Three spheres are constructed so that the edges $[AB], [BC], [AD]$ of the tetrahedron $ABCD$ are their respective diameters. Prove that the spheres cover all the tetrahedron.

The following diagram shows equilateral triangle $\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\vartriangle ABC$ a distance $\frac18 AB$ along a side of $\vartriangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. The shaded region inside all four of the triangles has area $300$. Find the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/a/8d724563c7411547d3161076015b247e882122.png[/img]

Starting from an equilateral triangle with perimeter $P_0$, we carry out the following iterations: the first iteration consists of dividing each side of the triangle into three segments of equal length, construct an exterior equilateral triangle on each of the middle segments, and then remove these segments (bases of each new equilateral triangle formed). The second iteration consists of apply the same process of the first iteration on each segment of the resulting figure after the first iteration. Successively, follow the other iterations. Let $A_n$ be the area of the figure after the $n$- th iteration, and let $P_n$ the perimeter of the same figure. If $A_n = P_n$, find the value of $P_0$ (in its simplest form).

The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent.

In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\vartriangle BEC$ is $100$, find the area of the quadrilateral $AFED$.

What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?

Suppose $Z, Y$ , and $W$ are points on a circle such that lengths $ZY = Y W$. Extend $ZY$ and let $X$ be a point on $ZY$ where $ZY = Y X$. If $XW$ is a tangent of the circle, what is $\angle W XY$ ?

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

Let $A$ be the sum of all $10$ distinct products of the sides of a convex pentagon, $S$ be the area of the pentagon. a) Prove thas $S \le \frac15 A$. b) Does there exist a constant $c<\frac15$ such that $S \le cA$ ? I.Voronovich

For each vertex of triangle $ABC$, the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle.

Given triangle $ABC$. Let $BPCQ$ be a parallelogram ($P$ is not on $BC$). Let $U$ be the intersection of $CA$ and $BP$, $V$ be the intersection of $AB$ and $CP$, $X$ be the intersection of $CA$ and the circumcircle of triangle $ABQ$ distinct from $A$, and $Y$ be the intersection of $AB$ and the circumcircle of triangle $ACQ$ distinct from $A$. Prove that $\overline{BU} = \overline{CV}$ if and only if the lines $AQ$, $BX$, and $CY$ are concurrent. [i]Proposed by Li4.[/i]

Let $ ABCD$ be a convex quadrilateral. We have that $ \angle BAC\equal{}3\angle CAD$, $ AB\equal{}CD$, $ \angle ACD\equal{}\angle CBD$. Find angle $ \angle ACD$

[b]p1.[/b] Find the unknown angle $a$ of the triangle inscribed in the square. [img]https://cdn.artofproblemsolving.com/attachments/b/1/4aab5079dea41637f2fa22851984f886f034df.png[/img] [b]p2.[/b] Draw a polygon in the plane and a point outside of it with the following property: no edge of the polygon is completely visible from that point (in other words, the view is obstructed by some other edge). [b]p3.[/b] This problem has two parts. In each part, $2022$ real numbers are given, with some additional property. (a) Suppose that the sum of any three of the given numbers is an integer. Show that the total sum of the $2022$ numbers is also an integer. (b) Suppose that the sum of any five of the given numbers is an integer. Show that 5 times the total sum of the $2022$ numbers is also an integer, but the sum itself is not necessarily an integer. [b]p4.[/b] Replace stars with digits so that the long multiplication in the example below is correct. [img]https://cdn.artofproblemsolving.com/attachments/9/7/229315886b5f122dc0675f6d578624e83fc4e0.png[/img] [b]p5.[/b] Five nodes of a square grid paper are marked (called marked points). Show that there are at least two marked points such that the middle point of the interval connecting them is also a node of the square grid paper [b]p6.[/b] Solve the system $$\begin{cases} \dfrac{xy}{x+y}=\dfrac{8}{3} \\ \dfrac{yz}{y+z}=\dfrac{12}{5} \\\dfrac{xz}{x+z}=\dfrac{24}{7} \end{cases}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.

In space, $n$ wire triangles are situated so that any two of them have a common vertex and each vertex is the vertex of $k$ triangles. Find all $n$ and $k$ for which this is possible.

Sandy likes to eat waffles for breakfast. To make them, she centers a circle of wafflebatter of radius $3$ cm at the origin of the coordinate plane and her waffle iron imprints non-overlapping unit-square holes centered at each lattice point. How many of these holes are contained entirely within the area of the waffle?

