A $ 20\times20\times20$ block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $ 1\times20\times20$ slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices.

In an acute-angled triangle $ABC$, $CF$ is an altitude, with $F$ on $AB$, and $BM$ is a median, with $M$ on $CA$. Given that $BM=CF$ and $\angle MBC=\angle FCA$, prove that triangle $ABC$ is equilateral.

[b]p1.[/b] One hundred hobbits sit in a circle. The hobbits realize that whenever a hobbit and his two neighbors add up their total rubles, the sum is always $2007$. Prove that each hobbit has $669$ rubles. [b]p2.[/b] There was a young lady named Chris, Who, when asked her age, answered this: "Two thirds of its square Is a cube, I declare." Now what was the age of the miss? (a) Find the smallest possible age for Chris. You must justify your answer. (Note: ages are positive integers; "cube" means the cube of a positive integer.) (b) Find the second smallest possible age for Chris. You must justify your answer. (Ignore the word "young.") [b]p3.[/b] Show that $$\sum_{n=1}^{2007}\frac{1}{n^3+3n^2+2n}<\frac14$$ [b]p4.[/b] (a) Show that a triangle $ABC$ is isosceles if and only if there are two distinct points $P_1$ and $P_2$ on side $BC$ such that the sum of the distances from $P_1$ to the sides $AB$ and $AC$ equals the sum of the distances from $P_2$ to the sides $AB$ and $AC$. (b) A convex quadrilateral is such that the sum of the distances of any interior point to its four sides is constant. Prove that the quadrilateral is a parallelogram. (Note: "distance to a side" means the shortest distance to the line obtained by extending the side.) [b]p5.[/b] Each point in the plane is colored either red or green. Let $ABC$ be a fixed triangle. Prove that there is a triangle $DEF$ in the plane such that $DEF$ is similar to $ABC$ and the vertices of $DEF$ all have the same color. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

The altitudes through the vertices $ A,B,C$ of an acute-angled triangle $ ABC$ meet the opposite sides at $ D,E, F,$ respectively. The line through $ D$ parallel to $ EF$ meets the lines $ AC$ and $ AB$ at $ Q$ and $ R,$ respectively. The line $ EF$ meets $ BC$ at $ P.$ Prove that the circumcircle of the triangle $ PQR$ passes through the midpoint of $ BC.$

Let $ABCD$ be a tangential and cyclic quadrilateral. Let $S$ be the intersection point of diagonals $AC$ and $BD$ of the quadrilateral. Let $I$, $I_1$, and $I_2$ be the incenters of quadrilateral $ABCD$ and triangles $ACD$ and $BCS$, respectively. Let the ray $II_2$ intersect the circumcircle of quadrilateral $ABCD$ at point $E$. Prove that the points $D$, $E$, $I_1$, and $I_2$ are collinear or concyclic. [i]Proposed by Teodor von Burg[/i]

Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_a$ passes in the points $A$ and $H$ and is tangent to the circumcircle of $ABC$. Define $X_b, X_c$ analogously, let $O_a, O_b, O_c$ the symmetric of $O$ to the sides $BC, AC$ and $AB$, respectively. Prove that the lines $O_aX_a, O_bX_b, O_cX_c$ are concurrents.

Let $D$ and $E$ be points on $[BC]$ and $[AC]$ of acute $\triangle ABC$, respectively. $AD$ and $BE$ meet at $F$. If $|AF|=|CD|=2|BF|=2|CE|$, and $Area(\triangle ABF) = Area(\triangle DEC)$, then $Area(\triangle AFC)/Area(\triangle BFC) = ?$ $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt2 \qquad \textbf{(E)}\ 1$

Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$. (a) Prove that $P$ is independent of the choice of $A$. (b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.

Prove that diagonals of a convex quadrilateral are perpendicular if and only if inside of the quadrilateral there is a point, whose orthogonal projections on sides of the quadrilateral are vertices of a rectangle.

A right triangle $ABC$ ($\angle A=90$) is inscribed in a circle $\omega$. Tangent to $\omega$ at $A$ intersects $BC$ at $P$, $B$ lies between $P$ and $C$. Let $M$ be the midpoint of the minor arc $AB$. $MP$ intersects $\omega$ at $Q$. Point $X$ lies on a ray $PA$ such that $\angle XCB=90$. Prove that line $XQ$ passes through the orthocenter of the triangle $ABO$ [i]Mayya Golitsyna[/i]

A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.

Let $SABCD$ be a pyramid with the vertex $S$ whose all edges are equal. Points $M$ and $N$ on the edges $SA$ and $BC$ respectively are such that $MN$ is perpendicular to both $SA$ and $BC$. Find the ratios $SM:MA$ and $BN:NC$.

Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.

Consider a solid hemisphere of radius $1$. Find the distance from its center of mass to the base.

Each side of the convex quadrilateral was continued into both sides and on all eight extensions set aside equal segments. It turned out that the resulting $8$ points are the outer ends of the construction the given segments are different and lie on the same circle. Prove that the original quadrilateral is a square.

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

The circle $\omega_1$ passes through the center $O$ of the circle $\omega_2$ and meets it at points $A$ and $B$. The circle $\omega_3$ centered at $A$ with radius $AB$ meets $\omega_1$ and $\omega_2$ at points $C$ and $D$ (distinct from $B$). Prove that $C, O, D$ are collinear.

For a given natural number $k{}$, a convex polygon is called $k{}$[i]-triangular[/i] if it is the intersection of some $k{}$ triangles. [list=a] [*]What is the largest $n{}$ for which there exist a $k{}$-triangular $n{}$-gon? [*]What is the largest $n{}$ for which any convex $n{}$-gon is $k{}$-triangular? [/list] [i]Proposed by P. Kozhevnikov[/i]

Let $\omega$ be the $A$-excircle of triangle $ABC$ and $M$ the midpoint of side $BC$. $G$ is the pole of $AM$ w.r.t $\omega$ and $H$ is the midpoint of segment $AG$. Prove that $MH$ is tangent to $\omega$.

Consider a circle of center $O$ and a chord $AB$ of it (not a diameter). Take a point $T$ on the ray $OB$. The perpendicular at $T$ onto $OB$ meets the chord $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto the chord $AB$. Prove that $AS \cdot BC = T E \cdot TD$.

In acute triangle $ABC$ bisector of angle $BAC$ intersects side $BC$ in point $D$. Bisector of line segment $AD$ intersects circumcircle of triangle $ABC$ in points $E$ and $F$. Show that circumcircle of triangle $DEF$ is tangent to line $BC$.

Four swallows are catching a fly. At first, the swallows are at the four vertices of a tetrahedron, and the fly is in its interior. Their maximal speeds are equal. Prove that the swallows can catch the fly.

Let $A,B,C,D$ be points in space. If for every point $M$ on the segment $AB$ the sum \[S_{AMC}+S_{CMD}+S_{DMB}\] Is constant show that the points $A,B,C,D$ lie in the same plane. [hide="Note."] [i]Note. $S_X$ denotes the area of triangle $X.$[/i][/hide]

Given an equilateral triangle $ABC$ in the plane, how many points $P$ in the plane are such that the three triangles $AP B, BP C $ and $CP A$ are isosceles and not degenerate?