Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.

[b]a)[/b] Among the $21$ pairwise distances between the $7$ points of the plane, prove that one and the same number occurs not more than $12$ times. [b]b)[/b] Find a maximum number of times may meet the same number among the $15$ pairwise distances between $6$ points of the plane.

Let $ABC$ be an acute triange with $B,C$ fixed. Let $D$ be the midpoint of $BC$ and $E,F$ be the foot of the perpendiculars from $D$ to $AB,AC$, respectively. a) Let $O$ be the circumcenter of triangle $ABC$ and $M=EF\cap AO, N=EF\cap BC$. Prove that the circumcircle of triangle $AMN$ passes through a fixed point; b) Assume that tangents of the circumcircle of triangle $AEF$ at $E,F$ are intersecting at $T$. Prove that $T$ is on a fixed line.

Given a square with area $A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $\frac{A}{B}.$

Triangle $ABC$ has area $1$. Points $E$, $F$, and $G$ lie, respectively, on sides $BC$, $CA$, and $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$.

As shown in the figure, let point $E$ be the intersection of the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$. The circumcenter of triangle $ABE$ is denoted as $K$. Point $X$ is the reflection of point $B$ with respect to the line $CD$, and point $Y$ is the point on the plane such that quadrilateral $DKEY$ is a parallelogram. Prove that the points $D,E,X,Y$ are concyclic. [img]https://cdn.artofproblemsolving.com/attachments/3/4/df852f90028df6f09b4ec1342f5330fc531d12.jpg[/img]

Let $M$ be the midpoint of side $BC$ of acute triangle $ABC$. The circle centered at $M$ passing through $A$ intersects the lines $AB$ and $AC$ again at $P$ and $Q$, respectively. The tangents to this circle at $P$ and $Q$ meet at $D$. Prove that the perpendicular bisector of $BC$ bisects segment $AD$.

The point $ M$ is chosen inside parallelogram $ ABCD$. Show that $ \angle MAB$ is congruent to $ \angle MCB$, if and only if $ \angle MBA$ and $ \angle MDA$ are congruent.

There are $2019$ segments $[a_1, b_1]$, $...$, $[a_{2019}, b_{2019}]$ on the line. It is known that any two of them intersect. Prove that they all have a point in common.

Cut the $9 \times 10$ grid rectangle along the grid lines into several squares so that there are exactly two of them with odd sidelengths.

(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle $\alpha$. Determine the smallest number of lines for which this is possible. (b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections.

[b]p1.[/b] Find the number of ordered triples $(a, b, c)$ satisfying $\bullet$ $a, b, c$ are are single-digit positive integers, and $\bullet$ $a \cdot b + c = a + b \cdot c$. [b]p2.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form an increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.) [b]p3.[/b] Suppose that $a + \frac{1}{b} = 2$ and $b +\frac{1}{a} = 3$. If$ \frac{a}{b} + \frac{b}{a}$ can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$. [b]p4.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that $$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B_1B_1B_1...}) + 1$$ Find the sum of all possible values of $10A + B$. [b]p5.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $AC$ and $BC$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE = EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a, b$ positive integers and a squarefree, then find $a + b$. [b]p6.[/b] Five bowling pins $P_1$, $P_2$,..., $P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is a b where a and b are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i -j| = 1$.) [b]p7.[/b] Let triangle $ABC$ have side lengths $AB = 10$, $BC = 24$, and $AC = 26$. Let $I$ be the incenter of $ABC$. If the maximum possible distance between $I$ and a point on the circumcircle of $ABC$ can be expressed as $a +\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$. (The incenter of any triangle $XY Z$ is the intersection of the angle bisectors of $\angle Y XZ$, $\angle XZY$, and $\angle ZY X$.) [b]p8.[/b] How many terms in the expansion of $$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$ have coefficients equal to $1011$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an acute triangle $ABC$ with circumcenter $O$. The circumcircle of $BCH$ and a circle with diameter of $AC$ intersect at $P (P \neq C)$. A point $Q$ on segment of $PC$ such that $PB = PQ$. Prove that $\angle ABC = \angle AQP$

[hide=Note][i]The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.[/i][/hide] For each integer $n\ge2$, consider a regular polygon with $2n$ sides, all of length $1$. Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length $1$. Determine the limit inferior and the limit superior of the sequence defined by $$\frac1{n^2}\log_2C(n).$$

Let $ABCD$ be a isosceles trapezoid. Base $AD$ is $11$ cm long while the other three sides are each $5$ cm long. We draw the line that is perpendicular to $BD$ and contains $C$ and the line that is perpendicular to $AC$ and contains$ B$. We mark the intersection of these two lines with $E$. What is the distance between point $E$ and line $AD$?

Triangle $ABC$ has a right angle at $B$. Point $D$ lies on side $BC$ such that $3\angle BAD = \angle BAC$. Given $AC=2$ and $CD=1$, compute $BD$.

The circle $\omega{}$ is inscribed in the quadrilateral $ABCD$. Prove that the diameter of the circle $\omega{}$ does not exceed the length of the segment connecting the midpoints of the sides $BC$ and $AD$. [i]Proposed by O. Yuzhakov[/i]

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$

Circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively lie on a plane such that that the circle $C_2$ passes through $O_1$. The ratio of radius of circle $C_1$ to $O_1O_2$ is $\sqrt{2+\sqrt3}$. a) Prove that the circles $C_1$ and $C_2$ intersect at two distinct points. b) Let $A,B$ be these points of intersection. What proportion of the area of circle is $C_1$ is the area of the sector $AO_1B$ ?

Let $ [t]$ denote the greatest integer $ \leq t$ where $ t \geq 0$ and $ S \equal{} \{(x,y): (x\minus{}T)^2 \plus{} y^2 \leq T^2 \text{ where } T \equal{} t \minus{} [t]\}$. Then we have $ \textbf{(A)}\ \text{the point } (0,0) \text{ does not belong to } S \text{ for any } t \qquad$ $ \textbf{(B)}\ 0 \leq \text{Area } S \leq \pi \text{ for all } t \qquad$ $ \textbf{(C)}\ S \text{ is contained in the first quadrant for all } t \geq 5 \qquad$ $ \textbf{(D)}\ \text{the center of } S \text{ for any } t \text{ is on the line } y\equal{}x \qquad$ $ \textbf{(E)}\ \text{none of the other statements is true}$

In the overlapping triangles $ \triangle{ABC}$ and $ \triangle{ABE}$ sharing common side $ AB$, $ \angle{EAB}$ and $ \angle{ABC}$ are right angles, $ AB \equal{} 4$, $ BC \equal{} 6$, $ AE \equal{} 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle{ADE}$ and $ \triangle{BDC}$? [asy] defaultpen(linewidth(0.8)+fontsize(10));size(200); unitsize(5mm) ; pair A=(0,0), B=(4,0), C=(4,6), D=(0,8), H=intersectionpoint(C--A, D--B); draw(A--B--C--cycle) ; draw(A--B--D--cycle) ; label("E",(0,8), N) ; label("8",(0,4),W) ; label("A",A,S) ; label("B",B,SE) ; label("C",C,NE) ; label("6",(4,3),E) ; label("4",(2,0),S) ; label("D",H,2*dir(85)) ;[/asy] $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

Prove that the bases of the altitudes and medians of an acute-angled triangle lie on the same circle.

The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$.

In triangle $ABC$, angle $B$ is different from a right angle, $AB : BC = k$. Let $M$ be the midpoint of $AC$. Lines symmetric to $BM$ wrt $AB$ and $BC$ intersect line $AC$ at points $D$ and $E$. Find $BD : BE$.