Found problems: 1581
2022 Sharygin Geometry Olympiad, 18
The products of the opposite sidelengths of a cyclic quadrilateral $ABCD$ are
equal. Let $B'$ be the reflection of $B$ about $AC$. Prove that the circle passing through $A,B', D$ touches $AC$
2013 Uzbekistan National Olympiad, 1
Let real numbers $a,b$ such that $a\ge b\ge 0$. Prove that \[ \sqrt{a^2+b^2}+\sqrt[3]{a^3+b^3}+\sqrt[4]{a^4+b^4} \le 3a+b .\]
2010 Tuymaada Olympiad, 3
Let $ABC$ be a triangle, $I$ its incenter, $\omega$ its incircle, $P$ a point such that $PI\perp BC$ and $PA\parallel BC$, $Q\in (AB), R\in (AC)$ such that $QR\parallel BC$ and $QR$ tangent to $\omega$.
Show that $\angle QPB = \angle CPR$.
2007 F = Ma, 32
A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$.
The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$.
Find an expression for $\beta$ in terms of $k$.
$ \textbf{(A)}\ 1+k^2$
$ \textbf{(B)}\ \sqrt{1+k^2}$
$ \textbf{(C)}\ \sqrt{\frac{k}{1+k}}$
$ \textbf{(D)}\ \sqrt{\frac{k^2}{1+k}}$
$ \textbf{(E)}\ \text{none of the above}$
2018 USA TSTST, 9
Show that there is an absolute constant $c < 1$ with the following property: whenever $\mathcal P$ is a polygon with area $1$ in the plane, one can translate it by a distance of $\frac{1}{100}$ in some direction to obtain a polygon $\mathcal Q$, for which the intersection of the interiors of $\mathcal P$ and $\mathcal Q$ has total area at most $c$.
[i]Linus Hamilton[/i]
2011 Canadian Students Math Olympiad, 4
Circles $\Gamma_1$ and $\Gamma_2$ have centers $O_1$ and $O_2$ and intersect at $P$ and $Q$. A line through $P$ intersects $\Gamma_1$ and $\Gamma_2$ at $A$ and $B$, respectively, such that $AB$ is not perpendicular to $PQ$. Let $X$ be the point on $PQ$ such that $XA=XB$ and let $Y$ be the point within $AO_1 O_2 B$ such that $AYO_1$ and $BYO_2$ are similar. Prove that $2\angle{O_1 AY}=\angle{AXB}$.
[i]Author: Matthew Brennan[/i]
1999 APMO, 3
Let $\Gamma_1$ and $\Gamma_2$ be two circles intersecting at $P$ and $Q$. The common tangent, closer to $P$, of $\Gamma_1$ and $\Gamma_2$ touches $\Gamma_1$ at $A$ and $\Gamma_2$ at $B$. The tangent of $\Gamma_1$ at $P$ meets $\Gamma_2$ at $C$, which is different from $P$, and the extension of $AP$ meets $BC$ at $R$.
Prove that the circumcircle of triangle $PQR$ is tangent to $BP$ and $BR$.
2008 Gheorghe Vranceanu, 3
If the circumradius of any three consecutive vertices of a convex polygon is at most $ 1, $ show that the discs of radius $ 1 $ centered at each vertex cover the polygon and its interior.
2005 Iran Team Selection Test, 2
Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that:
\[PX || AC \ , \ PY ||AB \]
Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$
1997 China Team Selection Test, 1
Given a real number $\lambda > 1$, let $P$ be a point on the arc $BAC$ of the circumcircle of $\bigtriangleup ABC$. Extend $BP$ and $CP$ to $U$ and $V$ respectively such that $BU = \lambda BA$, $CV = \lambda CA$. Then extend $UV$ to $Q$ such that $UQ = \lambda UV$. Find the locus of point $Q$.
1985 Federal Competition For Advanced Students, P2, 3
A line meets the lines containing sides $ BC,CA,AB$ of a triangle $ ABC$ at $ A_1,B_1,C_1,$ respectively. Points $ A_2,B_2,C_2$ are symmetric to $ A_1,B_1,C_1$ with respect to the midpoints of $ BC,CA,AB,$ respectively. Prove that $ A_2,B_2,$ and $ C_2$ are collinear.
2009 Ukraine National Mathematical Olympiad, 3
In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$
2007 China Team Selection Test, 1
Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively.
A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$
Prove that: $ X,\,Y,\,Z$ are collinear.
