Found problems: 85335
2006 AMC 8, 6
The letter T is formed by placing two $ 2\times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches?
[asy]size(150);
draw((0,6)--(4,6)--(4,4)--(3,4)--(3,0)--(1,0)--(1,4)--(0,4)--cycle, linewidth(1));[/asy]
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 16 \qquad
\textbf{(C)}\ 20 \qquad
\textbf{(D)}\ 22 \qquad
\textbf{(E)}\ 24$
2007 Peru Iberoamerican Team Selection Test, P2
Find all positive integer solutions of the equation
$n^5+n^4=7^{m}-1$
2007 Pre-Preparation Course Examination, 21
Find all primes $p,q$ such that
\[p^q-q^p=pq^2-19\]
1989 All Soviet Union Mathematical Olympiad, 509
$N$ is the set of positive integers. Does there exist a function $f: N \to N$ such that $f(n+1) = f( f(n) ) + f( f(n+2) )$ for all $n$?
1951 Kurschak Competition, 1
$ABCD$ is a square. $E$ is a point on the side $BC$ such that $BE =1/3 BC$, and $F$ is a point on the ray $DC$ such that $CF =1/2 DC$. Prove that the lines $AE$ and $BF$ intersect on the circumcircle of the square.
[img]https://cdn.artofproblemsolving.com/attachments/e/d/09a8235d0748ce4479e21a3bb09b0359de54b5.png[/img]
2015 Belarus Team Selection Test, 2
In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.
2005 China Girls Math Olympiad, 2
Find all ordered triples $ (x, y, z)$ of real numbers such that
\[ 5 \left(x \plus{} \frac{1}{x} \right) \equal{} 12 \left(y \plus{} \frac{1}{y} \right) \equal{} 13 \left(z \plus{} \frac{1}{z} \right),\]
and \[ xy \plus{} yz \plus{} zy \equal{} 1.\]
2017 Iberoamerican, 3
Consider the configurations of integers
$a_{1,1}$
$a_{2,1} \quad a_{2,2}$
$a_{3,1} \quad a_{3,2} \quad a_{3,3}$
$\dots \quad \dots \quad \dots$
$a_{2017,1} \quad a_{2017,2} \quad a_{2017,3} \quad \dots \quad a_{2017,2017}$
Where $a_{i,j} = a_{i+1,j} + a_{i+1,j+1}$ for all $i,j$ such that $1 \leq j \leq i \leq 2016$.
Determine the maximum amount of odd integers that such configuration can contain.
2005 Italy TST, 3
Let $N$ be a positive integer. Alberto and Barbara write numbers on a blackboard taking turns, according to the following rules. Alberto starts writing $1$, and thereafter if a player has written $n$ on a certain move, his adversary is allowed to write $n+1$ or $2n$ as long as he/she does not obtain a number greater than $N$. The player who writes $N$ wins.
$(a)$ Determine which player has a winning strategy for $N=2005$.
$(b)$ Determine which player has a winning strategy for $N=2004$.
$(c)$ Find for how many integers $N\le 2005$ Barbara has a winning strategy.
2009 Junior Balkan Team Selection Test, 1
Given are natural numbers $ a,b$ and $ n$ such that $ a^2\plus{}2nb^2$ is a complete square. Prove that the number $ a^2\plus{}nb^2$ can be written as a sum of squares of $ 2$ natural numbers.
1987 Canada National Olympiad, 5
For every positive integer $n$ show that
\[[\sqrt{4n + 1}] = [\sqrt{4n + 2}] = [\sqrt{4n + 3}] = [\sqrt{n} + \sqrt{n + 1}]\]
where $[x]$ is the greatest integer less than or equal to $x$ (for example $[2.3] = 2$, $[\pi] = 3$, $[5] = 5$).
2023 ELMO Shortlist, G4
Let \(D\) be a point on segment \(PQ\). Let \(\omega\) be a fixed circle passing through \(D\), and let \(A\) be a variable point on \(\omega\). Let \(X\) be the intersection of the tangent to the circumcircle of \(\triangle ADP\) at \(P\) and the tangent to the circumcircle of \(\triangle ADQ\) at \(Q\). Show that as \(A\) varies, \(X\) lies on a fixed line.
