Let $C$ be a circunference, $O$ is your circumcenter, $AB$ is your diameter and $R$ is any point in $C$ ($R$ is different of $A$ and $B$) Let $P$ be the foot of perpendicular by $O$ to $AR$, in the line $OP$ we match a point $Q$, where $QP$ is $\frac{OP}{2}$ and the point $Q$ isn't in the segment $OP$. In $Q$, we will do a parallel line to $AB$ that cut the line $AR$ in $T$. Denote $H$ the point of intersections of the line $AQ$ and $OT$. Show that $H$, $B$ and $R$ are collinears.

For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?

Let $ABCD$ be a cyclic quadrilateral with two diagonals intersect at $E$. Let $ M$, $N$, $P$, $Q$ be the reflections of $ E $ in midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Prove that the Euler lines of $ \triangle MAB$, $\triangle NBC$, $\triangle PCD,$ $\triangle QDA$ are concurrent. Trần Quang Hùng

A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal. (M. Volchkevich)

Consider a point $ B $ on a segment $ AC. $ Find the locus of the points $ M $ that have the property that the circumcircles of $ ABM $ and $ BCM $ have equal radii. [i]Nicolae Soare[/i]

Let $M$ be the midpoint of the side $BC$ of a acute triangle $ABC$. Incircle of the triangle $ABM$ is tangent to the side $AB$ at the point $D$. Incircle of the triangle $ACM$ is tangent to the side $AC$ at the point $E$. Let $F$ be the such point, that the quadrilateral $DMEF$ is a parallelogram. Prove that $F$ lies on the bisector of $\angle BAC$.

Let $ABC$ be an acute angles triangle with $O$ the center of the circumscribed circle. Point $Q$ lies on the circumscribed circle of $\vartriangle BOC$ so that $OQ$ is a diameter. Point $M$ lies on $CQ$ and point $N$ lies internally on line segment $BC$ so that $ANCM$ is a parallelogram. Prove that the circumscribed circle of $\vartriangle BOC$ and the lines $AQ$ and $NM$ pass through the same point.

Given an integer $ n\ge 2$ and a reular 2n-gon. Color all verices of the 2n-gon with n colors such that: [b](i)[/b] Each vertice is colored by exactly one color. [b](ii)[/b] Two vertices don't have the same color. Two ways of coloring, satisfying the conditions above, are called equilavent if one obtained from the other by a rotation whose center is the center of polygon. Find the total number of mutually non-equivalent ways of coloring. [i]Alternative statement:[/i] In how many ways we can color vertices of an regular 2n-polygon using n different colors such that two adjent vertices are colored by different colors. Two colorings which can be received from each other by rotation are considered as the same.

Points $A$ and $B$ move with a constant speed along lines $a$ and $b$. Two corresponding positions of these points $A_1,B_1$, and $A_2,B_2$ are known. Find the position of $A$ and $B$ for which the length of $AB$ is minimal.

Arpon chooses a positive real number $k$. For each positive integer $n$, he places a marker at the point $(n,nk)$ in the $(x,y)$ plane. Suppose that two markers whose $x$-coordinates differ by $4$ have distance $31$. What is the distance between the markers $(7,7k)$ and $(19,19k)$?

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

Let $ABCD$ be a quadrilateral, let $O$ be the intersection point of diagonals $AC$ and $BD$, and let $P$ be the intersection point of sides $AB$ and $CD$. Consider the parallelograms $AODE$ and $BOCF$. Prove that $E, F$ and $P$ are collinear.

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{16-p}{p^2+2p}$ where $p$ is the perimeter of the rectangle.

Let $\triangle ABC$ be a triangle with circumcenter $O$ satisfying $AB=13$, $BC = 15$, and $AC = 14$. Suppose there is a point $P$ such that $PB \perp BC$ and $PA \perp AB$. Let $X$ be a point on $AC$ such that $BX \perp OP$. What is the ratio $AX/XC$? [i]Proposed by Thomas Lam[/i]

1. In right triangle $ABC$ with $\angle{A}= \textdegree{90}$, point $P$ is chosen. $D \in BC$ such that $PD \perp BC$. Let the intersection of $PD$ with $AB$ and $AC$ be $E$ and $F$ respectively. Denote by $X$ and $Y$ as the intersection of $(APE)$ and $(APF)$ with $BP$ and $CP$ respectively. Prove that $CX,BY,PD$ are concurrent.