2013 Harvard-MIT Mathematics Tournament, 16
The walls of a room are in the shape of a triangle $ABC$ with $\angle ABC = 90^\circ$, $\angle BAC = 60^\circ$, and $AB=6$. Chong stands at the midpoint of $BC$ and rolls a ball toward $AB$. Suppose that the ball bounces off $AB$, then $AC$, then returns exactly to Chong. Find the length of the path of the ball.
1992 China Team Selection Test, 2
A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.
2011 ELMO Shortlist, 2
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.
[i]David Yang.[/i]
2005 CentroAmerican, 3
Let $ABC$ be a triangle. $P$, $Q$ and $R$ are the points of contact of the incircle with sides $AB$, $BC$ and $CA$, respectively. Let $L$, $M$ and $N$ be the feet of the altitudes of the triangle $PQR$ from $R$, $P$ and $Q$, respectively.
a) Show that the lines $AN$, $BL$ and $CM$ meet at a point.
b) Prove that this points belongs to the line joining the orthocenter and the circumcenter of triangle $PQR$.
[i]Aarón Ramírez, El Salvador[/i]
2008 National Olympiad First Round, 17
Let the vertices $A$ and $C$ of a right triangle $ABC$ be on the arc with center $B$ and radius $20$. A semicircle with diameter $[AB]$ is drawn to the inner region of the arc. The tangent from $C$ to the semicircle touches the semicircle at $D$ other than $B$. Let $CD$ intersect the arc at $F$. What is $|FD|$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ \frac 52
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 5
$
2010 Today's Calculation Of Integral, 576
For a function $ f(x)\equal{}(\ln x)^2\plus{}2\ln x$, let $ C$ be the curve $ y\equal{}f(x)$. Denote $ A(a,\ f(a)),\ B(b,\ f(b))\ (a<b)$ the points of tangency of two tangents drawn from the origin $ O$ to $ C$ and the curve $ C$. Answer the following questions.
(1) Examine the increase and decrease, extremal value and inflection point , then draw the approximate garph of the curve $ C$.
(2) Find the values of $ a,\ b$.
(3) Find the volume by a rotation of the figure bounded by the part from the point $ A$ to the point $ B$ and line segments $ OA,\ OB$ around the $ y$-axis.
2005 Sharygin Geometry Olympiad, 23
Envelop the cube in one layer with five convex pentagons of equal areas.
2011 Estonia Team Selection Test, 1
Two circles lie completely outside each other.Let $A$ be the point of intersection of internal common tangents of the circles and let $K$ be the projection of this point onto one of their external common tangents.The tangents,different from the common tangent,to the circles through point $K$ meet the circles at $M_1$ and $M_2$.Prove that the line $AK$ bisects angle $M_1 KM_2$.
2005 MOP Homework, 2
Let $ABC$ be a triangle, and let $D$ be a point on side $AB$. Circle $\omega_1$ passes through $A$ and $D$ and is tangent to line $AC$ at $A$. Circle $\omega_2$ passes through $B$ and $D$ and is tangent to line $BC$ at $B$. Circles $\omega_1$ and $\omega_2$ meet at $D$ and $E$. Point $F$ is the reflection of $C$ across the perpendicular bisector of $AB$. Prove that points $D$, $E$, and $F$ are collinear.
2006 Turkey MO (2nd round), 2
$ABC$ be a triangle. Its incircle touches the sides $CB, AC, AB$ respectively at $N_{A},N_{B},N_{C}$. The orthic triangle of $ABC$ is $H_{A}H_{B}H_{C}$ with $H_{A}, H_{B}, H_{C}$ are respectively on $BC, AC, AB$. The incenter of $AH_{C}H_{B}$ is $I_{A}$; $I_{B}$ and $I_{C}$ were defined similarly.
Prove that the hexagon $I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}$ has all sides equal.
2019 CMIMC, 8
Consider the following three lines in the Cartesian plane: $$\begin{cases}
\ell_1: & 2x - y = 7\\
\ell_2: & 5x + y = 42\\
\ell_3: & x + y = 14
\end{cases}$$
and let $f_i(P)$ correspond to the reflection of the point $P$ across $\ell_i$. Suppose $X$ and $Y$ are points on the $x$ and $y$ axes, respectively, such that $f_1(f_2(f_3(X)))= Y$. Let $t$ be the length of segment $XY$; what is the sum of all possible values of $t^2$?
2014 AMC 10, 18
A square in the coordinate plane has vertices whose $y$-coordinates are $0$, $1$, $4$, and $5$. What is the area of the square?
$ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 17\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 26\qquad\textbf{(E)}\ 27 $