On the sides $BC$ and $AB$ of the triangle $ABC$ the points ${{A} _ {1}}$ and ${{C} _ {1}} $ are selected accordingly so that the segments $A {{A} _ {1}}$ and $C {{C} _ {1}}$ are equal and perpendicular. Prove that if $\angle ABC = 45 {} ^ \circ$, then $AC = A {{A} _ {1}} $. (Gogolev Andrew)

Determine all bounded closed subsets $F$ of the plane with the following property: $F$ consists of at least two points and always contains two points $A$ and $B$ as well as at least one of the two semicircular arcs over the segment $AB$. Definitions: A subset of the $F$ of the plane is said to be closed if: For every point $P$ of the plane that is not an element of $F$ , there is a (non-degenerate) disc with center $P$ that has no elements of $F$.

Let $ABCD$ be a regular triangular pyramid with base $ABC$ (this means that $ABC$ is a regular triangle, and edges $AD$, $BD$ and $CD$ are equal) and plane angles at the opposite vertex equal to $a$. A plane parallel to $ABC$ intersects $AD$, $BD$ and $CD$, respectively, at points $A_1$, $B_1$ and $C_1$. The surface of the polyhedron $ABCA_1B_1C_1$ is cut along five edges: $A_1B_1$, $B_1C_1$, $C_1C$, $CA$ and $AB$, after which this surface is turned onto a plane. At what values of $a$ will the resulting scan necessarily cover itself?

Given a segment of a circle, consisting of a straight edge and an arc. The length of the straight edge is $24$. The length between the midpoint of the straight edge and the midpoint of the arc is $6$. Find the radius of the circle.

Prove that there do not exist $4$ points in the plane such that the distances between any pair of them is an odd integer.

Csenge has a yellow and a red foil on her rectangular window which look beautiful in the morning light. Where the two foils overlap, they look orange. The window is $80$ cm tall, $120$ cm wide and its corners are denoted by $A, B, C$ and $D$ in the figure. The two foils are triangular and both have two of their vertices at the two bottom corners of the window, A and $B$. The third vertex of the yellow foil is $S$, the trisecting point of side $DC$ closer to $D$, whereas the third vertex of the red foil is $P$, which is one fourth on the way on segment $SC$, closer to $C$. The red region (i.e. triangle $BPE$) is of area $16$ dm$^2$. What is the total area of the regions not covered by foil? [img]https://cdn.artofproblemsolving.com/attachments/b/c/ea371aeafde6968506da6f3456e88fa0bddc6d.png[/img]

A checkered polygon $A$ is drawn on the checkered plane. We call a cell of $A$ [i]internal[/i] if all $8$ of its adjacent cells belong to $A$. All other (non-internal) cells of $A$ we call [i]boundary[/i]. It is known that $1)$ each boundary cell has exactly two common sides with no boundary cells; and 2) the union of all boundary cells can be divided into isosceles trapezoid of area $2$ with vertices at the grid nodes (and acute angles of the trapezoids are equal $45^\circ$). Prove that the area of the polygon $A$ is congruent to $1$ modulo $4$.

A square with a side 1 is colored in 3 colors. What’s the greatest real number $a$ such that there can always be found 2 points of the same color at a distance $a$?

In an acute triangle $\triangle ABC$ the points $C_m, A_m, B_m$ are the midpoints of $AB, BC, CA$ respectively. Inside the triangle $\triangle ABC$ a point $P$ is chosen so that $\angle PCB = \angle B_mBC$ and $\angle PAB = \angle ABB_m.$ A line passing through $P$ and perpendicular to $AC$ meets the median $BB_m$ at $E.$ Prove that $E$ lies on the circumcircle of the triangle $\triangle A_mB_mC_m.$ [i](K. Ivanov )[/i]

Let $ABCDEF$ be a convex hexagon, such that $FA = AB$, $BC = CD$, $DE = EF$, and $\angle FAB = 2\angle EAC$. Suppose that the area of $ABC$ is $25$, the area of $CDE$ is $10$, the area of $EF A$ is $25$, and the area of $ACE$ is $x$. Find, with proof, all possible values of $x$.

Moe uses a mower to cut his rectangular $ 90$-foot by $ 150$-foot lawn. The swath he cuts is $ 28$ inches wide, but he overlaps each cut by $ 4$ inches to make sure that no grass is missed. He walks at the rate of $ 5000$ feet per hour while pushing the mower. Which of the following is closest to the number of hours it will take Moe to mow his lawn? $ \textbf{(A)}\ 0.75 \qquad \textbf{(B)}\ 0.8 \qquad \textbf{(C)}\ 1.35 \qquad \textbf{(D)}\ 1.5 \qquad \textbf{(E)}\ 3$

