Found problems: 85335
2016 Latvia National Olympiad, 5
Prove that every triangle can be cut into three pieces so that every piece has axis of symmetry. Show how to cut it (a) using three line segments, (b) using two line segments.
1986 AMC 12/AHSME, 21
In the configuration below, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are line segments and $AB$ is tangent to the circle at $A$.
[asy]
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair A=(0,-1), E=(0,1), C=(0,0), D=dir(10), F=dir(190), B=(-1/sin(10*pi/180))*dir(10);
fill(Arc((0,0),1,10,90)--C--D--cycle,mediumgray);
fill(Arc((0,0),1,190,270)--B--F--cycle,mediumgray);
draw(unitcircle);
draw(A--B--D^^A--E);
label("$A$",A,S);
label("$B$",B,W);
label("$C$",C,SE);
label("$\theta$",C,SW);
label("$D$",D,NE);
label("$E$",E,N);
[/asy]
A necessary and sufficient condition for the equality of the two shaded areas, given $0 < \theta < \frac{\pi}{2}$, is
$ \textbf{(A)}\ \tan \theta = \theta\qquad\textbf{(B)}\ \tan \theta = 2\theta\qquad\textbf{(C)}\ \tan \theta = 4\theta\qquad\textbf{(D)}\ \tan 2\theta = \theta\qquad \\ \textbf{(E)}\ \tan \frac{\theta}{2} = \theta$
2005 India IMO Training Camp, 1
Let $ABCD$ be a convex quadrilateral. The lines parallel to $AD$ and $CD$ through the orthocentre $H$ of $ABC$ intersect $AB$ and $BC$ Crespectively at $P$ and $Q$. prove that the perpendicular through $H$ to th eline $PQ$ passes through th eorthocentre of triangle $ACD$
2020 LIMIT Category 2, 18
Evaluate the following sum: $n \choose 1$ $\sin (a) +$ $n \choose 2$ $\sin (2a) +...+$ $n \choose n$ $\sin (na)$
(A) $2^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$
(B) $2^n \sin^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$
(C) $2^n \sin^n \left(\frac{a}{2}\right)\sin \left(\frac{na}{2}\right)$
(D) $2^n \cos^n \left(\frac{a}{2}\right)\cos \left(\frac{na}{2}\right)$
2012 NZMOC Camp Selection Problems, 2
Let $ABCD$ be a trapezoid, with $AB \parallel CD$ (the vertices are listed in cyclic order). The diagonals of this trapezoid are perpendicular to one another and intersect at $O$. The base angles $\angle DAB$ and $\angle CBA$ are both acute. A point $M$ on the line sgement $OA$ is such that $\angle BMD = 90^o$, and a point $N$ on the line segment $OB$ is such that $\angle ANC = 90^o$. Prove that triangles $OMN$ and $OBA$ are similar.
2022 Putnam, B6
Find all continuous functions $f:\mathbb{R}^+\rightarrow \mathbb{R}^+$ such that $$f(xf(y))+f(yf(x))=1+f(x+y)$$ for all $x, y>0.$
2023 Durer Math Competition (First Round), 3
In a Greek village of $100$ inhabitants in the beginning there lived $12$ Olympians and $88$ humans, they were the first generation. The Olympians are $100\%$ gods while humans are $0\%$ gods. In each generation they formed $50$ couples and each couple had $2$ children, who form the next generation (also consisting of $100$ members). From the second generation onwards, every person’s percentage of godness is the average of the percentages of his/her parents’ percentages. (For example the children of $25\%$ and $12.5\% $gods are $18.75\%$ gods.)
a) Which is the earliest generation in which it is possible that there are equally many $100\%$ gods as $ 0\%$ gods?
b) At most how many members of the fifth generation are at least 25% gods?
2016 Peru Cono Sur TST, P1
How many multiples of $11$ of four digits, of the form $\overline{abcd}$, satisfy that $a\neq b, b\neq c$ and $c\neq a$?
2018 AMC 12/AHSME, 8
Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
$\textbf{(A)} \text{ 25} \qquad \textbf{(B)} \text{ 38} \qquad \textbf{(C)} \text{ 50} \qquad \textbf{(D)} \text{ 63} \qquad \textbf{(E)} \text{ 75}$
2000 Harvard-MIT Mathematics Tournament, 1
How many different ways are there to paint the sides of a tetrahedron with exactly $4$ colors? Each side gets its own color, and two colorings are the same if one can be rotated to get the other.
