Let $ABCDEF$ be a convex hexagon such that $BCEF$ is a parallelogram and $ABF$ an equilateral triangle. Given that $BC = 1, AD = 3, CD+DE = 2$, compute the area of $ABCDEF$

A point $A_0$ and two lines $d_1$ and $d_2$ are given in the space. For each nonnegative integer $n$ we denote by $B_n$ the projection of $A_n$ on $d_2,$ and by $A_{n+1}$ the projection of $B_n$ on $d_1.$ Prove that there exist two segments $[A'A''] \subset d_1$ and $[B'B''] \subset d_2$ of length $0.001$ and a nonnegative integer $N$ such that $A_n \in [A'A'']$ and $B_n \in [B'B'']$ for any $n \ge N.$

$ABC$ is acute-angled triangle. $AA_1,BB_1,CC_1$ are altitudes. $X,Y$ - midpoints of $AC_1,A_1C$. $XY=BB_1$. Prove that one side of $ABC$ in $\sqrt{2}$ greater than other side.

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

Let $ABC$ be an isosceles triangle, $AB = AC$, and let $M$ and $N$ be points on the sides $BC$ and $CA$, respectively, such that $\angle BAM=\angle CNM$. The lines $AB$ and $MN$ meet at $P$. Show that the internal angle bisectors of the angles $BAM$ and $BPM$ meet at a point on the line $BC$.

Let $\triangle ABC$ be an acute angled triangle. Let $G$ be the centroid and let $D$ be the foot of the altitude from $A$ onto $BC$. Let ray $GD$ intersect $(ABC)$ at $X$ and let $AG$ intersect nine point circle at $Y$ not on $BC$. Let $Z$ be the intersection of the $\text{A-tangent}$ to $(ABC)$ and $\text{A-midline}$. Prove that perpendicular from $Z$ to the Euler line, $AX$ and $DY$ concur. The line joining the midpoints of $AB$ and $AC$ is called the $\text{A-midline}$. $(ABC)$ denotes the circumcircle of $\triangle ABC$

Let $\Omega$ and $O$ be the circumcircle of acute triangle $ABC$ and its center, respectively. $M\ne O$ is an arbitrary point in the interior of $ABC$ such that $AM$, $BM$, and $CM$ intersect $\Omega$ at $A_{1}$, $B_{1}$, and $C_{1}$, respectiuvely. Let $A_{2}$, $B_{2}$, and $C_{2}$ be the circumcenters of $MBC$, $MCA$, and $MAB$, respectively. It is to be proven that $A_{1}A_{2}$, $B_{1}B_{2}$, $C_{1}C{2}$ concur.

Let $ABCD$ be a rectangle and $P$ be a point on the side$ AD$ such that $\angle BPC = 90^o$. The perpendicular from $A$ on $BP$ cuts $BP$ at $M$ and the perpendicular from $D$ on $CP$ cuts $CP$ in $N$. Show that the center of the rectangle lies in the $MN$ segment.

Let two circles $A$ and $B$ with unequal radii $r$ and $R$, respectively, be tangent internally at the point $A_0$. If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$, and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$, prove that \[\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.\]

The acuteangled triangle $ABC$ with circumcenter $O$ is given. The midpoints of the sides $BC, CA$ and $AB$ are $D, E$ and $F$ respectively. An arbitrary point $M$ on the side $BC$, different of $D$, is choosen. The straight lines $AM$ and $EF$ intersects at the point $N$ and the straight line $ON$ cut again the circumscribed circle of the triangle $ODM$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $DP$ belongs on the nine points circle of the triangle $ABC$.

On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$.

