Prove that the volume of a tire (torus) is equal to the volume of a cylinder whose base is a meridian section of that and whose height is the length of the circumference formed by the centers of the meridian sections.

Triangle $ABC$ has side lengths $AB=2\sqrt{5}$, $BC=1$, and $CA=5$. Point $D$ is on side $AC$ such that $CD=1$, and $F$ is a point such that $BF=2$ and $CF=3$. Let $E$ be the intersection of lines $AB$ and $DF$. Find the area of $CDEB$.

A circle is inscribed in an equilateral triangle, and a square is inscribed in the circle. The ratio of the area of the triangle to the area of the square is: $\textbf{(A)}\ \sqrt{3}:1 \qquad \textbf{(B)}\ \sqrt{3}:\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{3}:2 \qquad \textbf{(D)}\ 3:\sqrt{2} \qquad \textbf{(E)}\ 3:2\sqrt{2}$

Let $\Gamma_1$ be a circle and $P$ a point outside of $\Gamma_1$. The tangents from $P$ to $\Gamma_1$ touch the circle at $A$ and $B$. Let $M$ be the midpoint of $PA$ and $\Gamma_2$ the circle through $P$, $A$ and $B$. Line $BM$ cuts $\Gamma_2$ at $C$, line $CA$ cuts $\Gamma_1$ at $D$, segment $DB$ cuts $\Gamma_2$ at $E$ and line $PE$ cuts $\Gamma_1$ at $F$, with $E$ in segment $PF$. Prove lines $AF$, $BP$, and $CE$ are concurrent.

Let $ABC$ a triangle with $AB<AC$ and let $I$ it´s incenter. Let $\Gamma$ the circumcircle of $\triangle BIC$. $AI$ intersect $\Gamma$ again in $P$. Let $Q$ a point in side $AC$ such that $AB=AQ$ and let $R$ a point in $AB$ with $B$ between $A$ and $R$ such that $AR=AC$. Prove that $IQPR$ is cyclic.

The circles $k_1, k_2$, centered at $O_1, O_2$, meet at two points, one of which is $A$. Let $P, Q$ lie on $AO_1, AO_2$, respectively, so that $PQ \parallel O_1O_2$. The tangents from $P$ to $k_2$ touch it at $X, Y$ and the tangents from $Q$ to $k_1$ touch it at $Z, T$. Show that $X, Y, Z, T$ are collinear or concyclic.

Construct a quadrangle along the given sides $a, b, c$, and $d$ and the distance $I$ between the midpoints of its diagonals.

Prove that there exists an ellipsoid touching all edges of an octahedron if and only if the octahedron's diagonals intersect. (Here an octahedron is a polyhedron consisting of eight triangular faces, twelve edges, and six vertices such that four faces meat at each vertex. The diagonals of an octahedron are the lines connecting pairs of vertices not connected by an edge).

Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.

The bisector of the exterior angle at vertex $C$ of the triangle $ABC$ intersects the bisector of the interior angle at vertex $B$ in point $K$. Consider the diameter of the circumcircle of the triangle $BCK$ whose one endpoint is $K$. Prove that $A$ lies on this diameter.

Points $A$ and $B$ move on circles centered at $O_A$ and $O_B$ such that $O_AA$ and $O_BB$ rotate at the same speed. Prove that vertex $C$ of the equilateral triangle $ABC$ moves along a certain circle at the same angular velocity. (The vertices of $ABC$ are oriented clockwise.)

Two parallel chords in a circle have lengths $10$ and $14$, and the distance between them is $6$. The chord parallel to these chords and midway between them is of length $\sqrt{a}$ where $a$ is [asy] // note: diagram deliberately not to scale -- azjps void htick(pair A, pair B, real r){ D(A--B); D(A-(r,0)--A+(r,0)); D(B-(r,0)--B+(r,0)); } size(120); pathpen = linewidth(0.7); pointpen = black+linewidth(3); real min = -0.6, step = 0.5; pair[] A, B; D(unitcircle); for(int i = 0; i < 3; ++i) { A.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[0]); B.push(intersectionpoints((-9,min+i*step)--(9,min+i*step),unitcircle)[1]); D(D(A[i])--D(B[i])); } MP("10",(A[0]+B[0])/2,N); MP("\sqrt{a}",(A[1]+B[1])/2,N); MP("14",(A[2]+B[2])/2,N); htick((B[1].x+0.1,B[0].y),(B[1].x+0.1,B[2].y),0.06); MP("6",(B[1].x+0.1,B[0].y/2+B[2].y/2),E);[/asy] $\textbf{(A)}\ 144 \qquad \textbf{(B)}\ 156 \qquad \textbf{(C)}\ 168 \qquad \textbf{(D)}\ 176 \qquad \textbf{(E)}\ 184$

We say that a triangle of sides $a,b,c$ is [i] virtual[/i] if such measures satisfy $$\begin{cases} a^{2024}+b^{2024}> c^{2024},\\ b^{2024}+c^{2024}> a^{2024},\\ c^{2024}+a^{2024}> b^{2024} \end{cases}$$ Find the number of ordered triples $(a,b,c)$ such that $a,b,c$ are integers between $1$ and $2024$ (inclusive) and $a,b,c$ are the sides of a [i]virtual [/i] triangle.

