Found problems: 1679
1969 IMO Longlists, 27
$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?
2013 NIMO Problems, 2
At a certain school, the ratio of boys to girls is $1:3$. Suppose that:
$\bullet$ Every boy has most $2013$ distinct girlfriends.
$\bullet$ Every girl has at least $n$ boyfriends.
$\bullet$ Friendship is mutual.
Compute the largest possible value of $n$.
[i]Proposed by Evan Chen[/i]
2005 All-Russian Olympiad, 3
We have an acute-angled triangle $ABC$, and $AA',BB'$ are its altitudes. A point $D$ is chosen on the arc $ACB$ of the circumcircle of $ABC$. If $P=AA'\cap BD,Q=BB'\cap AD$, show that the midpoint of $PQ$ lies on $A'B'$.
2015 AMC 10, 6
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller?
$\textbf{(A) }\dfrac54\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }\dfrac95\qquad\textbf{(D) }2\qquad\textbf{(E) }\dfrac52$
Estonia Open Senior - geometry, 2003.2.4
Consider the points $D, E$ and $F$ on the respective sides $BC, CA$ and $AB$ of the triangle $ABC$ in a way that the segments $AD, BE$ and $CF$ have a common point $P$. Let $\frac{|AP|}{|PD|}= x,$ $\frac{|BP|}{|PE|}= y$ and $\frac{|CP|}{|PF|}= z$. Prove that $xyz - (x + y + z) = 2$.
1953 AMC 12/AHSME, 46
Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal of the field and saved a distance equal to $ \frac{1}{2}$ the longer side. The ratio of the shorter side of the rectangle to the longer side was:
$ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{2}{3} \qquad\textbf{(C)}\ \frac{1}{4} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{2}{5}$
Estonia Open Senior - geometry, 2020.2.5
The bisector of the interior angle at the vertex $B$ of the triangle $ABC$ and the perpendicular line on side $BC$ passing through the vertex $C$ intersects at $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $BD$, respectively, with $N$ on the side $AC$. Find all possibilities of the angles of the triangles $ABC$, if it is known that $\frac{| AM |}{| BC |}=\frac{|CD|}{|BD|}$.
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2000 Harvard-MIT Mathematics Tournament, 28
What is the smallest possible volume to surface ratio of a solid cone with height = $1$ unit?
2006 Purple Comet Problems, 1
The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the students are freshmen?
2006 IMO Shortlist, 2
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
2010 Romanian Masters In Mathematics, 3
Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent.
[i]Pavel Kozhevnikov, Russia[/i]
2004 Purple Comet Problems, 5
The number $2.5081081081081 \ldots$ can be written as $m/n$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.
2003 Junior Balkan Team Selection Tests - Moldova, 3
The quadrilateral $ABCD$ with perpendicular diagonals is inscribed in the circle with center $O$, the points $M,N$ are the midpoints of $[BC]$ and $[CD]$ respectively. Find the ratio of areas of the figures $OMCN$ and $ABCD$
2011 Mediterranean Mathematics Olympiad, 2
Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$.
Show that $|A|\cdot|B|\le|C|^2$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
2010 Iran MO (3rd Round), 6
In a triangle $ABC$, $\angle C=45$. $AD$ is the altitude of the triangle. $X$ is on $AD$ such that $\angle XBC=90-\angle B$ ($X$ is in the triangle). $AD$ and $CX$ cut the circumcircle of $ABC$ in $M$ and $N$ respectively. if tangent to circumcircle of $ABC$ at $M$ cuts $AN$ at $P$, prove that $P$,$B$ and $O$ are collinear.(25 points)
the exam time was 4 hours and 30 minutes.
2011 Paraguay Mathematical Olympiad, 2
In a triangle $ABC$, let $D$ and $E$ be the midpoints of $AC$ and $BC$ respectively. The distance from the midpoint of $BD$ to the midpoint of $AE$ is $4.5$. What is the length of side $AB$?
