Found problems: 85335
2024 Malaysia IMONST 2, 6
Rui Xuen has a circle $\omega$ with center $O$, and a square $ABCJ$ with vertices on $\omega$. Let $M$ be the midpoint of $AB$, and let $\Gamma$ be the circle passing through the points $J$, $O$, $M$. Suppose $\Gamma$ intersect line $AJ$ at a point $P \neq J$, and suppose $\Gamma$ intersect $\omega$ at a point $Q \neq J$. A point $R$ lies on side $BC$ so that $RC = 3RB$.
Help Rui Xuen prove that the points $P$, $Q$, $R$ are collinear.
1974 IMO Longlists, 30
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
II Soros Olympiad 1995 - 96 (Russia), 9.3
Is there a convex pentagon in which each diagonal is equal to some side?
2007 Peru IMO TST, 3
Let $T$ a set with 2007 points on the plane, without any 3 collinear points.
Let $P$ any point which belongs to $T$.
Prove that the number of triangles that contains the point $P$ inside and
its vertices are from $T$, is even.
1995 AMC 12/AHSME, 8
In $\triangle ABC$, $\angle C = 90^\circ, AC = 6$ and $BC = 8$. Points $D$ and $E$ are on $\overline{AB}$ and $\overline{BC}$, respectively, and $\angle BED = 90^\circ$. If $DE = 4$, then $BD =$
[asy]
size(100); pathpen = linewidth(0.7); pointpen = black+linewidth(3);
pair A = (0,0), C = (6,0), B = (6,8), D = (2*A+B)/3, E = (2*C+B)/3; D(D("A",A,SW)--D("B",B,NW)--D("C",C,SE)--cycle); D(D("D",D,NW)--D("E",E,plain.E)); D(rightanglemark(D,E,B,16)); D(rightanglemark(A,C,B,16));[/asy]
$\mathbf{(A)}\;5\qquad
\mathbf{(B)}\;\frac{16}{3}\qquad
\mathbf{(C)}\; \frac{20}{3}\qquad
\mathbf{(D)}\; \frac{15}{2}\qquad
\mathbf{(E)}\; 8$
2023 Durer Math Competition Finals, 2
[b]a)[/b] Find all solutions of the equation $p^2+q^2+r^2=pqr$, where $p,q,r$ are positive primes.\\
[b]b)[/b] Show that for every positive integer $N$, there exist three integers $a,b,c\geq N$ with $a^2+b^2+c^2=abc$.
2007 Iran MO (3rd Round), 5
Look at these fractions. At firs step we have $ \frac{0}{1}$ and $ \frac{1}{0}$, and at each step we write $ \frac{a\plus{}b}{c\plus{}d}$ between $ \frac{a}{b}$ and $ \frac{c}{d}$, and we do this forever
\[ \begin{array}{ccccccccccccccccccccccccc}\frac{0}{1}&&&&&&&&\frac{1}{0}\\ \frac{0}{1}&&&&\frac{1}{1}&&&&\frac{1}{0}\\ \frac{0}{1}&&\frac{1}{2}&&\frac{1}{1}&&\frac{2}{1}&&\frac{1}{0}\\ \frac{0}{1}&\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{1}{1}&\frac{3}{2}&\frac{2}{1}&\frac{3}{1}&\frac{1}{0}\\ &&&&\dots\end{array}\]
a) Prove that each of these fractions is irreducible.
b) In the plane we have put infinitely many circles of diameter 1, over each integer on the real line, one circle. The inductively we put circles that each circle is tangent to two adjacent circles and real line, and we do this forever. Prove that points of tangency of these circles are exactly all the numbers in part a(except $ \frac{1}{0}$).
[img]http://i2.tinypic.com/4m8tmbq.png[/img]
c) Prove that in these two parts all of positive rational numbers appear.
If you don't understand the numbers, look at [url=http://upload.wikimedia.org/wikipedia/commons/2/21/Arabic_numerals-en.svg]here[/url].
1970 IMO Shortlist, 10
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.
[b]a.)[/b] Prove that $0\le b_n<2$.
[b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.
2009 Stanford Mathematics Tournament, 8
Simplify $\sum_{k=1}^{n}\frac{k^2(k - n)}{n^4}$
1998 AMC 12/AHSME, 19
How many triangles have area $ 10$ and vertices at $ (\minus{}5,0)$, $ (5,0)$, and $ (5\cos \theta, 5\sin \theta)$ for some angle $ \theta$?
