Inside square $ABCD$ consider a point $M$. Prove that the points of intersection of the medians of triangles $ABM, BCM, CDM$ and $DAM$ form a square. (V Prasolov)

2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.

In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$. [asy] size(250); defaultpen (linewidth (0.7) + fontsize (10)); pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3); draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0)); label("$A$", (.5,.5)); label("$M$", (7/6, 1/6)); label("$N$", (1/3, 4/3));[/asy] [i]Proposed by Aaron Lin[/i]

One right pyramid has a base that is a regular hexagon with side length $1$, and the height of the pyramid is $8$. Two other right pyramids have bases that are regular hexagons with side length $4$, and the heights of those pyramids are both $7$. The three pyramids sit on a plane so that their bases are adjacent to each other and meet at a single common vertex. A sphere with radius $4$ rests above the plane supported by these three pyramids. The distance that the center of the sphere is from the plane can be written as $\frac{p\sqrt{q}}{r}$ , where $p, q$, and $r$ are relatively prime positive integers, and $q$ is not divisible by the square of any prime. Find $p+q+r$.

Fold the next seven corners into a rectangle. [img]https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png[/img]

Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?

Given a trihedral angle Sxyz. A plane (P) not through S cuts Sx,Sy,Sz respectively at A,B,C. On the plane (P), outside triangle ABC, construct triangles DAB,EBC,FCA which are confruent to the triangles SAB,SBC,SCA respectively. Let (T) be the sphere lying inside Sxyz, but not inside the tetrahedron SABC, toucheing the planes containing the faces of SABC. Prove that (T) touches the plane (P) at the circumcenter of triangle DEF.

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

Is it possible to arrange natural numbers 1 to 8 on vertices of a cube such that each number divides sum of the three numbers sharing an edge with it?

In the quadrilateral $ABCD$, $AD$ is parallel to $BC$. $K$ is a point on $AB$. Draw the line through $A$ parallel to $KC$ and the line through $B$ parallel to $KD$. Prove that these two lines intersect at some point on $CD$.

In the $3\times5$ grid shown, fill in each empty box with a two-digit positive integer such that: [list][*]no number appears in more than one box, and [*] for each of the $9$ lines in the grid consisting of three boxes connected by line segments, the box in the middle of the line contains the least common multiple of the numbers in the two boxes on the line.[/list] You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] import graph; size(7cm); real labelscalefactor = 0.5; pen dps = linewidth(0.8) + fontsize(14); defaultpen(dps); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((2,0)--(3,0)--(3,1)--(2,1)--cycle); draw((4,0)--(5,0)--(5,1)--(4,1)--cycle); draw((6,0)--(7,0)--(7,1)--(6,1)--cycle); draw((8,0)--(9,0)--(9,1)--(8,1)--cycle); draw((0,2)--(1,2)--(1,3)--(0,3)--cycle); draw((0,4)--(1,4)--(1,5)--(0,5)--cycle); draw((2,2)--(3,2)--(3,3)--(2,3)--cycle); draw((2,4)--(3,4)--(3,5)--(2,5)--cycle); draw((4,4)--(5,4)--(5,5)--(4,5)--cycle); draw((4,2)--(5,2)--(5,3)--(4,3)--cycle); draw((6,2)--(7,2)--(7,3)--(6,3)--cycle); draw((6,4)--(7,4)--(7,5)--(6,5)--cycle); draw((8,4)--(9,4)--(9,5)--(8,5)--cycle); draw((8,2)--(9,2)--(9,3)--(8,3)--cycle); draw((0.5,1)--(0.5,2)); draw((0.5,3)--(0.5,4)); draw((1,4)--(2,3)); draw((2.5,1)--(2.5,2)); draw((2.5,3)--(2.5,4)); draw((3,4)--(4,3)); draw((3,2)--(4,1)); draw((4.5,1)--(4.5,2)); draw((4.5,3)--(4.5,4)); draw((5,4.5)--(6,4.5)); draw((7,4.5)--(8,4.5)); draw((5,4)--(6,3)); draw((7,2)--(8,1)); draw((5,2)--(6,1)); draw((5,0.5)--(6,0.5)); draw((7,0.5)--(8,0.5)); draw((8.5,1)--(8.5,2)); draw((8.5,3)--(8.5,4)); label("$4$",(4.5, 0.5)); label("$9$",(8.5, 4.5)); [/asy]

