Points $X$ , $Y$, $Z$ lie on a line $k$ in this order. Let $\omega_1$, $\omega_2$, $\omega_3$ be three circles of diameters $XZ$, $XY$ , $YZ$ , respectively. Line $\ell$ passing through point $Y$ intersects $\omega_1$ at points $A$ and $D$, $\omega_2$ at $B$ and $\omega_3$ at $C$ in such manner that points $A, B, Y, X, D$ lie on $\ell$ in this order. Prove that $AB =CD$.

Given a regular polygon with $p$ sides, where $p$ is a prime number. After rotating this polygon about its center by an integer number of degrees it coincides with itself. What is the maximal possible number for $p$?

Two segments slide along two skew lines. On each straight line there is a segment. Consider the tetrahedron with vertices at the endpoints of the segments. Prove that the volume of the tetrahedron does not depend on the position of the segments

[u]Set 8[/u] [b]p22.[/b] For monic quadratic polynomials $P = x^2 + ax + b$ and $Q = x^2 + cx + d$, where $1 \le a, b, c, d \le 10$ are integers, we say that $P$ and $Q$ are friends if there exists an integer $1 \le n \le 10$ such that $P(n) = Q(n)$. Find the total number of ordered pairs $(P, Q)$ of such quadratic polynomials that are friends. [b]p23.[/b] A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length $1$, $\vartriangle A_1A_2A_3$ and $\vartriangle B_1B_2B_3$. The other six faces are isosceles right triangles — $\vartriangle A_1B_2A_3$, $\vartriangle A_2B_3A_1$, $\vartriangle A_3B_1A_2$, $\vartriangle B_1A_2B_3$, $\vartriangle B_2A_3B_1$, $\vartriangle B_3A_1B_2$ — each with a right angle at the second vertex listed (so for instace $\vartriangle A_1B_2A_3$ has a right angle at $B_2$). Find the volume of this solid. [b]p24.[/b] The digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ are each colored red, blue, or green. Find the number of colorings such that any integer $ n \ge 2$ has that (a) If $n$ is prime, then at least one digit of $n$ is not blue. (b) If $n$ is composite, then at least one digit of $n$ is not green. PS. You should use hide for answers.

In my isosceles triangle $\vartriangle ABC$ with $AB = CA$, we draw $D$ the midpoint of $BC$. Let $E$ be a point on $AC$ such that $\angle CDE = 60^o$ and $M$ the midpoint of $DE$. Prove that $\angle AME = \angle BMD$.

Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa. [url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]

Let $ABCD$ be a convex quadrilateral. Is it possible to find a point $P$ such that the segments drawn between $P$ and the midpoints of the sides of $ABCD$ divide the quadrilateral into four sections of equal area? If $P$ exists, is it unique?

In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones. For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column. Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$. We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row). (a) Show that $f(C) \in \mathcal F$. (b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.

Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. A circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. A point $F$ on this circle is chosen so that $EF\perp BC$. If $BC = x$, $CF = y$, and $BF = z$, find the length of $DF$ in terms of $x, y, z$.

As shown in the following figure, point $ P$ lies on the circumcicle of triangle $ ABC.$ Lines $ AB$ and $ CP$ meet at $ E,$ and lines $ AC$ and $ BP$ meet at $ F.$ The perpendicular bisector of line segment $ AB$ meets line segment $ AC$ at $ K,$ and the perpendicular bisector of line segment $ AC$ meets line segment $ AB$ at $ J.$ Prove that \[ \left(\frac{CE}{BF} \right)^2 \equal{} \frac{AJ \cdot JE}{AK \cdot KF}.\]

Let $ABC$ be an acute-angled triangle whose inscribed circle touches $AB$ and $AC$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the points of intersection of the bisectors of the angles $\angle ACB$ and $\angle ABC$ with the line $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $XYZ$ is equilateral if and only if $\angle A = 60^\circ$.

