Found problems: 85335
2011 Iran Team Selection Test, 4
Define a finite set $A$ to be 'good' if it satisfies the following conditions:
[list][*][b](a)[/b] For every three disjoint element of $A,$ like $a,b,c$ we have $\gcd(a,b,c)=1;$
[*][b](b)[/b] For every two distinct $b,c\in A,$ there exists an $a\in A,$ distinct from $b,c$ such that $bc$ is divisible by $a.$[/list]
Find all good sets.
2001 Stanford Mathematics Tournament, 14
Find the prime factorization of $\textstyle\sum_{1\le i < j \le 100}ij$.
2001 Tournament Of Towns, 5
On a square board divided into $15 \times 15$ little squares there are $15$ rooks that do not attack each other. Then each rook makes one move like that of a knight. Prove that after this is done a pair of rooks will necessarily attack each other.
2020 CMIMC Geometry, Estimation
Gunmay picks $6$ points uniformly at random in the unit square. If $p$ is the probability that their convex hull is a hexagon, estimate $p$ in the form $0.abcdef$ where $a,b,c,d,e,f$ are decimal digits. (A [i]convex combination[/i] of points $x_1, x_2, \dots, x_n$ is a point of the form $\alpha_1x_1 + \alpha_2x_2 + \dots + \alpha_nx_n$ with $0 \leq \alpha_i \leq 1$ for all $i$ and $\alpha_1 + \alpha_2 + \dots + \alpha_n = 1$. [i]The convex hull[/i] of a set of points $X$ is the set of all possible convex combinations of all subsets of $X$.)
2002 Moldova National Olympiad, 2
Can a square of side $ 1024$ be partitioned into $ 31$ squares?Can a square of side $ 1023$ be partitioned into $ 30$ squares, one of which has a s side lenght not exceeding $ 1$?
2008 Bosnia And Herzegovina - Regional Olympiad, 3
Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.
2016 Turkey Team Selection Test, 5
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .
2013 Gheorghe Vranceanu, 2
Given a number $ a $ and natural number $ n\ge 3 $ having the property that $ x^n-x $ and $ x^2-x $ are integers, prove that $ x $ is integer.
2022 BMT, 20
The game Boddle uses eight cards numbered $6, 11, 12, 14, 24, 47, 54$, and $n$, where $0 \le n \le 56$. An integer D is announced, and players try to obtain two cards, which are not necessarily distinct, such that one of their differences (positive or negative) is congruent to $D$ modulo $57$. For example, if $D = 27$, then the pair $24$ and $54$ would work because $24 - 54 \equiv 27$ mod $57$. Compute $n$ such that this task is always possible for all $D$.
2023 CMIMC Team, 5
$1296$ CMU Students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the `score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students?
[i]Proposed by Dilhan Salgado[/i]
1994 Korea National Olympiad, Problem 2
Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that
$csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$
and find the conditions for equality.
2022 Purple Comet Problems, 16
The sum of the solutions to the equation $$x^{\log_2 x} =\frac{64}{x}$$ can be written as$ \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1974 Spain Mathematical Olympiad, 2
In a metallic disk, a circular sector is removed, so that with the remaining can form a conical glass of maximum volume. Calculate, in radians, the angle of the sector that is removed.
[hide=original wording]En un disco metalico se quita un sector circular, de modo que con la parte restante se pueda formar un vaso c´onico de volumen maximo. Calcular, en radianes, el angulo del sector que se quita.[/hide]
2011 IFYM, Sozopol, 4
There are $n$ points in a plane. Prove that there exist a point $O$ (not necessarily from the given $n$) such that on each side of an arbitrary line, through $O$, lie at least $\frac{n}{3}$ points (including the points on the line).
2019 India IMO Training Camp, P1
Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.
2017 India Regional Mathematical Olympiad, 4
Consider \(n^2\) unit squares in the \(xy\) plane centered at point \((i,j)\) with integer coordinates, \(1 \leq i \leq n\), \(1 \leq j \leq n\). It is required to colour each unit square in such a way that whenever \(1 \leq i < j \leq n\) and \(1 \leq k < l \leq n\), the three squares with centres at \((i,k),(j,k),(j,l)\) have distinct colours. What is the least possible number of colours needed?
2006 Iran MO (3rd Round), 5
$M$ is midpoint of side $BC$ of triangle $ABC$, and $I$ is incenter of triangle $ABC$, and $T$ is midpoint of arc $BC$, that does not contain $A$. Prove that \[\cos B+\cos C=1\Longleftrightarrow MI=MT\]
2018 CCA Math Bonanza, I9
What is the area of the smallest possible square that can be drawn around a regular hexagon of side length $2$ such that the hexagon is contained entirely within the square?
[i]2018 CCA Math Bonanza Individual Round #9[/i]
2015 AMC 8, 9
On her first day of work, Janabel sold one widget. On day two, she sold three widgets. On day three, she sold five widgets, and on each succeeding day, she sold two more widgets than she had sold on the previous day. How many widgets in total had Janabel sold after working $20$ days?
$\textbf{(A) }39\qquad\textbf{(B) }40\qquad\textbf{(C) }210\qquad\textbf{(D) }400\qquad \textbf{(E) }401$
2008 Bulgaria Team Selection Test, 3
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.
2023 JBMO Shortlist, A1
Prove that for all positive real numbers $a,b,c,d$,
$$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$
and determine when equality occurs.
2005 Germany Team Selection Test, 3
Let $b$ and $c$ be any two positive integers. Define an integer sequence $a_n$, for $n\geq 1$, by $a_1=1$, $a_2=1$, $a_3=b$ and $a_{n+3}=ba_{n+2}a_{n+1}+ca_n$.
Find all positive integers $r$ for which there exists a positive integer $n$ such that the number $a_n$ is divisible by $r$.
2017-IMOC, A2
Find all functions $f:\mathbb N\to\mathbb N$ such that
\begin{align*}
x+f(y)&\mid f(y+f(x))\\
f(x)-2017&\mid x-2017\end{align*}
2003 Rioplatense Mathematical Olympiad, Level 3, 2
Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Prove that $AA'$, $BB'$ and $CC'$ are concurrent.
2022 Costa Rica - Final Round, 2
Find all functions $f$, of the form $f(x) = x^3 +px^2 +qx+r$ with $p$, $q$ and $r$ integers, such that $f(s) = 506$ for some integer $s$ and $f(\sqrt3) = 0$.