Found problems: 1679
2014 Singapore Senior Math Olympiad, 18
Given that in the expansion of $(2+3x)^n$, the coefficients of $x^3$ and $x^4$ are in the ratio $8:15$. Find the value of $n$.
2005 China Team Selection Test, 2
In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.
1999 Poland - Second Round, 3
Let $ABCD$ be a cyclic quadrilateral and let $E$ and $F$ be the points on the sides $AB$ and $CD$ respectively such that $AE : EB = CF : FD$. Point $P$ on the segment EF satsfies $EP : PF = AB : CD$. Prove that the ratio of the areas of $\vartriangle APD$ and $\vartriangle BPC$ does not depend on the choice of $E$ and $F$.
2014 South East Mathematical Olympiad, 2
Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against $a_{j}$ for any $1\leq i<j\leq 4$
2012 Finnish National High School Mathematics Competition, 1
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.
1971 IMO Longlists, 38
Let $A,B,C$ be three points with integer coordinates in the plane and $K$ a circle with radius $R$ passing through $A,B,C$. Show that $AB\cdot BC\cdot CA\ge 2R$, and if the centre of $K$ is in the origin of the coordinates, show that $AB\cdot BC\cdot CA\ge 4R$.
1985 AMC 12/AHSME, 6
One student in a class of boys and girls is chosen to represent the class. Each student is equally likely to be chosen and the probability that a boy is chosen is $ \frac23$ of the probability that a girl is chosen. The ratio of the number of boys to the total number of boys and girls is
$ \textbf{(A)}\ \frac13 \qquad \textbf{(B)}\ \frac25 \qquad \textbf{(C)}\ \frac12 \qquad \textbf{(D)}\ \frac35 \qquad \textbf{(E)}\ \frac23$
1996 AIME Problems, 4
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$
PEN A Problems, 26
Let $m$ and $n$ be arbitrary non-negative integers. Prove that \[\frac{(2m)!(2n)!}{m! n!(m+n)!}\] is an integer.
2006 IberoAmerican Olympiad For University Students, 2
Prove that for any positive integer $n$ and any real numbers $a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n$ we have that the equation
\[a_1 \sin(x) + a_2 \sin(2x) +\cdots+a_n\sin(nx)=b_1 \cos(x)+b_2\cos(2x)+\cdots +b_n \cos(nx)\]
has at least one real root.
2009 South africa National Olympiad, 2
Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?
1955 AMC 12/AHSME, 14
The length of rectangle R is $ 10$ percent more than the side of square S. The width of the rectangle is $ 10$ percent less than the side of the square. The ratio of the areas, R:S, is:
$ \textbf{(A)}\ 99: 100 \qquad
\textbf{(B)}\ 101: 100 \qquad
\textbf{(C)}\ 1: 1 \qquad
\textbf{(D)}\ 199: 200 \qquad
\textbf{(E)}\ 201: 200$
2002 Iran MO (3rd Round), 21
Excircle of triangle $ABC$ corresponding vertex $A$, is tangent to $BC$ at $P$. $AP$ intersects circumcircle of $ABC$ at $D$. Prove \[r(PCD)=r(PBD)\] whcih $r(PCD)$ and $r(PBD)$ are inradii of triangles $PCD$ and $PBD$.
1992 AMC 12/AHSME, 11
The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is
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draw(A--B--C--A);
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[/asy]
$ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 26 $
2008 Irish Math Olympiad, 5
A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$.
Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.
2001 Brazil National Olympiad, 3
$ABC$ is a triangle
$E, F$ are points in $AB$, such that $AE = EF = FB$
$D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$
$AD$ is perpendicular to $CF$.
The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$)
Determine the ratio $\frac{DB}{DC}$.
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1957 Moscow Mathematical Olympiad, 365
(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.$$
(b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$
2024 Sharygin Geometry Olympiad, 15
The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.
2006 China Team Selection Test, 1
Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy:
(a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$.
(b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$.
Find the general term of $\{ a_{n}\}$.
2015 Switzerland Team Selection Test, 12
Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.
1986 Bundeswettbewerb Mathematik, 3
The points $S$ lie on side $AB$, $T$ on side $BC$, and $U$ on side $CA$ of a triangle so that the following applies: $\overline{AS} : \overline{SB} = 1 : 2$, $\overline{BT} : \overline{TC} = 2 : 3$ and $\overline{CU} : \overline{UA} = 3 : 1$. Construct the triangle $ABC$ if only the points $S, T$ and $U$ are given.
2009 Purple Comet Problems, 14
Let $ABCD$ be a trapezoid with $AB$ parallel to $CD, AB$ has length $1,$ and $CD$ has length $41.$ Let points $X$ and $Y$ lie on sides $AD$ and $BC,$ respectively, such that $XY$ is parallel to $AB$ and $CD,$ and $XY$ has length $31.$ Let $m$ and $n$ be two relatively prime positive integers such that the ratio of the area of $ABYX$ to the area of $CDXY$ is $\tfrac{m}{n}.$ Find $m+2n.$
2007 AMC 12/AHSME, 9
Yan is somewhere between his home and the stadium. To get to the stadium he can walk directly to the stadium, or else he can walk home and then ride his bicycle to the stadium. He rides $ 7$ times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the stadium?
$ \textbf{(A)}\ \frac {2}{3}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {4}{5}\qquad \textbf{(D)}\ \frac {5}{6}\qquad \textbf{(E)}\ \frac {6}{7}$
1987 Romania Team Selection Test, 2
Find all positive integers $A$ which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \]
where $m\geq n\geq p \geq 1$ are integer numbers.
[i]Ioan Bogdan[/i]
1972 Dutch Mathematical Olympiad, 5
Given is an acute-angled triangle $ABC$ with angles $\alpha$, $\beta$ and $\gamma$. On side $AB$ lies a point $P$ such that the line connecting the feet of the perpendiculars from $P$ on $AC$ and $BC$ is parallel to $AB$. Express the ratio $\frac{AP}{BP}$ in terms of $\alpha$ and $\beta$.