Found problems: 25757
1976 Bulgaria National Olympiad, Problem 5
It is given a tetrahedron $ABCD$ and a plane $\alpha$ intersecting the three edges passing through $D$. Prove that $\alpha$ divides the surface of the tetrahedron into two parts proportional to the volumes of the bodies formed if and only if $\alpha$ is passing through the center of the inscribed tetrahedron sphere.
1999 All-Russian Olympiad Regional Round, 9.2
In triangle $ABC$, on side $AC$ there are points $D$ and $E$, that $AB = AD$ and $BE = EC$ ($E$ between $A$ and $D$). Point $F$ is midpoint of arc $BC$ of circumcircle of triangle $ABC$. Prove that the points $B, E, D, F$ lie on the same circle.
2007 F = Ma, 32
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$.
The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$.
Find an expression for $\beta$ in terms of $k$.
$ \textbf{(A)}\ 1+k^2$
$ \textbf{(B)}\ \sqrt{1+k^2}$
$ \textbf{(C)}\ \sqrt{\frac{k}{1+k}}$
$ \textbf{(D)}\ \sqrt{\frac{k^2}{1+k}}$
$ \textbf{(E)}\ \text{none of the above}$
2009 AMC 10, 10
Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));
pair D=foot(B,A,C);
pair[] ps={B,C,A,D};
draw(A--B--C--cycle);
draw(B--D);
draw(rightanglemark(B,D,C));
dot(ps);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$3$",midpoint(A--D),NE);
label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad
\textbf{(B)}\ 7\sqrt3 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 14\sqrt3 \qquad
\textbf{(E)}\ 42$
1985 ITAMO, 2
When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?
1997 Poland - Second Round, 6
Let eight points be given in a unit cube. Prove that two of these points are on a distance not greater than $1$.
2014 Harvard-MIT Mathematics Tournament, 8
Let $ABC$ be a triangle with sides $AB = 6$, $BC = 10$, and $CA = 8$. Let $M$ and $N$ be the midpoints of $BA$ and $BC$, respectively. Choose the point $Y$ on ray $CM$ so that the circumcircle of triangle $AMY$ is tangent to $AN$. Find the area of triangle $NAY$.
Geometry Mathley 2011-12, 15.2
Let $O$ be the centre of the circumcircle of triangle $ABC$. Point $D$ is on the side $BC$. Let $(K)$ be the circumcircle of $ABD$. $(K)$ meets $AO$ at $E$ that is distinct from $A$.
(a) Prove that $B,K,O,E$ are on the same circle that is called $(L)$.
(b) $(L)$ intersects $AB$ at $F$ distinct $B$. Point $G$ is on $(L)$ such that $EG \parallel OF$. $GK$ meets $AD$ at $S, SO$ meets $BC$ at $T$ . Prove that $O,E, T,C$ are on the same circle.
Trần Quang Hùng
1962 AMC 12/AHSME, 39
Two medians of a triangle with unequal sides are $ 3$ inches and $ 6$ inches. Its area is $ 3 \sqrt{15}$ square inches. The length of the third median in inches, is:
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 3 \sqrt{3} \qquad
\textbf{(C)}\ 3 \sqrt{6} \qquad
\textbf{(D)}\ 6 \sqrt{3} \qquad
\textbf{(E)}\ 6 \sqrt{6}$
2018 USA TSTST, 9
Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$ has total area at most $c$.
[i]Linus Hamilton[/i]
2010 HMNT, 5
Circle $O$ has chord $AB$. A circle is tangent to $O$ at $T$ and tangent to$ AB$ at $X$ such that $AX = 2XB$. What is $\frac{AT}{BT}$ ?
1961 IMO, 5
Construct a triangle $ABC$ if $AC=b$, $AB=c$ and $\angle AMB=w$, where $M$ is the midpoint of the segment $BC$ and $w<90$. Prove that a solution exists if and only if \[ b \tan{\dfrac{w}{2}} \leq c <b \] In what case does the equality hold?
1994 National High School Mathematics League, 3
Circumcircle of $\triangle ABC$ is $\odot O$, incentre of $\triangle ABC$ is $I$. $\angle B=60^{\circ}.\angle A<\angle C$. Bisector of outer angle $\angle A$ intersects $\odot O$ at $E$. Prove:
[b](a)[/b] $IO=AE$.
[b](b)[/b] The radius of $\odot O$ is $R$, then $2R<IO+IA+IC<(1+\sqrt3)R$.
