Found problems: 25757
2003 AMC 12-AHSME, 19
A parabola with equation $ y \equal{} ax^2 \plus{} bx \plus{} c$ is reflected about the $ x$-axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of $ y \equal{} f(x)$ and $ y \equal{} g(x)$, respectively. Which of the following describes the graph of $ y \equal{} (f \plus{} g)(x)$?
$ \textbf{(A)}\ \text{a parabola tangent to the }x\text{ \minus{} axis}$
$ \textbf{(B)}\ \text{a parabola not tangent to the }x\text{ \minus{} axis} \qquad \textbf{(C)}\ \text{a horizontal line}$
$ \textbf{(D)}\ \text{a non \minus{} horizontal line} \qquad \textbf{(E)}\ \text{the graph of a cubic function}$
2017 Romania Team Selection Test, P3
Let $ABCD$ be a convex quadrilateral with $\angle ABC = \angle ADC < 90^{\circ}$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $E$ and $F$ respectively, and meet each other at point $P$. Let $M$ be the midpoint of $AC$ and let $\omega$ be the circumcircle of triangle $BPD$. Segments $BM$ and $DM$ intersect $\omega$ again at $X$ and $Y$ respectively. Denote by $Q$ the intersection point of lines $XE$ and $YF$. Prove that $PQ \perp AC$.
2012 AMC 10, 16
Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?
[asy]
filldraw((0,0)--(2,0)--(1,sqrt(3))--cycle,gray,gray);
filldraw(circle((1,sqrt(3)),1),gray);
filldraw(circle((0,0),1),gray);
filldraw(circle((2,0),1),grey);
[/asy]
$ \textbf{(A)}\ 10\pi+4\sqrt3\qquad\textbf{(B)}\ 13\pi-\sqrt3\qquad\textbf{(C)}\ 12\pi+\sqrt3\qquad\textbf{(D)}\ 10\pi+9\qquad\textbf{(E)}\ 13\pi$
Estonia Open Junior - geometry, 2000.2.4
In the plane, there is an acute angle $\angle AOB$ . Inside the angle points $C$ and $D$ are chosen so that $\angle AOC = \angle DOB$. From point $D$ the perpendicular on $OA$ intersects the ray $OC$ at point $G$ and from point C the perpendicular on $OB$ intersects the ray $OD$ at point $H$. Prove that the points $C, D, G$ and $H$ are conlyclic.
2009 Germany Team Selection Test, 3
In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$.
[i]Proposed by Davood Vakili, Iran[/i]
1998 Slovenia National Olympiad, Problem 3
A rectangle $ABCD$ with $AB>AD$ is given. The circle with center $B$ and radius $AB$ intersects the line $CD$ at $E$ and $F$.
(a) Prove that the circumcircle of triangle $EBF$ is tangent to the circle with diameter $AD$. Denote the tangency point by $G$.
(b) Prove that the points $D,G,$ and $B$ are collinear.
2011 Bundeswettbewerb Mathematik, 4
Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.
2018 JHMT, 4
Equilateral triangle $OAB$ of side length $1$ lies in the $xy$-plane ($O$ is the origin). Let $\ell, m$ be the vertical lines passing through $A,B$, respectively. Let $P,Q$ be on $\ell, m$ respectively such that the ratio $\overline{OP} : \overline{OQ} : \overline{PQ} = 3 : 3 : 5$. Let $Q = (x, y, z)$. If $z^2 = \frac{p}{q}$ . where $p, q$ are relatively prime positive integers, find $p + q$.
2012 IFYM, Sozopol, 7
Let $\Delta ABC$ be a triangle with orthocenter $H$ and midpoints $M_a,M_b$, and $M_c$ of $BC$, $AC$, and $AB$ respectively. A circle with center $H$ intersects the lines $M_bM_a$, $M_bM_c$, and $M_cM_a$ in points $U_1,U_2,V_1,V_2,W_1,W_2$ respectively. Prove that $CU_1=CU_2=AV_1=AV_2=BW_1=BW_2$.
2005 Sharygin Geometry Olympiad, 11.5
The angle and the point $K$ inside it are given on the plane. Prove that there is a point $M$ with the following property:
if an arbitrary line passing through intersects the sides of the angle at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.
