$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that $$ TB\cdot TC=TI_b^2.$$

Solve in positive numbers the system $ x_1\plus{}\frac{1}{x_2}\equal{}4, x_2\plus{}\frac{1}{x_3}\equal{}1, x_3\plus{}\frac{1}{x_4}\equal{}4, ..., x_{99}\plus{}\frac{1}{x_{100}}\equal{}4, x_{100}\plus{}\frac{1}{x_1}\equal{}1$

Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.

Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.

$ ABC$ and $ A'B'C'$ are two triangles in the same plane such that the lines $ AA',BB',CC'$ are mutually parallel. Let $ [ABC]$ denotes the area of triangle $ ABC$ with an appropriate $ \pm$ sign, etc.; prove that \[ 3([ABC] \plus{} [A'B'C']) \equal{} [AB'C'] \plus{} [BC'A'] \plus{} [CA'B'] \plus{} [A'BC] \plus{} [B'CA] \plus{} [C'AB].\]

Let $\mathcal{S}$ be a set of points in the plane such that for each subset $\mathcal{T}$ of $\mathcal{S}$, there exists a convex $2025$-gon which contains all of the points in $\mathcal{T}$ and none of the rest of the points in $\mathcal{S}$ but not $\mathcal{T}$. Determine the greatest possible number of points in $\mathcal{S}$. [i]Jason Lee[/i]

Let $p$ be an odd positive integer. Find all values of the natural numbers $n\ge 2$ for which holds $$\sum_{i=1}^{n} \prod_{j\ne i} (x_i-x_j)^p\ge 0$$ where $x_1,x_2,..,x_n$ are any real numbers.

Let $r_1, r_2, ..., r_{47}$ be the roots of $x^{47} - 1 = 0$. Compute $$\sum^{47}_{i=1}r^{2020}_i .$$

A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?

For real numbers, $a,b,c$ with $bc \ne 0$ we have to $\frac{1-c^2}{bc} \ge 0$. Prove that $$5( a^2+b^2+c^2 -bc^3) \ge ab.$$

The following diagram shows four vertices connected by six edges. Suppose that each of the edges is randomly painted either red, white, or blue. The probability that the three edges adjacent to at least one vertex are colored with all three colors is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/6/4/de0a2a1a659011a30de1859052284c696822bb.png[/img]

Define a [i]growing spiral[/i] in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\dots,P_n$ such that $n\ge 2$ and: • The directed line segments $P_0P_1,P_1P_2,\dots,P_{n-1}P_n$ are in successive coordinate directions east (for $P_0P_1$), north, west, south, east, etc. • The lengths of these line segments are positive and strictly increasing. \[\begin{picture}(200,180) \put(20,100){\line(1,0){160}} \put(100,10){\line(0,1){170}} \put(0,97){West} \put(180,97){East} \put(90,0){South} \put(90,180){North} \put(100,100){\circle{1}}\put(100,100){\circle{2}}\put(100,100){\circle{3}} \put(115,100){\circle{1}}\put(115,100){\circle{2}}\put(115,100){\circle{3}} \put(115,130){\circle{1}}\put(115,130){\circle{2}}\put(115,130){\circle{3}} \put(40,130){\circle{1}}\put(40,130){\circle{2}}\put(40,130){\circle{3}} \put(40,20){\circle{1}}\put(40,20){\circle{2}}\put(40,20){\circle{3}} \put(170,20){\circle{1}}\put(170,20){\circle{2}}\put(170,20){\circle{3}} \multiput(100,99.5)(0,.5){3}{\line(1,0){15}} \multiput(114.5,100)(.5,0){3}{\line(0,1){30}} \multiput(40,129.5)(0,.5){3}{\line(1,0){75}} \multiput(39.5,20)(.5,0){3}{\line(0,1){110}} \multiput(40,19.5)(0,.5){3}{\line(1,0){130}} \put(102,90){P0} \put(117,90){P1} \put(117,132){P2} \put(28,132){P3} \put(30,10){P4} \put(172,10){P5} \end{picture}\] How many of the points $(x,y)$ with integer coordinates $0\le x\le 2011,0\le y\le 2011$ [i]cannot[/i] be the last point, $P_n,$ of any growing spiral?

Let $P(x) = x^2 + ax + b$. The two zeros of $P$, $r_1$ and $r_2$, satisfy the equation $|r_1^2 -r_2^2| = 17$. Give that $a, b > 1$ and are both integers, find $P(1)$.

The plane angles at vertex $D$ of the pyramid $ABCD$ are equal to $\alpha$,$\beta$ and $\gamma$ ($\angle CDB = a$). An arbitrary point $M$ is taken on edge $CB$. A ball is inscribed in each of the pyramids $ABDM$ and $ACDM$. Let us draw through $D$ a plane distinct from $BCD$, tangent to both balls and not intersecting the segment connecting the centers of the balls. Let this plane intersect the segment $AM$ at point $P$. What is $\angle ADP$ equal to?

The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$.

The function $f(x)$ is defined for all $x\ge 0$ and is always nonnegative. It has the additional property that if any line is drawn from the origin with any positive slope $m$, it intersects the graph $y=f(x)$ at precisely one point, which is $\frac{1}{\sqrt{m}}$ units from the origin. Let $a$ be the unique real number for which $f$ takes on its maximum value at $x=a$ (you may assume that such an $a$ exists). Find $\int_{0}^{a}f(x) \, dx$.

A point $P$ lies in the interior of the triangle $ABC$. The lines $AP, BP$, and $CP$ intersect $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Prove that if two of the quadrilaterals $ABDE, BCEF, CAFD, AEPF, BFPD$, and $CDPE$ are concyclic, then all six are concyclic.

Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as the product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?

For each positive integer $n$, let $S_n = \sum_{k=1}^nk^3$, and let $d(n)$ be the number of positive divisors of $n$. For how many positive integers $m$, where $m\leq 25$, is there a solution $n$ to the equation $d(S_n) = m$?

Given a line $k$ and three distinct points on it. Each of these points is the beginning of a pair of the rays - all the rays lie on the same side of the halfplane of the edge $k$. Each of these three pairs form a convex quadrilateral with another pair (so we have three quadrilaterals formed by these pairs of the rays). Prove that if it is possible to inscribe a circle in two of these quadrilaterals, then it is possible to inscribe a circle in the third one as well.

[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted? [b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once) [b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden? [b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors? [b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest? [b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels) [b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$. [b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$. [b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two real numbers $x, y$ are chosen randomly and independently on the interval $(1, r)$ where $r$ is some real number between $1024$ and $2048$. Let $P$ be the probability that $\lfloor \log_2x \rfloor > \lfloor \log_2y \rfloor .$ The value of $P$ is maximized when $r = \frac{p}{q}$ where $p,q$ are relatively prime positive integers. Find $p+q.$

Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?