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 8$
2016 Romania National Olympiad, 4
Let $K$ be a finite field with $q$ elements, $q \ge 3.$ We denote by $M$ the set of polynomials in $K[X]$ of degree $q-2$ whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in $M$ that have $q-2$ distinct roots in $K.$
[i]Marian Andronache[/i]
Ukraine Correspondence MO - geometry, 2007.9
In triangle $ABC$, the lengths of all sides are integers, $\angle B=2 \angle A$ and $\angle C> 90^o$. Find the smallest possible perimeter of this triangle.
2017 Purple Comet Problems, 19
Find the sum of all values of $a + b$, where $(a, b)$ is an ordered pair of positive integers and $a^2+\sqrt{2017-b^2}$ is a perfect square.
2000 JBMO ShortLists, 4
Find all the integers written as $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base $7$.
2021 Harvard-MIT Mathematics Tournament., 9
Let scalene triangle $ABC$ have circumcenter $O$ and incenter $I$. Its incircle $\omega$ is tangent to sides $BC,CA,$ and $AB$ at $D,E,$ and $F$, respectively. Let $P$ be the foot of the altitude from $D$ to $EF$, and let line $DP$ intersect $\omega$ again at $Q \ne D$. The line $OI$ intersects the altitude from $A$ to$ BC$ at $T$. Given that $OI \|BC,$ show that $PQ=PT$.
2015 All-Russian Olympiad, 5
It is known that a cells square can be cut into $n$ equal figures of $k$ cells.
Prove that it is possible to cut it into $k$ equal figures of $n$ cells.
2022 Princeton University Math Competition, 14
Let $\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\vartriangle ABC$, and let $X, Y,Z$ denote the feet of the altitudes from $Q$ to sides $BC$, $CA$, $AB$, respectively. Suppose that $BC = 15$, $\angle ABC = 60^o$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^o$. Let line $QX$ intersect the circumcircle of $\vartriangle XY Z$ at the point $W\ne X$. If the ratio $\frac{ WY}{WZ}$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.
1981 Bundeswettbewerb Mathematik, 2
A [b] bijective[/b] mapping from a plane to itself maps every circle to a circle.
Prove that it maps every line to a line.
2006 Petru Moroșan-Trident, 2
Let be the sequence of sets $ \left(\left\{ A\in\mathcal{M}_2\left(\mathbb{R} \right) | A^{n+1} =2007^nA\right\}\right)_{n\ge 1} . $
[b]a)[/b] Prove that each term of the above sequence hasn't a finite cardinal.
[b]b)[/b] Determine the intersection of the fourth element of the above sequence with the $ 2007\text{th} $ element.
[i]Gheorghe Iurea[/i]
[hide=Note]Similar with [url]https://artofproblemsolving.com/community/c7h1928039p13233629[/url].[/hide]
2008 Balkan MO Shortlist, A3
Let $(a_m)$ be a sequence satisfying $a_n \geq 0$, $n=0,1,2,\ldots$ Suppose there exists $A >0$, $a_m - a_{m+1}$ $\geq A a_m ^2$ for all $m \geq 0$. Prove that there exists $B>0$ such that
\begin{align*} a_n \le \frac{B}{n} \qquad \qquad \text{for }1 \le n \end{align*}
1999 Gauss, 25
In a softball league, after each team has played every other team 4 times, the total accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team received 3 points for a win, 1 point for a tie and no points for a loss, how many games ended in a tie?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 10$
2021 Brazil National Olympiad, 4
Let $d(n)$ be the quantity of positive divisors of $n$, for example $d(1)=1,d(2)=2,d(10)=4$. The [b]size[/b] of $n$ is $k$ if $k$ is the least positive integer, such that $d^k(n)=2$. Note that $d^s(n)=d(d^{s-1}(n))$.
a) How many numbers in the interval $[3,1000]$ have size $2$ ?
b) Determine the greatest size of a number in the interval $[3,1000]$.
2014 Contests, 2
A convex quadrilateral $ABCD$ is inscribed into a circle $\omega$ . Suppose that there is a point $X$ on the segment $AC$ such that the $XB$ and $XD$ tangents to the circle $\omega$ . Tangent of $\omega$ at $C$, intersect $XD$ at $Q$. Let $E$ ($E\ne A$) be the intersection of the line $AQ$ with $\omega$ . Prove that $AD, BE$, and $CQ$ are concurrent.
1983 IMO Longlists, 43
Given a square $ABCD$, let $P, Q, R$, and $S$ be four variable points on the sides $AB, BC, CD$, and $DA$, respectively. Determine the positions of the points $P, Q, R$, and $S$ for which the quadrilateral $PQRS$ is a parallelogram, a rectangle, a square, or a trapezoid.
2011 Sharygin Geometry Olympiad, 1
The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles