Found problems: 85335
2016 BMT Spring, 3
Let $S$ be the set of all non-degenerate triangles with integer sidelengths, such that two of the sides are $20$ and $16$. Suppose we pick a triangle, at random, from this set. What is the probability that it is acute?
2005 Paraguay Mathematical Olympiad, 1
With the digits $1, 2, 3,. . . . . . , 9$ three-digit numbers are written such that the sum of the three digits is $17$. How many numbers can be written?
2010 Dutch BxMO TST, 5
For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.
2023 Romania National Olympiad, 3
We say that a natural number $n$ is interesting if it can be written in the form
\[
n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor,
\] where $a,b,c$ are positive real numbers such that $a + b + c = 1.$
Determine all interesting numbers. ( $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.)
2022 CHMMC Winter (2022-23), 6
Let $A$ be a set of $8$ elements, and $B := (B_1,...,B_7)$ be an ordered $7$-tuple of subsets of $A$. Let $N$ be the number of such $7$-tuples $B$ such that there exists a unique $4$-element subset $I \subseteq \{1,2,...,7\}$ for which the intersection $\cap _{ i\in I} B_i$ is nonempty. Find the remainder when $N$ is divided by $67$.
2025 Bulgarian Winter Tournament, 10.4
The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.
Kvant 2020, M2616
Let $p>5$ be a prime number. Prove that the sum \[\left(\frac{(p-1)!}{1}\right)^p+\left(\frac{(p-1)!}{2}\right)^p+\cdots+\left(\frac{(p-1)!}{p-1}\right)^p\]is divisible by $p^3$.
2020 AMC 10, 15
A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 23$
2018 Israel National Olympiad, 6
In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$, and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$, and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius.
[img]https://i.imgur.com/bYuBabS.png[/img]
2007 Pre-Preparation Course Examination, 17
For a positive integer $n$, denote $rad(n)$ as product of prime divisors of $n$. And also $rad(1)=1$. Define the sequence $\{a_i\}_{i=1}^{\infty}$ in this way: $a_1 \in \mathbb N$ and for every $n \in \mathbb N$, $a_{n+1}=a_n+rad(a_n)$.
Prove that for every $N \in \mathbb N$, there exist $N$ consecutive terms of this sequence which are in an arithmetic progression.
1988 ITAMO, 6
The edge lengths of the base of a tetrahedron are $a,b,c$, and the lateral edge lengths are $x,y,z$. If $d$ is the distance from the top vertex to the centroid of the base, prove that $x+y+z \le a+b+c+3d$.
2012 Bundeswettbewerb Mathematik, 2
On a round table, $n$ bowls are arranged in a circle. Anja walks around the table clockwise, placing marbles in the bowls according to the following rule:
She places a marble in any first bowl, then goes one bowl further and puts a marble in there. Then she goes two shells before putting another marble, then she goes three shells, etc. If there is at least one marble in each shell, she stops.
For which $n$ does this occur?
1996 Baltic Way, 4
$ABCD$ is a trapezium where $AD\parallel BC$. $P$ is the point on the line $AB$ such that $\angle CPD$ is maximal. $Q$ is the point on the line $CD$ such that $\angle BQA$ is maximal. Given that $P$ lies on the segment $AB$, prove that $\angle CPD=\angle BQA$.
2007 Nicolae Coculescu, 4
Let $ n\in{N^*}$,$ n\ge{3}$ and $ a_1,a_2,...,a_n\in{R^*}$, so that $ |a_i|\neq{|a_j|}$, for every $ i,j\in{\{1,2,...,n\}}, i\neq{j}$.
Find $ p\in{S_n}$ with the property:
$ a_ia_j < \equal{} a_{p(i)}a_{p(j)}$, for every $ i,j\in{\{1,2,....n\}}$,$ i\neq{j}$ (Teodor Radu)
2023 Peru MO (ONEM), 2
For each positive real number $x$, let $f(x)=\frac{x}{1+x}$ . Prove that if $a$, $b,$ $c$ are the sidelengths of a triangle, then $f(a)$, $f(b),$ $f(c)$ are sidelengths of a triangle.
2022 Korea National Olympiad, 7
Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions:
[list]
[*]$a_i \leq a_j$ for every positive integers $i <j$.
[*]For any positive integer $k \geq 3$, the following inequality holds:
$$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$
[/list]
Prove that $\{a_n\}$ is constant.
2023 Philippine MO, 4
In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board [i]can be attacked[/i] by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less.
In a $10 \times 10$ chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?
2020 CMIMC Team, 13
Given $10$ points arranged in a equilateral triangular grid of side length $4$, how many ways are there to choose two distinct line segments, with endpoints on the grid, that intersect in exactly one point (not necessarily on the grid)?
1977 IMO Longlists, 21
Given that $x_1+x_2+x_3=y_1+y_2+y_3=x_1y_1+x_2y_2+x_3y_3=0,$ prove that:
\[ \frac{x_1^2}{x_1^2+x_2^2+x_3^2}+\frac{y_1^2}{y_1^2+y_2^2+y_3^2}=\frac{2}{3}\]
2004 Poland - First Round, 4
4.Given is $n \in \mathbb Z$ and positive reals a,b. Find possible maximal value of the sum: $x_1y_1 + x_2y_2 + ... + x_ny_n$ when $x_1,x_2,...,x_n$ and $y_1,y_2,...,y_n$ are in $<0;1>$ and satisfies: $x_1 + x_2 + ... + x_n \leq a$ and $y_1 + y_2 + ... + y_n \leq b$
2003 Romania Team Selection Test, 11
In a square of side 6 the points $A,B,C,D$ are given such that the distance between any two of the four points is at least 5. Prove that $A,B,C,D$ form a convex quadrilateral and its area is greater than 21.
[i]Laurentiu Panaitopol[/i]
2012 Bogdan Stan, 2
For any $ a\in\mathbb{Z}_{\ge 0} $ make the notation $ a\mathbb{Z}_{\ge 0} =\{ an| n\in\mathbb{Z}_{\ge 0} \} . $ Prove that the following relations are equivalent:
$ \text{(1)} a\mathbb{Z}_{\ge 0} \setminus b\mathbb{Z}_{\ge 0}\subset c\mathbb{Z}_{\ge 0} \setminus d\mathbb{Z}_{\ge 0} $
$ \text{(2)} b|a\text{ or } (c|a\text{ and } \text{lcm} (a,b) |\text{lcm} (a,d)) $
[i]Marin Tolosi[/i] and [i]Cosmin Nitu[/i]
2014 IMO Shortlist, C8
A card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared.
Determine all possible first moves of the first player after which he has a winning strategy.
[i]Proposed by Ilya Bogdanov & Vladimir Bragin, Russia[/i]
1973 Czech and Slovak Olympiad III A, 1
Consider a triangle such that \[\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2.\] Show that the triangle is right.
1985 IMO Shortlist, 13
Let $m$ boxes be given, with some balls in each box. Let $n < m$ be a given integer. The following operation is performed: choose $n$ of the boxes and put $1$ ball in each of them. Prove:
[i](a) [/i]If $m$ and $n$ are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls.
[i](b)[/i] If $m$ and $n$ are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.