2025 China Team Selection Test, 8
Let quadrilateral $A_1A_2A_3A_4$ be not cyclic and haves edges not parallel to each other.
Denote $B_i$ as the intersection of the tangent line at $A_i$ with respect to circle $A_{i-1}A_iA_{i+1}$ and the $A_{i+2}$-symmedian with respect to triangle $A_{i+1}A_{i+2}A_{i+3}$ and $C_i$ as the intersection of lines $A_iA_{i+1}$ and $B_iB_{i+1}$, where all indexes taken cyclically.
Prove that $C_1$, $C_2$, $C_3$, and $C_4$ are collinear.
2016 Nordic, 2
Let $ABCD$ be a cyclic quadrilateral satysfing $AB=AD$ and $AB+BC=CD$. Determine $\measuredangle CDA$.
2022-2023 OMMC, 16
Let $ABCD$ be an isosceles trapezoid with $AB=5$, $CD = 8$, and $BC = DA = 6$. There exists an angle $\theta$ such that there is only one point $X$ satisfying $\angle AXD = 180^{\circ} - \angle BXC = \theta$. Find $\sin(\theta)^2$.
2010 IFYM, Sozopol, 5
We are given $\Delta ABC$, for which the excircle to side $BC$ is tangent to the continuations of $AB$ and $AC$ in points $E$ and $F$ respectively. Let $D$ be the reflection of $A$ in line $EF$. If it is known that $\angle BAC=2\angle BDC$, then determine $\angle BAC$.
2014 Saudi Arabia Pre-TST, 2.4
Let $ABC$ be an acute triangle with $\angle A < \angle B \le \angle C$, and $O$ its circumcenter. The perpendicular bisector of side $AB$ intersects side $AC$ at $D$. The perpendicular bisector of side $AC$ intersects side $AB$ at $E$. Express the angles of triangle $DEO$ in terms of the angles of triangle $ABC$.
2022 IFYM, Sozopol, 5
Let $\Delta ABC$ be an acute scalene triangle with $AC<BC$, an orthocenter $H$ and altitudes $AE$, $BF$. The points $E'$ and $F'$ are symmetrical to $E$ and $F$ with respect to $A$ and $B$ respectively. Point $O$ is the center of the circumscribed circle of $ABC$ and $M$ is the midpoint of $AB$. Let $N$ be the midpoint of $OM$. Prove that the tangent through $H$ to the circumscribed circle of $\Delta E'HF'$ is perpendicular to line $CN$.
1998 Harvard-MIT Mathematics Tournament, 6
circle is inscribed in an equilateral triangle of side length $1$. Tangents to the circle are drawn that cut off equilateral triangles at each corner. Circles are inscribed in each of these equilateral triangles. If this process is repeated infinitely many times, what is the sum of the areas of all the circles?
[img]https://cdn.artofproblemsolving.com/attachments/c/e/ef4000989155708db8cfa674dd00857afb9919.png[/img]
1999 Chile National Olympiad, 2
In an acute triangle $ABC$, let $ \overline {AK}, \overline {BL}, \overline {CM} $ be the altitudes of the triangle concurrent at the point $ H $ and let $ P $ the midpoint of $ \overline {AH} $. Let's define $ S = \overline {BH} \cap \overline {MK} $ and $ T = \overline {LP} \cap \overline {AB} $. Show that $ \overline {TS} \perp \overline {BC} $
1999 Harvard-MIT Mathematics Tournament, 11
Circles $C_1$, $C_2$, $C_3$ have radius $ 1$ and centers $O, P, Q$ respectively. $C_1$ and $C_2$ intersect at $A$, $C_2$ and $C_3$ intersect at $B$, $C_3$ and $C_1$ intersect at $C$, in such a way that $\angle APB = 60^o$ , $\angle BQC = 36^o$ , and $\angle COA = 72^o$ . Find angle $\angle ABC$ (degrees).
2004 IMO, 1
1. Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.
2021 Yasinsky Geometry Olympiad, 5
In triangle $ABC$, point $I$ is the center of the inscribed circle. $AT$ is a segment tangent to the circle circumscribed around the triangle $BIC$ . On the ray $AB$ beyond the point$ B$ and on the ray $AC$ beyond the point $C$, we draw the segments $BD$ and $CE$, respectively, such that $BD = CE = AT$. Let the point $F$ be such that $ABFC$ is a parallelogram. Prove that points $D, E$ and $F$ lie on the same line.
(Dmitry Prokopenko)
2017 Princeton University Math Competition, A3/B5
A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.