For what values of $n$ is it possible to paint the edges of a prism whose base is an $n$-gon so that there are edges of all three colours at each vertex and all the faces (including the upper and lower bases) have edges of all three colours? (AV Shapovelov)

Let $\mathcal K$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt3$ along a common hypotenuse. Eight copies of $\mathcal K$ are used to form the polygon shown below. What is the area of triangle $\Delta ABC$? [img]https://cdn.artofproblemsolving.com/attachments/1/3/03abbd4df2932f4a1d16a34c2b9e15b683dedb.png[/img] $\textbf{(A) }2+3\sqrt3\qquad\textbf{(B) }\dfrac92\sqrt3\qquad\textbf{(C) }\dfrac{10+8\sqrt3}{3}\qquad\textbf{(D) }8\qquad\textbf{(E) }5\sqrt3$

Let $N$ be midpoint of the side $AB$ of a triangle $ABC$ with $\angle A$ greater than $\angle B$. Let $D$ be a point on the ray $AC$ such that $CD=BC$ and $P$ be a point on the ray $DN$ which lies on the same side of $BC$ as $A$ and satisfies the condition $\angle PBC =\angle A$. The lines $PC$ and $AB$ intersect at $E$, and the lines $BC$ and $DP$ intersect at $T$. Determine the value of $\frac{BC}{TC} - \frac{EA}{EB}$.

p1. Four kite-shaped shapes as shown below ($a > b$, $a$ and $b$ are natural numbers less than $10$) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of $16$ units of length. What is the possible perimeter of the outermost square formed if it is also known that $a$ and $b$ are numbers coprime? [img]https://cdn.artofproblemsolving.com/attachments/4/1/fa95f5f557aa0ca5afb9584d5cee74743dcb10.png[/img] p2. If $a = 3^p$, $b = 3^q$, $c = 3^r$, and $d = 3^s$ and if $p, q, r$, and $s$ are natural numbers, what is the smallest value of $p\cdot q\cdot r\cdot s$ that satisfies $a^2 + b^3 + c^5 = d^7$ 3. Ucok intends to compose a key code (password) consisting of 8 numbers and meet the following conditions: i. The numbers used are $1, 2, 3, 4, 5, 6, 7, 8$, and $9$. ii. The first number used is at least $1$, the second number is at least $2$, third digit-at least $3$, and so on. iii. The same number can be used multiple times. a) How many different passwords can Ucok compose? b) How many different passwords can Ucok make, if provision (iii) is replaced with: no numbers may be used more than once. p 4. For any integer $a, b$, and $c$ applies $a\times (b + c) = (a\times b) + (a\times c)$. a) Look for examples that show that $a + (b\times c)\ne (a + b)\times (a + c)$. b) Is it always true that $a + (b\times c) = (a + b)\times(a + c)$? Justify your answer. p5. The results of a survey of $N$ people with the question whether they maintain dogs, birds, or cats at home are as follows: $50$ people keep birds, $61$ people don't have dogs, $13$ people don't keep a cat, and there are at least $74$ people who keep the most a little two kinds of animals in the house. What is the maximum value and minimum of possible value of $N$ ?

Three straight lines $\ell_1$, $\ell_2$ and $\ell_3$, forming a triangle, divide the plane into $7$ parts. Each of the points $M_1$, $M_2$ and $M_3$ lies in one of the angles, vertical to some angle of the triangle. The distance from $M_1$ to straight lines $\ell_1$, $\ell_2$ and $\ell_3$ are equal to $7,3$ and $1$ respectively The distance from $M_2$ to the same lines are $4$, $1$ and $3$ respectively. For $M_3$ these distances are $3$, $5$ and $2$. What is the radius of the circle inscribed in the triangle? [hide=second sentence in Russian]Каждая из точек М_1, М_2 и М_з лежит в одном из углов, вертикальном по отношению к какому-то углу треугольника.[/hide]

Let $O$ be the centre of the circumcircle of triangle $ABC$. Point $D$ is on the side $BC$. Let $(K)$ be the circumcircle of $ABD$. $(K)$ meets $AO$ at $E$ that is distinct from $A$. (a) Prove that $B,K,O,E$ are on the same circle that is called $(L)$. (b) $(L)$ intersects $AB$ at $F$ distinct $B$. Point $G$ is on $(L)$ such that $EG \parallel OF$. $GK$ meets $AD$ at $S, SO$ meets $BC$ at $T$ . Prove that $O,E, T,C$ are on the same circle. Trần Quang Hùng

At Zoom University, people’s faces appear as circles on a rectangular screen. The radius of one’s face is directly proportional to the square root of the area of the screen it is displayed on. Haydn’s face has a radius of $2$ on a computer screen with area $36$. What is the radius of his face on a $16 \times 9$ computer screen?

Given a triangle $ABC$, let $D$ be the midpoint of the side $AC$ and let $M$ be the point that divides the segment $BD$ in the ratio $1/2$; that is, $MB/MD=1/2$. The rays $AM$ and $CM$ meet the sides $BC$ and $AB$ at points $E$ and $F$, respectively. Assume the two rays perpendicular: $AM\perp CM$. Show that the quadrangle $AFED$ is cyclic if and only if the median from $A$ in triangle $ABC$ meets the line $EF$ at a point situated on the circle $ABC$.

The point $M$ inside a convex quadrilateral $ABCD$ is equidistant from the lines $AB$ and $CD$ and is equidistant from the lines $BC$ and $AD$. The area of $ABCD$ occurred to be equal to $MA\cdot MC +MB \cdot MD$. Prove that the quadrilateral $ABCD$ is a) tangential (circumscribed), b) cyclic (inscribed). (Nairi Sedrakyan)

Let $ABC$ be a triangle right in $C$ and the points $D, E$ on the sides $BC$ and $CA$ respectively, such that $\frac{BD}{AC} =\frac{AE}{CD} = k$. Lines $BE$ and $AD$ intersect at $O$. Show that the angle $\angle BOD = 60^o$ if and only if $k =\sqrt3$.

