Found problems: 1679
2012 Math Prize For Girls Problems, 9
Bianca has a rectangle whose length and width are distinct primes less than 100. Let $P$ be the perimeter of her rectangle, and let $A$ be the area of her rectangle. What is the least possible value of $\frac{P^2}{A}$?
1993 AMC 12/AHSME, 17
Amy painted a dart board over a square clock face using the "hour positions" as boundaries. [See figure.] If $t$ is the area of one of the eight triangular regions such as that between $12$ o'clock and $1$ o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between $1$ o'clock and $2$ o'clock, then $\frac{q}{t}=$
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draw((0,3.1)--(4,.9)--(4,3.1)--(0,.9)--(0,2)--(4,2));
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$ \textbf{(A)}\ 2\sqrt{3}-2 \qquad\textbf{(B)}\ \frac{3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{5}+1}{2} \qquad\textbf{(D)}\ \sqrt{3} \qquad\textbf{(E)}\ 2 $
2007 AMC 10, 23
A pyramid with a square base is cut by a plane that is parallel to its base and is $ 2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 2 \plus{} \sqrt{2}\qquad
\textbf{(C)}\ 1 \plus{} 2\sqrt{2}\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ 4 \plus{} 2\sqrt{2}$
1993 All-Russian Olympiad Regional Round, 10.7
Points $ M,N$ are taken on sides $ BC,CD$ respectively of parallelogram $ ABCD$. Let $ E\equal{}BD\cap AM, F\equal{}BD\cap AN$. Diagonal $ BD$ cuts triangle $ AMN$ into two parts. Prove that these two parts have equal area if and only if the point $ K$ given by $ EK\parallel{}AD, FK\parallel{}AB$ lies on segment $ MN$.
2013 Sharygin Geometry Olympiad, 7
Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Find the ratio of the greatest and the smallest radii of the spheres.
2015 AMC 10, 11
The ratio of the length to the width of a rectangle is $4:3$. If the rectangle has diagonal of length $d$, then the area may be expressed as $kd^2$ for some constant $k$. What is $k$?
$\textbf{(A) }\dfrac27\qquad\textbf{(B) }\dfrac37\qquad\textbf{(C) }\dfrac{12}{25}\qquad\textbf{(D) }\dfrac{16}{25}\qquad\textbf{(E) }\dfrac34$
2005 Slovenia National Olympiad, Problem 3
Suppose that a triangle $ABC$ with incenter $I$ satisfies $CA+AI=BC$. Find the ratio between the measures of the angles $\angle BAC$ and $\angle CBA$.
2020 Ukrainian Geometry Olympiad - December, 2
Let $ABCD$ be a cyclic quadrilateral such that $AC =56, BD = 65, BC>DA$ and $AB: BC =CD: DA$. Find the ratio of areas $S (ABC): S (ADC)$.
2013 Today's Calculation Of Integral, 875
Evaluate $\int_0^1 \frac{x^2+x+1}{x^4+x^3+x^2+x+1}\ dx.$
1963 AMC 12/AHSME, 22
Acute-angled triangle $ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 120$ and $\stackrel \frown {BC} = 72$. A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Then the ratio of the magnitudes of angles $OBE$ and $BAC$ is:
$\textbf{(A)}\ \dfrac{5}{18} \qquad
\textbf{(B)}\ \dfrac{2}{9} \qquad
\textbf{(C)}\ \dfrac{1}{4} \qquad
\textbf{(D)}\ \dfrac{1}{3} \qquad
\textbf{(E)}\ \dfrac{4}{9}$
2006 Tuymaada Olympiad, 4
For a positive integer, we define it's [i]set of exponents[/i] the unordered list of all the exponents of the primes, in it`s decomposition. For example, $18=2\cdot 3^{2}$ has it`s set of exponents $1,2$ and $300=2^{2}\cdot 3\cdot 5^{2}$ has it`s set of exponents $1,2,2$. There are given two arithmetical progressions $\big(a_{n}\big)_{n}$ and $\big(b_{n}\big)_{n}$, such that for any positive integer $n$, $a_{n}$ and $b_{n}$ have the same set of exponents. Prove that the progressions are proportional (that is, there is $k$ such that $a_{n}=kb_{n}$ for any $n$).
[i]Proposed by A. Golovanov[/i]
2013 Moldova Team Selection Test, 3
Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.
1998 Harvard-MIT Mathematics Tournament, 4
Given that $r$ and $s$ are relatively prime positive integers such that $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$, find $r$ and $s$.
2008 Korea Junior Math Olympiad, 1
In a $\triangle XYZ$, points $A,B$ lie on segment $ZX, C,D$ lie on segment $XY , E, F$ lie on segment $YZ$. $A, B, C, D$ lie on a circle, and $\frac{AZ \cdot EY \cdot ZB \cdot Y F}{EZ \cdot CY \cdot ZF \cdot Y D}= 1$ . Let $L = ZX \cap DE$, $M = XY \cap AF$, $N = Y Z \cap BC$. Prove that $L,M,N$ are collinear.
2008 USAMO, 5
Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.
Croatia MO (HMO) - geometry, 2019.7
On the side $AB$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that diagonal $AC$ bisects the segment $DX$, and the diagonal $BD$ bisects the segment $CX$. What is the smallest possible ratio $|AB | : |CD|$ in such a quadrilateral ?
2016 Romania National Olympiad, 2
In a cube $ABCDA'B'C'D' $two points are considered, $M \in (CD')$ and $N \in (DA')$. Show that the $MN$ is common perpendicular to the lines $CD'$ and $DA'$ if and only if $$\frac{D'M}{D'C}=\frac{DN}{DA'} =\frac{1}{3}.$$
2006 Switzerland Team Selection Test, 3
An airport contains 25 terminals which are two on two connected by tunnels. There is exactly 50 main tunnels which can be traversed in the two directions, the others are with single direction. A group of four terminals is called [i]good[/i] if of each terminal of the four we can arrive to the 3 others by using only the tunnels connecting them. Find the maximum number of good groups.
2009 USA Team Selection Test, 4
Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \angle PAB \equal{} \angle CQB \equal{} 45^\circ$, $ \angle ABP \equal{} \angle QBC \equal{} 75^\circ$, $ \angle RAC \equal{} 105^\circ$, and $ RQ^2 \equal{} 6CM^2$, compute $ AC^2/AR^2$.
[i]Zuming Feng.[/i]
Ukrainian TYM Qualifying - geometry, I.10
Given a circle of radius $R$. Find the ratio of the largest area of the circumscribed quadrilateral to the smallest area of the inscribed one.
2006 China Team Selection Test, 1
$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$).
Prove that $l_1 || l_2$.
2014 Bosnia And Herzegovina - Regional Olympiad, 3
Excircle of triangle $ABC$ to side $AB$ of triangle $ABC$ touches side $AB$ in point $D$. Determine ratio $AD : BD$ if $\angle CAB = 2 \angle ADC$
2008 Harvard-MIT Mathematics Tournament, 6
Determine the value of $ \lim_{n\rightarrow\infty}\sum_{k \equal{} 0}^n\binom{n}{k}^{ \minus{} 1}$.
1998 Korea - Final Round, 1
Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.
2025 Euler Olympiad, Round 1, 4
Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$, such that $$CD = DB = 2AC.$$ Find the ratio $\frac{CM}{MD}$, where $M$ is a point on arc $AB$, such that $\angle CMD$ is maximized.
[img]https://i.imgur.com/NfjRpgP.png[/img]
[i]
Proposed by Andria Gvaramia, Georgia [/i]