Two equal circles $\Omega_1$ and $\Omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Two rays were drawn from $M$, lying in the same half-plane wrt $AB$ (see figure). The first ray intersects the circles $\Omega_1$ and $\Omega_2$ at points $X_1$ and $X_2$, and the second ray intersects them at points $Y_1$ and $Y_2$, respectively. Let $C$ be the intersection point of straight lines $AX_1$ and $BY_2$, and let $D$ be the intersection point of straight lines $AX_2$ and $BY_1$. Prove that $CD \parallel AB$. [img]https://cdn.artofproblemsolving.com/attachments/4/a/fae047c3956d8b30f15a9d88e8d12e5f4d48ec.png[/img]

[b]p1.[/b] A band of pirates unloaded some number of treasure chests from their ship. The number of pirates was between $60$ and $69$ (inclusive). Each pirate handled exactly $11$ treasure chests, and each treasure chest was handled by exactly $7$ pirates. Exactly how many treasure chests were there? Show that your answer is the only solution. [b]p2.[/b] Let $a$ and $b$ be the lengths of the legs of a right triangle, let $c$ be the length of the hypotenuse, and let $h$ be the length of the altitude drawn from the vertex of the right angle to the hypotenuse. Prove that $c+h>a+b$. [b]p3.[/b] Prove that $$\frac{1}{70}< \frac{1}{2} \frac{3}{4} \frac{5}{6} ... \frac{2001}{2002} < \frac{1}{40}$$ [b]p4.[/b] Given a positive integer $a_1$ we form a sequence $a_1 , a_2 , a _3,...$ as follows: $a_2$ is obtained from $a_1$ by adding together the digits of $a_1$ raised to the $2001$-st power; $a_3$ is obtained from $a_2$ using the same rule, and so on. For example, if $a_1 =25$, then $a_2 =2^{2001}+5^{2001}$, which is a $1399$-digit number containing $106$ $0$'s, $150$ $1$'s, 4124$ 42$'s, $157$ $3$'s, $148$ $4$'s, $141$ $5$'s, $128$ $6$'s, $1504 47$'s, $152$ $8$'s, $143$ $9$'s. So $a_3 = 106 \times 0^{2001}+ 150 \times 1^{2001}+ 124 \times 2^{2001}+ 157 \times 3^{2001}+ ...+ 143 \times 9^{2001}$ which is a $1912$-digit number, and so forth. Prove that if any positive integer $a_1$ is chosen to start the sequence, then there is a positive integer $M$ (which depends on $a_1$ ) that is so large that $a_n < M$ for all $n=1,2,3,...$ [b]p5.[/b] Let $P(x)$ be a polynomial with integer coefficients. Suppose that there are integers $a$, $b$, and $c$ such that $P(a)=0$, $P(b)=1$, and $P(c)=2$. Prove that there is at most one integer $n$ such that $P(n)=4$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given an equilateral triangle lattice composed of $\frac{n(n+1)}{2}$ points (as shown in the figure), record the number of equilateral triangles with three points in the lattice as vertices as $f(n)$. Find an expression for $f(n)$. [img]https://cdn.artofproblemsolving.com/attachments/7/f/1de27231e8ef9c1c6a3dfd590a7c71adc508d6.png[/img]

Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.

Let $ABCD$ be a parallelogram with center $O$ such that $\angle BAD <90^o$ and $\angle AOB> 90^o$. Consider points $A_1$ and $B_1$ on the rays $OA$ and $OB$ respectively, such that $A_1B_1$ is parallel to $AB$ and $\angle A_1B_1C = \frac12 \angle ABC$. Prove that $A_1D$ is perpendicular to $B_1C$.

Let $ R $ be the circumradius of a triangle $ ABC. $ The points $ B,C, $ lie on a circle of radius $ \rho $ that intersects $ AB,AC $ at $ E,D, $ respectively. $ \rho' $ is the circumradius of $ ADE. $ Show that there exists a triangle with sides $ R,\rho ,\rho' , $ and having an angle whose value doesn't depend on $ \rho . $ [i]Laurențiu Panaitopol[/i]

[b]p1.[/b] Let $A$ be the point $(-1, 0)$, $B$ be the point $(0, 1)$ and $C$ be the point $(1, 0)$ on the $xy$-plane. Assume that $P(x, y)$ is a point on the $xy$-plane that satisfies the following condition $$d_1 \cdot d_2 = (d_3)^2,$$ where $d_1$ is the distance from $P$ to the line $AB$, $d_2$ is the distance from $P$ to the line $BC$, and $d_3$ is the distance from $P$ to the line $AC$. Find the equation(s) that must be satisfied by the point $P(x, y)$. [b]p2.[/b] On Day $1$, Peter sends an email to a female friend and a male friend with the following instructions: $\bullet$ If you’re a male, send this email to $2$ female friends tomorrow, including the instructions. $\bullet$ If you’re a female, send this email to $1$ male friend tomorrow, including the instructions. Assuming that everyone checks their email daily and follows the instructions, how many emails will be sent from Day $1$ to Day $365$ (inclusive)? [b]p3.[/b] For every rational number $\frac{a}{b}$ in the interval $(0, 1]$, consider the interval of length $\frac{1}{2b^2}$ with $\frac{a}{b}$ as the center, that is, the interval $\left( \frac{a}{b}- \frac{1}{2b^2}, \frac{a}{b}+\frac{1}{2b^2}\right)$ . Show that $\frac{\sqrt2}{2}$ is not contained in any of these intervals. [b]p4.[/b] Let $a$ and $b$ be real numbers such that $0 < b < a < 1$ with the property that $$\log_a x + \log_b x = 4 \log_{ab} x - \left(\log_b (ab^{-1} - 1)\right)\left(\log_a (ab^{-1} - 1) + 2 log_a ab^{-1} \right)$$ for some positive real number $x \ne 1$. Find the value of $\frac{a}{b}$. [b]p5.[/b] Find the largest positive constant $\lambda$ such that $$\lambda a^2 b^2 (a - b)^2 \le (a^2 - ab + b^2)^3$$ is true for all real numbers $a$ and $b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Points $K$ and $L$ are on the sides $BC$ and $CD$, respectively of the parallelogram $ABCD$, such that $AB + BK = AD + DL$. Prove that the bisector of angle $BAD$ is perpendicular to the line $KL$.

Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)

Given triangle $ABC$ in which $\angle CAB= 30^{\circ}$ and $\angle ACB=60^{\circ}$. On the ray $AB$ a point $D$ is chosen, and on the ray $CB$ a point $E$ is chosen such that $\angle BDE=60^{\circ}$. Lines $AC$ and $DE$ intersect at $F$. Prove that the circumcircle of $AEF$ passes through a fixed point, which is different from $A$ and does not depend on $D$.

In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.

In triangle $\triangle ABC, AB=AC.$ Let $D$ be on segment $AC$ and $E$ be a point on the extended line $BC$ such that $C$ is located between $B$ and $E$ and $\frac{AD}{DC}=\frac{BC}{2CE}$. Let $\omega$ be the circle with diameter $AB,$ and $\omega$ intersects segment $DE$ at $F.$ Prove that $B,C,F,D$ are concyclic.

Determine a right parallelepiped with minimal area, if its volume is strictly greater than $1000$, and the lengths of it sides are integer numbers.

Let $O$ be the centre of a parallelogram $ABCD$ and $P$ be any point in the plane. Let $M, N$ be the midpoints of $AP, BP$, respectively and $Q$ be the intersection of $MC$ and $ND$. Prove that $O, P$ and $Q$ are collinear.

Three distinct points $A$, $B$, and $N$ are marked on the line $l$, with $B$ lying between $A$ and $N$. For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$, points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$, $C$, and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$; and $A$, $B$, $C$, and $D$ are concyclic. Find the locus of the intersection points of the diagonals of $ABCD$ as $\alpha$ varies between $0$ and $\frac{\pi}{2}$.

Let $ABCDEF$ be a convex hexagon satisfying $AC = DF, CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

In triangle $ABC$, $AB = AC$. Point $D$ is the midpoint of side $BC$. Point $E$ lies outside the triangle $ABC$ such that $CE \perp AB$ and $BE = BD$. Let $M$ be the midpoint of segment $BE$. Point $F$ lies on the minor arc $\widehat{AD}$ of the circumcircle of triangle $ABD$ such that $MF \perp BE$. Prove that $ED \perp FD.$ [asy] defaultpen(fontsize(10)); size(6cm); pair A = (3,10), B = (0,0), C = (6,0), D = (3,0), E = intersectionpoints( Circle(B, 3), C--(C+100*dir(B--A)*dir(90)) )[1], M = midpoint(B--E), F = intersectionpoints(M--(M+50*dir(E--B)*dir(90)), circumcircle(A,B,D))[0]; dot(A^^B^^C^^D^^E^^M^^F); draw(B--C--A--B--E--D--F--M^^circumcircle(A,B,D)); pair point = extension(M,F,A,D); pair[] p={A,B,C,D,E,F,M}; string s = "A,B,C,D,E,F,M"; int size = p.length; real[] d; real[] mult; for(int i = 0; i<size; ++i) { d[i] = 0; mult[i] = 1;} d[4] = -50; string[] k= split(s,","); for(int i = 0;i<p.length;++i) { label("$"+k[i]+"$",p[i],mult[i]*dir(point--p[i])*dir(d[i])); }[/asy]

$ABCD$ is a parallelogram satisfying $AB = 7$, $BC = 2$, and $\angle DAB = 120^o$. Parallelogram $ECFA$ is contained in $ABCD$ and is similar to it. Find the ratio of the area of $ECFA$ to the area of $ABCD$.

In triangle $ABC$, with orthocenter $H$ and circumcircle $\Gamma$, the bisector of angle $BAC$ meets $\overline{BC}$ at $K$. Point $Q$ lies on $\Gamma$ such that $\overline{AQ} \perp \overline{QK}$. Circumcircle of $\triangle AQH$ meets $\overline{AC}$ at $Y$ and $\overline{AB}$ at $Z$. Let $\overline{BY}$ and $\overline{CZ}$ meet at $T$. Prove that $\overline{TH} \perp \overline{KA}$

A round pencil has length $ 8$ when unsharpened, and diameter $ \frac {1}{4}$. It is sharpened perfectly so that it remains $ 8$ inches long, with a $ 7$ inch section still cylindrical and the remaining $ 1$ inch giving a conical tip. What is its volume?

A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?

Let $\triangle ABC$ be a triangle with $AB=10$ and $AC=16,$ and let $I$ be the intersection of the internal angle bisectors of $\triangle ABC.$ Suppose the tangents to the circumcircle of $\triangle BIC$ at $B$ and $C$ intersect at a point $P$ with $PA=8.$ Compute the length of ${BC}.$ [i]Proposed by Kyle Lee[/i]

In an acute $\triangle ABC$ ($AB \neq AC$) with angle $\alpha$ at the vertex $A$, point $E$ is the nine-point center, and $P$ a point on the segment $AE$. If $\angle ABP = \angle ACP = x$, prove that $x = 90$° $ -2 \alpha $. [i]Proposed by Dusan Djukic[/i]

In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.

Given two circles of radii $r$ and $r'$ exterior to each other, construct a line parallel to a given line and intersecting the two circles in chords with the sum of lengths $\ell$.