[b]p1.[/b] We say that integers $a$ and $b$ are [i]friends [/i] if their product is a perfect square. Prove that if $a$ is a friend of $b$, then $a$ is a friend of $gcd (a, b)$. [b]p2.[/b] On the island of knights and liars, a traveler visited his friend, a knight, and saw him sitting at a round table with five guests. "I wonder how many knights are among you?" he asked. " Ask everyone a question and find out yourself" advised him one of the guests. "Okay. Tell me one: Who are your neighbors?" asked the traveler. This question was answered the same way by all the guests. "This information is not enough!" said the traveler. "But today is my birthday, do not forget it!" said one of the guests. "Yes, today is his birthday!" said his neighbor. Now the traveler was able to find out how many knights were at the table. Indeed, how many of them were there if [i]knights always tell the truth and liars always lie[/i]? [b]p3.[/b] A rope is folded in half, then in half again, then in half yet again. Then all the layers of the rope were cut in the same place. What is the length of the rope if you know that one of the pieces obtained has length of $9$ meters and another has length $4$ meters? [b]p4.[/b] The floor plan of the palace of the Shah is a square of dimensions $6 \times 6$, divided into rooms of dimensions $1 \times 1$. In the middle of each wall between rooms is a door. The Shah orders his architect to eliminate some of the walls so that all rooms have dimensions $2 \times 1$, no new doors are created, and a path between any two rooms has no more than $N$ doors. What is the smallest value of $N$ such that the order could be executed? [b]p5.[/b] There are $10$ consecutive positive integers written on a blackboard. One number is erased. The sum of remaining nine integers is $2011$. Which number was erased? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given two points $A,B$ outside of a given plane $P,$ find the positions of points $M$ in the plane $P$ for which the ratio $\frac{MA}{MB}$ takes a minimum or maximum.

In quadrilateral $ABCD,$ suppose that $\overline{CD}$ is perpendicular to $\overline{BC}$ and $\overline{DA}$. Point $E$ is chosen on segment $\overline{CD}$ such that $\angle AED = \angle BEC$. If $AB = 6$, $AD = 7$, and $\angle ABC = 120^o$ , compute $AE + EB$.

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

2. The quadrilateral $A B C D$ is convex. Its sides $A B$ and $C D$ are parallel. It is known that the angles $D A C$ and $A B D$ are equal. Furthermore the angles $C A B$ and $D B C$ are equal. Is $A B C D$ necessarily a square? Alexandr Terteryan

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 $A_i.$

In acute triangle $ABC$ bisector of angle $BAC$ intersects side $BC$ in point $D$. Bisector of line segment $AD$ intersects circumcircle of triangle $ABC$ in points $E$ and $F$. Show that circumcircle of triangle $DEF$ is tangent to line $BC$.

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A straight line is drawn through point $B$, which again intersects circles $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Point $E$, located on circle $\omega_1$ , satisfies the relation $CE = CB$ , and point $F$, located on circle $\omega_2$, satisfies the relation $DB = DF$. The line $BF$ intersects again the circle $\omega_1$ at the point $P$, and the line $BE$ intersects again the circle $\omega_2$ at the point $Q$. Prove that the points $A, P$, and $Q$ are collinear.

A trapezoid is divided into seven strips of equal width as shown. What fraction of the trapezoid’s area is shaded?

Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$

$6$ pepperoni circles will exactly fit across the diameter of a $12$-inch pizza when placed. If a total of $24$ circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered with pepperoni? $ \textbf{(A)}\ \frac 12 \qquad\textbf{(B)}\ \frac 23 \qquad\textbf{(C)}\ \frac 34 \qquad\textbf{(D)}\ \frac 56 \qquad\textbf{(E)}\ \frac 78 $

Let ${ABC}$ be an acute angled triangle, let ${O}$ be its circumcentre, and let ${D,E,F}$ be points on the sides ${BC,CA,AB}$, respectively. The circle ${(c_1)}$ of radius ${FA}$, centered at ${F}$, crosses the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centered at $D$, crosses the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$. Finally, the circle ${(c_3)}$ of radius $EC$, centered at $E$, crosses the segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$. Prove that the quadrilaterals $BKF{A}',CLD{B}'$ and $AME{C}'$ are all cyclic, and their circumcircles share a common point. Evangelos Psychas (Greece)

Let $A$ and $B$ be points on a circle $k$. Suppose that an arc $k'$ of another circle, $\ell$, connects $A$ with $B$ and divides the area inside the circle $k$ into two equal parts. Prove that arc $k'$ is longer than the diameter of $k$.

In $\triangle ABC$ with $AB=10$, $AC=13$, and $\measuredangle ABC = 30^\circ$, $M$ is the midpoint of $\overline{BC}$ and the circle with diameter $\overline{AM}$ meets $\overline{CB}$ and $\overline{CA}$ again at $D$ and $E$, respectively. The area of $\triangle DEM$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Compute $100m + n$. [i]Based on a proposal by Matthew Babbitt[/i]