Found problems: 25757
1996 Tournament Of Towns, (497) 4
Is it possible to tile space using a combination of regular tetrahedra and regular octahedra?
(A Belov)
2005 Tuymaada Olympiad, 7
Let $I$ be the incentre of triangle $ABC$. A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$. Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$.
[i]Proposed by S. Berlov[/i]
2001 Tournament Of Towns, 7
The vertices of a triangle have coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$. For any integers $h$ and $k$, not both 0, both triangles whose vertices have coordinates $(x_1+h,y_1+k),(x_2+h,y_2+k)$ and $(x_3+h,y_3+k)$ has no common interior points with the original triangle.
(a) Is it possible for the area of this triangle to be greater than $\tfrac{1}{2}$?
(b) What is the maximum area of this triangle?
2011 Princeton University Math Competition, B1
Let triangle $ABC$ have $\angle A = 70^\circ, \angle B = 60^\circ$, and $\angle C = 50^\circ$. Extend altitude $BH$ past $H$ to point $D$ so that $BD = BC$. Find $\angle BDA$ in degrees.
2017 Azerbaijan EGMO TST, 3
In $\bigtriangleup$$ABC$ $BL$ is bisector. Arbitrary point $M$ on segment $CL$ is chosen. Tangent to $\odot$$(ABC)$ at $B$ intersects $CA$ at $P$. Tangents to $\odot$$BLM$ at $B$ and $M$ intersect at point $Q$. Prove that $PQ$$\parallel$$BL$.
1998 AMC 8, 15
Problems $15, 16$, and $17$ all refer to the following:
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In $1998$ the number of people on these islands is only 200, but the population triples every $25$ years. Queen Irene has decreed that there must be at least $1.5$ square miles for every person living in the Isles. The total area of the Nisos Isles is $24,900$ square miles.
15. Estimate the population of Nisos in the year $2050$.
$ \text{(A)}\ 600\qquad\text{(B)}\ 800\qquad\text{(C)}\ 1000\qquad\text{(D)}\ 2000\qquad\text{(E)}\ 3000 $
2016 Abels Math Contest (Norwegian MO) Final, 3b
Let $ABC$ be an acute triangle with $AB < AC$. The points $A_1$ and $A_2$ are located on the line $BC$ so that $AA_1$ and $AA_2$ are the inner and outer angle bisectors at $A$ for the triangle $ABC$. Let $A_3$ be the mirror image $A_2$ with respect to $C$, and let $Q$ be a point on $AA_1$ such that $\angle A_1QA_3 = 90^o$. Show that $QC // AB$.
2024 Bulgarian Autumn Math Competition, 11.2
Let $ABC$ be a triangle with $\angle ABC = 60^{\circ}$. Find the angles of the triangle if $\angle BHI = 60^{\circ}$, where $H$ and $I$ are the orthocenter and incenter of $ABC$
2009 AMC 12/AHSME, 23
A region $ S$ in the complex plane is defined by \[ S \equal{} \{x \plus{} iy: \minus{} 1\le x\le1, \minus{} 1\le y\le1\}.\] A complex number $ z \equal{} x \plus{} iy$ is chosen uniformly at random from $ S$. What is the probability that $ \left(\frac34 \plus{} \frac34i\right)z$ is also in $ S$?
$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac23\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac79\qquad \textbf{(E)}\ \frac78$
2021 Bundeswettbewerb Mathematik, 3
Consider a triangle $ABC$ with $\angle ACB=120^\circ$. Let $A’, B’, C’$ be the points of intersection of the angular bisector through $A$, $B$ and $C$ with the opposite side, respectively.
Determine $\angle A’C’B’$.
Russian TST 2018, P2
Inside the acute-angled triangle $ABC$, the points $P{}$ and $Q{}$ are chosen so that $\angle ACP = \angle BCQ$ and $\angle CBP =\angle ABQ$. The point $Z{}$ is the projection of $P{}$ onto the line $BC$. The point $Q'$ is symmetric to $Q{}$ with respect to $Z{}$. The points $K{}$ and $L{}$ are chosen on the rays $AB$ and $AC$ respectively, so that $Q'K \parallel QC$ and $Q'L \parallel QB$. Prove that $\angle KPL=\angle BPC$.
