Found problems: 25757
2023 Vietnam National Olympiad, 7
Let $\triangle{ABC}$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Incircle $(I)$ of the $\triangle{ABC}$ is tangent to the sides $BC,CA,AB$ at $M,N,P$ respectively. Denote $\Omega_A$ to be the circle passing through point $A$, external tangent to $(I)$ at $A'$ and cut again $AB,AC$ at $A_b,A_c$ respectively. The circles $\Omega_B,\Omega_C$ and points $B',B_a,B_c,C',C_a,C_b$ are defined similarly.
$a)$ Prove $B_cC_b+C_aA_c+A_bB_a \ge NP+PM+MN$.
$b)$ Suppose $A',B',C'$ lie on $AM,BN,CP$ respectively. Denote $K$ as the circumcenter of the triangle formed by lines $A_bA_c,B_cB_a,C_aC_b.$ Prove $OH//IK$.
2014 Poland - Second Round, 5.
Circles $o_1$ and $o_2$ tangent to some line at points $A$ and $B$, respectively, intersect at points $X$ and $Y$ ($X$ is closer to the line $AB$). Line $AX$ intersects $o_2$ at point $P\neq X$. Tangent to $o_2$ at point $P$ intersects line $AB$ at point $Q$. Prove that $\sphericalangle XYB = \sphericalangle BYQ$.
1980 Tournament Of Towns, (006) 3
We are given $30$ non-zero vectors in $3$ dimensional space.
Prove that among these there are two such that the angle between them is less than $45^o$.
2018 China Team Selection Test, 3
In isosceles $\triangle ABC$, $AB=AC$, points $D,E,F$ lie on segments $BC,AC,AB$ such that $DE\parallel AB$, $DF\parallel AC$. The circumcircle of $\triangle ABC$ $\omega_1$ and the circumcircle of $\triangle AEF$ $\omega_2$ intersect at $A,G$. Let $DE$ meet $\omega_2$ at $K\neq E$. Points $L,M$ lie on $\omega_1,\omega_2$ respectively such that $LG\perp KG, MG\perp CG$. Let $P,Q$ be the circumcenters of $\triangle DGL$ and $\triangle DGM$ respectively. Prove that $A,G,P,Q$ are concyclic.
1973 AMC 12/AHSME, 4
Two congruent $ 30^{\circ}$-$ 60^{\circ}$-$ 90^{\circ}$ are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is
$ \textbf{(A)}\ 6\sqrt3 \qquad
\textbf{(B)}\ 8\sqrt3 \qquad
\textbf{(C)}\ 9\sqrt3 \qquad
\textbf{(D)}\ 12\sqrt3 \qquad
\textbf{(E)}\ 24$
2023 CMIMC Geometry, 8
Let $\omega$ be a unit circle with center $O$ and diameter $AB$. A point $C$ is chosen on $\omega$. Let $M$, $N$ be the midpoints of arc $AC$, $BC$, respectively, and let $AN,BM$ intersect at $I$. Suppose that $AM,BC,OI$ concur at a point. Find the area of $\triangle ABC$.
[i]Proposed by Kevin You[/i]
2007 APMO, 1
Let $S$ be a set of $9$ distinct integers all of whose prime factors are at most $3.$ Prove that $S$ contains $3$ distinct integers such that their product is a perfect cube.
2017 Tournament Of Towns, 7
$1\times 2$ dominoes are placed on an $8 \times 8$ chessboard without overlapping. They may partially
stick out from the chessboard but the center of each domino must be strictly inside the
chessboard (not on its border). Place on the chessboard in such a way:
a) at least $40$ dominoes, (3 points)
b) at least $41$ dominoes, (3 points)
c) more than $41$ dominoes. (6 points)
[i](Mikhail Evdokimov)[/i]
2018 Purple Comet Problems, 8
On side $AE$ of regular pentagon $ABCDE$ there is an equilateral triangle $AEF$, and on side $AB$ of the pentagon there is a square $ABHG$ as shown. Find the degree measure of angle $AFG$.
[img]https://cdn.artofproblemsolving.com/attachments/7/7/0d689d2665e67c9f9afdf193fb0a2db6dddb3d.png[/img]
2005 Iran MO (3rd Round), 1
From each vertex of triangle $ABC$ we draw 3 arbitary parrallell lines, and from each vertex we draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals is triangle's side. We draw their other diagnals and call them $\ell_1$, $\ell_2$ and $\ell_3$.
a) Prove that $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent at a point $P$.
b) Find the locus of $P$ as we move the 3 arbitary lines.
2004 Cuba MO, 3
In the non-isosceles $\vartriangle ABC$, the interior bisectors of vertices $B$ and $C$ are drawn, which cut the sides $AC$ and $AB$ at $E$ and $F$ respectively.The line $EF$ cuts the extension of side $BC$ at $T$. In the side$ BC$ a point D is located, so that $\frac{DB}{DC} = \frac{TB}{TC}$. Prove that $AT$ is the exterior bisector of angle $A$.
