Found problems: 1581
Take two points $A\ (-1,\ 0),\ B\ (1,\ 0)$ on the $xy$-plane. Let $F$ be the figure by which the whole points $P$ on the plane satisfies $\frac{\pi}{4}\leq \angle{APB}\leq \pi$ and the figure formed by $A,\ B$.
Answer the following questions:
(1) Illustrate $F$.
(2) Find the volume of the solid generated by a rotation of $F$ around the $x$-axis.
Equilateral $\triangle ABC$ is inscribed in a circle of radius 2. Extend $\overline{AB}$ through $B$ to point $D$ so that $AD=13$, and extend $\overline{AC}$ through $C$ to point $E$ so that $AE=11$. Through $D$, draw a line $l_1$ parallel to $\overline{AE}$, and through $E$, draw a line ${l}_2$ parallel to $\overline{AD}$. Let $F$ be the intersection of ${l}_1$ and ${l}_2$. Let $G$ be the point on the circle that is collinear with $A$ and $F$ and distinct from $A$. Given that the area of $\triangle CBG$ can be expressed in the form $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p+q+r$.
In $\triangle ABC$, $\angle ABC=90^{\circ}$. Points $D,G$ lie on side $AC$. Points $E, F$ lie on segment $BD$, such that $AE \perp BD $ and $GF \perp BD$. Show that if $BE=EF$, then $\angle ABG=\angle DFC$.
Triangle $ABC$ lies in the first quadrant. Points $A$, $B$, and $C$ are reflected across the line $y=x$ to points $A'$, $B'$, and $C'$, respectively. Assume that none of the vertices of the triangle lie on the line $y=x$. Which of the following statements is [u][i]not[/i][/u] always true?
$(A)$ Triangle $A'B'C'$ lies in the first quadrant.
$(B)$ Triangles $ABC$ and $A'B'C'$ have the same area.
$(C)$ The slope of line $AA'$ is $-1$.
$(D)$ The slopes of lines $AA'$ and $CC'$ are the same.
$(E)$ Lines $AB$ and $A'B'$ are perpendicular to each other.
$ M$ is the midpoint of base $ BC$ in a trapezoid $ ABCD$. A point $ P$ is chosen on the base $ AD$. The line $ PM$ meets the line $ CD$ at a point $ Q$ such that $ C$ lies between $ Q$ and $ D$. The perpendicular to the bases drawn through $ P$ meets the line $ BQ$ at $ K$. Prove that $ \angle QBC \equal{} \angle KDA$.
[i]Proposed by S. Berlov[/i]
In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.
A ball moves endlessly on a circular billiard table. When it hits the edge it is reflected. Show that if it passes through a point on the table three times, then it passes through it infinitely many times.
Let AOB and COD be angles which can be identified by a rotation of the plane (such that rays OA and OC are identified). A circle is inscribed in each of these angles; these circles intersect at points E and F. Show that angles AOE and DOF are equal.
Triangle $ABC$ has sides $AB = 25$, $BC = 30$, and $CA=20$. Let $P,Q$ be the points on segments $AB,AC$, respectively, such that $AP=5$ and $AQ=4$. Suppose lines $BQ$ and $CP$ intersect at $R$ and the circumcircles of $\triangle{BPR}$ and $\triangle{CQR}$ intersect at a second point $S\ne R$. If the length of segment $SA$ can be expressed in the form $\frac{m}{\sqrt{n}}$ for positive integers $m,n$, where $n$ is not divisible by the square of any prime, find $m+n$.
[i]Victor Wang[/i]
Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.
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 $AB+AC=PQ+PR$.
Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2 \angle ADB$, $\angle ABD = 2 \angle CDB$ and $AB = CB$.
Prove that $AD = CD$.
In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.
Ten distinct points are placed on a circle. All ten of the points are paired so that the line segments connecting the pairs do not intersect. In how many different ways can this pairing be done?