Let us say that a circle intersects a quadrilateral [i]properly[/i] if it intersects each of the quadrilateral’s sides at two distinct interior points. Is it true that for each convex quadrilateral there exists a circle which intersects it properly? [i]Alexandr Perepechko[/i]

$\angle A,\angle B,\angle C$ are angles of a triangle such that $\sin^2A+\sin^2B=\sin^2C$, then $\angle C$ in degrees is equal to $\textbf{(A)}~30$ $\textbf{(B)}~90$ $\textbf{(C)}~45$ $\textbf{(D)}~\text{none of the above}$

The figure below shows a $9\times7$ arrangement of $2\times2$ squares. Alternate squares of the grid are split into two triangles with one of the triangles shaded. Find the area of the shaded region. [asy] size(5cm); defaultpen(linewidth(.6)); fill((0,1)--(1,1)--(1,0)--cycle^^(0,3)--(1,3)--(1,2)--cycle^^(1,2)--(2,2)--(2,1)--cycle^^(2,1)--(3,1)--(3,0)--cycle,rgb(.76,.76,.76)); fill((0,5)--(1,5)--(1,4)--cycle^^(1,4)--(2,4)--(2,3)--cycle^^(2,3)--(3,3)--(3,2)--cycle^^(3,2)--(4,2)--(4,1)--cycle^^(4,1)--(5,1)--(5,0)--cycle,rgb(.76,.76,.76)); fill((0,7)--(1,7)--(1,6)--cycle^^(1,6)--(2,6)--(2,5)--cycle^^(2,5)--(3,5)--(3,4)--cycle^^(3,4)--(4,4)--(4,3)--cycle^^(4,3)--(5,3)--(5,2)--cycle^^(5,2)--(6,2)--(6,1)--cycle^^(6,1)--(7,1)--(7,0)--cycle,rgb(.76,.76,.76)); fill((2,7)--(3,7)--(3,6)--cycle^^(3,6)--(4,6)--(4,5)--cycle^^(4,5)--(5,5)--(5,4)--cycle^^(5,4)--(6,4)--(6,3)--cycle^^(6,3)--(7,3)--(7,2)--cycle^^(7,2)--(8,2)--(8,1)--cycle^^(8,1)--(9,1)--(9,0)--cycle,rgb(.76,.76,.76)); fill((4,7)--(5,7)--(5,6)--cycle^^(5,6)--(6,6)--(6,5)--cycle^^(6,5)--(7,5)--(7,4)--cycle^^(7,4)--(8,4)--(8,3)--cycle^^(8,3)--(9,3)--(9,2)--cycle,rgb(.76,.76,.76)); fill((6,7)--(7,7)--(7,6)--cycle^^(7,6)--(8,6)--(8,5)--cycle^^(8,5)--(9,5)--(9,4)--cycle,rgb(.76,.76,.76)); fill((8,7)--(9,7)--(9,6)--cycle,rgb(.76,.76,.76)); draw((0,0)--(0,7)^^(1,0)--(1,7)^^(2,0)--(2,7)^^(3,0)--(3,7)^^(4,0)--(4,7)^^(5,0)--(5,7)^^(6,0)--(6,7)^^(7,0)--(7,7)^^(8,0)--(8,7)^^(9,0)--(9,7)); draw((0,0)--(9,0)^^(0,1)--(9,1)^^(0,2)--(9,2)^^(0,3)--(9,3)^^(0,4)--(9,4)^^(0,5)--(9,5)^^(0,6)--(9,6)^^(0,7)--(9,7)); draw((0,1)--(1,0)^^(0,3)--(3,0)^^(0,5)--(5,0)^^(0,7)--(7,0)^^(2,7)--(9,0)^^(4,7)--(9,2)^^(6,7)--(9,4)^^(8,7)--(9,6)); [/asy]

Two circles $C_{1}$ and $C_{2}$ are (externally or internally) tangent at a point $P$. The straight line $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is an interior or exterior bisector of the angle $\angle BPC$.

Let the points $O$ and $H$ be the circumcenter and orthocenter of an acute angled triangle $ABC.$ Let $D$ be the midpoint of $BC.$ Let $E$ be the point on the angle bisector of $\angle BAC$ such that $AE\perp HE.$ Let $F$ be the point such that $AEHF$ is a rectangle. Prove that $D,E,F$ are collinear.

On the side $BC$ of the triangle $ABC$ there are two points $D$ and $E$ such that $BD < BE$. Denote by $p_1$ and $p_2$ the perimeters of triangles $ABC$ and $ADE$ respectively. Prove that \[p_1 > p_2 + 2\cdot \min\{BD, EC\}.\]

The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge expressed as a number of feet is $\textbf{(A) }6\qquad\textbf{(B) }864\qquad\textbf{(C) }1728\qquad\textbf{(D) }6\times 1728\qquad \textbf{(E) }2304$

Assume that $ x$ is a positive real number. Which is equivalent to $ \sqrt[3]{x\sqrt{x}}$? $ \textbf{(A)}\ x^{1/6} \qquad \textbf{(B)}\ x^{1/4} \qquad \textbf{(C)}\ x^{3/8} \qquad \textbf{(D)}\ x^{1/2} \qquad \textbf{(E)}\ x$

$60$ points lie on the plane, such that no three points are collinear. Prove that one can divide the points into $20$ groups, with $3$ points in each group, such that the triangles ( $20$ in total) consist of three points in a group have a non-empty intersection.