Found problems: 821
Given is the acute triangle $ABC$. Let us denote $k$ a circle with diameter $AB$. Another circle, tangent to $AB$ at point $A$ and passing through point $C$ intersects the circle $k$ at point $P, P \ne A$. Another circle which touches AB at point $B$ and passes point $C$, intersects the circle $k$ at point $Q, Q \ne B$. Prove that the intersection of the line $AQ$ and $BP$ lies on one of the sides of angle $ACB$.
(Peter Novotný)
A square, with an area of $40$, is inscribed in a semicircle. The area of a square that could be inscribed in the entire circle with the same radius, is:
$ \textbf{(A)}\ 80 \qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 120\qquad\textbf{(D)}\ 160\qquad\textbf{(E)}\ 200 $
Let skew lines $PM, QN$ be given such that $PM\perp PQ\perp QN$. Let a plane $\sigma\perp PQ$ containing the midpoint $O$ of segment $PQ$ be given and in it a circle $k$ with center $O$ and given radius $r$. Consider all segments $XY$ with endpoint $X, Y$ on lines $PM, QN$, respectively, which contain a point of $k$. Show that segments $XY$ have the same length. Find the locus of all such points $X$.
Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
[img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]
Two circles ${{c} _ {1}}, \, \, {{c} _ {2}}$ pass through the center $O$ of the circle $c$ and touch it internally in points $A$ and $B$, respectively. Prove that the line $AB$ passes though a common point of circles ${{c} _ {1}}, \, \, {{c} _ {2}} $.
Let $ABC$ be an acute triangle with $AB<AC<BC$, inscribed in circle $c(O,R)$ (with center $O$ and radius $R$). Let $O_1$ be the symmetric point of $O$ wrt $AC$. Circle $c_1(O_1,R)$ intersects $BC$ at $Z$. If the extension of the altitude $AD$ intersects the cicrumscribed circle $c(O,R)$ at point $E$, prove that $EC$ is perpendicular on $AZ$.
Given a regular hexagon, whose sidelength is $ 1$ . What is the largest number of circles of radius $\frac{\sqrt3}{4}$ can be placed without overlapping inside such a hexagon? (Circles can touch each other and the sides of the hexagon.)
The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that
a) there are three consecutive numbers with the sum being at least $15$,
b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.
Two circles $\omega_1$ and $\omega_2$, which have centers $O_1$ and $O_2$, respectively, intersect at $A$ and $B$. A line $\ell$ that passes through $B$ cuts to $\omega_1$ again at $C$ and cuts to $\omega_2$ again in $D$, so that points $C, B, D$ appear in that order. The tangents of $\omega_1$ and $\omega_2$ in $C$ and $D$, respectively, intersect in $E$. Line $AE$ intersects again to the circumscribed circumference of the triangle $AO_1O_2$ in $F$. Try that the length of the $EF$ segment is constant, that is, it does not depend on the choice of $\ell$.
A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than $3.4$.
Is it possible to assert that it is more than $3.5$?
Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.
Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.
[i]Proposed by Horst Sewerin, Germany[/i]
Diameter $AB$ is drawn to a circle with radius $1$. Two straight lines $s$ and $t$ touch the circle at points $A$ and $B$, respectively. Points $P$ and $Q$ are chosen on the lines $s$ and $t$, respectively, so that the line $PQ$ touches the circle. Find the smallest possible area of the quadrangle $APQB$.
On the inner surface of a fixed circle, rolls a wheel half the radius of the circle, without slipping. We marked a point red on the wheel. Prove that while the wheel makes a turn, the point moves on a line.
[img]https://1.bp.blogspot.com/-PhgUWk0eU2c/X9j1gNJ7w3I/AAAAAAAAMzo/gP13TIZq7YsvNDBGVISkMQSdjwCgk_zwQCLcBGAsYHQ/s0/2014%2BDurer%2BD2.png[/img]
Consider the equilateral triangle $ABC$ with $|BC| = |CA| = |AB| = 1$.
On the extension of side $BC$, we define points $A_1$ (on the same side as B) and $A_2$ (on the same side as C) such that $|A_1B| = |BC| = |CA_2| = 1$. Similarly, we define $B_1$ and $B_2$ on the extension of side $CA$ such that $|B_1C| = |CA| =|AB_2| = 1$, and $C_1$ and $C_2$ on the extension of side $AB$ such that $|C_1A| = |AB| = |BC_2| = 1$. Now the circumcentre of 4ABC is also the centre of the circle that passes through the points $A_1,B_2,C_1,A_2,B_1$ and $C_2$.
Calculate the radius of the circle through $A_1,B_2,C_1,A_2,B_1$ and $C_2$.
