Found problems: 25757
Estonia Open Junior - geometry, 2015.1.5
Let $ABC$ be an acute triangle. The arcs $AB$ and $AC$ of the circumcircle of the triangle are reflected over the lines AB and $AC$, respectively. Prove that the two arcs obtained intersect in another point besides $A$.
2005 AIME Problems, 10
Triangle $ABC$ lies in the Cartesian Plane and has an area of 70. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20)$, respectively, and the coordinates of $A$ are $(p,q)$. The line containing the median to side $BC$ has slope $-5$. Find the largest possible value of $p+q$.
2017 CMIMC Geometry, 2
Triangle $ABC$ has an obtuse angle at $\angle A$. Points $D$ and $E$ are placed on $\overline{BC}$ in the order $B$, $D$, $E$, $C$ such that $\angle BAD=\angle BCA$ and $\angle CAE=\angle CBA$. If $AB=10$, $AC=11$, and $DE=4$, determine $BC$.
1989 Polish MO Finals, 2
Three circles of radius $a$ are drawn on the surface of a sphere of radius $r$. Each pair of circles touches externally and the three circles all lie in one hemisphere. Find the radius of a circle on the surface of the sphere which touches all three circles.
2007 Today's Calculation Of Integral, 177
On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin.
Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$
Let $S_{1}$ be the area of the region surrounded by the line passing through two points $P,\ Q$ and $K,$ or $S_{2}$ be the area of the region surrounded by the line which is passing through $P$ and parallel to $x$ axis and $K.$ Find the value of $\frac{S_{1}}{S_{2}}.$
1981 All Soviet Union Mathematical Olympiad, 308
Given real $a$. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities $$y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a$$
2019 Taiwan TST Round 3, 1
For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that [b]for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime. [/b] Find the minimal possible value of the maximum of all numbers.
2008 Harvard-MIT Mathematics Tournament, 28
Let $ P$ be a polyhedron where every face is a regular polygon, and every edge has length $ 1$. Each vertex of $ P$ is incident to two regular hexagons and one square. Choose a vertex $ V$ of the polyhedron. Find the volume of the set of all points contained in $ P$ that are closer to $ V$ than to any other vertex.
2021 Latvia Baltic Way TST, P11
Incircle of $\triangle ABC$ has centre $I$ and touches sides $AC, AB$ at $E,F$, respectively. The perpendicular bisector of segment $AI$ intersects side $AC$ at $P$. On side $AB$ a point $Q$ is chosen so that $QI \perp FP$. Prove that $EQ \perp AB$.
2002 Moldova National Olympiad, 3
In a triangle $ ABC$, the angle bisector at $ B$ intersects $ AC$ at $ D$ and the circumcircle again at $ E$. The circumcircle of the triangle $ DAE$ meets the segment $ AB$ again at $ F$. Prove that the triangles $ DBC$ and $ DBF$ are congruent.
1998 Harvard-MIT Mathematics Tournament, 10
Lukas is playing pool on a table shaped like an equilateral triangle.
The pockets are at the corners of the triangle and are labeled $C$, $H$, and $T$. Each side of the table is $16$ feet long. Lukas shoots a ball from corner $C$ of the table in such a way that on the second bounce, the ball hits $2$ feet away from him along side $CH$.
a. How many times will the ball bounce before hitting a pocket?
b. Which pocket will the ball hit?
c. How far will the ball travel before hitting the pocket?
Indonesia MO Shortlist - geometry, g5
Two circles intersect at points $A$ and $B$. The line $\ell$ through A intersects the circles at $C$ and $D$, respectively. Let $M, N$ be the midpoints of arc $BC$ and arc $BD$. which does not contain $A$, and suppose that $K$ is the midpoint of the segment $CD$ . Prove that $\angle MKN=90^o$.
