Let $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively. Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.

The diagram below shows a large triangle with area $72$. Each side of the triangle has been trisected, and line segments have been drawn between these trisection points parallel to the sides of the triangle. Find the area of the shaded region. [asy] size(4cm); pair A,B1,B2,B3,C1,C2,C3,M,I,J; A=origin; dotfactor=4; B1=dir(49); B2=2*B1; B3=3*B1; C1=1.35*dir(127); C2=2*C1; C3=3*C1; M=(B2+C2)/2; I=B1+C2; J=C1+B2; pair d[] = {A,B1,B2,B3,C1,C2,C3,M,I,J}; filldraw(C1--B1--B2--J--I--C2--cycle,rgb(.76,.76,.76)); draw(A--C3--B3--cycle); draw(C1--J^^C2--B2^^B1--I); for(int i=0;i<10;++i){ dot(d[i]); } [/asy]

Given $a, \theta \in \mathbb R, m \in \mathbb N$, and $P(x) = x^{2m}- 2|a|^mx^m \cos \theta +a^{2m}$, factorize $P(x)$ as a product of $m$ real quadratic polynomials.

In a convex qualrilateral $ABCD$, let $P$ be the intersection point of $AD$ and $BC$. Suppose that $I_1$ and $I_2$ are the incenters of triangles $PAB$ and $PDC$,respectively. Let $O$ be the circumcenter of $PAB$, and $H$ the orthocenter of $PDC$. Show that the circumcircles of triangles $AI_1B$ and $DHC$ are tangent together if and only if the circumcircles of triangles $AOB$ and $DI_2C$ are tangent together. Proposed by Hooman Fattahimoghaddam

Q. A light source at the point $(0, 16)$ in the co-ordinate plane casts light in all directions. A disc(circle along ith it's interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m, n$ are positive integers and $n$ is squarefree. Find $m + n$.

Let $g_1$ and $g_2$ be some lines, which intersect in point $A$. A circle $k_1$ is tangent to $g_1$ at point $A$ and intersects $g_2$ for a second time in $C$. A circle $k_2$ is tangent to $g_2$ at point $A$ and intersects $g_1$ for a second time in $D$. The circles $k_1$ and $k_2$ intersect for a second time in point $B$. Prove that, if $\frac{AC}{AD}=\sqrt{2}$, then $\frac{BC}{BD}=2$.

Let $ABC$ be a triangle. Distinct points $D$, $E$, $F$ lie on sides $BC$, $AC$, and $AB$, respectively, such that quadrilaterals $ABDE$ and $ACDF$ are cyclic. Line $AD$ meets the circumcircle of $\triangle ABC$ again at $P$. Let $Q$ denote the reflection of $P$ across $BC$. Show that $Q$ lies on the circumcircle of $\triangle AEF$. [i]Proposed by Ankan Bhattacharya[/i]

Let $\triangle ABC$ be a scalene triangle, $O$ be the midpoint of $BC$, and $M$ and $N$ be the intersections of the circle with diameter $BC$ and $AB$ and $BC$, respectively. The bisectors of $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of $\triangle BMR$ and $\triangle CNR$ intersect on $BC$.

In a triangle $ABC$, let $D,E, F$ be interior points of sides $BC,CA,AB$ respectively. Let $AD,BE,CF$ meet the circumcircle of triangle $ABC$ in $K, L,M$ respectively. Prove that $\frac{AD}{DK} + \frac{BE}{EL} + \frac{CF}{FM} \ge 9$. When does the equality hold?

Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there?

The diagram below shows a $1\times2\times10$ duct with $2\times2\times2$ cubes attached to each end. The resulting object is empty, but the entire surface is solid sheet metal. A spider walks along the inside of the duct between the two marked corners. There are positive integers $m$ and $n$ so that the shortest path the spider could take has length $\sqrt{m}+\sqrt{n}$. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); draw(origin--(43,0)--(61,20)--(18,20)--cycle--(0,-43)--(43,-43)--(43,0)^^(43,-43)--(61,-23)--(61,20)); draw((43,-43)--(133,57)--(90,57)--extension((90,57),(0,-43),(61,20),(18,20))); draw((0,-43)--(0,-65)--(43,-65)--(43,-43)^^(43,-65)--(133,35)--(133,57)); draw((133,35)--(133,5)--(119.5,-10)--(119.5,20)^^(119.5,-10)--extension((119.5,-10),(100,-10),(43,-65),(133,35))); dot(origin^^(133,5)); [/asy]

Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$. Suppose that the tangent line at $C$ to the circle passing through $A,B,C$ intersects the line $AB$ at $D$. Let $E$ be the midpoint of $CD$ and let $F$ be a point on $EB$ such that $AF$ is parallel to $CD$. Prove that the lines $AB$ and $CF$ are perpendicular.

