Found problems: 25757
2019 Hong Kong TST, 3
Let $\Gamma_1$ and $\Gamma_2$ be two circles with different radii, with $\Gamma_1$ the smaller one. The two circles meet at distinct points $A$ and $B$. $C$ and $D$ are two points on the circles $\Gamma_1$ and $\Gamma_2$, respectively, and such that $A$ is the midpoint of $CD$. $CB$ is extended to meet $\Gamma_2$ at $F$, while $DB$ is extended to meet $\Gamma_1$ at $E$. The perpendicular bisector of $CD$ and the perpendicular bisector of $EF$ meet at $P$.
(a) Prove that $\angle{EPF} = 2\angle{CAE}$.
(b) Prove that $AP^2 = CA^2 + PE^2$.
2003 Turkey Team Selection Test, 2
Let $K$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Let $L\in [AD]$, $M \in [AC]$, $N \in [BC]$ such that $KL\parallel AB$, $LM\parallel DC$, $MN\parallel AB$. Show that \[\dfrac{Area(KLMN)}{Area(ABCD)} < \dfrac {8}{27}.\]
2006 AMC 10, 6
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure $ 2/\pi $, as shown. What is the perimeter of this region?
[asy]
size(90); defaultpen(linewidth(0.7));
filldraw((0,0)--(2,0)--(2,2)--(0,2)--cycle,gray(0.5));
filldraw(arc((1,0),1,180,0, CCW)--cycle,gray(0.7));
filldraw(arc((0,1),1,90,270)--cycle,gray(0.7));
filldraw(arc((1,2),1,0,180)--cycle,gray(0.7));
filldraw(arc((2,1),1,270,90, CCW)--cycle,gray(0.7));[/asy]
$ \textbf{(A) }\frac {4}\pi\qquad\textbf{(B) }2\qquad\textbf{(C) }\frac {8}\pi\qquad\textbf{(D) }4\qquad\textbf{(E) }\frac{16}{\pi} $
1967 AMC 12/AHSME, 15
The difference in the areas of two similar triangles is $18$ square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triange, in square feet, is an integer, and one of its sides is $3$ feet. The corresponding side of the larger triangle, in feet, is:
$\textbf{(A)}\ 12\quad
\textbf{(B)}\ 9\qquad
\textbf{(C)}\ 6\sqrt{2}\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 3\sqrt{2}$
Kyiv City MO Juniors Round2 2010+ geometry, 2013.7.3
In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.
2018 Purple Comet Problems, 2
A triangle with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so that the total area inside the three circles but outside of the triangle is $n\pi$.
[img]https://4.bp.blogspot.com/-dpCi7Gai3ZE/XoEaKo3C5wI/AAAAAAAALl8/KAuCVDT9R5MiIA_uTfRyoQmohEVw9cuVACK4BGAYYCw/s200/2018%2Bpc%2Bhs2.png[/img]
2014 CHMMC (Fall), 8
What’s the greatest pyramid volume one can form using edges of length $2, 3, 3, 4, 5, 5$, respectively?
2006 Portugal MO, 2
In the equilateral triangle $[ABC], D$ is the midpoint of $[AC], E$ and the orthogonal projection of $D$ over $[CB]$ and $F$ is the midpoint of $[DE]$. Prove that $[FB]$ and $[AE]$ are perpendicular.
[img]https://1.bp.blogspot.com/-TjSyQotGIOM/X4XMolaXHvI/AAAAAAAAMng/cVsHfl-lrXAFE5LMdosE6vqK1Tf-8WOQgCLcBGAsYHQ/s0/2006%2Bportugal%2Bp2.png[/img]
2010 Kazakhstan National Olympiad, 1
Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.
2014 Contests, 1
The four congruent circles below touch one another and each has radius 1.
[asy]
unitsize(30);
fill(box((-1,-1), (1, 1)), gray);
filldraw(circle((1, 1), 1), white);
filldraw(circle((1, -1), 1), white);
filldraw(circle((-1, 1), 1), white);
filldraw(circle((-1, -1), 1), white);
[/asy]
What is the area of the shaded region?
ICMC 5, 1
Let $S$ be a set of $2022$ lines in the plane, no two parallel, no three concurrent. $S$ divides the plane into finite regions and infinite regions. Is it possible for all the finite regions to have integer area?