2007 Pre-Preparation Course Examination, 3
$ABC$ is an arbitrary triangle. $A',B',C'$ are midpoints of arcs $BC, AC, AB$. Sides of triangle $ABC$, intersect sides of triangle $A'B'C'$ at points $P,Q,R,S,T,F$. Prove that \[\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}\]
2016 Saudi Arabia BMO TST, 2
Let $ABC$ be a triangle with $AB \ne AC$. The incirle of triangle $ABC$ is tangent to $BC, CA, AB$ at $D, E, F$, respectively. The perpendicular line from $D$ to $EF$ intersects $AB$ at $X$. The second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX \perp T F$
2002 ITAMO, 2
The plan of a house has the shape of a capital $L$, obtained by suitably placing side-by-side four squares whose sides are $10$ metres long. The external walls of the house are $10$ metres high. The roof of the house has six faces, starting at the top of the six external walls, and each face forms an angle of $30^\circ$ with respect to a horizontal plane.
Determine the volume of the house (that is, of the solid delimited by the six external walls, the six faces of the roof, and the base of the house).
2007 Croatia Team Selection Test, 3
Let $ABC$ be a triangle such that $|AC|>|AB|$. Let $X$ be on line $AB$ (closer to $A$) such that $|BX|=|AC|$ and let $Y$ be on the segment $AC$ such that $|CY|=|AB|$. Intersection of lines $XY$ and bisector of $BC$ is point $P$. Prove that $\angle BPC+\angle BAC = 180^\circ$.
1987 Traian Lălescu, 2.2
Construct a convex quadrilateral given two opposite angles and sides.
1975 All Soviet Union Mathematical Olympiad, 216
For what $k$ is it possible to construct a cube $k\times k\times k$ of the black and white cubes $1\times 1\times 1$ in such a way that every small cube has the same colour, that have exactly two his neighbours. (Two cubes are neighbours, if they have the common face.)
2023 BMT, 9
Let triangle $\vartriangle ABC$ be acute, and let point $M$ be the midpoint of $\overline{BC}$. Let $E$ be on line segment $\overline{AB}$ such that $\overline{AE} \perp \overline{EC}$. Then, suppose $T$ is a point on the other side of $\overleftrightarrow{BC}$ as $A$ is such that $\angle BTM = \angle ABC$ and $\angle TCA = \angle BMT$. If $AT = 14$, $AM = 9,$ and $\frac{AE}{AC} =\frac27$ , compute $BC$.
2009 ISI B.Stat Entrance Exam, 3
Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR, APQ$ and $PQCR$. Find the minimum possible value of $M$.
Kyiv City MO Juniors 2003+ geometry, 2010.89.4
Point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AO$ intersects the side $BC$ at point $D$ so that $OD = BD = 1/3 BC$ . Find the angles of the triangle $ABC$. Justify the answer.
2013 Czech-Polish-Slovak Junior Match, 5
Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Point $P$ lies on the segment $AB$, and points $S_1$ and $S_2$ are the centers of the circumcircles of $APC$ and $BPC$, respectively. Show that the midpoint of segment $S_1S_2$ lies on the perpendicular bisector of segment $CM$.
2018 Middle European Mathematical Olympiad, 3
Let $ABC$ be an acute-angled triangle with $AB<AC,$ and let $D$ be the foot of its altitude from$A.$ Let $R$ and $Q$ be the centroids of triangles $ABD$ and $ACD$, respectively. Let $P$ be a point on the line segment $BC$ such that $P \neq D$ and points $P$ $Q$ $R$ and $D$ are concyclic .Prove that the lines $AP$ $BQ$ and $CR$ are concurrent.
2007 Ukraine Team Selection Test, 9
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
2022 Romania EGMO TST, P3
Let be given a parallelogram $ ABCD$ and two points $ A_1$, $ C_1$ on its sides $ AB$, $ BC$, respectively. Lines $ AC_1$ and $ CA_1$ meet at $ P$. Assume that the circumcircles of triangles $ AA_1P$ and $ CC_1P$ intersect at the second point $ Q$ inside triangle $ ACD$. Prove that $ \angle PDA \equal{} \angle QBA$.