Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.

Given a triangle $ABC$ such that $AB<AC$ . On sides $AB$ and $BC$, points $P$ and $Q$ are marked respectively such that the lines $AQ$ and $CP$ are perpendicular, and the circle inscribed in the triangle $ABC$ touches the length $PQ$. The line $CP$ intersects the circle circumscribed around the triangle $ABC$ at the points $C$ and $T$. If the lines $CA,PQ$ and $BT$ intersect at one point, prove that the angle $\angle CAB$ is right.

Let $ABC$ and $A'B'C'$ be two triangles so that the midpoints of $\overline{AA'}, \overline{BB'}, \overline{CC'}$ form a triangle as well. Suppose that for any point $X$ on the circumcircle of $ABC$, there exists exactly one point $X'$ on the circumcircle of $A'B'C'$ so that the midpoints of $\overline{AA'}, \overline{BB'}, \overline{CC'}$ and $\overline{XX'}$ are concyclic. Show that $ABC$ is similar to $A'B'C'$. [i]Proposed by usjl[/i]

One of the sides of a triangle is divided into segments of $ 6$ and $ 8$ units by the point of tangency of the inscribed circle. If the radius of the circle is $ 4$, then the length of the shortest side of the triangle is: $ \textbf{(A)}\ 12\text{ units} \qquad\textbf{(B)}\ 13\text{ units} \qquad\textbf{(C)}\ 14\text{ units} \qquad\textbf{(D)}\ 15\text{ units} \qquad\textbf{(E)}\ 16\text{ units}$

Let be given triangle $ABC$. Find all points $M$ such that area of $\vartriangle MAB$= area of $\vartriangle MAC$

A rectangle is said to be inscribed in a parallelogram if its vertices lie one on each side of the parallelogram. On the larger side $AB$ of the parallelogram $ABCD$, find all those points $K$ that are the vertices of the rectangles inscribed in $ABCD$.

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

Let $ABC$ be an isosceles triangle with $AC = BC$, and let $M$ be the midpoint of $AB$. Let $P$ be a point inside the triangle such that $\angle PAB =\angle PBC$. Prove that $\angle APM + \angle BPC = 180^o$.

Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB, Q, R$ on $BC$ and $S$ on $AC$. If the area of the square $PQRS$ is $m +n\sqrt{k}$ where $m, n$ are integers and $k$ is a prime number then determine the value of $\sqrt{\frac{m+n}{k^2}}$.

Let $I$ be the incenter of triangle $ABC$. Lines $AI$, $BI$, $CI$ meet the sides of $\triangle ABC$ at $D$, $E$, $F$ respectively. Let $X$, $Y$, $Z$ be arbitrary points on segments $EF$, $FD$, $DE$, respectively. Prove that $d(X, AB) + d(Y, BC) + d(Z, CA) \leq XY + YZ + ZX$, where $d(X, \ell)$ denotes the distance from a point $X$ to a line $\ell$.

Jonathan and Eric are standing one kilometer apart on a large, flat, empty field. Jonathan rotates an angle of $\theta = 120^o$ counterclockwise around Eric, then Eric moves half of the distance to Jonathan. They keep repeating the previous two movements in this order. After a very long time, their locations approach a point $P$ on the field. What is the distance, in kilometers, from Jonathan’s starting location to $P$?

Let $ABCD$ be a trapezoid with $AB\parallel CD$. Its diagonals intersect at a point $P$. The line passing through $P$ parallel to $AB$ intersects $AD$ and $BC$ at $Q$ and $R$, respectively. Exterior angle bisectors of angles $DBA$, $DCA$ intersect at $X$. Let $S$ be the foot of $X$ onto $BC$. Prove that if quadrilaterals $ABPQ$, $CDQP$ are circumcribed, then $PR=PS$. [i]Proposed by Dominik Burek, Poland[/i]

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

In an acute-angled and not isosceles triangle $ABC,$ we draw the median $AM$ and the height $AH.$ Points $Q$ and $P$ are marked on the lines $AB$ and $AC$, respectively, so that the $QM \perp AC$ and $PM \perp AB$. The circumcircle of $PMQ$ intersects the line $BC$ for second time at point $X.$ Prove that $BH = CX.$ M. Didin

How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$? [i]Proposed by Joshua Hsieh[/i]

A line in the plane is called [i]strange[/i] if it passes through $(a,0)$ and $(0,10-a)$ for some $a$ in the interval $[0,10]$. A point in the plane is called [i]charming[/i] if it lies in the first quadrant and also lies [b]below[/b] some strange line. What is the area of the set of all charming points?

