On the circle 6 distinct points $ A $, $ B $, $ C $, $ D $, $ E $, $ F $ are chosen in such a way that $ AB $ is parallel to $ DE $, and $ DC $ is parallel to $ AF $. Prove that $ BC $ is parallel to $ EF $

In the triangle $ ABC,$ with $ \angle A \equal{} 60 ^{\circ},$ a parallel $ IF$ to $ AC$ is drawn through the incenter $ I$ of the triangle, where $ F$ lies on the side $ AB.$ The point $ P$ on the side $ BC$ is such that $ 3BP \equal{} BC.$ Show that $ \angle BFP \equal{} \frac{\angle B}{2}.$

A circle with center $O$ is tangent to the sides of the angle with the vertex $A$ at the points B and C. Let M be a point on the larger of the two arcs $BC$ of this circle (different from $B$ and $C$) such that $M$ does not lie on the line $AO$. Lines $BM$ and $CM$ intersect the line $AO$ at the points $P$ and $Q$ respectively. Let $K$ be the foot of the perpendicular drawn from $P$ to $AC$ and $L$ be the foot of the perpendicular drawn from $Q$ to $AB$. Prove that the lines $OM$ and $KL$ are perpendicular.

[b]p1. [/b]Ed, Fred and George are playing on a see-saw that is slightly off center. When Ed sits on the left side and George, who weighs $100$ pounds, on the right side, they are perfectly balanced. Similarly, if Fred, who weighs $400$ pounds, sits on the left and Ed sits on the right, they are also perfectly balanced. Assuming the see-saw has negligible weight, what is the weight of Ed, in pounds? [b]p2.[/b] How many digits does the product $2^{42}\cdot 5^{38}$ have? [b]p3.[/b] Square $ABCD$ has equilateral triangles drawn external to each side, as pictured. If each triangle is folded upwards to meet at a point $E$, then a square pyramid can be made. If the center of square $ABCD$ is $O$, what is the measure of $\angle OEA$? [img]https://cdn.artofproblemsolving.com/attachments/9/a/39c0096ace5b942a9d3be1eafe7aa7481fbb9f.png[/img] [b]p4.[/b] How many solutions $(x, y)$ in the positive integers are there to $3x + 7y = 1337$ ? [b]p5.[/b] A trapezoid with height $12$ has legs of length $20$ and $15$ and a larger base of length $42$. What are the possible lengths of the other base? [b]p6.[/b] Let $f(x) = 6x + 7$ and $g(x) = 7x + 6$. Find the value of a such that $g^{-1}(f^{-1}(g(f(a)))) = 1$. [b]p7.[/b] Billy and Cindy want to meet at their favorite restaurant, and they have made plans to do so sometime between $1:00$ and $2:00$ this Sunday. Unfortunately, they didn’t decide on an exact time, so they both decide to arrive at a random time between $1:00$ and $2:00$. Silly Billy is impatient, though, and if he has to wait for Cindy, he will leave after $15$ minutes. Cindy, on the other hand, will happily wait for Billy from whenever she arrives until $2:00$. What is the probability that Billy and Cindy will be able to dine together? [b]p8.[/b] As pictured, lines are drawn from the vertices of a unit square to an opposite trisection point. If each triangle has legs with ratio $3 : 1$, what is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/e/9/35a6340018edcddfcd7e085f8f6e56686a8e07.png[/img] [b]p9.[/b] For any positive integer $n$, let $f_1(n)$ denote the sum of the squares of the digits of $n$. For $k \ge 2$, let $f_k(n) = f_{k-1}(f_1(n))$. Then, $f_1(5) = 25$ and $f_3(5) = f_2(25) = 85$. Find $f_{2012}(15)$. [b]p10.[/b] Given that $2012022012$ has $ 8$ distinct prime factors, find its largest prime factor. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let the two tangents from a point $A$ outside a circle $k$ touch $k$ at $M$ and $N$. A line $p$ through $A$ intersects $k$ at $B$ and $C$, and $D$ is the midpoint of $MN$. Prove that $MN$ bisects the angle $BDC$

Consider the convex lattice quadrilateral $PQRS$ whose diagonals intersect at $E$. Prove that if $\angle P+\angle Q<180^\circ$, then the $\triangle PQE$ contains inside it or on one of its sides a lattice point other than $P$ and $Q$.