[b]7.1[/b] A rectangle that is not a square is inscribed in a square. Prove that its semi-perimeter is equal to the diagonal of the square. [b]7.2[/b] Find five numbers whose pairwise sums are 0, 2, 4,5, 7, 9, 10, 12, 14, 17. [b]7.3 [/b] In a $1000$-digit number, all but one digit is a five. Prove that this number is not a perfect square. [b]7.4 / 6.5[/b] Several teams took part in the volleyball tournament. Team A is considered stronger than team B if either A beat B or there is a team C such that A beat C, and C beat B. Prove that if team T is the winner of the tournament, then it is the strongest the rest of the teams. [b]7.5[/b] In a pentagon $ABCDE$, $K$ is the midpoint of $AB$, $L$ is the midpoint of $BC$, $M$ is the midpoint of $CD$, $N$ is the midpoint of $DE$, $P$ is the midpoint of $KM$, $Q$ is the midpoint of $LN$. Prove that the segment $ PQ$ is parallel to side $AE$ and is equal to its quarter. [img]https://cdn.artofproblemsolving.com/attachments/2/5/be8e9b0692d98115dbad04f960e8a856dc593f.png[/img] [b]7.6 / 8.4[/b] Several circles are arbitrarily placed in a circle of radius $3$, the sum of their radii is $25$. Prove that there is a straight line that intersects at least $9$ of these circles. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988084_1968_leningrad_math_olympiad]here[/url].

Let square $ABCD$ and circle $\Omega$ be on the same plane, and $AA'$, $BB'$, $CC'$, $DD'$ be tangents to $\Omega$. Let $WXY Z$ be a convex quadrilateral with side lengths $WX = AA'$, $XY = BB'$, $Y Z = CC'$, and $ZW = DD'$. If $WXY Z$ has an inscribed circle, prove that the diagonals $WY$ and $XZ$ are perpendicular to each other.

Let $ABC$ be an acute triangle with circumcircle $\Omega$ and orthocenter $H$. Points $D$ and $E$ lie on segments $AB$ and $AC$ respectively, such that $AD = AE$. The lines through $B$ and $C$ parallel to $\overline{DE}$ intersect $\Omega$ again at $P$ and $Q$, respectively. Denote by $\omega$ the circumcircle of $\triangle ADE$. [list=a] [*] Show that lines $PE$ and $QD$ meet on $\omega$. [*] Prove that if $\omega$ passes through $H$, then lines $PD$ and $QE$ meet on $\omega$ as well. [/list] [i]Merlijn Staps[/i]

Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$. Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that \[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]

You are given some equilateral triangles and squares, all with side length 1, and asked to form convex $n$ sided polygons using these pieces. If both types must be used, what are the possible values of $n$, assuming that there is sufficient supply of the pieces?

Pentagon $ABCDE$ with $CD\parallel BE$ is inscribed in circle $\omega$. Tangent to $\omega$ through $B$ intersects line $AC$ at $F$ in a way that $A$ lies between $C$ and $F$. Lines $BD$ and $AE$ intersect at $G$. Prove that $FG$ is tangent to the circumcircle of $\triangle ADG$.

Let $ABC$ be a triangle, and let $BCDE$, $CAFG$, $ABHI$ be squares that do not overlap the triangle with centers $X$, $Y$, $Z$ respectively. Given that $AX=6$, $BY=7$, and $CA=8$, find the area of triangle $XYZ$.

In his triangle $ABC$ Serge made some measurements and informed Ilias about the lengths of median $AD$ and side $AC$. Based on these data Ilias proved the assertion: angle $CAB$ is obtuse, while angle $DAB$ is acute. Determine a ratio $AD/AC$ and prove Ilias' assertion (for any triangle with such a ratio).

In trapezoid $ABCD$ with bases $AD, BC$, $AD = 2BC$ and $M$ is midpoint of $AB$. Prove that line $BD$ passes through the midpoint of segment $CM$.

Suppose that we have a countable set $A$ of balls and a unit cube in $\mathbb R^3$. Assume that for every finite subset $B$ of $A$ it is possible to put all balls of $B$ into the cube in such a way that they have disjoint interiors. Show that it is possible to arrange all the balls in the cube so that all of them have pairwise disjoint interiors.

In an acute triangle $ABC$ the angle bisector $AL$, $L \in BC$, intersects its circumcircle at $N$. Let $K$ and $M$ be the projections of $L$ onto sides $AB$ and $AC$. Prove that triangle $ABC$ and quadrilateral $A K N M$ have equal areas.

A parabola has focus $F$ and vertex $V$ , where $VF = 1$0. Let $AB$ be a chord of length $100$ that passes through $F$. Determine the area of $\vartriangle VAB$.

A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is $\textbf{(A) }\frac{\sqrt{3}}{2}\qquad\textbf{(B) }1\qquad\textbf{(C) }\frac{2}{\sqrt{3}}\qquad\textbf{(D) }\frac{3}{2}\qquad\textbf{(E) }\sqrt{3}$

Three vertices $KLM$ of the rhombus (diamond) $KLMN$ lays on the sides $[AB], [BC]$ and $[CD]$ of the given unit square. Find the area of the set of all the possible vertices $N$.