[b]7.1 . [/b] The area of the quadrilateral is $3$ cm$^2$ , and the lengths of its diagonals are $6$ cm and $2$ cm. Find the angle between the diagonals. [b]7.2[/b] Prove that the number $1 + 2^{3456789}$ is composite. [b]7.3[/b] $20$ people took part in the chess tournament. The participant who took clear (undivided) $19$th place scored $9.5$ points. How could they distribute points among other participants? [b]7.4[/b] The sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semi-perimeter. Prove that this quadrilateral is a parallelogram. [b]7.5[/b] $40$ people travel on a bus without a conductor passengers carrying only coins in denominations of $10$, $15$ and $20$ kopecks. Total passengers have $ 49$ coins. Prove that passengers will not be able to pay the required amount of money to the ticket office and pay each other correctly. (Cost of a bus ticket in 1963 was 5 kopecks.) [b]7.6[/b] Some natural number $a$ is divided with a remainder by all natural numbers less than $a$. The sum of all the different (!) remainders turned out to be equal to $a$. Find $a$. [b]7.7[/b] Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form $2\times 1$) without overlapping? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

For an acute triangle $ ABC $ let $ H $ be the point of intersection of the altitudes $ AA_1 $, $ BB_1 $, $ CC_1 $. Let $ M $ and $ N $ be the midpoints of the $ BC $ and $ AH $ segments, respectively. Show that $ MN $ is the perpendicular bisector of segment $ B_1C_1 $.

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Two circles $c_1$ and $c_2$ with centres $O_1$ and $O_2$, respectively, are touching externally at $P$. On their common tangent at $P$, point $A$ is chosen, rays drawn from which touch the circles $c_1$ and $c_2$ at points $P_1$ and $P_2$ both different from $P$. It is known that $\angle P_1AP_2 = 120^o$ and angles $P_1AP$ and $P_2AP$ are both acute. Rays $AP_1$ and $AP_2$ intersect line $O_1O_2$ at points $G_1$ and $G_2$, respectively. The second intersection between ray $AO_1$ and $c_1$ is $H_1$, the second intersection between ray $AO_2$ and $c_2$ is $H_2$. Lines $G_1H_1$ and $AP$ intersect at $K$. Prove that if $G_1K$ is a tangent to circle $c_1$, then line $G_2A$ is tangent to circle $c_2$ with tangency point $H_2$.

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

The altitude, angle bisector and median coming from one vertex of the triangle are equal to $\sqrt3$, $2$ and $\sqrt6$, respectively. Find the radius of the circle circumscribed round this triangle.

In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long. [asy] draw(Circle((0,0),10)); draw((0,0)--(10,0)--(8.5,5.3)--(-8.5,5.3)--(-3,9.5)--(3,9.5)); dot((0,0)); dot((10,0)); dot((8.5,5.3)); dot((-8.5,5.3)); dot((-3,9.5)); dot((3,9.5)); label("1", (5,0), S); label("s", (8,2.6)); label("d", (0,4)); label("s", (-5,7)); label("s", (0,8.5)); label("O", (0,0),W); label("R", (10,0), E); label("M", (-8.5,5.3), W); label("N", (8.5,5.3), E); label("P", (-3,9.5), NW); label("Q", (3,9.5), NE); [/asy] Of the three equations \[ \textbf{I.}\ d-s=1, \qquad \textbf{II.}\ ds=1, \qquad \textbf{III.}\ d^2-s^2=\sqrt{5} \]those which are necessarily true are $\textbf{(A)}\ \textbf{I}\ \text{only} \qquad\textbf{(B)}\ \textbf{II}\ \text{only} \qquad\textbf{(C)}\ \textbf{III}\ \text{only} \qquad\textbf{(D)}\ \textbf{I}\ \text{and}\ \textbf{II}\ \text{only} \qquad\textbf{(E)}\ \textbf{I, II}\ \text{and} \textbf{III}$

Given regular pentagon $ABCDE$. $M$ is an arbitrary point inside $ABCDE$ or on its side. Let the distances $|MA|, |MB|, ... , |ME|$ be renumerated and denoted with $$r_1\le r_2\le r_3\le r_4\le r_5.$$ Find all the positions of the $M$, giving $r_3$ the minimal possible value. Find all the positions of the $M$, giving $r_3$ the maximal possible value.

Let $AD$ and $BC$ be the parallel sides of a trapezium $ABCD$. Let $P$ and $Q$ be the midpoints of the diagonals $AC$ and $BD$. If $AD = 16$ and $BC = 20$, what is the length of $PQ$?

The incircle of the triangle $\triangle{ABC}$ has center at $O$ and it is tangent to the sides $BC$, $AC$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. The lines $BO$ and $CO$ intersect the line $YZ$ at the points $P$ and $Q$, respectively. Show that if the segments $XP$ and $XQ$ has the same length, then the triangle $\triangle ABC$ is isosceles.