Let $\overline{AD}$ be a diameter of a circle. Let point $B$ be on the circle, point $C$ on $\overline{AD}$ such that $A, B, C$ form a right triangle at $C$. The value of the hypotenuse of the triangle is $4$ times the square root of its area. If $\overline{BC}$ has length $30$, what is the length of the radius of the circle?

We say that a pile is a set of four or more nuts. Two persons play the following game. They start with one pile of $n \geq 4$ nuts. During a move a player takes one of the piles that they have and split it into two nonempty sets (these sets are not necessarily piles, they can contain arbitrary number of nuts). If the player cannot move, he loses. For which values of $n$ does the first player have a winning strategy?

Let $\triangle ABC$ be a triangle with $AB =AC$ and $\angle BAC = 20^{\circ}$. Let $D$ be point on the side $AB$ such that $\angle BCD = 70^{\circ}$. Let $E$ be point on the side $AC$ such that $\angle CBE = 60^{\circ}$. Determine the value of angle $\angle CDE$.

Given two circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively intesrecting at two points $A$ and $B$. Let $X$ and $Y$ be points on $\omega_1$. The lines $XA$ and $YA$ intersect $\omega_2$ again in $Z$ and $W$, respectively, such that $A$ is between $X,Z$ and $A$ is between $Y,W$. Let $M$ be the midpoint of $O_1O_2$, S be the midpoint of $XA$ and $T$ be the midpoint of $WA$. Prove that $MS = MT$ if, and only if, the points $X, Y, Z$ and $W$ are concyclic.

Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that \[ \angle ECB+180^{\circ }=2\angle EBC. \]

Distinct points $A$, $B$ and $C$ lie on a line in this order. Point $D$ lies on the perpendicular bisector of the segment $BC$. Denote by $M$ the midpoint of the segment $BC$. Let $r$ be the radius of the incircle of the triangle $ABD$ and let $R$ be the radius of the circle with center lying outside the triangle $ACD$, tangent to $CD$, $AC$ and $AD$. Prove that $DM=r+R$.

Billy is standing at $(1,0)$ in the coordinate plane as he watches his Aunt Sydney go for her morning jog starting at the origin. If Aunt Sydney runs into the First Quadrant at a constant speed of $1$ meter per second along the graph of $x=\frac25y^2$, find the rate, in radians per second, at which Billy’s head is turning clockwise when Aunt Sydney passes through $x=1$.

A semicircle of diameter $ 1$ sits at the top of a semicircle of diameter $ 2$, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune. Determine the area of this lune. [asy]unitsize(2.5cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); filldraw(Circle((0,.866),.5),grey,black); label("1",(0,.866),S); filldraw(Circle((0,0),1),white,black); draw((-.5,.866)--(.5,.866),linetype("4 4")); clip((-1,0)--(1,0)--(1,2)--(-1,2)--cycle); draw((-1,0)--(1,0)); label("2",(0,0),S);[/asy]$ \textbf{(A)}\ \frac {1}{6}\pi \minus{} \frac {\sqrt {3}}{4} \qquad \textbf{(B)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{12}\pi \qquad \textbf{(C)}\ \frac {\sqrt {3}}{4} \minus{} \frac {1}{24}\pi\qquad\textbf{(D)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{24}\pi$ $ \textbf{(E)}\ \frac {\sqrt {3}}{4} \plus{} \frac {1}{12}\pi$

Let $ABC$ be a triangle satisfying $BC < CA$. Let $P$ be an arbitrary point on the side $AB$ (different from $A$ and $B$), and let the line $CP$ meet the circumcircle of triangle $ABC$ at a point $S$ (apart from the point $C$). Let the circumcircle of triangle $ASP$ meet the line $CA$ at a point $R$ (apart from $A$), and let the circumcircle of triangle $BPS$ meet the line $CB$ at a point $Q$ (apart from $B$). Prove that the excircle of triangle $APR$ at the side $AP$ is identical with the excircle of triangle $PQB$ at the side $PQ$ if and only if the point $S$ is the midpoint of the arc $AB$ on the circumcircle of triangle $ABC$.

In a right circular cone with the radius of the base $R$ and the height $h$ are $n$ spheres of the same radius $r$ ($n \ge 3$). Each ball touches the base of the cone, its side surface and other two balls. Determine $r$.

Let $ABC$ be a triangle and let $AD,BE,CF$ be its altitudes . $FA_{1},DB_{1},EC_{1}$ are perpendicular segments to $BC,AC,AB$ respectively. Prove that : $ABC$~$A_{1}B_{1}C_{1}$

From point $P$ outside a circle, with a circumference of $10$ units, a tangent is drawn. Also from $P$ a secant is drawn dividing the circle into unequal arcs with lengths $m$ and $n$. It is found that $t_1$, the length of the tangent, is the mean proportional between $m$ and $n$. If $m$ and $t$ are integers, then $t$ may have the following number of values: $ \textbf{(A)}\ \text{zero} \qquad\textbf{(B)}\ \text{one} \qquad\textbf{(C)}\ \text{two} \qquad\textbf{(D)}\ \text{three} \qquad\textbf{(E)}\ \text{infinitely many} $

Given a square $ABCD$. Find the locus of points $M$ such that $\angle AMB = \angle CMD$.

