$ABCD$ is a tetrahedron such that we can find a sphere $k(A,B,C)$ through $A, B, C$ which meets the plane $BCD$ in the circle diameter $BC$, meets the plane $ACD$ in the circle diameter $AC$, and meets the plane $ABD$ in the circle diameter $AB$. Show that there exist spheres $k(A,B,D)$, $k(B,C,D)$ and $k(C,A,D)$ with analogous properties.

Let $M$ be an arbitrary point on side $BC$ of triangle $ABC$. $W$ is a circle which is tangent to $AB$ and $BM$ at $T$ and $K$ and is tangent to circumcircle of $AMC$ at $P$. Prove that if $TK||AM$, circumcircles of $APT$ and $KPC$ are tangent together.

Let the line $ z\equal{}x, \, y\equal{}0$ rotate at a constant speed about the $ z$-axis; let at the same time the point of intersection of this line with the $ z$-axis be displaced along the $ z$-axis at constant speed. (a) Determine that surface of rotation upon which the resulting helical surface can be developed (i.e. isometrically mapped). (b) Find those lines of the surface of rotation into which the axis and the generators of the helical surface will be mapped by this development.

Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.

Decompose the space of homogeneous polynomials of degree $5$ in $(x, y, z)$ into irreducible subspaces invariant with respect to the rotation group $SO(3)$.

Let $ABC$ be an acute scalene triangle with altitudes $AE$ $(E \in BC)$ and $BD$ $(D \in AC)$. Point $M$ lies on $AC$, such that $AM = AE$ and $C,A$ and $M$ lie in this order. Point $L$ lies on $BC$, such that $BL=BD$ and $C,B$ and $L$ lie in this order. Let $P$ be the midpoint of $DE$. Prove that $EM,DL$ and the perpendicular from $P$ to $AB$ are concurrent.

Given is a regular tetrahedron of volume $1$. We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?

Let $\vartriangle ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ and orthocenter $H$. Circle $\odot V$ is the circumcircle of $\vartriangle DE F$. Let segments $FD$, $BH$ intersect at point $P$. Let segments $ED$, $HC$ intersect at point $Q$. Let $K$ be a point on $AC$ such that $VK \perp CF$. a) Prove that $\vartriangle PQH \sim \vartriangle AKV$. b) Let line $PQ$ intersect $\odot V$ at points $I,G$. Prove that points $B,I,H,G,C$ are concyclic [hide]with center the symmetric point $X$ of circumcenter $O$ of $\vartriangle ABC$ wrt $BC$.[/hide] [hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png[/img] [url=https://www.geogebra.org/m/cjduebke]geogebra file[/url]

In a convex quadrilateral $ABCD$ sides $AB$ and $DC$ are both divided into $m$ equal parts by points $A, S_1 , S_2 , \ldots , S_{m-1} ,B$ and $D,T_1, T_2, \ldots , T_{m-1},C,$ respectively (in this order). Similarly, sides $BC$ and $AD$ are divided into $n$ equal parts by points $B,U_1,U_2, \ldots, U_{n-1},C$ and $A,V_1,V_2, \ldots,V_{n-1}, D$. Prove that for $1 \leq i \leq m-1$ each of the segments $S_i T_i$ is divided by the segments $U_j V_j$ ($1\leq j \leq n-1$) into $n$ equal parts

$ \overline{AB}$ is the hypotenuse of a right triangle $ ABC$. Median $ \overline{AD}$ has length $ 7$ and median $ \overline{BE}$ has length $ 4$. The length of $ \overline{AB}$ is: $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 5\sqrt{3}\qquad \textbf{(C)}\ 5\sqrt{2}\qquad \textbf{(D)}\ 2\sqrt{13}\qquad \textbf{(E)}\ 2\sqrt{15}$

Let $E$ be a point outside of square $ABCD$. If the distance of $E$ to $AC$ is $6$, to $BD$ is $17$, and to the nearest vertex of the square is $10$, what is the area of the square? $ \textbf{(A)}\ 200 \qquad\textbf{(B)}\ 196 \qquad\textbf{(C)}\ 169 \qquad\textbf{(D)}\ 162 \qquad\textbf{(E)}\ 144 $

Let $I$ be the center of the circle inscribed in triangle $ABC$. The inscribed circle is tangent to side $BC$ at point $K$. Let $X$ and $Y$ be points on segments $BI$ and $CI$ respectively, such that $KX \perp AB $ and $KY\perp AC$. The circumscribed circle around triangle $XYK$ intersects line $BC$ at point $D$. Prove that $AD \perp BC$. (Matthew Kurskyi)

As shown in figure , in the pentagon $ABCDE$, $BC = DE$, $CD \parallel BE$, $AB>AE$. If $\angle BAC = \angle DAE$ and $\frac{AB}{BD}=\frac{AE}{ED}$. Prove that $AC$ bisects the line segment $BE$. [img]https://cdn.artofproblemsolving.com/attachments/3/2/5ce44f1e091786b865ae4319bda56c3ddbb8d7.png[/img]

In a convex quadrilateral $PQRS$, the areas of triangles $PQS$, $QRS$ and $PQR$ are in the ratio $3:4:1$. A line through $Q$ cuts $PR$ at $A$ and $RS$ at $B$ such that $PA:PR=RB:RS$. Prove that $A$ is the midpoint of $PR$ and $B$ is the midpoint of $RS$.

The figure shows a glass prism which is partially filled with liquid. The surface of the prism consists of two isosceles right triangles, two squares with side length $10$ cm and a rectangle. The prism can lie in three different ways. If the prism lies as shown in figure $1$, the height of the liquid is $5$ cm. [img]https://cdn.artofproblemsolving.com/attachments/4/2/cda98a00f8586132fe519855df123534516b50.png[/img] a) What is the height of the liquid when it lies as shown in figure $2$? b) What is the height of the liquid when it lies as shown in figure$ 3$?

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

The diagonals of a convex quadrilateral $ABCD$ intersect at $E$. The triangles $ABE, BCE, CDE$ and $DAE$ have centroids $K,L,M$ and $N$, and orthocentres $Q,R,S$ and $T$. Show that the quadrilaterals $QRST$ and $LMNK$ are similar.

In triangle $ABC$ length of altitude $CH$, with $H \in AB$, is equal to half of side $AB$. If $\angle BAC = 45^{\circ}$ find $\angle ABC$

Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.

The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.

Take any point $P_1$ on the side $BC$ of a triangle $ABC$ and draw the following chain of lines: $P_1P_2$ parallel to $AC$; $P_2P_3$ parallel to $BC$; $P_3P_4$ parallel to $AB$ ; $P_4P_5$ parallel to $CA$; and $P_5P_6$ parallel to $BC$, Here, $P_2,P_5$ lie on $AB$; $P_3,P_6$ lie on $CA$ and $P_4$ on $BC$> Show that $P_6P_1$ is parallel to $AB$.

Determine all finite sets $S$ of points in the plane with the following property: if $x,y,x',y'\in S$ and the closed segments $xy$ and $x'y'$ intersect in only one point, namely $z$, then $z\in S$.

A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.

Let $A_1A_2A_3A_4A_5$ be a convex 5-gon in which the coordinates of all of it's vertices are rational. For each $1\leq i \leq 5$ define $B_i$ the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i+3}A_{i+4}$. ($A_i=A_{i+5}$) Prove that at most 3 lines from the lines $A_iB_i$ ($1\leq i \leq 5$) are concurrent. Time allowed for this problem was 75 minutes.

In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$. (RK Gordin)