$.
2006 Kyiv Mathematical Festival, 5
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
All the positive integers from 1 till 1000 are written on the blackboard in some order and there is a collection of cards each containing 10 numbers. If there is a card with numbers $1\le a_1<a_2<\ldots<a_{10}\le1000$ in collection then it is allowed to arrange in increasing order the numbers at places $a_1, a_2, \ldots, a_{10},$ counting from left to right. What is the smallest amount of cards in the collection which enables us to arrange in
increasing order all the numbers for any initial arrangement of them?
2013 CentroAmerican, 2
Around a round table the people $P_1, P_2,..., P_{2013}$ are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none); there are a total of $10000$ coins. Starting with $P_1$ and proceeding in clockwise order, each person does the following on their turn:
[list][*]If they have an even number of coins, they give all of their coins to their neighbor to the left.
[*]If they have an odd number of coins, they give their neighbor to the left an odd number of coins (at least $1$ and at most all of their coins) and keep the rest.[/list]
Prove that, repeating this procedure, there will necessarily be a point where one person has all of the coins.
2007 Hanoi Open Mathematics Competitions, 15
Let $p = \overline{abc}$ be the 3-digit prime number. Prove that the equation $ax^2 + bx + c = 0$ has no rational roots.
2007 Today's Calculation Of Integral, 215
For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$.
Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$.
2024 Sharygin Geometry Olympiad, 17
Let $ABC$ be a non-isosceles triangle, $\omega$ be its incircle. Let $D, E, $ and $F$ be the points at which the incircle of $ABC$ touches the sides $BC, CA, $ and $AB$ respectively. Let $M$ be the point on ray $EF$ such that $EM = AB$. Let $N$ be the point on ray $FE$ such that $FN = AC$. Let the circumcircles of $\triangle BFM$ and $\triangle CEN$ intersect $\omega$ again at $S$ and $T$ respectively. Prove that $BS, CT, $ and $AD$ concur.
2006 CentroAmerican, 6
Let $ABCD$ be a convex quadrilateral. $I=AC\cap BD$, and $E$, $H$, $F$ and $G$ are points on $AB$, $BC$, $CD$ and $DA$ respectively, such that $EF \cap GH= I$. If $M=EG \cap AC$, $N=HF \cap AC$, show that \[\frac{AM}{IM}\cdot \frac{IN}{CN}=\frac{IA}{IC}.\]
1963 Bulgaria National Olympiad, Problem 1
Find all three-digit numbers whose remainders after division by $11$ give quotient, equal to the sum of the squares of its digits.