Found problems: 1581
2014 ELMO Shortlist, 11
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$.
[i]Proposed by Yang Liu[/i]
2002 All-Russian Olympiad, 2
A quadrilateral $ABCD$ is inscribed in a circle $\omega$. The tangent to $\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\omega$ at $B$ intersects the ray $DA$ at $M$. Prove that if $AM=AD$ and $BK=BC$, then $ABCD$ is a trapezoid.
2010 Sharygin Geometry Olympiad, 22
A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.
2012 AMC 10, 20
A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
$ \textbf{(A)}\ \dfrac{49}{512}
\qquad\textbf{(B)}\ \dfrac{7}{64}
\qquad\textbf{(C)}\ \dfrac{121}{1024}
\qquad\textbf{(D)}\ \dfrac{81}{512}
\qquad\textbf{(E)}\ \dfrac{9}{32}
$
2023 Macedonian Team Selection Test, Problem 2
Let $ABC$ be an acute triangle such that $AB<AC$ and $AB<BC$. Let $P$ be a point on the segment $BC$ such that $\angle APB = \angle BAC$. The tangent to the circumcircle of triangle $ABC$ at $A$ meets the circumcircle of triangle $APB$ at $Q \neq A$. Let $Q'$ be the reflection of $Q$ with respect to the midpoint of $AB$. The line $PQ$ meets the segment $AQ'$ at $S$. Prove that
$$\frac{1}{AB}+\frac{1}{AC} > \frac{1}{CS}.$$
[i]Authored by Nikola Velov[/i]
2006 AIME Problems, 10
This is the one with the 8 circles?
I made each circle into the square in which the circle is inscribed, then calculated it with that. It got the right answer but I don't think that my method is truly valid...
1999 National Olympiad First Round, 21
$ ABC$ is a triangle with $ \angle BAC \equal{} 10{}^\circ$, $ \angle ABC \equal{} 150{}^\circ$. Let $ X$ be a point on $ \left[AC\right]$ such that $ \left|AX\right| \equal{} \left|BC\right|$. Find $ \angle BXC$.
$\textbf{(A)}\ 15^\circ \qquad\textbf{(B)}\ 20^\circ \qquad\textbf{(C)}\ 25^\circ \qquad\textbf{(D)}\ 30^\circ \qquad\textbf{(E)}\ 35^\circ$
PEN N Problems, 8
An integer sequence $\{a_{n}\}_{n \ge 1}$ is given such that \[2^{n}=\sum^{}_{d \vert n}a_{d}\] for all $n \in \mathbb{N}$. Show that $a_{n}$ is divisible by $n$ for all $n \in \mathbb{N}$.
2008 China National Olympiad, 1
Suppose $\triangle ABC$ is scalene. $O$ is the circumcenter and $A'$ is a point on the extension of segment $AO$ such that $\angle BA'A = \angle CA'A$. Let point $A_1$ and $A_2$ be foot of perpendicular from $A'$ onto $AB$ and $AC$. $H_{A}$ is the foot of perpendicular from $A$ onto $BC$. Denote $R_{A}$ to be the radius of circumcircle of $\triangle H_{A}A_1A_2$. Similiarly we can define $R_{B}$ and $R_{C}$. Show that:
\[\frac{1}{R_{A}} + \frac{1}{R_{B}} + \frac{1}{R_{C}} = \frac{2}{R}\]
where R is the radius of circumcircle of $\triangle ABC$.
2016 Korea - Final Round, 1
In a acute triangle $\triangle ABC$, denote $D, E$ as the foot of the perpendicular from $B$ to $AC$ and $C$ to $AB$.
Denote the reflection of $E$ with respect to $AC, BC$ as $S, T$.
The circumcircle of $\triangle CST$ hits $AC$ at point $X (\not= C)$.
Denote the circumcenter of $\triangle CST$ as $O$. Prove that $XO \perp DE$.
2010 Postal Coaching, 2
Suppose $\triangle ABC$ has circumcircle $\Gamma$, circumcentre $O$ and orthocentre $H$. Parallel lines $\alpha, \beta, \gamma$ are drawn through the vertices $A, B, C$, respectively. Let $\alpha ', \beta ', \gamma '$ be the reflections of $\alpha, \beta, \gamma$ in the sides $BC, CA, AB$, respectively.
