Found problems: 85335
2018 Tuymaada Olympiad, 6
The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values.
[i]Proposed by A. Golovanov[/i]
1999 Romania Team Selection Test, 5
Let $x_1,x_2,\ldots,x_n$ be distinct positive integers. Prove that
\[ x_1^2+x_2^2 + \cdots + x_n^2 \geq \frac {2n+1}3 ( x_1+x_2+\cdots + x_n). \]
[i]Laurentiu Panaitopol[/i]
2023 Purple Comet Problems, 7
Elijah went on a four-mile journey. He walked the first mile at $3$ miles per hour and the second mile at $4$ miles per hour. Then he ran the third mile at $5$ miles per hour and the fourth mile at $6$ miles per hour. Elijah’s average speed for this journey in miles per hour was $\frac{m}{n}$, where m and $n$ are relatively prime positive integers. Find $m + n$.
1987 AMC 12/AHSME, 30
In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$ and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC$.) The ratio $\frac{AD}{AB}$ is
[asy]
size((220));
draw((0,0)--(20,0)--(7,6)--cycle);
draw((6,6)--(10,-1));
label("A", (0,0), W);
label("B", (20,0), E);
label("C", (7,6), NE);
label("D", (9.5,-1), W);
label("E", (5.9, 6.1), SW);
label("$45^{\circ}$", (2.5,.5));
label("$60^{\circ}$", (7.8,.5));
label("$30^{\circ}$", (16.5,.5));
[/asy]
$ \textbf{(A)}\ \frac{1}{\sqrt{2}} \qquad\textbf{(B)}\ \frac{2}{2+\sqrt{2}} \qquad\textbf{(C)}\ \frac{1}{\sqrt{3}} \qquad\textbf{(D)}\ \frac{1}{\sqrt[3]{6}} \qquad\textbf{(E)}\ \frac{1}{\sqrt[4]{12}} $
2020 LMT Fall, B13
Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles
2016 CMIMC, 4
Kevin colors three distinct squares in a $3\times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.
2014 Cuba MO, 9
The triangle $ABC$ is inscribed in circle $\Gamma$. The points X, Y, Z are the midpoints of the arcs $BC$, $CA$ and $AB$ respectively in $\Gamma$ (those that do not contain the third vertex, in each case). The intersection points of the sides of the triangles $\vartriangle ABC$ and $\vartriangle XY Z$ form the hexagon $DEFGHK$. Prove that the diagonals $DG$, $EH$ and $FK$ are concurrent
2013 Turkey Team Selection Test, 1
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
2002 District Olympiad, 2
Let $ ABCD $ be an inscriptible quadrilateral and $ M $ be a point on its circumcircle, distinct from its vertices. Let $ H_1,H_2,H_3,H_4 $ be the orthocenters of $ MAB,MBC, MCD, $ respectively, $ MDA, $ and $ E,F, $ the midpoints of the segments $ AB, $ respectivley, $ CD. $ Prove that:
[b]a)[/b] $ H_1H_2H_3H_4 $ is a parallelogram.
[b]b)[/b] $ H_1H_3=2\cdot EF. $
1992 Turkey Team Selection Test, 1
Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.
1996 AMC 8, 10
When Walter drove up to the gasoline pump, he noticed that his gasoline tank was $\frac{1}{8}$ full. He purchased $7.5$ gallons of gasoline for $ \$10$. With this additional gasoline, his gasoline tank was then $\frac{5}{8}$ full. The number of gallons of gasoline his tank holds when it is full is
$\text{(A)}\ 8.75 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 11.5 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 22.5$
2020 USA IMO Team Selection Test, 2
Two circles $\Gamma_1$ and $\Gamma_2$ have common external tangents $\ell_1$ and $\ell_2$ meeting at $T$. Suppose $\ell_1$ touches $\Gamma_1$ at $A$ and $\ell_2$ touches $\Gamma_2$ at $B$. A circle $\Omega$ through $A$ and $B$ intersects $\Gamma_1$ again at $C$ and $\Gamma_2$ again at $D$, such that quadrilateral $ABCD$ is convex.
Suppose lines $AC$ and $BD$ meet at point $X$, while lines $AD$ and $BC$ meet at point $Y$. Show that $T$, $X$, $Y$ are collinear.
[i]Merlijn Staps[/i]
1998 IMO Shortlist, 7
Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that \[ \angle ECB+180^{\circ }=2\angle EBC. \]
2024 May Olympiad, 3
Ana writes an infinite list of numbers using the following procedure. The first number of the list is a positive integer $a$ chosen by Ana. From there, each number in the list is obtained by calculating the sum of all the integers from $1$ to the last number written. For example, if $a = 3$, Ana's list starts as $3, 6, 21, 231, \dots$ because $1 + 2 + 3 = 6$, $1 + 2 + 3 + 4 + 5 + 6 = 21$ and $1 + 2 + 3 + \dots + 21 = 231$. Is it possible for all the numbers in Ana's list to be even?
2001 Kazakhstan National Olympiad, 8
There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.
2001 USA Team Selection Test, 1
Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$. Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$.
2008 China Northern MO, 3
Prove that:
(1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$
(2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.
2015 AoPS Mathematical Olympiad, 5
Let $ABC$ be a triangle with orthocenter $h$. Let $AH$, $BH$, and $CH$ intersect the circumcircle of $\triangle ABC$ at points $D$, $E$, and $F$. Find the maximum value of $\frac{[DEF]}{[ABC]}$. (Here $[X]$ denotes the area of $X$.)
[i]Proposed by tkhalid.[/i]
2023 HMNT, 6
A function $g$ is [i]ever more[/i] than a function $h$ if, for all real numbers $x$, we have $g(x) \ge h(x)$. Consider all quadratic functions $f(x)$ such that $f(1) = 16$ and $f(x)$ is ever more than both $(x + 3)^2$ and $x^2 + 9$. Across all such quadratic functions $f$, compute the minimum value of $f(0)$.
1960 Putnam, A6
A player repeatedly throwing a die is to play until their score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n,$ and find the value of $\lim_{n \to \infty} p(n).$
2010 Korea National Olympiad, 3
Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.
2000 Belarus Team Selection Test, 4.1
Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$
2003 Rioplatense Mathematical Olympiad, Level 3, 1
Let $x$, $y$, and $z$ be positive real numbers satisfying $x^2+y^2+z^2=1$. Prove that \[x^2yz+xy^2z+xyz^2\le\frac{1}{3}.\]
1996 IMO Shortlist, 9
Let the sequence $ a(n), n \equal{} 1,2,3, \ldots$ be generated as follows with $ a(1) \equal{} 0,$ and for $ n > 1:$
\[ a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}.\]
1.) Determine the maximum and minimum value of $ a(n)$ over $ n \leq 1996$ and find all $ n \leq 1996$ for which these extreme values are attained.
2.) How many terms $ a(n), n \leq 1996,$ are equal to 0?
2016 NIMO Summer Contest, 14
Find the smallest positive integer $n$ such that $n^2+4$ has at least four distinct prime factors.
[i]Proposed by Michael Tang[/i]