A bicentric quadrilateral is one that is both inscribable in and circumscribable about a circle, i.e. both the incircle and circumcircle exists. Show that for such a quadrilateral, the centers of the two associated circles are collinear with the point of intersection of the diagonals.

Let $ABC$ be a triangle where$\angle$[b]B=55[/b] and $\angle$ [b]C = 65[/b]. [b]D[/b] is the mid-point of [b]BC[/b]. Circumcircle of [b]ACD[/b] and[b] ABD[/b] cuts [b]AB[/b] and[b] AC[/b] at point [b]F[/b] and [b]E[/b] respectively. Center of circumcircle of [b]AEF[/b] is[b] O[/b]. $\angle$[b]FDO[/b] = ?

Given is an acute triangle $ABC$ with $AB<AC$ with altitudes $BD$ and $CE$. Let the tangents to the circumcircle at $B$ and $C$ meet at $Y$. Let $\omega_1$ be the circle through $A$ tangent to $DE$ at $E$; define $\omega_2$ similarly, and let their intersection point be $X$. Prove that $A, X, Y$ are colinear. $\textit{Proposed by Nikola Velov}$

A cube $A_1A_2A_3A_4A_5A_6A_7A_8$ is given in space. We will mark its center with the letter $S$ (intersection of solid diagonals). Find all natural numbers $k$ for which there exists a plane not containing the point $S$ and intersecting just $k$ of the rays $SA_1, SA_2, .. SA_8$

An acute angled triangle \(ABC\) has circumcenter \(O\). The lines \(AO\) and \(BC\) intersect at \(D\), while \(BO\) and \(AC\) intersect at \(E\) and \(CO\) and \(AB\) intersect at \(F\). Show that if the triangles \(ABC\) and \(DEF\) are similar(with vertices in that order), than \(ABC\) is equilateral.

In triangle $ABC$ the median $BM$ is equal to half of the side $BC$. Show that $\angle ABM = \angle BCA + \angle BAC$. [i](Proposed by Anton Trygub)[/i]

Find all positive integers $ n$ such that acute-angled $ \triangle ABC$ with $ \angle BAC<\frac{\pi}{4}$ could be divided into $ n$ quadrilateral. Every quadrilateral is inscribed in circle and radiuses of circles are in geometric progression. [hide] be carefull ! :lol: [/hide]

Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le3$, let $D_i $, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$. Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.

[u]Round 9[/u] [b]p25.[/b] What is the largest integer that cannot be expressed as the sum of nonnegative multiples of $7$, $11$, and $13$? [b]p26.[/b] Evaluate $12{3 \choose3}+ 11{4\choose 3}+ 10{5\choose 3}+ ...+ 2{13\choose 3}+{14 \choose 3}$. [b]p27.[/b] Worker Bob drives to work at $30$ mph half the time and $60$ mph half the time. He returns home along the same route at $30$ mph half the distance and $60$ mph half the distance. What is his average speed along the entire trip, in mph? [u]Round 10[/u] [b]p28.[/b] In quadrilateral $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$ with $BP = 4$, $P D = 6$, $AP = 8$, $P C = 3$, and $AB = 6$. What is the length of $AD$? [b]p29.[/b] Find all positive integers $x$ such that$ x^2 + 17x + 17$ is a square number. [b]p30.[/b] Zach has ten weighted coins that turn up heads with probabilities $\frac{2}{11^2}$ ,$\frac{2}{10^2}$ ,$\frac{2}{9^2}$ $, . . $.,$\frac{2}{2^2}$ . If he flips all ten coins simultaneously, then what is the probability that he will get an even number of heads? [u]Round 11[/u] [b]p31.[/b] Given a sequence $a_1, a_2, . . .$ such that $a_1 = 3$ and $a_{n+1} = a^2_n - 2a_n + 2$ for $n \ge 1$, find the remainder when the product a1a2 · · · a2012 is divided by 100. [b]p32.[/b] Let $ABC$ be an equilateral triangle and let $O$ be its circumcircle. Let $D$ be a point on $\overline{BC}$, and extend $\overline{AD}$ to intersect $O$ at $P$. If $BP = 5$ and $CP = 4$, then what is the value of $DP$? [b]p33.[/b] Surya and Hao take turns playing a game on a calendar. They start with the date January $1$ and they can either increase the month to a later month or increase the day to a later day in that month but not both. The first person to adjust the date to December $31$ is the winner. If Hao goes first, then what is the first date that he must choose to ensure that he does not lose? [u]Round 12[/u] [b]p34.[/b] On May $5$, $1868$, exactly $144$ years before today, Memorial Day in the United States was officially proclaimed. The first Memorial Day took place that year on May $30$ at Waterloo, New York. On May $5$, $2012$, at $12:00$ PM, how many results did the search “memorial day” on Google return? The search phrase is in quotes, so Google will only return sites that have the words memorial and day next to each other in that order. Let $N = max-\{0, \rfloor 15.5 \times \frac{ Your\,\,\, Answer}{Actual \,\,\,Answer} \rfloor \}$. You will earn the number of points equal to $min\{N, max\{0, 30 - N\}\}$. [b]p35.[/b] Estimate the side length of a regular pentagon whose area is $2012$. You will earn the number of points equal to $max\{0, 15 - \lfloor 5 \times |Your \,\,\,Answer - Actual \,\,\,Answer| \rfloor \}$. [b]p36.[/b] Write down one integer between $1$ and $15$, inclusive. (If you do not, then you will receive $0$ points.) Let the number that you submit be $x$. Let $\overline{x}$ be the arithmetic mean of all of the valid numbers submitted by all of the teams. If $x > \overline{x}$, then you will receive $0$ points; otherwise, you will receive $x$ points. PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 6-8 [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Inside a square of side $16$ are placed $1000$ points. Show that it is possible to put a equilateral triangle of side $2\sqrt3$ in the plane so that it covers at least $16$ of these points.