In a triangle, a symmedian is a line through a vertex that is symmetric to the median with the respect to the internal bisector (all relative to the same vertex). In the triangle $ABC$, the median $m_a$ meets $BC$ at $A'$ and the circumcircle again at $A_1$. The symmedian $s_a$ meets $BC$ at $M$ and the circumcircle again at $A_2$. Given that the line $A_1A_2$ contains the circumcenter $O$ of the triangle, prove that: [i](a) [/i]$\frac{AA'}{AM} = \frac{b^2+c^2}{2bc} ;$ [i](b) [/i]$1+4b^2c^2 = a^2(b^2+c^2)$

Let $S$ be the circle of center $O$ and radius $R$, and let $A, A'$ be two diametrically opposite points in $S$. Let $P$ be the midpoint of $OA'$ and $\ell$ a line passing through $P$, different from $AA '$ and from the perpendicular on $AA '$. Let $B$ and $C$ be the intersection points of $\ell$ with $S$ and let $M$ be the midpoint of $BC$. a) Let $H$ be the foot of the altitude from $A$ in the triangle $ABC$. Let $D$ be the intersection point of the line $A'M$ with $AH$. Determine the locus of point $D$ while $\ell$ varies . b) Line $AM$ intersects $OD$ at $I$. Prove that $2 OI = ID$ and determine the locus of point $I$ while $\ell$ varies .

Let $\triangle{ABC}$ be a triangle with circumcircle $\tau$. The tangentlines to $\tau$ through $A$ and $B$ intersect at $T$. The circle through $A$, $B$ and $T$ intersects $BC$ and $AC$ again at $D$ and $E$, respectively; $CT$ and $BE$ intersect at $F$. Suppose $D$ is the midpoint of $BC$. Calculate the ratio $BF:BE$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 5)[/i]

In a regular hexagon, segments with lengths from $1$ to $6$ were drawn as shown in the right figure (the segments go sequentially in increasing length, all the angles between them are right). Find the side length of this hexagon. [img]https://cdn.artofproblemsolving.com/attachments/3/1/82e4225b56d984e897a43ba1f403d89e5f4736.png[/img]

We will say that a set $ A $ of points is [i]disastrous [/i] if it meets the following conditions: $\bullet$ There are no $ 3 $ collinear points $\bullet$ There is not a trio of mutually equal distances between points. If $ P $ and $ Q $ are points in $ A $, then there are $ M $, $ N $, $ R $ and $ T $ in $ A $ such that: $$ d (P, Q) = \frac {d (M, N) + d (R, T)} {2} $$ Show that all disastrous sets are infinite. [hide=original wording of second condition]No existe ni un trío de distancias entre puntos mutuamente iguales. [/hide]

A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.

On the hypotenuse $AB$ of the right-angled triangle $ABC$, a point $K$ is chosen such that $BK = BC$. Let $P$ be a point on the perpendicular line from point $K$ to the line $CK$, equidistant from the points $K$ and $B$. Also let $L$ denote the midpoint of the segment $CK$. Prove that line $AP$ is tangent to the circumcircle of the triangle $BLP$.

Let $ABC$ be a non-isosceles triangle. Two isosceles triangles $AB'C$ with base $AC$ and $CA'B$ with base $BC$ are constructed outside of triangle $ABC$. Both triangles have the same base angle $\varphi$. Let $C_1$ be a point of intersection of the perpendicular from $C$ to $A'B'$ and the perpendicular bisector of the segment $AB$. Determine the value of $\angle AC_1B.$

Is it possible to disassemble and reassemble a $4\times 4\times 4$ Rubik's Cuble in at least $577,800$ non-equivalent ways? Notes: 1. When we reassemble the cube, a corner cube has to go to a corner cube, an edge cube must go to an edge cube and a central cube must go to a central cube. 2. Two positions of the cube are called equivalent if they can be obtained from one two another by rotating the faces or layers which are parallel to the faces.

Let $ABCD$ be a convex quadrilateral, let $E$ and $F$ be the midpoints of the sides $AD$ and $BC$, respectively. The segment $CE$ meets $DF$ in $O$. Show that if the lines $AO$ and $BO$ divide the side $CD$ in 3 equal parts, then $ABCD$ is a parallelogram.

In the triangle $ABC$ the edge $BC$ has length $a$, the edge $AC$ length $b$, and the edge $AB$ length $c$. Extend all the edges at both ends – by the length $a$ from the vertex $A, b$ from $B$, and $c$ from $C$. Show that the six endpoints of the extended edges all lie on a common circle. [img]https://cdn.artofproblemsolving.com/attachments/8/7/14c8c6a4090d4fade28893729a510d263e7abb.png[/img]

We divide up the plane into disjoint regions using a circle, a rectangle and a triangle. What is the greatest number of regions that we can get?

Let \(D\) be the midpoint of side \(BC\) of triangle \(ABC\). Let \(E, F\) be points on sides \(AB, AC\) respectively such that \(DE = DF\). Prove that \(AE + AF = BE + CF \iff \angle EDF = \angle BAC\).

Let $ABC$ be a triangle. Let $X$ be the point on side $AB$ such that $\angle{BXC} = 60^{\circ}$. Let $P$ be the point on segment $CX$ such that $BP\bot AC$. Given that $AB = 6, AC = 7,$ and $BP = 4,$ compute $CP$.

Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.