[i]Proposed by Elliott Liu and Anthony Wang[/i]
2023 Chile Junior Math Olympiad, 3
Let $\vartriangle ABC$ be an equilateral triangle with side $1$. Four points are marked $P_1$, $P_2$, $P_3$, $P_4$ on side $AC$ and four points $Q_1$, $Q_2$, $Q_3$, $Q_4$ on side $AB$ (see figure) of such a way to generate $9$ triangles of equal area. Find the length of segment $AP_4$.
[img]https://cdn.artofproblemsolving.com/attachments/5/f/29243932262cb963b376244f4c981b1afe87c6.png[/img]
PS. Easier version of [url=https://artofproblemsolving.com/community/c6h3323141p30741525]2023 Chile NMO L2 P3[/url]
2010 Contests, 1
Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.
2006 Germany Team Selection Test, 2
Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle.
[i]Proposed by Dimitris Kontogiannis, Greece[/i]
2010 Slovenia National Olympiad, 5
Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties.
[hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]
JOM 2015 Shortlist, C7
Navi and Ozna are playing a game where Ozna starts first and the two take turn making moves. A positive integer is written on the waord. A move is to (i) subtract any positive integer at most 2015 from it or (ii) given that the integer on the board is divisible by $2014$, divide by $2014$. The first person to make the integer $0$ wins. To make Navi's condition worse, Ozna gets to pick integers $a$ and $b$, $a\ge 2015$ such that all numbers of the form $an+b$ will not be the starting integer, where $n$ is any positive integer.
Find the minimum number of starting integer where Navi wins.
2012 Iran MO (2nd Round), 2
Suppose $n$ is a natural number. In how many ways can we place numbers $1,2,....,n$ around a circle such that each number is a divisor of the sum of it's two adjacent numbers?
2000 Tournament Of Towns, 1
The diagonals of a convex quadrilateral $ABCD$ meet at $P$. The sum of the areas of triangles $PAB$ and $PCD$ is equal to the sum of areas of triangles $PAD$ and $PCB$. Prove that $P$ is the midpoint of either $AC$ or $BD$.
(Folklore)
2017 Purple Comet Problems, 7
Find the number of positive integers less than 100 that are divisors of 300.
2000 ITAMO, 3
A pyramid with the base $ABCD$ and the top $V$ is inscribed in a sphere. Let $AD = 2BC$ and let the rays $AB$ and $DC$ intersect in point $E$. Compute the ratio of the volume of the pyramid $VAED$ to the volume of the pyramid $VABCD$.
2017 Mathematical Talent Reward Programme, MCQ: P 8
How many finite sequances $x_1,x_2,\cdots,x_m$ are there such that $x_i=1$ or 2 and $\sum \limits_{i=1}^mx_i=10$ ?
[list=1]
[*] 89
[*] 73
[*] 107
[*] 119
[/list]
2021 Taiwan TST Round 3, 6
Let $ ABCD $ be a rhombus with center $ O. $ $ P $ is a point lying on the side $ AB. $ Let $ I, $ $ J, $ and $ L $ be the incenters of triangles $ PCD, $ $ PAD, $ and $PBC, $ respectively. Let $ H $ and $ K $ be orthocenters of triangles $ PLB $ and $ PJA, $ respectively.
Prove that $ OI \perp HK. $
[i]Proposed by buratinogigle[/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?
$\textbf{(A) }39\qquad\textbf{(B) }40\qquad\textbf{(C) }210\qquad\textbf{(D) }400\qquad \textbf{(E) }401$
2021 Bosnia and Herzegovina Team Selection Test, 2
Let $p > 2$ be a prime number. Prove that there is a permutation $k_1, k_2, ..., k_{p-1}$ of numbers $1,2,...,p-1$ such that the number $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$ is divisible by $p$.
Note: The numbers $k_1, k_2, ..., k_{p-1}$ are a permutation of the numbers $1,2,...,p-1$ if each of of numbers $1,2,...,p-1$ appears exactly once among the numbers $k_1, k_2, ..., k_{p-1}$.