Let $ABCD$ be a cyclic quadrilateral with $|AB|=26$, $|BC|=10$, $m(\widehat{ABD})=45^\circ$,$m(\widehat{ACB})=90^\circ$. What is the area of $\triangle DAC$ ? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 108 \qquad\textbf{(C)}\ 90 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ 80 $

A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon $5$ centimeters of rain falls in the open field partially filling the previously empty container. Find the depth in centimeters of the rainwater in the bottom of the container after the rain.

We are given a convex pentagon $ABCDE$ in the coordinate plane such that $A$, $B$, $C$, $D$, $E$ are lattice points. Let $Q$ denote the convex pentagon bounded by the five diagonals of the pentagon $ABCDE$ (so that the vertices of $Q$ are the interior points of intersection of diagonals of the pentagon $ABCDE$). Prove that there exists a lattice point inside of $Q$ or on the boundary of $Q$.

Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.

There are some ink-blots on a white paper square with side length $a$. The area of each blot is not greater than $1$ and every line parallel to any one of the sides of the square intersects no more than one blot. Prove that the total area of the blots is not greater than $a$. (A. Razborov, Moscow)

Let $F=A_0A_1...A_n$ be a convex polygon in the plane. Define for all $1 \leq k \leq n-1$ the operation $f_k$ which replaces $F$ with a new polygon $f_k(F)=A_0A_1..A_{k-1}A_k^\prime A_{k+1}...A_n$ where $A_k^\prime$ is the symmetric of $A_k$ with respect to the perpendicular bisector of $A_{k-1}A_{k+1}$. Prove that $(f_1\circ f_2 \circ f_3 \circ...\circ f_{n-1})^n(F)=F$.

Two non-congruent triangles are called [i]analogous [/i] if they can be denoted as $ABC$ and $A'B'C'$ such that $AB = A'B', AC = A'C'$ and $\angle B = \angle B'$ . Do there exist three mutually [i]analogous[/i] triangles?

Let $ABCDE$ be a convex pentagon and area of $\vartriangle ABC =$ area of $\vartriangle BCD =$ area of $\vartriangle CDE=$ area of $\vartriangle DEA =$ area of $\vartriangle EAB$. Given that area of $\vartriangle ABCDE = 2$. Evaluate the area of area of $\vartriangle ABC$.

If $A$ and $B$ are points of the plane, then by $A * B$ we denote a point symmetric to $A$ with respect to $B$. Is it possible, by applying the operation $*$ several times, to obtain from the three vertices of a given square its fourth vertex?

Each of two regular polygons $P$ and $Q$ was divided by a line into two parts. One part of $P$ was attached to one part of $Q$ along the dividing line so that the resulting polygon was regular and not congruent to $P$ or $Q$. How many sides can it have?

Let $ABC$ be an acute triangle and $A_{1}, B_{1}, C_{1}$ be the touchpoints of the excircles with the segments $BC, CA, AB$ respectively. Let $O_{A}, O_{B}, O_{C}$ be the circumcenters of $\triangle AB_{1}C_{1}, \triangle BC_{1}A_{1}, \triangle CA_{1}B_{1}$ respectively. Prove that the lines through $O_{A}, O_{B}, O_{C}$ respectively parallel to the internal angle bisectors of $\angle A,\angle B, \angle C$ are concurrent.

A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a>b>0$. The area of $PQRS$ is $16$. What is $a+b$? $\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 13$

Consider a circle $k$ with center $S$ and radius $r$. Denote $\mathsf M$ the set of all triangles with incircle $k$ such that the largest inner angle is twice bigger than the smallest one. For a triangle $\mathcal T\in\mathsf M$ denote its vertices $A,B,C$ in way that $SA\ge SB\ge SC$. Find the locus of points $\{B\mid\mathcal T\in\mathsf M\}$.

The point $O$ is the centre of the circumscribed circle of the acute-angled triangle $ABC$. The line $AO$ cuts the side $BC$ in point $N$, and the line $BO$ cuts the side $AC$ at point $M$. Prove that if $CM=CN$, then $AC=BC$.