$(a)$ Show that $\alpha ', \beta ', \gamma '$ are concurrent if and only if $\alpha, \beta, \gamma$ are parallel to the Euler line $OH$.
$(b)$ Suppose that $\alpha ', \beta ' , \gamma '$ are concurrent at the point $P$ . Show that $\Gamma$ bisects $OP$ .
1992 Putnam, A6
Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points?
2013 Kazakhstan National Olympiad, 2
Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.
1989 Iran MO (2nd round), 3
A line $d$ is called [i]faithful[/i] to triangle $ABC$ if $d$ be in plane of triangle $ABC$ and the reflections of $d$ over the sides of $ABC$ be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one [i]faithful[/i] line to both of them, or there exist infinitely [i]faithful[/i] lines to them.
2014 AMC 12/AHSME, 18
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is [i]bad[/i] if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there?
$ \textbf {(A) } 1 \qquad \textbf {(B) } 2 \qquad \textbf {(C) } 3 \qquad \textbf {(D) } 4 \qquad \textbf {(E) } 5 $
2021 Taiwan Mathematics Olympiad, 4.
Let $I$ be the incenter of triangle $ABC$ and let $D$ the foot of altitude from $I$ to $BC$. Suppose the reflection point $D’$ of $D$ with respect to $I$ satisfying $\overline{AD’} = \overline{ID’}$. Let $\Gamma$ be the circle centered at $D’$ that passing through $A$ and $I$, and let $X$, $Y\neq A$ be the intersection of $\Gamma$ and $AB$, $AC$, respectively. Suppose $Z$ is a point on $\Gamma$ so that $AZ$ is perpendicular to $BC$.
Prove that $AD$, $D’Z$, $XY$ concurrent at a point.
1983 Tournament Of Towns, (036) O5
A version of billiards is played on a right triangular table, with a pocket in each of the three corners, and one of the acute angles being $30^o$. A ball is played from just in front of the pocket at the $30^o$. vertex toward the midpoint of the opposite side. Prove that if the ball is played hard enough, it will land in the pocket of the $60^o$ vertex after $8$ reflections.
2011 China Second Round Olympiad, 2
For any integer $n\ge 4$, prove that there exists a $n$-degree polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$
satisfying the two following properties:
[b](1)[/b] $a_i$ is a positive integer for any $i=0,1,\ldots,n-1$, and
[b](2)[/b] For any two positive integers $m$ and $k$ ($k\ge 2$) there exist distinct positive integers $r_1,r_2,...,r_k$, such that $f(m)\ne\prod_{i=1}^{k}f(r_i)$.
2011 Indonesia TST, 3
Let $ABC$ and $PQR$ be two triangles such that
[list]
[b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$.
[b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$
[/list]
Prove that $AB+AC=PQ+PR$.
2010 China Team Selection Test, 1
Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.
1953 AMC 12/AHSME, 49
The coordinates of $ A,B$ and $ C$ are $ (5,5),(2,1)$ and $ (0,k)$ respectively. The value of $ k$ that makes $ \overline{AC}\plus{}\overline{BC}$ as small as possible is:
$ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4\frac{1}{2} \qquad\textbf{(C)}\ 3\frac{6}{7} \qquad\textbf{(D)}\ 4\frac{5}{6} \qquad\textbf{(E)}\ 2\frac{1}{7}$
2021 AMC 12/AHSME Spring, 5
The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$?
$\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$
2014 PUMaC Individual Finals A, 1
Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.
2006 MOP Homework, 3
Let $ABC$ be a triangle with $AB\neq AC$, and let $A_{1}B_{1}C_{1}$ be the image of triangle $ABC$ through a rotation $R$ centered at $C$.
Let $M,E , F$ be the midpoints of the segments $BA_{1}, AC, BC_{1}$ respectively
Given that $EM = FM$, compute $\angle EMF$.
1999 Federal Competition For Advanced Students, Part 2, 2
Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.