The figure below shows a barn in the shape of two congruent pentagonal prisms that intersect at right angles and have a common center. The ends of the prisms are made of a 12 foot by 7 foot rectangle surmounted by an isosceles triangle with sides 10 feet, 10 feet, and 12 feet. Each prism is 30 feet long. Find the volume of the barn in cubic feet. [center][img]https://snag.gy/Ox9CUp.jpg[/img][/center]

Let $ABC$ be an equilateral triangle and $\mathcal{E}$ the set of all points contained in the three segments $AB$, $BC$, and $CA$ (including $A$, $B$, and $C$). Determine whether, for every partition of $\mathcal{E}$ into two disjoint subsets, at least one of the two subsets contains the vertices of a right-angled triangle.

[u]Round 5[/u] [b]p13.[/b] A unit square is rotated $30^o$ counterclockwise about one of its vertices. Determine the area of the intersection of the original square with the rotated one. [b]p14.[/b] Suppose points $A$ and $B$ lie on a circle of radius $4$ with center $O$, such that $\angle AOB = 90^o$. The perpendicular bisectors of segments $OA$ and $OB$ divide the interior of the circle into four regions. Find the area of the smallest region. [b]p15.[/b] Let $ABCD$ be a quadrilateral such that $AB = 4$, $BC = 6$, $CD = 5$, $DA = 3$, and $\angle DAB = 90^o$. There is a point $I$ inside the quadrilateral that is equidistant from all the sides. Find $AI$. [u]Round 6[/u] [i]The answer to each of the three questions in this round depends on the answer to one of the other questions. There is only one set of correct answers to these problems; however, each question will be scored independently, regardless of whether the answers to the other questions are correct. [/i] [b]p16.[/b] Let $C$ be the answer to problem $18$. Compute $$\left( 1 - \frac{1}{2^2} \right) \left( 1 - \frac{1}{3^2} \right) ... \left( 1 - \frac{1}{C^2} \right).$$ [b]p17.[/b] Let $A$ be the answer to problem $16$. Let $PQRS$ be a square, and let point $M$ lie on segment $PQ$ such that $MQ = 7PM$ and point $N$ lie on segment $PS$ such that $NS = 7PN$. Segments $MS$ and $NQ$ meet at point $X$. Given that the area of quadrilateral $PMXN$ is $A - \frac12$, find the side length of the square. [b]p18.[/b] Let $B$ be the answer to problem $17$ and let $N = 6B$. Find the number of ordered triples $(a, b, c)$ of integers between $0$ and $N - 1$, inclusive, such that $a + b + c$ is divisible by $N$. [u]Round 7[/u] [b]p19.[/b] Let $k$ be the units digit of $\underbrace{7^{7^{7^{7^{7^{7^{7}}}}}}}_{Seven \,\,7s}$ . What is the largest prime factor of the number consisting of $k$ $7$’s written in a row? [b]p20.[/b] Suppose that $E = 7^7$ , $M = 7$, and $C = 7·7·7$. The characters $E, M, C, C$ are arranged randomly in the following blanks. $$... \times ... \times ... \times ... $$ Then one of the multiplication signs is chosen at random and changed to an equals sign. What is the probability that the resulting equation is true? [b]p21[/b]. During a recent math contest, Sophy Moore made the mistake of thinking that $133$ is a prime number. Fresh Mann replied, “To test whether a number is divisible by $3$, we just need to check whether the sum of the digits is divisible by $3$. By the same reasoning, to test whether a number is divisible by $7$, we just need to check that the sum of the digits is a multiple of $7$, so $133$ is clearly divisible by $7$.” Although his general principle is false, $133$ is indeed divisible by $7$. How many three-digit numbers are divisible by $7$ and have the sum of their digits divisible by $7$? [u]Round 8[/u] [b]p22.[/b] A [i]look-and-say[/i] sequence is defined as follows: starting from an initial term $a_1$, each subsequent term $a_k$ is found by reading the digits of $a_{k-1}$ from left to right and specifying the number of times each digit appears consecutively. For example, $4$ would be succeeded by $14$ (“One four.”), and $31337$ would be followed by $13112317$ (“One three, one one, two three, one seven.”) If $a_1$ is a random two-digit positive integer, find the probability that $a_4$ is at least six digits long. [b]p23.[/b] In triangle $ABC$, $\angle C = 90^o$. Point $P$ lies on segment $BC$ and is not $B$ or $C$. Point $I$ lies on segment $AP$, and $\angle BIP = \angle PBI = \angle CAB$. If $\frac{AP}{BC} = k$, express $\frac{IP}{CP}$ in terms of $k$. [b]p24.[/b] A subset of $\{1, 2, 3, ... , 30\}$ is called [i]delicious [/i] if it does not contain an element that is $3$ times another element. A subset is called super delicious if it is delicious and no delicious set has more elements than it has. Determine the number of super delicious subsets. PS. You sholud use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2784267p24464980]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] An iscoceles triangle has one angle equal to $100$ degrees, what is the degree measure of one of the two remaining angles. [b]p2.[/b] Tanmay picks four cards from a standard deck of $52$ cards at random. What is the probability he gets exactly one Ace, exactly exactly one King, exactly one Queen, exactly one Jack and exactly one Ten? [b]p3.[/b] What is the sum of all the factors of $2014$? [u]Round 2[/u] [b]p4.[/b] Which number under $1000$ has the greatest number of factors? [b]p5.[/b] How many $10$ digit primes have all distinct digits? [b]p6.[/b] In a far o universe called Manhattan, the distance between two points on the plane $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ is defined as $d(P,Q) = |x_1-x_2|+|y_1-y_2|$. Let $S$ be the region of points that are a distance of $\le 7$ away from the origin $(0, 0)$. What is the area of $S$? [u]Round 3[/u] [b]p7.[/b] How many factors does $13! + 14! + 15!$ have? [b]p8.[/b] How many zeroes does $45!$ have consecutively at the very end in its representation in base $45$? [b]p9.[/b] A sequence of circles $\omega_0$, $\omega_1$, $\omega_2$, ... is drawn such that: $\bullet$ $\omega_0$ has a radius of $1$. $\bullet$ $\omega_{i+1}$ has twice the radius of $\omega_i$. $\bullet$ $\omega_i$ is internally tangent to $\omega_{i+1}$. Let $A$ be a point on $\omega_0$ and $B$ be a point on $\omega_{10}$. What is the maximum possible value of $AB$? [u]Round 4[/u] [b]p10.[/b] A $3-4-5$ triangle is constructed. Then a similar triangle is constructed with the shortest side of the first triangle being the new hypotenuse for the second triangle. This happens an infinite amount of times. What is the maximum area of the resulting figure? [b]p11.[/b] If an unfair coin is flipped $4$ times and has a $3/64$ chance of coming heads exactly thrice, what is the probability the coin comes tails on a single flip. [b]p12.[/b] Find all triples of positive integers $(a, b, c)$ that satisfy $2a = 1+bc$, $2b = 1+ac$, and $2c = 1 + ab$. [u]Round 5[/u] [b]p13.[/b] $6$ numbered points on a plane are placed so that they can create a regular hexagon $P_1P_2P_3P_4P_5P_6$ if connected. If a triangle is drawn to include a certain amount of points in it, how many triangles are there that hold a different set of points? (note: the triangle with $P_1$ and $P_2$ is not the same as the one with $P_3$ and $P_4$). [b]p14.[/b] Let $S$ be the set of all numbers of the form $n(2n + 1)(3n + 2)(4n + 3)(5n + 4)$ for $n \ge 1$. What is the largest number that divides every member of $S$? [b]p15. [/b]Jordan tosses a fair coin until he gets heads at least twice. What is the expected number of flips of the coin that he will make? PS. You should use hide for answers. Rounds 6-10 have been posted [url=https://artofproblemsolving.com/community/c3h3156859p28695035]here[/url].. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that every polygon with perimeter $ 2a $ can be covered by a disk with diameter $ a $.

Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]

Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$? Remark: Two cells are called connected if they have a common edge.

The point $ (\minus{}3, 2)$ is rotated $ 90^\circ$ clockwise around the origin to point $ B$. Point $ B$ is then reflected over the line $ y \equal{} x$ to point $ C$. What are the coordinates of $ C$? $ \textbf{(A)}\ ( \minus{} 3, \minus{} 2)\qquad \textbf{(B)}\ ( \minus{} 2, \minus{} 3)\qquad \textbf{(C)}\ (2, \minus{} 3)\qquad \textbf{(D)}\ (2,3)\qquad \textbf{(E)}\ (3,2)$

The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.

A trapezoid $ABCD$ with bases $BC$ and $AD$ is given. The points $K$ and $L$ are chosen on the sides $AB$ and $CD$, respectively, so that $KL \parallel AD$. It turned out that the areas of the quadrilaterals $AKLD$ and $KBCL$ are equal. Find the length $KL$ if $BC = 3, AD = 5$.

In the plan $6$ points were located such that the distance between two damages of them is greater than or equal to $1$. Prove that it is possible to choose two of those points such that their distance is greater than or equal to $2 \cos{18}$ Observation: It might help you to know that $\cos{18} = 0.95105\ldots$ and $\cos{24} = 0.91354\ldots$

Let $PQRS$ be a cyclic quadrilateral such that the segments $PQ$ and $RS$ are not parallel. Consider the set of circles through $P$ and $Q$, and the set of circles through $R$ and $S$. Determine the set $A$ of points of tangency of circles in these two sets.

Let $ABCD$ be a square inscribed in circle $K$. Let $P$ be a point on the small arc $CD$ of circle $K$. The line $PB$ intersects $AC$ in $E$. The line $PA$ intersects $DB$ in $F$. The circle circumscribed to triangle $PEF$ intersects for second time $K$ in $Q$. Prove that $PQ$ is parallel to $CD$.