2021 Lusophon Mathematical Olympiad, 5
There are 3 lines $r, s$ and $t$ on a plane. The lines $r$ and $s$ intersect perpendicularly at point $A$. the line $t$ intersects the line $r$ at point $B$ and the line $s$ at point $C$. There exist exactly 4 circumferences on the plane that are simultaneously tangent to all those 3 lines.
Prove that the radius of one of those circumferences is equal to the sum of the radius of the other three circumferences.
2017 Taiwan TST Round 2, 6
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$.
[list=a]
[*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
[*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$.
[/list]
2005 Vietnam Team Selection Test, 3
Find all functions $f: \mathbb{Z} \mapsto \mathbb{Z}$ satisfying the condition: $f(x^3 +y^3 +z^3 )=f(x)^3+f(y)^3+f(z)^3.$
1988 Bundeswettbewerb Mathematik, 3
Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.
2022 MOAA, 14
Find the greatest prime number $p$ for which there exists a prime number $q$ such that $p$ divides $4^q + 1$ and $q$ divides $4^p + 1$.
2017 Taiwan TST Round 2, 1
Determine all surjective functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right) $$ for all $ x,y,z $ in $ \mathbb{Z} $
2018 BMT Spring, 5
Alice and Bob play a game where they start from a complete graph with $n$ vertices and take turns removing a single edge from the graph, with Alice taking the fi rst turn. The first player to disconnect the graph loses. Compute the sum of all $n$ between $2$ and $100$ inclusive such that Alice has a winning strategy. (A complete graph is one where there is an edge between every pair of vertices.)
2021 LMT Spring, A4 B11
Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Proposed by Alex Li[/i]
2008 National Olympiad First Round, 25
Let $C$ and $D$ be points on the circle with center $O$ and diameter $[AB]$ where $C$ and $D$ are on different semicircles with diameter $[AB]$. Let $H$ be the foot perpendicular from $B$ to $[CD]$. If $|AO|=13$, $|AC|=24$, and $|HD|=12$, what is $\widehat{DCB}$ in degrees?
$
\textbf{(A)}\ 30
\qquad\textbf{(B)}\ 45
\qquad\textbf{(C)}\ 60
\qquad\textbf{(D)}\ 75
\qquad\textbf{(E)}\ 80
$
1997 Hungary-Israel Binational, 3
Can a closed disk can be decomposed into a union of two congruent parts having no common point?
2022 CCA Math Bonanza, L3.3
Determine the sum of all positive integers $n<100$ satisfying the following expression. \[\sum_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod \;{10^{k+1})}-n \;(\bmod \;{10^k)}\right)=\prod_{k=0}^{\lfloor{\log_{10} n}\rfloor}\frac{1}{10^k}\left(n \; (\bmod\; 10^{k+1})-n \;(\bmod\; 10^k)\right).\] Here, $\textstyle\sum$ and $\textstyle\prod$ represent sum and product, respectively.
[i]2022 CCA Math Bonanza Lightning Round 3.3[/i]
2013 Cuba MO, 9
Let ABC be a triangle with $\angle A = 90^o$, $\angle B = 75^o$, and $AB = 2$. Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$. Calculate the lenght of $QA$.
2000 Taiwan National Olympiad, 2
Let $n$ be a positive integer and $A=\{ 1,2,\ldots ,n\}$. A subset of $A$ is said to be connected if it consists of one element or several consecutive elements. Determine the maximum $k$ for which there exist $k$ distinct subsets of $A$ such that the intersection of any two of them is connected.
2022 AIME Problems, 10
Find the remainder when $$\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}$$ is divided by $1000$.
2016 Kyrgyzstan National Olympiad, 6
Given three pairwise tangent equal circles $\Omega_i (i=1,2,3)$ with radius $r$.
The circle $\Gamma $ touches the three circles internally (circumscribed about 3 circles).The three equal circles $\omega_i (i=1,2,3)$ with radius $x$ touches $\Omega_i$ and $\Omega_{i+1}$ externally ($\Omega_4= \Omega_1$) and touches $\Gamma$ internally.Find $x$ in terms of $r$