Let $ ABC$ be an acute triangle. Points $ D$, $ E$, and $ F$ lie on segments $ BC$, $ CA$, and $ AB$, respectively, and each of the three segments $ AD$, $ BE$, and $ CF$ contains the circumcenter of $ ABC$. Prove that if any two of the ratios $ \frac{BD}{DC}$, $ \frac{CE}{EA}$, $ \frac{AF}{FB}$, $ \frac{BF}{FA}$, $ \frac{AE}{EC}$, $ \frac{CD}{DB}$ are integers, then triangle $ ABC$ is isosceles.

The incircle with centre $I$ of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ at $D, E, F$ respectively. The line $ID$ intersects the segment $EF$ at $K$. Proof that $A, K, M$ collinear, where $M$ is the midpoint of $BC$.

All angles of the convex $n$-gon $A_1A_2\dots A_n$ are obtuse, where $n\ge5$. For all $1\le i\le n$, $O_i$ is the circumcenter of triangle $A_{i-1}A_iA_{i+1}$ (where $A_0=A_n$ and $A_{n+1}=A_1$). Prove that the closed path $O_1O_2\dots O_n$ doesn't form a convex $n$-gon.

The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8,15,8,8,8,6,8,$ and $29$ as shown. All of its angles are right angles. Turn this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$. [asy] size(200); defaultpen(linewidth(2)); pen qq=font("phvb"); pair rectangle[] = {origin,(0,-8),(15,-8),(15,-16),(23,-16),(23,-8),(29,-8),(29,0)}; string point[] = {"A","B","C","D","E","F","G","H"}; int dirlbl[] = {135,225,225,225,315,315,315,45}; string value[] = {"8","15","8","8","8","6","8","29"}; int direction[] = {0,90,0,90,180,90,180,270}; for(int i=0;i<=7;i=i+1) { draw(rectangle[i]--rectangle[(i+1) % 8]); label(point[i],rectangle[i],dir(dirlbl[i]),qq); label(value[i],(rectangle[i]+rectangle[(i+1) % 8])/2,dir(direction[i]),qq); } [/asy]

In a triangle $ABC$ with $\widehat{A}=80^{\circ}$ and $\widehat{B}=60^{\circ}$, the internal angle bisector of $\widehat{C}$ meets the side $AB$ at the point $D$. The parallel from $D$ to the side $AC$, meets the side $BC$ at the point $E$. Find the measure of the angle $\angle EAB$.

Consider a polyhedron having $100$ edges. (a) Find the maximal possible number of its edges which can be intersected by a plane (not containing any vertices of the polyhedron) if the polyhedron is convex. (b) Prove that for a non-convex polyhedron this number i. can be as great as $96$, ii. cannot be as great as $100$. (A Andjans, Riga

Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that a) $\triangle OMN \sim \triangle OBA$; b) $OE \perp AB$. [i]Claudiu-Stefan Popa[/i]

[b]7.1.[/b] Prove that from the sides of an arbitrary quadrilateral you can fold a trapezoid. [b]7.2 / 6.2[/b] The numbers $A$ and $B$ are relatively prime. What common divisors can have the numbers $A+B$ and $A-B$? [b]7.3. / 6.4[/b] $15$ magazines lie on the table, completely covering it. Prove that it is possible to remove eight of them so that the remaining magz cover at least $7/15$ of the table area. [b]7.4[/b] In a six-digit number that is divisible by $7$, the last digit has been moved to the beginning. Prove that the resulting number is also divisible at $7$. [url=https://artofproblemsolving.com/community/c6h3391057p32066818]7.5*[/url] (asterisk problems in separate posts) [b]7.6 [/b] On sides $AB$ and $ BC$ of triangle $ABC$ , are constructed squares $ABDE$ and $BCKL$ with centers $O_1$ and $O_2$. $M_1$ and $M_2$ are midpoints of segments $DL$ and $AC$. Prove that $O_1M_1O_2M_2$ is a square. [img]https://cdn.artofproblemsolving.com/attachments/8/1/8aa816a84c5ac9de78b396096cf718063de390.png[/img] PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].