A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.

A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising?

Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3 \text{cm}$ from $P$? $\text{(A)} \ 1 \qquad \text{(B)} \ 2 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 8$

We say that a finite set of points is [i]well scattered[/i] on the surface of a sphere if every open hemisphere (half the surface of the sphere without its boundary) contains at least one of the points. The set $\{ (1,0,0), (0,1,0), (0,0,1) \}$ is not well scattered on the unit sphere (the sphere of radius $1$ centered at the origin), but if you add the correct point $P$ it becomes well scattered. Find, with proof, all possible points $P$ that would make the set well scattered.

Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$. If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$, then what is the product of the lengths of the diagonals of $ABCD$ ?

Let $a,b,c$ be the side lengths of any triangle $\triangle ABC$ opposite to $A,B$ and $C,$ respectively. Let $x,y,z$ be the length of medians from $A,B$ and $C,$ respectively. If $T$ is the area of $\triangle ABC$, prove that $\frac{a^2}{x}+\frac{b^2}{y}+\frac{c^2}{z}\ge\sqrt{\sqrt{3}T}$

Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$.Let $\Gamma_1$be the circle centered at $A$ with radius $AB$,$\Gamma_2$ be the circle centered at $I$ with radius $BI$.A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$,$\Gamma_2$ again at $P,Q$ (different from $B$) respectively.Let $R$ be the intersection of $PI$ and $BQ$.Show that $BR \perp CR$.

When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$ Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$ 30 points

Two circles touch internally at point $P$. A line tangent to one of the circles at point $A$ intersects the other circle at points $B$ and $C$. Prove that the line $ PA $ is the bisector of the angle $ BPC $.

A right-triangulated prism made of benzene sits on a table. The hypotenuse makes an angle of $30^\circ$ with the horizontal table. An incoming ray of light hits the hypotenuse horizontally, and leaves the prism from the vertical leg at an acute angle of $ \gamma $ with respect to the vertical leg. Find $\gamma$, in degrees, to the nearest integer. The index of refraction of benzene is $1.50$. [i](Proposed by Ahaan Rungta)[/i]

In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$. [img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]

Determine the area of the region defined by [i]x[/i]²+[i]y[/i]²≤[i]π[/i]² and [i]y[/i] ≥ sin [i]x[/i].

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

In every $1\times1$ cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be $0$?

Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

Prove that for every integer $n\ge 3$ there exists $N(n)$ with the following property: whenever $P$ is a set of at least $N(n)$ points of the plane such that any three points of $P$ determines a nondegenerate triangle containing at most one point of $P$ in its interior, then $P$ contains the vertices of a convex $n$-gon whose interior does not contain any point of $P$.

Consider equilateral triangle $ABC$ and suppose that there exist three distinct points $X, Y,Z$ lie inside triangle $ABC$ such that i) $AX = BY = CZ$ ii) The triplets of points $(A,X,Z), (B,Y,X), (C,Z,Y )$ are collinear in that order. Prove that $XY Z$ is an equilateral triangle.

Point $P$ is in the interior of $\triangle ABC$. The side lengths of $ABC$ are $AB = 7$, $BC = 8$, $CA = 9$. The three foots of perpendicular lines from $P$ to sides $BC$, $CA$, $AB$ are $D$, $E$, $F$ respectively. Suppose the minimal value of $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$ can be written as $\frac{a}{b}\sqrt{c}$, where $\gcd(a,b) = 1$ and $c$ is square free, calculate $abc$. [asy] size(120); pathpen = linewidth(0.7); pointfontpen = fontsize(10); // pointpen = black; pair B=(0,0), C=(8,0), A=IP(CR(B,7),CR(C,9)), P = (2,1.6), D=foot(P,B,C), E=foot(P,A,C), F=foot(P,A,B); D(A--B--C--cycle); D(P--D); D(P--E); D(P--F); D(MP("A",A,N)); D(MP("B",B)); D(MP("C",C)); D(MP("D",D)); D(MP("E",E,NE)); D(MP("F",F,NW)); D(MP("P",P,SE)); [/asy]

Line passes the focal point $F$ of parabola $y^2=8(x+2)$ with bank angle of $60^{\circ}$ intersects the parabola at $A,B$. Perpendicular bisector of $AB$ intersects $x$-axis at $P$, then the length of $PF$ is $\text{(A)}\frac{16}{3}\qquad\text{(B)}\frac{8}{3}\qquad\text{(C)}\frac{16}{3}\sqrt3\qquad\text{(D)}8\sqrt3$

Three circles $\omega_1$, $\omega_2$ and $\omega_3$ pass through one common point, say $P$. The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ for the second time at points $P_{1,2}$ and $P_{1,3}$, respectively. Points $P_{2,1}$, $P_{2,3}$, $P_{3,1}$ and $P_{3,2}$ are similarly defined. Prove that the perpendicular bisector of segments $P_{1,2}P_{1,3}$, $P_{2,1}P_{2,3}$ and $P_{3,1}P_{3,2}$ are concurrent. [i]Proposed by Mahdi Etesamifard[/i]

On the chord $AB$ of the circle $\omega$ points $C$ and $D$ are chosen such that $AC=BD$ and point $C$ lies between $A$ and $D$. On $\omega$ point $E$ and $F$ are marked, they lie on different sides with respect to line $AB$ and lines $EC$ and $FD$ are perpendicular to $AB$. The angle bisector of $AEB$ intersects line $DF$ at $R$ Prove that the circle with center $F$ and radius $FR$ is tangent to $\omega$ [i]V. Kamenetskii, D. Bariev[/i]

Three cubes of volume $1, 8$ and $27$ are glued together at their faces. The smallest possible surface area of the resulting configuration is $ \textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74 $

A regular pentagon is inscribed in a circle of radius $r$. $P$ is any point inside the pentagon. Perpendiculars are dropped from $P$ to the sides, or the sides produced, of the pentagon. a) Prove that the sum of the lengths of these perpendiculars is constant. b) Express this constant in terms of the radius $r$.

Let $ABCD$ be a rectangle with $AB = a$ and $BC = b$. Suppose $r_1$ is the radius of the circle passing through $A$ and $B$ touching $CD$; and similarly $r_2$ is the radius of the circle passing through $B$ and $C$ and touching $AD$. Show that \[ r_1 + r_2 \geq \frac{5}{8} ( a + b) . \]

Let $\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be distinct circles such that $\Gamma_1,\Gamma_3$ are externally tangent at $P$, and $\Gamma_2,\Gamma_4$ are externally tangent at the same point $P$. Suppose that $\Gamma_1$ and $\Gamma_2$; $\Gamma_2$ and $\Gamma_3$; $\Gamma_3$ and $\Gamma_4$; $\Gamma_4$ and $\Gamma_1$ meet at $A,B,C,D,$ respectively, and that all of these points are different from $P$. Prove that $$\frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}$$