Let a triangle $ABC$ be given. Two circles passing through $A$ touch $BC$ at points $B$ and $C$ respectively. Let $D$ be the second common point of these circles ($A$ is closer to $BC$ than $D$). It is known that $BC = 2BD$. Prove that $\angle DAB = 2\angle ADB.$

An acute-angled triangle with one side equal to the altitude from the opposite vertex is cut from paper. Construct a point inside this triangle such that the square of the distance from it to one of the vertices equals the sum of the squares of distances to to the remaining two vertices. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines. Proposed by: M.Evdokimov

Quadrilateral $ABCD$ is inscribed on circle $O$. $AC\cap BD=P$. Circumcenters of $\triangle ABP,\triangle BCP,\triangle CDP,\triangle DAP$ are $O_1,O_2,O_3,O_4$. Prove that $OP,O_1O_3,O_2O_4$ share one point.

For each of $k=1,2,3,4,5$ find necessary and sufficient conditions on $a>0$ such that there exists a tetrahedron with $k$ edges length $a$ and the remainder length $1$.

Let $\vartriangle ABC$ be an acute triangle with angles $\angle BAC = 70^o$, $\angle ABC = 60^o$, let $D, E$ be the feet of perpendiculars from $B, C$ to $AC$, $AB$, respectively, and let $H$ be the orthocenter of $ABC$. Let $F$ be a point on the shorter arc $AB$ of circumcircle of $ABC$ satisfying $\angle F AB = 10^o$ and let $G$ be the foot of perpendicular from $H$ to $AF$. If $I = BF \cap EG$ and $J = CF \cap DG$, compute the angle $\angle GIJ$.

Suppose three circles of radius $5$ intersect at a common point. If the three (other) pairwise intersections between the circles form a triangle of area $ 8$, find the radius of the smallest possible circle containing all three circles.

Let $ABCD$ be a square of center $O$ and let $M$ be the symmetric of the point $B$ with respect to point $A$. Let $E$ be the intersection of $CM$ and $BD$, and let $S$ be the intersection of $MO$ and $AE$. Show that $SO$ is the angle bisector of $\angle ESB$.

Let $ Ax,By$ be two perpendicular semi-straight lines, being not complanar, (non-coplanar rays) such that $ AB$ is the their common perpendicular, and let $ M$ and $ N$ be the two variable points on $ Ax$ and $ Bx,$ respectively, such that $ AM \plus{} BN \equal{} MN.$ [b](a)[/b] Prove that there exist infinitely many lines being co-planar with each of the straight lines $ MN.$ [b](b)[/b] Prove that there exist infinitely many rotations around a fixed axis $ \delta$ mapping the line $ Ax$ onto a line coplanar with each of the lines $ MN.$

Let $A$, $B$ and $C$ be points lying on a circle $\Gamma$ with centre $O$. Assume that $\angle ABC > 90$. Let $D$ be the point of intersection of the line $AB$ with the line perpendicular to $AC$ at $C$. Let $l$ be the line through $D$ which is perpendicular to $AO$. Let $E$ be the point of intersection of $l$ with the line $AC$, and let $F$ be the point of intersection of $\Gamma$ with $l$ that lies between $D$ and $E$. Prove that the circumcircles of triangles $BFE$ and $CFD$ are tangent at $F$.

Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling? $\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$

Two circles $\Gamma_1, \Gamma_2$ with radii $r_i, r_2$, respectively, touch internally at the point $P$. A tangent parallel to the diameter through $P$ touches $ \Gamma_1$ at $R$ and intersects $\Gamma_2$ at $M$ and $N$. Prove that $PR$ bisects $\angle MPN$.

Let $\omega_1$ and $\omega_2$ be two circles with centers $C_1$ and $C_2$, respectively, which intersect at two points $P$ and $Q$. Suppose that the circumcircle of triangle $PC_1C_2$ intersects $\omega_1$ at $A \neq P$ and $\omega_2$ at $B \neq P$. Suppose further that $Q$ is inside the triangle $PAB$. Show that $Q$ is the incenter of triangle $PAB$.

The area of the largest triangle that can be inscribed in a semi-circle whose radius is $r$ is: $\textbf{(A)}\ r^2 \qquad \textbf{(B)}\ r^3 \qquad \textbf{(C)}\ 2r^2 \qquad \textbf{(D)}\ 2r^3 \qquad \textbf{(E)}\ \dfrac{1}{2}r^2$

Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.