A concrete sewer pipe fitting is shaped like a cylinder with diameter $48$ with a cone on top. A cylindrical hole of diameter $30$ is bored all the way through the center of the fitting as shown. The cylindrical portion has height $60$ while the conical top portion has height $20$. Find $N$ such that the volume of the concrete is $N \pi$. [asy] import three; size(250); defaultpen(linewidth(0.7)+fontsize(10)); pen dashes = linewidth(0.7) + linetype("2 2"); currentprojection = orthographic(0,-15,5); draw(circle((0,0,0), 15),dashes); draw(circle((0,0,80), 15)); draw(scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw(shift((0,0,60))*scale3(24)*((-1,0,0)..(0,-1,0)..(1,0,0))); draw((-24,0,0)--(-24,0,60)--(-15,0,80)); draw((24,0,0)--(24,0,60)--(15,0,80)); draw((-15,0,0)--(-15,0,80),dashes); draw((15,0,0)--(15,0,80),dashes); draw("48", (-24,0,-20)--(24,0,-20)); draw((-15,0,-20)--(-15,0,-17)); draw((15,0,-20)--(15,0,-17)); label("30", (0,0,-15)); draw("60", (50,0,0)--(50,0,60)); draw("20", (50,0,60)--(50,0,80)); draw((50,0,60)--(47,0,60));[/asy]

In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.

In a cube-shaped box with an edge equal to $5$, there are two balls. The radius of one of the balls is $2$. Find the radius of the other ball if one of the balls touches the base and two side faces of the cube, and the other ball touches the first ball, base and two other side faces of the cube.

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. It is given that there exist points $X$ and $Y$ on the circumference of $\omega$ such that $\angle BXC=\angle BYC=90^\circ$. Suppose further that $X$, $I$, and $Y$ are collinear. If $AB=80$ and $AC=97$, compute the length of $BC$.

A circle $\omega$ and a point $T$ outside the circle is given. Let a tangent from $T$ to $\omega$ touch $\omega$ at $A$, and take points $B,C$ lying on $\omega$ such that $T,B,C$ are colinear. The bisector of $\angle ATC$ intersects $AB$ and $AC$ at $P$ and $Q$,respectively. Prove that $PA=\sqrt{PB\cdot QC}$

Calculate the volume $ V $ of tetrahedron $ ABCD $ given the length $ d $ of edge $ AB $ and the area $ S $ of the projection of the tetrahedron on the plane perpendicular to the line $ AB $.

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent. [i]Proposed by Michael Kural[/i]

The Fibonacci sequence is given by equalities $$F_1=F_2=1, F_{k+2}=F_k+F_{k+1}, k\in N$$. a) Prove that for every $m \ge 0$, the area of ​​the triangle $A_1A_2A_3$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$ is equal to $0.5$. b) Prove that for every $m \ge 0$ the quadrangle $A_1A_2A_4$ with vertices $A_1(F_{m+1},F_{m+2})$, $A_2 (F_{m+3},F_{m+4})$, $A_3 (F_{m+5},F_{m+6})$, $A_4 (F_{m+7},F_{m+8})$ is a trapezoid, whose area is equal to $2.5$. c) Prove that the area of ​​the polygon $A_1A_2...A_n$ , $n \ge3$ with vertices does not depend on the choice of numbers $m \ge 0$, and find this area.

Two circles intersect at points $P$ and $Q$. Point $A$ lies on the first circle, but outside the second. Lines $AP$ and $AQ$ intersect the second circle at points $B$ and $C$, respectively. Indicate the position of point $A$ at which triangle $ABC$ has the largest area. (D. Prokopenko)

In a convex pentagon $ ABCDE$, let $ \angle EAB \equal{} \angle ABC \equal{} 120^{\circ}$, $ \angle ADB \equal{} 30^{\circ}$ and $ \angle CDE \equal{} 60^{\circ}$. Let $ AB \equal{} 1$. Prove that the area of the pentagon is less than $ \sqrt {3}$.

Suppose that points $A$ and $B$ lie on circle $\Omega$, and suppose that points $C$ and $D$ are the trisection points of major arc $AB$, with $C$ closer to $B$ than $A$. Let $E$ be the intersection of line $AB$ with the line tangent to $\Omega$ at $C$. Suppose that $DC = 8$ and $DB = 11$. If $DE = a\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$.

Let $C_1, C_2$ and $C_3$ be three circles, of radii $2, 4$ and $6$ respectively. It is known that each of them are tangent exteriorly with the other two circles. Let $\Omega_1$ and $\Omega_2$ be two more circles, each of them tangent to all of the $3$ circles above, of radius $\omega_1$ and $\omega_2$ respectively. Prove that $\omega_1 + \omega_2 = 2\omega_1\omega_2$.

In the complex plane consider the unit circle with the origin as its center. Furthermore, consider inscribed regular 17-gon with one of its vertices being $1+0i.$ How many of its vertices lie in the (open) unit disc centered in $\sqrt{3/2}(1+i)$?

$ABCD$ is a quadrilateral, $E, F, G, H$ are the midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Find the point $P$ such that area $(PHAE) =$ area $(PEBF) =$ area $(PFCG) =$ area $(PGDH).$

Find all real numbers with the following property: the difference of its cube and its square is equal to the square of the difference of its square and the number itself.

Let $DFGE$ be a cyclic quadrilateral. Line $DF$ intersects $EG$ at $C,$ and line $FE$ intersects $DG$ at $H.$ $J$ is the midpoint of $FG.$ The line $\ell$ is the reflection of the line $DE$ in $CH,$ and it intersects line $GF$ at $I.$ Prove that $C,J,H,I$ are concyclic.

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.

Let $ABC$ be an acute triangle and let $\ell$ be a line in the plane of triangle $ABC.$ We've drawn the reflection of the line $\ell$ over the sides $AB, BC$ and $AC$ and they intersect in the points $A', B'$ and $C'.$ Prove that the incenter of the triangle $A'B'C'$ lies on the circumcircle of the triangle $ABC.$

There is a square $ ABCD. $ $ P $ is on $\bar{AB}$ , and $Q$ is on $ \bar{AD} $ They follow $ \bar{AP}=\bar{AQ}=\frac{\bar{AB}}{5} $ Let $ H $ be the foot of the perpendicular point from $ A $ to $ \bar{PD} $ If $ |\triangle APH|=20 $, Find the area of $ \triangle HCQ $.

Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.

Prove that an arbitrary triangle can be cut into three polygons, one of which must be an obtuse triangle, so that they can then be folded into a rectangle. (Turning over parts is possible).

Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle

The lengths of the two legs of a right triangle are the two distinct roots of the quadratic $x^2 - 36x + 70$. What is the length of the triangle’s hypotenuse?

Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.

In a circle $S$, a chord $AB$ is drawn and let $M$ be the midpoint of the arc $AB$. Let $P$ be a point in segment $AB$ other than its midpoint. The extension of the segment $MP$ cuts $S$ in $Q$. Let $S_1$ be the circle that is tangent to the AP segments and $MP$, and also is tangent to $S$, and let $S_2$ be the circle that is tangent to the segments $BP$ and $MP$, and also tangent to $S$. The common outer tangent lines to the circles $S_1$ and $S_2$ are cut at $C$. Prove that $\angle MQC = 90^o$.