A $4*4*4$ cubical box contains 64 identical small cubes that exactly fill the box. How many of these small cubes touch a side or the bottom of the box? $ \text{(A)}\ 48\qquad\text{(B)}\ 52\qquad\text{(C)}\ 60\qquad\text{(D)}\ 64\qquad\text{(E)}\ 80 $

In a right triangle with sides $ a$ and $ b$, and hypotenuse $ c$, the altitude drawn on the hypotenuse is $ x$. Then: $ \textbf{(A)}\ ab \equal{} x^2 \qquad\textbf{(B)}\ \frac {1}{a} \plus{} \frac {1}{b} \equal{} \frac {1}{x} \qquad\textbf{(C)}\ a^2 \plus{} b^2 \equal{} 2x^2$ $ \textbf{(D)}\ \frac {1}{x^2} \equal{} \frac {1}{a^2} \plus{} \frac {1}{b^2} \qquad\textbf{(E)}\ \frac {1}{x} \equal{} \frac {b}{a}$

The circle is divided into $49$ areas so that no three areas touch at one point. The resulting “map” is colored in three colors so that no two adjacent areas have the same color. The border of two areas is considered to be colored in both colors. Prove that on the circle there are two diametrically opposite points, colored in one color.

Let $\overline{AD}$ be a diameter of a circle. Let point $B$ be on the circle, point $C$ on $\overline{AD}$ such that $A, B, C$ form a right triangle at $C$. The value of the hypotenuse of the triangle is $4$ times the square root of its area. If $\overline{BC}$ has length $30$, what is the length of the radius of the circle?

Let $ABCD$ be a parallelogram and $M$ is the intersection of $AC$ and $BD$. The point $N$ is inside of the $\triangle AMB$ such that $\angle AND=\angle BNC$. Prove that $\angle MNC=\angle NDA$ and $\angle MND=\angle NCB$.

In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.

We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated. It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?

The base of the pyramid $ABCD$ is an equilateral triangle $ABC$ with side length $1$. Additionally, \[\angle ADB=\angle BDC=\angle CDA=90^\circ\,.\] Calculate the volume of pyramid $ABCD$.

The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the triangle $ABC$ . $$\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$$ Prove that the perimeter $P$ of the triangle $ABC$ and the perimeter $p$ of the triangle $A_1B_1C_1$ , satisfy inequality $$\frac{P}{2} < p < \frac{3P}{4}$$

(A. Myakishev, 8--9) In the plane, given two concentric circles with the center $ A$. Let $ B$ be an arbitrary point on some of these circles, and $ C$ on the other one. For every triangle $ ABC$, consider two equal circles mutually tangent at the point $ K$, such that one of these circles is tangent to the line $ AB$ at point $ B$ and the other one is tangent to the line $ AC$ at point $ C$. Determine the locus of points $ K$.

In the triangle $ABC$, where $<$ $ACB$ $=$ $45$, $O$ and $H$ are the center the circumscribed circle, respectively, the orthocenter. The line that passes through $O$ and is perpendicular to $CO$ intersects $AC$ and $BC$ in $K$, respectively $L$. Show that the perimeter of $KLH$ is equal to the diameter of the circumscribed circle of triangle $ABC$.

Ali puts $5$ points on the plane such that no three of them are collinear. Ramtin adds a sixth point that is not collinear with any two of the former points.Ali wants to eventually construct two triangles from the six points such that one can be placed inside another. Can Ali put the 5 points in such a manner so that he would always be able to construct the desired triangles? (We say that triangle $T_1$ can be placed inside triangle $T_2$ if $T_1$ is congruent to a triangle that is located completely inside $T_2$.)

The incircle of a triangle $ABC$ is centered at $I$ and touches $AC,AB$ and $BC$ at points $K,L,M$, respectively. The median $BB_1$ of $\triangle ABC$ intersects $MN$ at $D$. Prove that the points $I,D,K$ are collinear.

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

Consider a rectangular sheet of paper $ABCD$ such that the lengths of $AB$ and $AD$ are respectively $7$ and $3$ centimetres. Suppose that $B'$ and $D'$ are two points on $AB$ and $AD$ respectively such that if the paper is folded along $B'D'$ then $A$ falls on $A'$ on the side $DC$. Determine the maximum possible area of the triangle $AB'D'$.

Let $O$ be an inner point of an acute-angled triangle $ABC$. Let $A_1, B_1$ and $C_1$ be the projections of $O$ on the sides $BC, AC$ and $AB$ respectively . Let $P$ be the intersection of the perpendiculars on $B_1C_1$ and $A_1C_1$ from points$ A$ and $B$ respectilvey. Let $H$ be the projection of $P$ on $AB$. Show that points $A_1, B_1, C_1$ and $H$ lie on a circle.

On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$.

Let $ABC$ be a triangle with $AB > AC$. The internal angle bisector of $\angle BAC$ intersects the side $BC$ at $D$. The circles with diameters $BD$ and $CD$ intersect the circumcircle of $\triangle ABC$ a second time at $P \not= B$ and $Q \not= C$, respectively. The lines $PQ$ and $BC$ intersect at $X$. Prove that $AX$ is tangent to the circumcircle of $\triangle ABC$.

Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$. (a) Prove that $ABMD$ is a rhombus. (b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.

Let $\vartriangle ABC$ be a triangle with an obtuse angle $\angle ACB$. The incircle of $\vartriangle ABC$ centered at $I$ is tangent to the sides $AB, BC, CA$ at $D, E, F$ respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$ respectively. Let $G$ be the midpoint of $AB$. Show that $M, N, G, D$ lie on a circle.

Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $. a) Prove that the lines $ EF $ and $ AB $ are perpendicular. b) Determine the locus of the point $ E $. c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.

Construct a right triangle given the lengths of segments of the medians $m_a,m_b$ corresponding on its legs.

On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram

The quadrilateral $ABCD$ can be inscribed in a circle and $\angle{ABD}$ is a right angle. $M$ is the midpoint of $BD$, where $CM$ is an altitude of $\triangle{BCD}$. If $AB=14$ and $CD=6\sqrt{11}$, what [is] the length of $AD$? $\text{(A) }36\qquad\text{(B) }38\qquad\text{(C) }41\qquad\text{(D) }42\qquad\text{(E) }44$

On circle $ O$, points $ C$ and $ D$ are on the same side of diameter $ \overline{AB}$, $ \angle AOC \equal{} 30^\circ$, and $ \angle DOB \equal{} 45^\circ$. What is the ratio of the area of the smaller sector $ COD$ to the area of the circle? [asy]unitsize(6mm); defaultpen(linewidth(0.7)+fontsize(8pt)); pair C = 3*dir (30); pair D = 3*dir (135); pair A = 3*dir (0); pair B = 3*dir(180); pair O = (0,0); draw (Circle ((0, 0), 3)); label ("$C$", C, NE); label ("$D$", D, NW); label ("$B$", B, W); label ("$A$", A, E); label ("$O$", O, S); label ("$45^\circ$", (-0.3,0.1), WNW); label ("$30^\circ$", (0.5,0.1), ENE); draw (A--B); draw (O--D); draw (O--C);[/asy]$ \textbf{(A)}\ \frac {2}{9} \qquad \textbf{(B)}\ \frac {1}{4} \qquad \textbf{(C)}\ \frac {5}{18} \qquad \textbf{(D)}\ \frac {7}{24} \qquad \textbf{(E)}\ \frac {3}{10}$