Found problems: 85335
2012 Korea National Olympiad, 2
Let $ w $ be the incircle of triangle $ ABC $. Segments $ BC, CA $ meet with $ w $ at points $ D, E$. A line passing through $ B $ and parallel to $ DE $ meets $ w $ at $ F $ and $ G $. ($ F $ is nearer to $ B $ than $ G $.) Line $ CG $ meets $ w $ at $ H ( \ne G ) $. A line passing through $ G $ and parallel to $ EH $ meets with line $ AC $ at $ I $. Line $ IF $ meets with circle $ w $ at $ J (\ne F ) $. Lines $ CJ $ and $ EG $ meets at $ K $. Let $ l $ be the line passing through $ K $ and parallel to $ JD $. Prove that $ l, IF, ED $ meet at one point.
1987 ITAMO, 7
A square paper of side $n$ is divided into $n^2$ unit square cells. A maze is drawn on the paper with unit walls between some cells in such a way that one can reach every cell from every other cell not crossing any wall. Find, in terms of $n$, the largest possible total length of the walls.
2019 ELMO Shortlist, C5
Given a permutation of $1,2,3,\dots,n$, with consecutive elements $a,b,c$ (in that order), we may perform either of the [i]moves[/i]:
[list]
[*] If $a$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $b,c,a$ (in that order)
[*] If $c$ is the median of $a$, $b$, and $c$, we may replace $a,b,c$ with $c,a,b$ (in that order)
[/list]
What is the least number of sets in a partition of all $n!$ permutations, such that any two permutations in the same set are obtainable from each other by a sequence of moves?
[i]Proposed by Milan Haiman[/i]
1987 Putnam, B3
Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and
\[
(x,y) = \left( \frac{r^2-1}{r^2+1}, \frac{2r}{r^2+1} \right)
\]
where $r$ runs through the elements of $F$ such that $r^2\neq -1$.
1926 Eotvos Mathematical Competition, 1
Prove that, if $a$ and $b$ are given integers, the system of equatìons
$$x + y + 2z + 2t = a$$
$$2x - 2y + z- t = b$$
has a solution in integers $x, y,z,t$.
2008 AMC 8, 17
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of $50$ units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
$\textbf{(A)}\ 76\qquad
\textbf{(B)}\ 120\qquad
\textbf{(C)}\ 128\qquad
\textbf{(D)}\ 132\qquad
\textbf{(E)}\ 136$
2012 AMC 10, 4
Let $\angle ABC=24^\circ$ and $\angle ABD =20^\circ$. What is the smallest possible degree measure for $\angle CBD$?
$ \textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 6
\qquad\textbf{(E)}\ 12
$
Estonia Open Senior - geometry, 2011.1.5
Given a triangle $ABC$ where $|BC| = a, |CA| = b$ and $|AB| = c$, prove that the equality $\frac{1}{a + b}+\frac{1}{b + c}=\frac{3}{a + b + c}$ holds if and only if $\angle ABC = 60^o$.
2013 Iran MO (3rd Round), 4
We have constructed a rhombus by attaching two equal equilateral triangles. By putting $n-1$ points on all 3 sides of each triangle we have divided the sides to $n$ equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into $2n^2$ equal triangles.
We write the numbers $1,2,\dots,2n^2$ on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments.
(25 points)
[i]Proposed by Amirali Moinfar[/i]
2024 LMT Fall, 8
The LHS Math Team is doing Karaoke. William sings every song, David sings every other song, Peter sings every third song, and Muztaba sings every fourth song. If they sing $600$ songs, find the average number of people singing each song.
2014 ELMO Shortlist, 1
You have some cyan, magenta, and yellow beads on a non-reorientable circle, and you can perform only the following operations:
1. Move a cyan bead right (clockwise) past a yellow bead, and turn the yellow bead magenta.
2. Move a magenta bead left of a cyan bead, and insert a yellow bead left of where the magenta bead ends up.
3. Do either of the above, switching the roles of the words ``magenta'' and ``left'' with those of ``yellow'' and ``right'', respectively.
4. Pick any two disjoint consecutive pairs of beads, each either yellow-magenta or magenta-yellow, appearing somewhere in the circle, and swap the orders of each pair.
5. Remove four consecutive beads of one color.
Starting with the circle: ``yellow, yellow, magenta, magenta, cyan, cyan, cyan'', determine whether or not you can reach
a) ``yellow, magenta, yellow, magenta, cyan, cyan, cyan'',
b) ``cyan, yellow, cyan, magenta, cyan'',
c) ``magenta, magenta, cyan, cyan, cyan'',
d) ``yellow, cyan, cyan, cyan''.