A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices? [i]Proposed by A. Golovanov[/i]

If in a convex quadrilateral $ ABCD, E$ and $ F$ are the midpoints of the sides $ BC$ and $ DA$ respectively. Show that the sum of the areas of the triangles $ EDA$ and $ FBC$ is equal to the area of the quadrangle.

On one of the sides of the $60$ degree angle with vertex $O$ a fixed point $F$ is marked. On the other side of the angle a point $A$ is chosen, and on the ray $OF$, but not the segment $OF$, a point $B$ such that $OA=FB$. On the segment $AB$ equilateral triangle $ABC$ and $ABD$ are built such that points $O$ and $C$ lie in the same half-plane with respect to $AB$, and $D$ in the other. a) Prove that the point $C$ does not depend on $A$. b) Prove that all points $D$ lie on a line.

Ants, named Anna and Bob, are located at vertices \(A\) and \(B\) respectively of a cube \(ABCD A_1 B_1 C_1 D_1\), with a sugar cube placed at vertex \(C_1\). It is known that Bob can move at a speed of $20$ meters per minute. Determine the minimum speed in integer meters per minute that Anna must be able to travel in order to reach the sugar cube at \(C_1\) before Bob. [i]Proposed by Tamar Turashvili, Georgia [/i]

In $\triangle ABC$, $\angle C=90^{\circ},\angle B=30^{\circ}, AC=2$. $M$ is the midpoint of $AB$. Fold up $\triangle ACM$ along $CM$, satisfying that $|AB|=2\sqrt2$. The volume of triangular pyramid $A-BCM$ is________.

Prove that if the two angles on the base of a trapezoid are different, then the diagonal starting from the smaller angle is longer than the other diagonal. [img]https://cdn.artofproblemsolving.com/attachments/7/1/77cf4958931df1c852c347158ff1e2bbcf45fd.png[/img]

The line joining the midpoints of the diagonals of a trapezoid has length $3$. If the longer base is $97$, then the shorter base is: $ \textbf{(A)}\ 94 \qquad\textbf{(B)}\ 92\qquad\textbf{(C)}\ 91\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 89 $

Given a quadrangular pyramid $SABCD$, the basis of which is a convex quadrilateral $ABCD$. It is known that the pyramid can be tangent to a sphere. Let $P$ be the point of contact of this sphere with the base $ABCD$. Prove that $\angle APB + \angle CPD = 180^o$.

Let $ABCD$ be an inscribed quadrilateral and $P$ be an inner point for it so that $\angle PAB=\angle PBC=\angle PCD=\angle PDA$. The lines $AD$ and $BC$ intersect in point $Q$ and lines $AB$ and $CD$ – in point $R$. Prove that $\angle (PQ,PR)=\angle (AC,BD)$.

We choose 5 points $A_1, A_2, \ldots, A_5$ on a circumference of radius 1 and centre $O.$ $P$ is a point inside the circle. Denote $Q_i$ as the intersection of $A_iA_{i+2}$ and $A_{i+1}P$, where $A_7 = A_2$ and $A_6 = A_1.$ Let $OQ_i = d_i, i = 1,2, \ldots, 5.$ Find the product $\prod^5_{i=1} A_iQ_i$ in terms of $d_i.$

A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown? [asy] defaultpen(linewidth(0.8)); path p=unitsquare; draw(p^^shift(0,1)*p^^shift(1,0)*p); draw(shift(4,0)*p^^shift(5,0)*p^^shift(5,1)*p); label("FRONT", (1,0), S); label("SIDE", (5,0), S);[/asy] $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Let $ABC$ be an acute-angled triangle. Let $A_1$ be the midpoint of the arc $BC$ which contain the point $A$. Let $A_2$ and $A_3$ be the foot(s) of the perpendicular(s) of the point $A_1$ to the lines $AB$ and $AC$, respectively. Define $B_2,B_3,C_2,C_3$ analogously. a) Prove that the line $A_2A_3$ cuts $BC$ in the midpoint. b) Prove that the lines $A_2A_3,B_2B_3$ and $C_2C_3$ are concurrents.