[i]Proposed by Tony Wang[/i]
Swiss NMO - geometry, 2019.1
Let $A$ be a point and let k be a circle through $A$. Let $B$ and $C$ be two more points on $k$. Let $X$ be the intersection of the bisector of $\angle ABC$ with $k$. Let $Y$ be the reflection of $A$ wrt point $X$, and $D$ the intersection of the straight line $YC$ with $k$. Prove that point $D$ is independent of the choice of $B$ and $C$ on the circle $k$.
2023 Federal Competition For Advanced Students, P2, 2
Given is a triangle $ABC$ with circumcentre $O$. The circumcircle of triangle $AOC$ intersects side $BC$ at $D$ and side $AB$ at $E$. Prove that the triangles $BDE$ and $AOC$ have circumradiuses of equal length.
2021 Stanford Mathematics Tournament, 1
What is the radius of the largest circle centered at $(2, 2)$ that is completely bounded within the parabola $y = x^2 - 4x + 5$?
2003 Junior Tuymaada Olympiad, 3
In the acute triangle $ ABC $, the point $ I $ is the center of the inscribed the circle, the point $ O $ is the center of the circumscribed circle and the point $ I_a $ is the center the excircle tangent to the side $ BC $ and the extensions of the sides $ AB $ and $ AC $. Point $ A'$ is symmetric to vertex $ A $ with respect to the line $ BC $. Prove that $ \angle IOI_a = \angle IA'I_a $.
1995 Poland - First Round, 8
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.
2014 Purple Comet Problems, 9
The diagram below shows a shaded region bounded by a semicircular arc of a large circle and two smaller semicircular arcs. The smallest semicircle has radius $8$, and the shaded region has area $180\pi$. Find the diameter of the large circle.
[asy]
import graph;
size(3cm);
fill((-22.5,0)..(0,22.5)..(22.5,0)--cycle,rgb(.76,.76,.76));
fill((6.5,0.1)..(14.5,-8)..(22.5,0.1)--cycle,rgb(.76,.76,.76));
fill((-22.5,-0.1)..(-8,14.5)..(6.5,-0.1)--cycle,white);
draw((-22.5,0)..(-8,14.5)..(6.5,0),linewidth(1.5));
draw((6.5,0)..(14.5,-8)..(22.5,0),linewidth(1.5));
draw(Circle((0,0),22.5),linewidth(1.5));
[/asy]
1972 IMO Shortlist, 2
We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$
2004 Tournament Of Towns, 1
Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)
1987 Greece National Olympiad, 4
Let $A,B$ be two points interior of circle $C(O,R)$ and $M$ a point on the circle. Let $A_1,B_1$ be the intersections of the circle with lines $MA$,$MB$ respectively. Let $G$ be the midpoint of $AB$and $G_1= C\cap MG$. Prove that$$\frac{MA}{AA_1}+ \frac{MB}{BB_1}> 2\frac{MG}{GG_1}$$
2008 Dutch IMO TST, 5
Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.
KoMaL A Problems 2019/2020, A. 774
Let $O$ be the circumcenter of triangle $ABC,$ and $D$ be an arbitrary point on the circumcircle of $ABC.$ Let points $X, Y$ and $Z$ be the orthogonal projections of point $D$ onto lines $OA, OB$ and $OC,$ respectively. Prove that the incenter of triangle $XYZ$ is on the Simson-Wallace line of triangle $ABC$ corresponding to point $D.$
Ukrainian TYM Qualifying - geometry, 2011.11
Let $BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC$, which intersect its angle bisector $AL$ at two different points $P$ and $Q$, respectively. Denote by $F$ such a point that $PF\parallel AB$ and $QF\parallel AC$, and by $T$ the intersection point of the tangents drawn at points $B$ and $C$ to the circumscribed circle of the triangle $ABC$. Prove that the points $A, F$ and $T$ lie on the same line.
2005 Iran MO (3rd Round), 5
Suppose $a,b,c \in \mathbb R^+$and \[\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2\]
Prove that $ab+ac+bc\leq \frac32$
2019 India PRMO, 28
In a triangle $ABC$, it is known that $\angle A=100^{\circ}$ and $AB=AC$. The internal angle bisector $BD$ has length $20$ units. Find the length of $BC$ to the nearest integer, given that $\sin 10^{\circ} \approx 0.174$