Found problems: 85335
2016 Dutch BxMO TST, 5
Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$
MOAA Gunga Bowls, 2023.24
Circle $\omega$ is inscribed in acute triangle $ABC$. Let $I$ denote the center of $\omega$, and let $D,E,F$ be the points of tangency of $\omega$ with $BC, CA, AB$ respectively. Let $M$ be the midpoint of $BC$, and $P$ be the intersection of the line through $I$ perpendicular to $AM$ and line $EF$. Suppose that $AP=9$, $EC=2EA$, and $BD=3$. Find the sum of all possible perimeters of $\triangle ABC$.
[i]Proposed by Harry Kim[/i]
2020 AMC 10, 18
An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color?
$\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$
1997 Israel National Olympiad, 8
Two equal circles are internally tangent to a larger circle at $A$ and $B$. Let $M$ be a point on the larger circle, and let lines $MA$ and $MB$ intersect the corresponding smaller circles at $A'$ and $B'$. Prove that $A'B'$ is parallel to $AB$.
2006 Federal Math Competition of S&M, Problem 1
In a convex quadrilateral $ABCD$, $\angle BAC=\angle DAC=55^\circ$, $\angle DCA=20^\circ$, and $\angle BCA=15^\circ$. Find the measure of $\angle DBA$.
2011 Bulgaria National Olympiad, 1
Point $O$ is inside $\triangle ABC$. The feet of perpendicular from $O$ to $BC,CA,AB$ are $D,E,F$. Perpendiculars from $A$ and $B$ respectively to $EF$ and $FD$ meet at $P$. Let $H$ be the foot of perpendicular from $P$ to $AB$. Prove that $D,E,F,H$ are concyclic.
2020 Vietnam Team Selection Test, 2
In acute $\triangle ABC$, $O$ is the circumcenter, $I$ is the incenter. The incircle touches $BC,CA,AB$ at $D,E,F$. And the points $K,M,N$ are the midpoints of $BC,CA,AB$ respectively.
a) Prove that the lines passing through $D,E,F$ in parallel with $IK,IM,IN$ respectively are concurrent.
b) Points $T,P,Q$ are the middle points of the major arc $BC,CA,AB$ on $\odot ABC$. Prove that the lines passing through $D,E,F$ in parallel with $IT,IP,IQ$ respectively are concurrent.
2017 Purple Comet Problems, 2
The gure below shows a large square divided into $9$ congruent smaller squares. A shaded square bounded by some of the diagonals of those smaller squares has area $14$. Find the area of the large square.
[img]https://cdn.artofproblemsolving.com/attachments/5/e/bad21be1b3993586c3860efa82ab27d340dbcb.png[/img]
2016 ASDAN Math Tournament, 1
Let $x$ and $y$ be positive real numbers such that $x+y=\tfrac{1}{x}+\tfrac{1}{y}=5$. Compute $x^2+y^2$.
2018 Dutch IMO TST, 3
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.
2024 Baltic Way, 7
A $45 \times 45$ grid has had the central unit square removed. For which positive integers $n$ is it possible to cut the remaining area into $1 \times n$ and $n\times 1$ rectangles?
2010 India IMO Training Camp, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
2014 ASDAN Math Tournament, 6
In triangle $ABC$, we have that $AB=AC$, $BC=16$, and that the area of $\triangle ABC$ is $120$. Compute the length of $AB$.
1986 All Soviet Union Mathematical Olympiad, 437
Prove that the sum of all numbers representable as $\frac{1}{mn}$, where $m,n$ -- natural numbers, $1 \le m < n \le1986$, is not an integer.
2017 China Team Selection Test, 1
Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.
2001 Bulgaria National Olympiad, 3
Let $p$ be a prime number congruent to $3$ modulo $4$, and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$.
Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$.
2001 Bundeswettbewerb Mathematik, 4
A square $ R$ of sidelength $ 250$ lies inside a square $ Q$ of sidelength $ 500$. Prove that: One can always find two points $ A$ and $ B$ on the perimeter of $ Q$ such that the segment $ AB$ has no common point with the square $ R$, and the length of this segment $ AB$ is greater than $ 521$.
2000 All-Russian Olympiad Regional Round, 8.6
The electric train traveled from platform A to platform B in $X$ minutes ($0< X<60$). Find $X$ if it is known that as at the moment departure from A, and at the time of arrival at B, the angle between hourly and the minute hand was equal to $X$ degrees.
2013 JBMO Shortlist, 4
A rectangle in xy Cartesian System is called latticed if all it's vertices have integer coordinates.
a) Find a latticed rectangle of area $2013$, whose sides are not parallel to the axes.
b) Show that if a latticed rectangle has area $2011$, then their sides are parallel to the axes.
2013 Paraguay Mathematical Olympiad, 5
Let $ABC$ be an obtuse triangle, with $AB$ being the largest side.
Draw the angle bisector of $\measuredangle BAC$. Then, draw the perpendiculars to this angle bisector from vertices $B$ and $C$, and call their feet $P$ and $Q$, respectively.
$D$ is the point in the line $BC$ such that $AD \perp AP$.
Prove that the lines $AD$, $BQ$ and $PC$ are concurrent.
1972 Bulgaria National Olympiad, Problem 4
Find maximal possible number of points lying on or inside a circle with radius $R$ in such a way that the distance between every two points is greater than $R\sqrt2$.
[i]H. Lesov[/i]
2013 AMC 10, 9
In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on $20\%$ of her three-point shots and $30\%$ of her two-point shots. Shenille attempted $30$ shots. How many points did she score?
$ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36 $
2023 Math Prize for Girls Problems, 9
The ring shown below is made out of 18 congruent regular hexagons. How many ways are there to tile the ring using tiles that consist of two hexagons, each congruent to any one of the 18 in the design, joined edge-to-edge? (The central hexagon, in black, is not to be covered with a tile and the ring cannot be rotated or reflected.)
2020 Polish Junior MO First Round, 1.
Determine all natural numbers $n$, such that it's possible to insert one digit at the right side of $n$ to obtain $13n$.
2018 PUMaC Algebra B, 4
If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$,
$$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$
find the smallest $n$ such that $a_n < \frac{1}{2018}$.