1977 IMO Longlists, 47
A square $ABCD$ is given. A line passing through $A$ intersects $CD$ at $Q$. Draw a line parallel to $AQ$ that intersects the boundary of the square at points $M$ and $N$ such that the area of the quadrilateral $AMNQ$ is maximal.
2016 Olympic Revenge, 3
Let $\Gamma$ a fixed circunference. Find all finite sets $S$ of points in $\Gamma$ such that:
For each point $P\in \Gamma$, there exists a partition of $S$ in sets $A$ and $B$
($A\cup B=S$, $A\cap B=\phi$) such that $\sum_{X\in A}PX = \sum_{Y\in B}PY$.
2014 Iran Team Selection Test, 6
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$.
let $X$ is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$.
prove that $\widehat{BAD}=\widehat{CAX}$
KoMaL A Problems 2018/2019, A. 745
A clock hand is attached to every face of a convex polyhedron. Each hand always points towards a neighboring face and every minute, exactly one of the hands turns clockwise to point at the next face. Suppose that the hands on neighboring faces never point towards one another. Show that one of the hands makes only finitely many turns.
1989 IMO Longlists, 6
Let $ E$ be the set of all triangles whose only points with integer coordinates (in the Cartesian coordinate system in space), in its interior or on its sides, are its three vertices, and let $ f$ be the function of area of a triangle. Determine the set of values $ f(E)$ of $ f.$
2004 AMC 10, 7
A grocer stacks oranges in a pyramid-like stack whose rectangular base is $ 5$ oranges by $ 8$ oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack?
$ \textbf{(A)}\ 96 \qquad
\textbf{(B)}\ 98 \qquad
\textbf{(C)}\ 100 \qquad
\textbf{(D)}\ 101 \qquad
\textbf{(E)}\ 134$
2004 IMO Shortlist, 2
Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.
[i]Proposed by Horst Sewerin, Germany[/i]
2010 Harvard-MIT Mathematics Tournament, 6
Three unit circles $\omega_1$, $\omega_2$, and $\omega_3$ in the plane have the property that each circle passes through the centers of the other two. A square $S$ surrounds three circles in such a way that each of its four sides is tangent to at least one of $\omega_1$, $\omega_2$, and $\omega_3$. Find the side length of the square $S$.
2024-25 IOQM India, 27
In a triangle $ABC$, a point $P$ in the interior of $ABC$ is such that $$ \angle BPC - \angle BAC = \angle CPA - \angle CBA = \angle APB - \angle ACB.$$ Suppose $\angle BAC = 30^{\circ}$ and $AP = 12$. Let $D,E,F$ be the feet of perpendiculars from $P$ on to $BC,CA,AB$ respectively. If $m \sqrt{n}$ is the area of the triangle DEF where $m,n$ are integers with $n$ prime, then what is the value of the product $mn$?
2024 Korea Junior Math Olympiad (First Round), 2
There is an isosceles triangle which follows the following:
$ \bar{AB}=\bar{AC}=5, \bar{BC}=6 $
D,E are points on $ \bar{AC} $ which follows $ \bar{AD}=1, \bar{EC}=2 $
If the extent of $ \triangle $ BDE = S, Find 15S.
1969 AMC 12/AHSME, 6
The area of the ring between two concentric circles is $12\tfrac12\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is:
$\textbf{(A) }\dfrac5{\sqrt2}\qquad
\textbf{(B) }5\qquad
\textbf{(C) }5\sqrt2\qquad
\textbf{(D) }10\qquad
\textbf{(E) }10\sqrt2$
2013 Today's Calculation Of Integral, 894
Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$
1969 IMO Longlists, 58
$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.
2003 Cono Sur Olympiad, 4
In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$.
2000 Tournament Of Towns, 1
The diagonals of a convex quadrilateral $ABCD$ meet at $P$. The sum of the areas of triangles $PAB$ and $PCD$ is equal to the sum of areas of triangles $PAD$ and $PCB$. Prove that $P$ is the midpoint of either $AC$ or $BD$.
(Folklore)