2009 Junior Balkan Team Selection Tests - Romania, 2
Consider a rhombus $ABCD$. Point $M$ and $N$ are given on the line segments $AC$ and $BC$ respectively, such that $DM = MN$. Lines $AC$ and $DN$ meet at point $P$ and lines $AB$ and $DM$ meet at point $R$. Prove that $RP = PD$.
2013 BMT Spring, 6
Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Because Bubble Boy loves Bubble Girl, he wants to reach her as quickly as possible, but he needs to bring a gift; luckily, there are plenty of gifts along the $x$-axis. Assuming that Bubble Girl remains stationary, find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then reach Bubble Girl (the bubble is a solid boundary, and anything the bubble can touch, Bubble Boy can touch too)
2000 Dutch Mathematical Olympiad, 5
Consider an infinite strip of unit squares. The squares are numbered "1", "2", "3", ... A pawn starts on one of the squares and it can move according to the following rules:
(1) from the square numbered "$n$" to the square numbered "$2n$", and vice versa;
(2) from the square numbered "$n$" to the square numbered "$3n + 1$", and vice versa.
Show that the pawn can reach the square numbered "$1$" in a finite number of moves.
2021/2022 Tournament of Towns, P2
A cube was split into 8 parallelepipeds by three planes parallel to its faces. The resulting parts were painted in a chessboard pattern. The volumes of the black parallelepipeds are 1, 6, 8, 12. Find the volumes of the white parallelepipeds.
[i]Oleg Smirnov[/i]
2006 AMC 8, 7
Circle $ X$ has a radius of $ \pi$. Circle $ Y$ has a circumference of $ 8\pi$. Circle $ Z$ has an area of $ 9\pi$. List the circles in order from smallest to largest radius.
$ \textbf{(A)}\ X, Y, Z \qquad
\textbf{(B)}\ Z, X, Y \qquad
\textbf{(C)}\ Y, X, Z \qquad
\textbf{(D)}\ Z, Y, X \qquad
\textbf{(E)}\ X, Z, Y$
1998 Bundeswettbewerb Mathematik, 3
A triangle $ABC$ satisfies $BC = AC +\frac12 AB$. Point $P$ on side $AB$ is taken so that $AP = 3PB$. Prove that $ \angle PAC = 2\angle CPA$.
1990 Tournament Of Towns, (244) 2
Two circles $c$ and $d$ are situated in the plane each outside the other. The points $C$ and $D$ are located on circles $c$ and $d$ respectively, so as to be as far apart as possible. Two smaller circles are constructed inside $c$ and $d$. Of these the first circle touches $c$ and the two tangents drawn from $C$ to $d$, while the second circle touches $d$ and the two tangents from $D$ to $c$. Prove that the small circles are equal.
(J. Tabov, Sofia)
1969 IMO Longlists, 20
$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$
1972 Spain Mathematical Olympiad, 3
Given a regular hexagonal prism. Find a polygonal line that, starting from a vertex of the base, runs through all the lateral faces and ends at the vertex of the face top, located on the same edge as the starting vertex, and has a minimum length.
2005 All-Russian Olympiad Regional Round, 9.4
9.4, 10.3 Let $I$ be an incenter of $ABC$ ($AB<BC$), $M$ is a midpoint of $AC$, $N$ is a midpoint of circumcircle's arc $ABC$. Prove that $\angle IMA=\angle INB$.
([i]A. Badzyan[/i])
2011 China National Olympiad, 2
On the circumcircle of the acute triangle $ABC$, $D$ is the midpoint of $ \stackrel{\frown}{BC}$. Let $X$ be a point on $ \stackrel{\frown}{BD}$, $E$ the midpoint of $ \stackrel{\frown}{AX}$, and let $S$ lie on $ \stackrel{\frown}{AC}$. The lines $SD$ and $BC$ have intersection $R$, and the lines $SE$ and $AX$ have intersection $T$. If $RT \parallel DE$, prove that the incenter of the triangle $ABC$ is on the line $RT.$
IV Soros Olympiad 1997 - 98 (Russia), 9.3
What is angle $B$ of triangle$ ABC$, if it is known that the altitudes drawn from $A$ and $C$ intersect inside the triangle and one of them is divided by of intersection point into equal parts, and the other one in the ratio of $2: 1$, counting from the vertex?
2014 Lithuania Team Selection Test, 1
Circle touches parallelogram‘s $ABCD$ borders $AB, BC$ and $CD$ respectively at points $K, L$ and $M$. Perpendicular is drawn from vertex $C$ to $AB$ . Prove, that the line $KL$ divides this perpendicular into two equal parts (with the same length).
2017 Singapore Junior Math Olympiad, 1
A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.