2013 Harvard-MIT Mathematics Tournament, 23
Let $ABCD$ be a parallelogram with $AB=8$, $AD=11$, and $\angle BAD=60^\circ$. Let $X$ be on segment $CD$ with $CX/XD=1/3$ and $Y$ be on segment $AD$ with $AY/YD=1/2$. Let $Z$ be on segment $AB$ such that $AX$, $BY$, and $DZ$ are concurrent. Determine the area of triangle $XYZ$.
2010 AIME Problems, 15
In triangle $ ABC$, $ AC \equal{} 13, BC \equal{} 14,$ and $ AB\equal{}15$. Points $ M$ and $ D$ lie on $ AC$ with $ AM\equal{}MC$ and $ \angle ABD \equal{} \angle DBC$. Points $ N$ and $ E$ lie on $ AB$ with $ AN\equal{}NB$ and $ \angle ACE \equal{} \angle ECB$. Let $ P$ be the point, other than $ A$, of intersection of the circumcircles of $ \triangle AMN$ and $ \triangle ADE$. Ray $ AP$ meets $ BC$ at $ Q$. The ratio $ \frac{BQ}{CQ}$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\minus{}n$.
2008 Germany Team Selection Test, 1
Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \left\{A,B,C\right\}$ can be mapped to the set $ \left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?
Indonesia MO Shortlist - geometry, g6.7
Given triangle $ ABC$ with sidelengths $ a,b,c$. Tangents to incircle of $ ABC$ that parallel with triangle's sides form three small triangle (each small triangle has 1 vertex of $ ABC$). Prove that the sum of area of incircles of these three small triangles and the area of incircle of triangle $ ABC$ is equal to
$ \frac{\pi (a^{2}\plus{}b^{2}\plus{}c^{2})(b\plus{}c\minus{}a)(c\plus{}a\minus{}b)(a\plus{}b\minus{}c)}{(a\plus{}b\plus{}c)^{3}}$
(hmm,, looks familiar, isn't it? :wink: )
1989 APMO, 3
Let $A_1$, $A_2$, $A_3$ be three points in the plane, and for convenience, let $A_4= A_1$, $A_5 = A_2$. For $n = 1$, $2$, and $3$, suppose that $B_n$ is the midpoint of $A_n A_{n+1}$, and suppose that $C_n$ is the midpoint of $A_n B_n$. Suppose that $A_n C_{n+1}$ and $B_n A_{n+2}$ meet at $D_n$, and that $A_n B_{n+1}$ and $C_n A_{n+2}$ meet at $E_n$.
Calculate the ratio of the area of triangle $D_1 D_2 D_3$ to the area of triangle $E_1 E_2 E_3$.
2011 Flanders Math Olympiad, 4
Given is a triangle $ABC$ and points $D$ and $E$, respectively on $] BC [$ and $] AB [$. $F$ it is intersection of lines $AD$ and $CE$. We denote as $| CD | = a, | BD | = b, | DF | = c$ and $| AF | = d$. Determine the ratio $\frac{| BE |}{|AE |}$ in terms of $a, b, c$ and $d$
[img]https://cdn.artofproblemsolving.com/attachments/5/7/856c97045db2d9a26841ad00996a2b809addaa.png[/img]
2008 China Team Selection Test, 1
Let $P$ be an arbitrary point inside triangle $ABC$, denote by $A_{1}$ (different from $P$) the second intersection of line $AP$ with the circumcircle of triangle $PBC$ and define $B_{1},C_{1}$ similarly. Prove that $\left(1 \plus{} 2\cdot\frac {PA}{PA_{1}}\right)\left(1 \plus{} 2\cdot\frac {PB}{PB_{1}}\right)\left(1 \plus{} 2\cdot\frac {PC}{PC_{1}}\right)\geq 8$.
Cono Sur Shortlist - geometry, 2018.G5
We say that a polygon $P$ is inscribed in another polygon $Q$ when all the vertices of $P$ belong to the perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $\ell$ be the largest side of a square inscribed in $T$ and $L$ is the shortest side of a square circumscribed to $T$ . Find the smallest possible value of the ratio $L/\ell$ .
2005 Bulgaria National Olympiad, 2
Consider two circles $k_{1},k_{2}$ touching externally at point $T$. a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$. Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that :
(a) the points $C,T,Y,I$ are concyclic.
(b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$.