[asy]
unitsize(1.5 cm);
pair[] A, B, C;
A[0] = (0,0);
B[0] = (1,0);
C[0] = dir(60);
A[1] = B[0] + dir(-60);
A[2] = C[0] + dir(120);
B[1] = C[0] + dir(60);
B[2] = A[0] + dir(240);
C[1] = A[0] + (-1,0);
C[2] = B[0] + (1,0);
draw(A[1]--A[2]);
draw(B[1]--B[2]);
draw(C[1]--C[2]);
draw(circumcircle(A[2],B[1],C[2]));
dot("$A$", A[0], SE);
dot("$A_1$", A[1], SE);
dot("$A_2$", A[2], NW);
dot("$B$", B[0], SW);
dot("$B_1$", B[1], NE);
dot("$B_2$", B[2], SW);
dot("$C$", C[0], N);
dot("$C_1$", C[1], W);
dot("$C_2$", C[2], E);
[/asy]
Let $ABC$ be a triangle with no right angle, $E$ on the line $BC$ such that $\angle AEB = \angle BAC$ and $\Delta_A$ the perpendicular to $BC$ at $E$. Let the circle $\gamma$ with diameter $BC$ intersect $BA$ again at $D$. For each point $M$ on $\gamma$ ($M$ is distinct from $B$), the line $BM$ meets $\Delta_A$ at $M'$ and the line $AM$ meets $\gamma$ again at $M''$.
(a) Show that $p(A) = AM' \times DM''$ is independent of the chosen $M$.
(b) Keeping $B,C$ fixed, and let $A$ vary. Show that $\frac{p(A)}{d(A,\Delta_A)}$ is independent of $A$.
Michel Bataille
Circles $\omega_1$, $\omega_2$, and $\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\omega_1$, $\omega_2$, and $\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\triangle P_1P_2P_3$ can be written in the form $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$?
[asy]
unitsize(12);
pair A = (0, 8/sqrt(3)), B = rotate(-120)*A, C = rotate(120)*A;
real theta = 41.5;
pair P1 = rotate(theta)*(2+2*sqrt(7/3), 0), P2 = rotate(-120)*P1, P3 = rotate(120)*P1;
filldraw(P1--P2--P3--cycle, gray(0.9));
draw(Circle(A, 4));
draw(Circle(B, 4));
draw(Circle(C, 4));
dot(P1);
dot(P2);
dot(P3);
defaultpen(fontsize(10pt));
label("$P_1$", P1, E*1.5);
label("$P_2$", P2, SW*1.5);
label("$P_3$", P3, N);
label("$\omega_1$", A, W*17);
label("$\omega_2$", B, E*17);
label("$\omega_3$", C, W*17);
[/asy]
$\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554$
Let $k$ be a circle with diameter $AB$. Let $C$ be a point on the straight line $AB$, so that $B$ between $A$ and $C$ lies. Let $T$ be a point on $k$ such that $CT$ is a tangent to $k$. Let $l$ be the parallel to $CT$ through $A$ and $D$ the intersection of $l$ and the perpendicular to $AB$ through $T$. Show that the line $DB$ bisects segment $CT$.
Circles $C_1$, $C_2$, $C_3$ have radius $ 1$ and centers $O, P, Q$ respectively. $C_1$ and $C_2$ intersect at $A$, $C_2$ and $C_3$ intersect at $B$, $C_3$ and $C_1$ intersect at $C$, in such a way that $\angle APB = 60^o$ , $\angle BQC = 36^o$ , and $\angle COA = 72^o$ . Find angle $\angle ABC$ (degrees).
Let $\Gamma_1$ and $\Gamma_2$ be two circles of unequal radii, with centres $O_1$ and $O_2$ respectively, intersecting in two distinct points $A$ and $B$. Assume that the centre of each circle is outside the other circle. The tangent to $\Gamma_1$ at $B$ intersects $\Gamma_2$ again in $C$, different from $B$; the tangent to $\Gamma_2$ at $B$ intersects $\Gamma_1$ again at $D$, different from $B$. The bisectors of $\angle DAB$ and $\angle CAB$ meet $\Gamma_1$ and $\Gamma_2$ again in $X$ and $Y$, respectively. Let $P$ and $Q$ be the circumcentres of triangles $ACD$ and $XAY$, respectively. Prove that $PQ$ is the perpendicular bisector of the line segment $O_1O_2$.
[i]Proposed by Prithwijit De[/i]
Given a point $A$ that lies on circle $c(o,R)$ (with center $O$ and radius $R$). Let $(e)$ be the tangent of the circle $c$ at point $A$ and a line $(d)$ that passes through point $O$ and intersects $(e)$ at point $M$ and the circle at points $B,C$ (let $B$ lie between $O$ and $A$). If $AM = R\sqrt3$ , prove that
a) triangle $AMC$ is isosceles.
b) circumcenter of triangle $AMC$ lies on circle $c$ .
In the plane we are given two circles intersecting at $ X$ and $ Y$. Prove that there exist four points with the following property:
(P) For every circle touching the two given circles at $ A$ and $ B$, and meeting the line $ XY$ at $ C$ and $ D$, each of the lines $ AC$, $ AD$, $ BC$, $ BD$ passes through one of these points.
Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that
a) points $T, \, \, X, \, \, B$ lie on one line;
b) points $K, \, \, X, \, \, H$ lie on one line.
(Vadym Mitrofanov)
$20.$ The circle $\omega$ touches the circle $\Omega$ internally at point $P.$ The centre $O$ of $\Omega$ is outside $\omega.$ Let $XY$ be a diameter of $\Omega$ which is also tangent to $\omega.$ Assume $PY>PX.$ Let $PY$ intersect $\omega$ at $z.$ If $YZ=2PZ,$ what is the magnitude of $\angle{PYX}$ in degrees $?$
A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$.
[i]Proposed by S. Berlov[/i]