Found problems: 25757
1971 IMO Longlists, 50
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
Indonesia Regional MO OSP SMA - geometry, 2004.5
The lattice point on the plane is a point that has coordinates in the form of a pair of integers.
Let $P_1, P_2, P_3, P_4, P_5$ be five different lattice points on the plane.
Prove that there is a pair of points $(P_i, P_j), i \ne j$, so that the line segment $P_iP_j$ contains a lattice point other than $P_i$ and $P_j$.
2020 Germany Team Selection Test, 2
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
(Nigeria)
2001 Mexico National Olympiad, 5
$ABC$ is a triangle with $AB < AC$ and $\angle A = 2 \angle C$. $D$ is the point on $AC$ such that $CD = AB$. Let L be the line through $B$ parallel to $AC$. Let $L$ meet the external bisector of $\angle A$ at $M$ and the line through $C$ parallel to $AB$ at $N$. Show that $MD = ND$.
2021 BMT, 7
The line $\ell$ passes through vertex$ B$ and the interior of regular hexagon $ABCDEF$. If the distances from $\ell$ to the vertices $A$ and $C$ are $7$ and $4$, respectively, compute the area of hexagon $ABCDEF$.
1998 French Mathematical Olympiad, Problem 1
A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$. Find the minimum possible value of
$$BC^6+BD^6-AC^6-AD^6.$$
1980 AMC 12/AHSME, 23
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}2$. The length of the hypotenuse is
$\text{(A)} \ \frac 43 \qquad \text{(B)} \ \frac 32 \qquad \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad \text{(D)} \ \frac{2\sqrt{5}}{3} \qquad \text{(E)} \ \text{not uniquely determined}$
2019 PUMaC Team Round, 8
The curves $y = x + 5$ and $y = x^2 - 3x$ intersect at points $A$ and $B$. $C$ is a point on the lower curve between $A$ and $B$. The maximum possible area of the quadrilateral $ABCO$ can be written as $A/B$ for coprime $A, B$. Find $A + B$.
2003 Regional Competition For Advanced Students, 3
Given are two parallel lines $ g$ and $ h$ and a point $ P$, that lies outside of the corridor bounded by $ g$ and $ h$. Construct three lines $ g_1$, $ g_2$ and $ g_3$ through the point $ P$. These lines intersect $ g$ in $ A_1,A_2, A_3$ and $ h$ in $ B_1, B_2, B_3$ respectively. Let $ C_1$ be the intersection of the lines $ A_1B_2$ and $ A_2B_1$, $ C_2$ be the intersection of the lines $ A_1B_3$ and $ A_3B_1$ and let $ C_3$ be the intersection of the lines $ A_2B_3$ and $ A_3B_2$. Show that there exists exactly one line $ n$, that contains the points $ C_1,C_2,C_3$ and that $ n$ is parallel to $ g$ and $ h$.
II Soros Olympiad 1995 - 96 (Russia), 9.3
It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.
2020 Bulgaria Team Selection Test, 6
In triangle $\triangle ABC$, $BC>AC$, $I_B$ is the $B$-excenter, the line through $C$ parallel to $AB$ meets $BI_B$ at $F$. $M$ is the midpoint of $AI_B$ and the $A$-excircle touches side $AB$ at $D$. Point $E$ satisfies $\angle BAC=\angle BDE, DE=BC$, and lies on the same side as $C$ of $AB$. Let $EC$ intersect $AB,FM$ at $P,Q$ respectively. Prove that $P,A,M,Q$ are concyclic.
2011 Today's Calculation Of Integral, 756
Let $a$ be real number. A circle $C$ touches the line $y=-x$ at the point $(a, -a)$ and passes through the point $(0,\ 1).$
Denote by $P$ the center of $C$. When $a$ moves, find the area of the figure enclosed by the locus of $P$ and the line $y=1$.
Kyiv City MO Juniors 2003+ geometry, 2004.9.7
The board depicts the triangle $ABC$, the altitude $AH$ and the angle bisector $AL$ which intersectthe inscribed circle in the triangle at the points $M$ and $N, P$ and $Q$, respectively. After that, the figure was erased, leaving only the points $H, M$ and $Q$. Restore the triangle $ABC$.
(Bogdan Rublev)
1951 Polish MO Finals, 6
Given a circle and a segment $ MN $. Find a point $ C $ on the circle such that the triangle $ ABC $, where $ A $ and $ B $ are the intersection points of the lines $ MC $ and $ NC $ with the circle, is similar to the triangle $ MNC $.
2001 National Olympiad First Round, 25
The circumradius of acute triangle $ABC$ is twice of the distance of its circumcenter to $AB$. If $|AC|=2$ and $|BC|=3$, what is the altitude passing through $C$?
$
\textbf{(A)}\ \sqrt {14}
\qquad\textbf{(B)}\ \dfrac{3}{7}\sqrt{21}
\qquad\textbf{(C)}\ \dfrac{4}{7}\sqrt{21}
\qquad\textbf{(D)}\ \dfrac{1}{2}\sqrt{21}
\qquad\textbf{(E)}\ \dfrac{2}{3}\sqrt{14}
$
2006 AMC 12/AHSME, 6
The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
[asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$
2002 Federal Competition For Advanced Students, Part 2, 3
Let $ABCD$ and $AEFG$ be two similar cyclic quadrilaterals (with the vertices denoted counterclockwise). Their circumcircles intersect again at point $P$. Prove that $P$ lies on line $BE$.
2000 CentroAmerican, 3
Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.
2002 AIME Problems, 2
Three vertices of a cube are $P=(7,12,10),$ $Q=(8,8,1),$ and $R=(11,3,9).$ What is the surface area of the cube?
Kyiv City MO 1984-93 - geometry, 1991.8.5
The diagonals of the convex quadrilateral $ABCD$ are mutually perpendicular. Through the midpoint of the sides $AB$ and $AD$ draw lines, which are perpendicular to the opposite sides. Prove that they intersect on line $AC$.
2008 China Team Selection Test, 3
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
2022 IOQM India, 9
Let $P_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \ge 0$ by $$x_{n+1} = - \frac{3x_n - y_n}{2}, y_{n+1} = - \frac{x_n + y_n}{2}$$Find the area of the quadrilateral formed by the points $P_{96}, P_{97}, P_{98}, P_{99}$.
1998 Belarusian National Olympiad, 6
Points $M$ and $N$ are marked on the straight line containing the side $AC$ of triangle $ABC$ so that $MA = AB$ and $NC = CB$ (the order of the points on the line: $M, A, C, N$). Prove that the center of the circle inscribed in triangle $ABC$ lies on the common chord of the circles circumscribed around triangles $MCB$ and $NAB$ .
2006 Sharygin Geometry Olympiad, 10.4
Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.
2013 Iran Team Selection Test, 14
we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.