Is it possible to tile space using a combination of regular tetrahedra and regular octahedra? (A Belov)

Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.

Let $\vartriangle ABC$ be a triangle. Determine all points $P$ in the plane such that the triangles $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$ all have the same area.

Four circles of radius $1$ are each tangent to two sides (line segments) of a square and externally tangent to a circle of radius $3$. What is the area of the space that is inside the square but not contained in any of the circles?

in $ABC$ let $E$ and $F$ be points on line $AC$ and $AB$ respectively such that $BE$ is parallel to $CF$. suppose that the circumcircle of $BCE$ meet $AB$ again at $F'$ and the circumcircle of $BCF$ meets $AC$ again at $E'$. show that $BE'$ Is parallel to $CF'$.

Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.

A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player loses, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

Let $B$ and $C$ be points on a semicircle with diameter $AD$ such that $B$ is closer to $A$ than $C$. Diagonals $AC$ and $BD$ intersect at point $E$. Let $P$ and $Q$ be points such that $\overline{PE} \perp \overline{BD}$ and $\overline{PB} \perp \overline{AD}$, while $\overline{QE} \perp \overline{AC}$ and $\overline{QC} \perp \overline{AD}$. If $BQ$ and $CP$ intersect at point $T$, prove that $\overline{TE} \perp \overline{BC}$. [i]Proposed by Andrew Wen[/i]

Two triangles are drawn on a plane in such a way that the area covered by their union is an n-gon (not necessarily convex). Find all possible values of the number of vertices n.

There are two pipes on the plane (the pipes are circular cylinders of equal size, $4$ m around). Two of them are parallel and, being tangent one to another in the common generatrix, form a tunnel over the plane. The third pipe is perpendicular to two others and cuts out a chamber in the tunnel. Determine the area of the surface of this chamber.

In a convex quadrilateral $ABCD$ suppose $AC \cap BD = O$ and $M$ is the midpoint of $BC$. Let $MO \cap AD = E$. Prove that $\frac{AE}{ED} = \frac{S_{\triangle ABO}}{S_{\triangle CDO}}$.

Let $AB$ and $FD$ be chords in circle, which does not intersect and $P$ point on arc $AB$ which does not contain chord $FD$. Lines $PF$ and $PD$ intersect chord $AB$ in $Q$ and $R$. Prove that $\frac{AQ* RB}{QR}$ is constant, while point $P$ moves along the ray $AB$.

Two not necessarily equal non-intersecting wooden disks, one gray and one black, are glued to a plane. An infinite angle with one gray side and one black side can be moved along the plane so that the disks remain outside the angle, while the colored sides of the angle are tangent to the disks of the same color (the tangency points are not the vertices). Prove that it is possible to draw a ray in the angle, starting from the vertex of the angle and such that no matter how the angle is positioned, the ray passes through some fixed point of the plane. (Egor Bakaev, Ilya Bogdanov, Pavel Kozhevnikov, Vladimir Rastorguev) (Junior version [url=https://artofproblemsolving.com/community/c6h2094701p15140671]here[/url]) [hide=note]There was a mistake in the text of the problem 3, we publish here the correct version. The solutions were estimated according to the text published originally.[/hide]

With a compass and a ruler, split a triangle into two smaller triangles with the same sum of squares of sides.

An ant starts at vertex $A$ in equilateral triangle $\triangle ABC$ and walks around the perimeter of the triangle from $A$ to $B$ to $C$ and back to $A$. When the ant is $42$ percent of its way around the triangle, it stops for a rest. Find the percent of the way from $B$ to $C$ the ant is at that point

Let ABC be a triangle. Let B' and C' denote the reflection of B and C in the internal angle bisector of angle A. Show that the triangles ABC and AB'C' have the same incenter.

A point $ (x,y)$ is randomly picked from inside the rectangle with vertices $ (0,0)$, $ (4,0)$, $ (4,1)$, and $ (0,1)$. What is the probability that $ x<y$? $ \textbf{(A)}\ \frac{1}{8} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{3}{8} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{3}{4}$

Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$. Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and $\angle DPC > \angle ABC$. Prove that $$AB+CD>DA+BC.$$

If P is a point inside a convex quadrilateral $ABCD$, let the angle bisectors of $\angle APB$, $\angle BPC$, $\angle CPD$ and $\angle DPA$ meet $AB$, $BC$, $CD$ and $DA$ at $K$, $L$, $M$ and $N$ respectively. (a) Find a point $P$ such that $KLMN$ is a parallelogram. (b) Find the locus of all such points $P$. (S Tokarev)

Is it possible to place some circles inside a square side length $1$, such that no two circles intersect and the sum of their radii is $2007$?

For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which $\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.

Let $ABCD$ be a trapezoid with bases $AB,CD$ such that $CD=k \cdot AB$ ($0<k<1$). Point $P$ is such that $\angle PAB=\angle CAD$ and $\angle PBA=\angle DBC$. Prove that $PA+PB \leq \dfrac{1}{\sqrt{1-k^2}} \cdot AB$.

A triangle $ ABC$ is inscribed in a circle $ C(O,R)$ and has incenter $ I$. Lines $ AI,BI,CI$ meet the circumcircle $ (O)$ of triangle $ ABC$ at points $ D,E,F$ respectively. The circles with diameter $ ID,IE,IF$ meet the sides $ BC,CA, AB$ at pairs of points $ (A_1,A_2), (B_1, B_2), (C_1, C_2)$ respectively. Prove that the six points $ A_1,A_2, B_1, B_2, C_1, C_2$ are concyclic. Babis

The altitude from a vertex of a given tetrahedron intersects the opposite face in its orthocenter. Prove that all four altitudes of the tetrahedron are concurrent.

Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.