[asy]
import graph; size(12cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
draw((2.46,0.12)--(3.05,-0.69));
draw((2.46,1.12)--(4,-1));
draw((5.54,0.12)--(4.95,-0.69));
draw((3.05,1.93)--(5.54,1.12));
draw((4.95,1.93)--(4,2.24));
draw((8.05,1.93)--(7.46,1.12));
draw((7.46,0.12)--(8.05,-0.69));
draw((9,2.24)--(9,-1));
draw((9.95,-0.69)--(9.95,1.93));
draw((10.54,1.12)--(10.54,0.12));
draw((15.54,1.12)--(15.54,0.12));
draw((14.95,-0.69)--(12.46,0.12));
draw((13.05,-0.69)--(14,-1));
draw((12.46,1.12)--(14.95,1.93));
draw((14,2.24)--(13.05,1.93));
label("1",(-1.08,2.03),SE*labelscalefactor);
label("2",(-0.3,1.7),SE*labelscalefactor);
label("3",(0.05,1.15),SE*labelscalefactor);
label("4",(0.00,0.38),SE*labelscalefactor);
label("5",(-0.33,-0.12),SE*labelscalefactor);
label("6",(-1.08,-0.4),SE*labelscalefactor);
label("7",(-1.83,-0.19),SE*labelscalefactor);
label("8",(-2.32,0.48),SE*labelscalefactor);
label("9",(-2.3,1.21),SE*labelscalefactor);
label("10",(-1.86,1.75),SE*labelscalefactor);
dot((-1,-1),dotstyle);
dot((-0.05,-0.69),dotstyle);
dot((0.54,0.12),dotstyle);
dot((0.54,1.12),dotstyle);
dot((-0.05,1.93),dotstyle);
dot((-1,2.24),dotstyle);
dot((-1.95,1.93),dotstyle);
dot((-2.54,1.12),dotstyle);
dot((-2.54,0.12),dotstyle);
dot((-1.95,-0.69),dotstyle);
dot((4,-1),dotstyle);
dot((4.95,-0.69),dotstyle);
dot((5.54,0.12),dotstyle);
dot((5.54,1.12),dotstyle);
dot((4.95,1.93),dotstyle);
dot((4,2.24),dotstyle);
dot((3.05,1.93),dotstyle);
dot((2.46,1.12),dotstyle);
dot((2.46,0.12),dotstyle);
dot((3.05,-0.69),dotstyle);
dot((9,-1),dotstyle);
dot((9.95,-0.69),dotstyle);
dot((10.54,0.12),dotstyle);
dot((10.54,1.12),dotstyle);
dot((9.95,1.93),dotstyle);
dot((9,2.24),dotstyle);
dot((8.05,1.93),dotstyle);
dot((7.46,1.12),dotstyle);
dot((7.46,0.12),dotstyle);
dot((8.05,-0.69),dotstyle);
dot((14,-1),dotstyle);
dot((14.95,-0.69),dotstyle);
dot((15.54,0.12),dotstyle);
dot((15.54,1.12),dotstyle);
dot((14.95,1.93),dotstyle);
dot((14,2.24),dotstyle);
dot((13.05,1.93),dotstyle);
dot((12.46,1.12),dotstyle);
dot((12.46,0.12),dotstyle);
dot((13.05,-0.69),dotstyle);[/asy]
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 $O_1 O_2$.
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?
Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$.
[i]Author: L. Emelyanov[/i]
[b]i.)[/b] Consider a circle $K$ with diameter $AB;$ with circle $L$ tangent to $AB$ and to $K$ and with a circle $M$ tangent to circle $K,$ circle $L$ and $AB.$ Calculate the ration of the area of circle $K$ to the area of circle $M.$
[b]ii.)[/b] In triangle $ABC, AB = AC$ and $\angle CAB = 80^{\circ}.$ If points $D,E$ and $F$ lie on sides $BC, AC$ and $AB,$ respectively and $CE = CD$ and $BF = BD,$ then find the size of $\angle EDF.$
Two congruent squares $Q$ and $Q'$ are given in the plane. Show that they can be divided into parts $T_1, T_2, \ldots , T_n$ and $T'_1 , T'_2 , \ldots , T'_n$, respectively, such that $T'_i$ is the image of $T_i$ under a translation for $i=1,2, \ldots, n.$
$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.
The ray of light starts from the center of a square and reflects from its sides with the principle that the angle of reflection is equal to the angle of incidence. After some time the ray returns to the center of the square. The ray never reached the vertex and has never returned to the center of the square before. Prove that the ray reflected from the sides of the square an odd number of times.
Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that \[ \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}.\] Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.
Let $ABCD$ be a tetrahedron; $B', C', D'$ be the midpoints of the edges $AB, AC, AD$; $G_A, G_B, G_C, G_D$ be the barycentres of the triangles $BCD, ACD, ABD, ABC$, and $G$ be the barycentre of the tetrahedron. Show that $A, G, G_B, G_C, G_D$ are all on a sphere if and only if $A, G, B', C', D'$ are also on a sphere.
[i]Dan Brânzei[/i]
Given are three points $A(2,\ 0,\ 2),\ B(1,\ 1,\ 0),\ C(0,\ 0,\ 3)$ in the coordinate space. Find the volume of the solid of a triangle $ABC$ generated by a rotation about $z$-axis.
Let $ a$, $ b$, $ c$ be three pairwise skew lines in space. Prove that they have a common perpendicular if and only if $ S_a \circ S_b \circ S_c$ is a reflection in a line, where $ S_x$ denotes the reflection in line $ x$.