On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.

Let $I$ be the center of a circle inscribed in triangle $ABC$, in which $\angle BAC = 60 ^o$ and $AB \ne AC$. The points $D$ and $E$ were marked on the rays $BA$ and $CA$ so that $BD = CE = BC$. Prove that the line $DE$ passes through the point $I$.

Find all numbers $n \geq 4$ such that there exists a convex polyhedron with exactly $n$ faces, whose all faces are right-angled triangles. (Note that the angle between any pair of adjacent faces in a convex polyhedron is less than $180^\circ$.) [i]Proposed by Hesam Rajabzadeh[/i]

We will say that an octagon is integral if its is equiangular, its vertices are lattice points (i.e., points with integer coordinates), and its area is an integer. For example, the figure on the right shows an integral octagon of area $21$. Determine, with proof, the smallest positive integer $K$ so that for every positive integer $k\geq K$, there is an integral octagon of area $k$. [asy] size(200); defaultpen(linewidth(0.8)); draw((-1/2,0)--(17/2,0)^^(0,-1/2)--(0,15/2)); for(int i=1;i<=6;++i){ draw((0,i)--(17/2,i),linetype("4 4")); } for(int i=1;i<=8;++i){ draw((i,0)--(i,15/2),linetype("4 4")); } draw((2,1)--(1,2)--(1,3)--(4,6)--(5,6)--(7,4)--(7,3)--(5,1)--cycle,linewidth(1)); label("$1$",(1,0),S); label("$2$",(2,0),S); label("$x$",(17/2,0),SE); label("$1$",(0,1),W); label("$2$",(0,2),W); label("$y$",(0,15/2),NW); [/asy]

Suppose we have an (infinite) cone $ \mathcal C$ with apex $ A$ and a plane $ \pi$. The intersection of $ \pi$ and $ \mathcal C$ is an ellipse $ \mathcal E$ with major axis $ BC$, such that $ B$ is closer to $ A$ than $ C$, and $ BC \equal{} 4$, $ AC \equal{} 5$, $ AB \equal{} 3$. Suppose we inscribe a sphere in each part of $ \mathcal C$ cut up by $ \mathcal E$ with both spheres tangent to $ \mathcal E$. What is the ratio of the radii of the spheres (smaller to larger)?

Let $\Gamma$ be the circumscribed circle of a triangle $ABC$ and let $D$ be a point at line segment $BC$. The circle passing through $B$ and $D$ tangent to $\Gamma$ and the circle passing through $C $and $D$ tangent to $\Gamma$ intersect at a point $E \ne D$. The line $DE$ intersects $\Gamma$ at two points $X$ and $Y$ . Prove that $|EX| = |EY|$.

Three circles are pairwise tangent, with none of them lying inside another. The centres of the circles are the corners of a triangle with circumference $1$. What is the smallest possible value for the sum of the areas of the circles?

Calculate the volume of the tetrahedron $ ABCD $ given the edges $ AB = b $, $ AC = c $, $ AD = d $ and the angles $ \measuredangle CAD = \beta $, $ \measuredangle DAB = \gamma $ and $ \measuredangle BAC = \delta$.

Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square. [b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.

Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?

A circle is inscribed in an angle with vertex $O$, touching its sides at points $M$ and $N$. On an arc $MN$ nearest to point $O$, an arbitrary point $P$ is selected. At point $P$, a tangent is drawn to the circle $P$, intersecting the sides of the angle at points $A$ and $B$. Prove that that the length of the segment $AB$ is the smallest when $P$ is its midpoint.

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$. Rays $\displaystyle \overrightarrow{OB}$ and $\displaystyle \overrightarrow{DC}$ intersect at $E$, and rays $\displaystyle \overrightarrow{OC}$ and $\displaystyle \overrightarrow{AB}$ intersect at $F$. Suppose that $AE = EC = CF = 4$, and the circumcircle of $ODE$ bisects $\overline{BF}$. Find the area of triangle $ADF$. [i]Proposed by Howard Halim[/i]

Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.

It is given a right-angled triangle $ABC$ and its circumcircle $k$. (a) prove that the radii of the circle $k_1$ tangent to the cathets of the triangle and to the circle $k$ is equal to the diameter of the incircle of the triangle ABC. (b) on the circle $k$ there may be found a point $M$ for which the sum $MA+MB+MC$ is as large as possible.

If a $5 \times n$ rectangle can be tiled using $n$ pieces like those shown in the diagram, prove that $n$ is even. Show that there are more than $2 \cdot 3^{k-1}$ ways to file a fixed $5 \times 2k$ rectangle $(k \geq 3)$ with $2k$ pieces. (symmetric constructions are supposed to be different.)