Fourteen white cubes are put together to form the figure on the right. The complete surface of the figure, including the bottom, is painted red. The figure is then separated into individual cubes. How many of the individual cubes have exactly four red faces? [asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(3/4,8/15,7/15); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(0,0,1)*unitcube, white, thick(), nolight); draw(shift(2,0,1)*unitcube, white, thick(), nolight); draw(shift(0,1,0)*unitcube, white, thick(), nolight); draw(shift(2,1,0)*unitcube, white, thick(), nolight); draw(shift(0,2,0)*unitcube, white, thick(), nolight); draw(shift(2,2,0)*unitcube, white, thick(), nolight); draw(shift(0,3,0)*unitcube, white, thick(), nolight); draw(shift(0,3,1)*unitcube, white, thick(), nolight); draw(shift(1,3,0)*unitcube, white, thick(), nolight); draw(shift(2,3,0)*unitcube, white, thick(), nolight); draw(shift(2,3,1)*unitcube, white, thick(), nolight);[/asy] $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12$

Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

$ABCD$ is a scalene tetrahedron and let $G$ be its baricentre. A plane $\alpha$ passes through $G$ such that it intersects neither the interior of $\Delta BCD$ nor its perimeter. Prove that $$\textnormal{dist}(A,\alpha)=\textnormal{dist}(B,\alpha)+\textnormal{dist}(C,\alpha)+\textnormal{dist}(D,\alpha).$$ [i] (Adapted from folklore)[/i]

The areas of two right-angled triangles have ratio equal to the squares of their hypotenuses. Show that these triangles are similar.

Let $A_0B_0C_0$ be a triangle. For a positive integer $n \geq 1$, we define $A_n$ on the segment $B_{n-1}C_{n-1}$ such that $B_{n-1}A_n:C_{n-1}A_n=2:1$ and $B_n, C_n$ are defined cyclically in a similar manner. Show that there exists an unique point $P$ that lies in the interior of all triangles $A_nB_nC_n$.

Let $A_1A_2A_3\ldots A_n$ be a regular $n$-gon. Let $B_1$ and $B_{n-1}$ be the midpoints of its sides $A_1A_2$ and $A_{n-1}A_n$. Also, for every $i\in\left\{2,3,4,\ldots ,n-2\right\}$. Let $S$ be the point of intersection of the lines $A_1A_{i+1}$ and $A_nA_i$, and let $B_i$ be the point of intersection of the angle bisector bisector of the angle $\measuredangle A_iSA_{i+1}$ with the segment $A_iA_{i+1}$. Prove that $\sum_{i=1}^{n-1} \measuredangle A_1B_iA_n=180^{\circ}$. [i]Proposed by Dusan Dukic, Serbia and Montenegro[/i]

We are given $100$ points. $N$ of these are vertices of a convex $N$-gon and the other $100 - N$ of these are inside this $N$-gon. The labels of these points make it impossible to tell whether or not they are vertices of the $N$-gon. It is known that no three points are collinear and that no $4$ points belong to two parallel lines. It has been decided to ask questions of the following type: What is the area of the triangle $XYZ$, where $X, Y$ and $Z$ are labels representing three of the $100$ given points? Prove that $300$ such questions are sufficient in order to clarify which points are vertices and to determine the area of the $N$-gon. (D. Fomin, Leningrad)

Let $ABC$ be an equilateral triangle with the side of $20$ units. Amir divides this triangle into $400$ smaller equilateral triangles with the sides of $1$ unit. Reza then picks $4$ of the vertices of these smaller triangles. The vertices lie inside the triangle $ABC$ and form a parallelogram with sides parallel to the sides of the triangle $ABC.$ There are exactly $46$ smaller triangles that have at least one point in common with the sides of this parallelogram. Find all possible values for the area of this parallelogram. [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 20; /* # of vertical lines, including BC */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(1)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); }[/asy] [Thanks azjps for drawing the diagram.] [hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]

Let $n$ be a positive integer. Let $P,Q,A_1,A_2,...,A_n$ be distinct points such that $A_1,A_2,...,A_n$ are not collinear. Suppose that $PA_1 + PA_2 + ...+PA_n$, and $QA_1 + QA_2 +...+ QA_n$, have a common value $s$ for some real number $s$. Prove that there exists a point $R$ such that $$RA_1 + RA_2 +... + RA_n < s.$$

Starting from a pyramid $T_0$ whose edges are all of length $2019$, we construct the Figure $T_1$ when considering the triangles formed by the midpoints of the edges of each face of $T_0$, building in each of these new pyramid triangles with faces identical to base. Then the bases of these new pyramids are removed. Figure $T_2$ is constructed by applying the same process from $T_1$ on each triangular face resulting from $T_1$, and so on for $T_3, T_4, ...$ Let $D_0= \max \{d(x,y)\}$, where $x$ and $y$ are vertices of $T_0$ and $d(x,y)$ is the distance between $x$ and $y$. Then we define $D_{n + 1} = \max \{d (x, y) |d (x, y) \notin \{D_0, D_1,...,D_n\}$, where $x, y$ are vertices of $T_{n+1}$. Find the value of $D_n$ for all $n$.