Three families of parallel lines are drawn,$10$ lines each, are drawn. What is the greatest number of triangles they can cut from plane?

Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I , K , H$ be feet of perpendiculars from $B , O , C$ to $AD$, respectively. Prove that $AD \times BI \times CH \le AC \times BD \times OK$.

Let $ABCD$ be a square, and let $E$ and $F$ be points on sides $\overline{AB}$ and $\overline{CD}$, respectively, such that $AE:EB=AF:FD=2:1$. Let $G$ be the intersection of $\overline{AF}$ and $\overline{DE}$, and let $H$ be the intersection of $\overline{BF}$ and $\overline{CE}$. Find the ratio of the area of quadrilateral $EGFH$ to the area of square $ABCD$. [asy] size(5cm,0); draw((0,0)--(3,0)); draw((3,0)--(3,3)); draw((3,3)--(0,3)); draw((0,3)--(0,0)); draw((0,0)--(2,3)); draw((1,0)--(3,3)); draw((0,3)--(1,0)); draw((2,3)--(3,0)); label("$A$",(0,3),NW); label("$B$",(3,3),NE); label("$C$",(3,0),SE); label("$D$",(0,0),SW); label("$E$",(2,3),N); label("$F$",(1,0),S); label("$G$",(0.66666667,1),E); label("$H$",(2.33333333,2),W); [/asy]

[u]Round 1[/u] [b]p1.[/b] Compute $(1 - 2(3 - 4(5 - 6)))(7 - (8 - 9))$. [b]p2.[/b] How many numbers are in the set $\{20, 21, 22, ..., 88, 89\}$? [b]p3.[/b] Three times the complement of the supplement of an angle is equal to $60$ degrees less than the angle itself. Find the measure of the angle in degrees. [u]Round 2[/u] [b]p4.[/b] A positive number is decreased by $10\%$, then decreased by $20\%$, and finally increased by $30\%$. By what percent has this number changed from the original? Give a positive answer for a percent increase and a negative answer for a percent decrease. [b]p5.[/b] What is the area of the triangle with vertices at $(2, 3)$, $(8, 11)$, and $(13, 3)$? [b]p6.[/b] There are three bins, each containing red, green, and/or blue pens. The first bin has $0$ red, $0$ green, and $3$ blue pens, the second bin has $0$ red, $2$ green, and $4$ blue pens, and the final bin has $1$ red, $5$ green, and $6$ blue pens. What is the probability that if one pen is drawn from each bin at random, one of each color pen will be drawn? [u]Round 3[/u] [b]p7.[/b] If a and b are positive integers and $a^2 - b^2 = 23$, what is the value of $a$? [b]p8.[/b] Find the prime factorization of the greatest common divisor of $2^3\cdot 3^2\cdot 5^5\cdot 7^4$ and $2^4\cdot 3^1\cdot 5^2\cdot 7^6$. [b]p9.[/b] Given that $$a + 2b + 3c = 5$$ $$2a + 3b + c = -2$$ $$3a + b + 2c = 3,$$ find $3a + 3b + 3c$. [u]Round 4[/u] [b]p10.[/b] How many positive integer divisors does $11^{20}$ have? [b]p11.[/b] Let $\alpha$ be the answer to problem $10$. Find the real value of $x$ such that $2^{x-5} = 64^{x/\alpha}$. [b]p12.[/b] Let $\beta$ be the answer to problem $11$. Triangle $LMT$ has a right angle at $M$, $LM = \beta$, and $LT = 4\beta - 3$. If $Z$ is the midpoint of $LT$, what is the length$ MZ$? PS. You should use hide for answers. Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134133p28400917]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.

Let $ABC$ be convex quadrilateral and $X$ lying inside it such that $XA \cdot XC^2 = XB \cdot XD^2$ and $\angle AXD + \angle BXC = \angle CXD$. Prove that $\angle XAD + \angle XCD = \angle XBC + \angle XDC$.

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at $P \neq A$. Let $PE$ intersect $\Gamma$ again at $Q \neq P$. Prove that $BQ$ bisects segment $\overline{EF}$. [i]Proposed by Luke Robitaille[/i]

Given a triangle $ABC$, show how to construct the point $P$ such that $\angle PAB= \angle PBC= \angle PCA$. Express this angle in terms of $\angle A,\angle B,\angle C$ using trigonometric functions.

$ABC$ is a triangle with $\angle A = 90^o$. For a point $D$ on the side $BC$, the feet of the perpendiculars to $AB$ and $AC$ are $E$ and$ F$. For which point $D$ is $ EF$ a minimum?

It is known that the graph of the function $y =\frac{a-6x}{2+x}$ is centrally summetric to the graph of the function $y = \frac{1}{x}$ with respect to some point. Find the value of the parameter $a$ and the coordinates of the center of symmetry.

Prove the inequality for all positive integer $n$ : \[ \left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}} \]

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\] Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.