Let $a>1,\ 0\leq x\leq \frac{\pi}{4}$. Find the volume of the solid generated by a rotation of the part bounded by two curves $y=\frac{\sqrt{2}\sin x}{\sqrt{\sin 2x+a}},\ y=\frac{1}{\sqrt{\sin 2x+a}}$ about the $x$-axis. [i]1993 Hiroshima Un iversity entrance exam/Science[/i]

Find the circumradius of the triangle with side lengths $104$, $112$, and $120$.

Points $A, B, C, D$ lie on a line $l$, in that order. Find the locus of points $P$ in the plane for which $\angle{APB}=\angle{CPD}$.

Prove that there is a similarity between a triangle $ABC$ and the triangle having as sides the medians of the triangle $ABC$ if and only if the squares of the lengths of the sides of triangle $ABC$ form an arithmetic sequence. [i]Marian Teler & Marin Ionescu[/i]

Let $ABC$ be a triangle, $I$ its incenter, and $D$ a point on the arc $BC$ of the circumcircle of $ABC$ not containing $A$. The bisector of the angle $\angle ADB$ intesects the segment $AB$ at $E$. The bisector of the angle $\angle CDA$ intesects the segment $AC$ at $F$. Prove that the points $E, F,I$ are collinear. (Malik Talbi)

Let $D$ be the center of the circumcircle of the acute triangle $ABC$. If the circumcircle of triangle $ADB$ intersects $AC$ (or its extension) at $M$ and also $BC$ (or its extension) at $N$, show that the radii of the circumcircles of $\triangle ADB$ and $\triangle MNC$ are equal.

Let $ABCD$ is a convex quadrilateral.The internal angle bisectors of $\angle {BAC}$ and $\angle {BDC}$ meets at $P$.$\angle {APB}=\angle {CPD}$.Prove that $AB+BD=AC+CD$.

Let $n\in\mathbb{N}^*$. $2n+3$ points on the plane are given so that no 3 lie on a line and no 4 lie on a circle. Is it possible to find 3 points so that the interior of the circle passing through them would contain exactly $n$ of the remaining points.

The sidelengths of a triangle are natural numbers multiples of $7$, smaller than $40$. How many triangles satisfy these conditions?

Give a construction by straight-edge and compass of a point $C$ on a line $\ell$ parallel to a segment $AB$, such that the product $AC \cdot BC$ is minimum.

Let $ABCD$ ba a square and let point $E$ be the midpoint of side $AD$. Points $G$ and $F$ are located on the segment $(BE)$ such that the lines $AG$ and $CF$ are perpendicular on the line $BE$. Prove that $DF= CG$.

Let $ABC$ be an isosceles right triangle with $\angle A=90^o$. Point $D$ is the midpoint of the side $[AC]$, and point $E \in [AC]$ is so that $EC = 2AE$. Calculate $\angle AEB + \angle ADB$ .

Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.

Square $ABCD$ has side length $36$. Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$. Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$.

Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$.

The width of a long winding river is not greater than $1$ km. This means by definition that from any point of each bank of the river one can reach the other bank swimming $1$ km or less. Is it true that a boat can move along the river so that its distances from both banks are never greater than (a) $0.7$ km? (b) $0.8$ km? (Grigory Kondakov, Moscow) You may assume that the banks consist of segments and arcs of circles.

The figure shows a rectangle, its circumscribed circle and four semicircles, which have the rectangle’s sides as diameters. Prove that the combined area of the four dark gray crescentshaped regions is equal to the area of the light gray rectangle. [img]https://1.bp.blogspot.com/-gojv6KfBC9I/XzT9ZMKrIeI/AAAAAAAAMVU/NB-vUldjULI7jvqiFWmBC_Sd8QFtwrc7wCLcBGAsYHQ/s0/2013%2BMohr%2Bp3.png[/img]

The inscribed and exscribed sphere of a triangular pyramid $ABCD$ touch her face $BCD$ at different points $X$ and $Y$. Prove that the triangle $AXY$ is obtuse triangle.

Let $\omega$ be a circle with the center $O$ and $A$ and $C$ be two different points on $\omega$. For any third point $P$ of the circle let $X$ and $Y$ be the midpoints of the segments $AP$ and $CP$. Finally, let $H$ be the orthocenter (the point of intersection of the altitudes) of the triangle $OXY$ . Prove that the position of the point H does not depend on the choice of $P$. (Artemiy Sokolov)