Let $\astrosun O_1$ and $\astrosun O_2$ intersect at $P$ and $Q$, their common external tangent touches $\astrosun O_1$ and $\astrosun O_2$ at $A$ and $B$ respectively. A circle $\Gamma$ passing through $A$ and $B$ intersects $\astrosun O_1$, $\astrosun O_2$ at $D$, $C$. Prove that $\displaystyle \frac{CP}{CQ}=\frac{DP}{DQ}$

Let $AD$ and $AE$ be the altitude and median of triangle $ABC$, in with $\angle B = 2\angle C$. Prove that $AB = 2DE$.

Let $ ABC$ be a triangle in which $ AB \equal{} AC$ and let $ I$ be its in-centre. Suppose $ BC \equal{} AB \plus{} AI$. Find $ \angle{BAC}$

Given is an acute angled triangle $ABC$. A circle going through $B$ and the triangle's circumcenter, $O$, intersects $BC$ and $BA$ at points $P$ and $Q$ respectively. Prove that the intersection of the heights of the triangle $POQ$ lies on line $AC$.

The Ultimate Question is a 10-part problem in which each question after the first depends on the answer to the previous problem. As in the Short Answer section, the answer to each (of the 10) problems is a nonnegative integer. You should submit an answer for each of the 10 problems you solve (unlike in previous years). In order to receive credit for the correct answer to a problem, you must also correctly answer $\textit{every one}$ $\textit{of the previous parts}$ $\textit{of the Ultimate Question}$.

A square is placed in the plane and a point $P$ is marked in this plane with invisible ink. A certain person can see this point through special glasses. One can draw a straight line and this person will say on which side of the line the point $P$ lies. If $P$ lies on the line, the person says so. What is the minimal number of questions one needs to find out if $P$ lies inside the square or not? (Folklore)

A regular 14-gon with side $a$ is inscribed in a circle of radius one. Prove \[ \frac{2-a}{2 \cdot a} > \sqrt{3 \cdot \cos \left( \frac{\pi}{7} \right)}. \]

$ \triangle ABC$ is a right isosceles triangle. Choose points $ K$ and $ M$ from the hypotenuse $ AB$, such that $ K \in AM$ and $ \angle KCM \equal{} 45$º. Prove that $ (AK)^2 \plus{} (MB)^2 \equal{} (KM)^2$ Thanks for any help.

5. Let $\triangle$ABC be a triangle with circumcentre [i]O[/i]. The perpendicular line from [i]O[/i] to [i]BC[/i] intersects line [i]BC[/i] at [i]M[/i] and line [i]AC[/i] at [i]P[/i], and the perpendicular line from [i]O[/i] to [i]AC[/i] intersects line [i]AC[/i] at [i]N[/i] and line [i]BC[/i] at [i]Q[/i]. Let [i]D[/i] be the intersection point of lines [i]PQ[/i] and [i]MN[/i]. construct the parallelogram [i]PCQJ[/i] with [i]PJ[/i] || [i]CQ[/i] and [i]QJ[/i] || [i]CP[/i]. Prove the following: a) The points [i]A[/i], [i]B[/i], [i]O[/i], [i]P[/i], [i]Q[/i], [i]J[/i] are all on the same circle. b) line [i]OD[/i] is perpendicular to line [i]CJ[/i].

Let $ABC$ be a right triangle $(AB<AC)$ with heights $AD, BE,$ and $CF$ and orthocenter $H$. Let $M$ denote the midpoint of $BC$ and let $X$ be the second intersection of the circle with diameter $HM$ and line $AM.$ Given that lines $HX$ and $BC$ intersect at $T,$ prove that the circumcircles of $\triangle TFD$ and $\triangle AEF$ are tangent.

As you know, the moon revolves around the earth. We assume that the Earth and the Moon are points, and the Moon rotates around the Earth in a circular orbit with a period of one revolution per month. The flying saucer is in the plane of the lunar orbit. It can be jumped through the Moon and the Earth - from the old place (point $A$), it instantly appears in the new (at point $A '$) so that either the Moon or the Earth is in the middle of segment $AA'$. Between the jumps, the flying saucer hangs motionless in outer space. 1) Determine the minimum number of jumps a flying saucer will need to jump from any point inside the lunar orbit to any other point inside the lunar orbit. 2) Prove that a flying saucer, using an unlimited number of jumps, can jump from any point inside the lunar orbit to any other point inside the lunar orbit for any period of time, for example, in a second.

There are two identical clocks on the wall, one showing the current Moscow time and the other showing current local time. The minimum distance between the ends of their hour hands equals $m$ and the maximum distance equals $M$. Find the distance between the centres of the clocks. (S Fomin, Leningrad)

What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\sqrt{13}, BD=2\sqrt{5},$ and $CD=5$? $\textbf{(A) }3 \qquad \textbf{(B) }2\sqrt{3} \qquad \textbf{(C) }4 \qquad \textbf{(D) }3\sqrt{3} \qquad \textbf{(E) }6$

A rectangular floor is covered by a certain number of equally large quadratic tiles. The tiles along the edge are red, and the rest are white. There are equally many red and white tiles. How many tiles can there be?

Let $AH_a $ and $BH_b$ be the altitudes of triangle $ABC$. Points $P$ and $Q$ are the projections of $H_a$ to $AB$ and $AC$. Prove that line $PQ $ bisects segment $H_aH_b$. (A.Akopjan, K.Savenkov)