[i]Proposed by Sammy Luo[/i]
2019 Switzerland Team Selection Test, 1
Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$. Lines $EF$ and $AD$ intersect at $G$. Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $BC$ in $M$. Prove that $GM$ and $HK$ are parallel.
1997 Federal Competition For Advanced Students, P2, 2
A positive integer $ K$ is given. Define the sequence $ (a_n)$ by $ a_1\equal{}1$ and $ a_n$ is the $ n$-th natural number greater than $ a_{n\minus{}1}$ which is congruent to $ n$ modulo $ K$.
$ (a)$ Find an explicit formula for $ a_n$.
$ (b)$ What is the result if $ K\equal{}2?$
2023 Ecuador NMO (OMEC), 6
Let $DE$ the diameter of a circunference $\Gamma$. Let $B, C$ on $\Gamma$ such that $BC$ is perpendicular to $DE$, and let $Q$ the intersection of $BC$ with $DE$. Let $P$ a point on segment $BC$ such that $BP=4PQ$. Let $A$ the second intersection of $PE$ with $\Gamma$. If $DE=2$ and $EQ=\frac{1}{2}$, find all possible values of the sides of triangle $ABC$.
2023 All-Russian Olympiad Regional Round, 11.10
Given is a simple connected graph with $2n$ vertices. Prove that its vertices can be colored with two colors so that if there are $k$ edges connecting vertices with different colors and $m$ edges connecting vertices with the same color, then $k-m \geq n$.
Ukrainian TYM Qualifying - geometry, 2018.17
Using a compass and a ruler, construct a triangle $ABC$ given the sides $b, c$ and the segment $AI$, where$ I$ is the center of the inscribed circle of this triangle.
2019 Irish Math Olympiad, 2
Jenny is going to attend a sports camp for $7$ days. Each day, she will play exactly one of three sports: hockey, tennis or camogie. The only restriction is that in any period of $4$ consecutive days, she must play all three sports. Find, with proof, the number of possible sports schedules for Jennys week.
2004 China National Olympiad, 1
For a given real number $a$ and a positive integer $n$, prove that:
i) there exists exactly one sequence of real numbers $x_0,x_1,\ldots,x_n,x_{n+1}$ such that
\[\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}\]
ii) the sequence $x_0,x_1,\ldots,x_n,x_{n+1}$ in i) satisfies $|x_i|\le |a|$ where $i=0,1,\ldots,n+1$.
[i]Liang Yengde[/i]
2005 France Pre-TST, 1
Let $I$ be the incenter of the triangle $ABC$, et let $A',B',C'$ be the symmetric of $I$ with respect to the lines $BC,CA,AB$ respectively. It is known that $B$ belongs to the circumcircle of $A'B'C'$.
Find $\widehat {ABC}$.
Pierre.
2023 Belarus Team Selection Test, 1.1
Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.
2013 Costa Rica - Final Round, 5
Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.
1999 Iran MO (2nd round), 3
Let $A_1,A_2,\cdots,A_n$ be $n$ distinct points on the plane ($n>1$). We consider all the segments $A_iA_j$ where $i<j\leq{n}$ and color the midpoints of them. What's the minimum number of colored points? (In fact, if $k$ colored points coincide, we count them $1$.)
2021-IMOC, G8
Let $P$ be an arbitrary interior point of $\triangle ABC$, and $AP$, $BP$, $CP$ intersect $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Suppose that $M$ be the midpoint of $BC$, $\odot(AEF)$ and $\odot(ABC)$ intersect at $S$, $SD$ intersects $\odot(ABC)$ at $X$, and $XM$ intersects $\odot(ABC)$ at $Y$. Show that $AY$ is tangent to $\odot(AEF)$.
2018 Online Math Open Problems, 30
Let $ABC$ be an acute triangle with $\cos B =\frac{1}{3}, \cos C =\frac{1}{4}$, and circumradius $72$. Let $ABC$ have circumcenter $O$, symmedian point $K$, and nine-point center $N$. Consider all non-degenerate hyperbolas $\mathcal H$ with perpendicular asymptotes passing through $A,B,C$. Of these $\mathcal H$, exactly one has the property that there exists a point $P\in \mathcal H$ such that $NP$ is tangent to $\mathcal H$ and $P\in OK$. Let $N'$ be the reflection of $N$ over $BC$. If $AK$ meets $PN'$ at $Q$, then the length of $PQ$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers such that $c$ is not divisible by the square of any prime. Compute $100a+b+c$.
[i]Proposed by Vincent Huang[/i]
2019 Iran MO (3rd Round), 2
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ :
$$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$.
[i]Proposed by Amirhossein Zolfaghari [/i]