2012 Tournament of Towns, 6
We attempt to cover the plane with an infi nite sequence of rectangles, overlapping allowed.
(a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$?
(b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?
2013 AMC 10, 16
In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$
[asy]
unitsize(75);
pathpen = black; pointpen=black;
pair A = MP("A", D((0,0)), dir(200));
pair B = MP("B", D((2,0)), dir(-20));
pair C = MP("C", D((1/2,1)), dir(100));
pair D = MP("D", D(midpoint(B--C)), dir(30));
pair E = MP("E", D(midpoint(A--B)), dir(-90));
pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013);
draw(A--B--C--cycle);
draw(A--D--E--C);
[/asy]
$\textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 13.5 \qquad
\textbf{(C)}\ 14 \qquad
\textbf{(D)}\ 14.5 \qquad
\textbf{(E)}\ 15 $
2015 Iran Geometry Olympiad, 5
Do there exist $6$ circles in the plane such that every circle passes through centers of exactly $3$ other circles?
by Morteza Saghafian
2014 PUMaC Geometry A, 2
Triangle $ABC$ has lengths $AB=20$, $AC=14$, $BC=22$. The median from $B$ intersects $AC$ at $M$ and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\dfrac pq=\dfrac{[AMPN]}{[ABC]}$ for positive integers $p$, $q$ coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p+q$.
the 7th XMO, 1
As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]
2024 ITAMO, 2
We are given a unit square in the plane. A point $M$ in the plane is called [i]median [/i]if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$.
Determine the geometric locus of all median points.
Durer Math Competition CD 1st Round - geometry, 2014.D3
$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$, different from $C$. What is the length of the segment $IF$?
2019 Sharygin Geometry Olympiad, 5
Let $AA_1, BB_1, CC_1$ be the altitudes of triangle $ABC$, and $A0, C0$ be the common points of the circumcircle of triangle $A_1BC_1$ with the lines $A_1B_1$ and $C_1B_1$ respectively. Prove that $AA_0$ and $CC_0$ meet on the median of ABC or are parallel to it
2005 Estonia Team Selection Test, 1
On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·
Durer Math Competition CD Finals - geometry, 2013.D3
The circle circumscribed to the triangle $ABC$ is $k$. The altitude $AT$ intersects circle $k$ at $P$. The perpendicular from $P$ on line $AB$ intersects is at $R$. Prove that line $TR$ is parallel to the tangent of the circle $k$ at point $A$.
2013 Harvard-MIT Mathematics Tournament, 1
Jarris the triangle is playing in the $(x, y)$ plane. Let his maximum $y$ coordinate be $k$. Given that he has side lengths $6$, $8$, and $10$ and that no part of him is below the $x$-axis, find the minimum possible value of $k$.
1977 Chisinau City MO, 139
Let $\beta$ be the length of the bisector of angle $B$, and $a', c'$ be the lengths of the segments into which this bisector divides the side $AC$ of the triangle $ABC$. Prove the relation $\beta^2 = ac-a'c'$ and derive from this the formula $\beta^2=ac-\frac{b^2ac}{(a+c)^2}$.
2016 Korea National Olympiad, 3
Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$. A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$. Let $BC \cap AP = D$, $CA \cap BP = E$, $AB \cap CP= F$.
Let $T$ be the area of $\triangle DEF$. Prove the following inequality.
$$ \left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2 $$
2010 Balkan MO Shortlist, G7
A triangle $ABC$ is given. Let $M$ be the midpoint of the side $AC$ of the triangle and $Z$ the image of point $B$ along the line $BM$. The circle with center $M$ and radius $MB$ intersects the lines $BA$ and $BC$ at the points $E$ and $G$ respectively. Let $H$ be the point of intersection of $EG$ with the line $AC$, and $K$ the point of intersection of $HZ$ with the line $EB$. The perpendicular from point $K$ to the line $BH$ intersects the lines $BZ$ and $BH$ at the points $L$ and $N$, respectively.
If $P$ is the second point of intersection of the circumscribed circles of the triangles $KZL$ and $BLN$, prove that, the lines $BZ, KN$ and $HP$ intersect at a common point.