$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$. Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

Let $p(x) = 3x^2 + 1$. Compute the largest prime divisor of $p(100) - p(3)$

If $n$ is a positive even number, find the last two digits of $(2^{6n}+26)-(6^{2n}-62)$.

Find the number of ordered tuples $\left(C,A,M,B\right)$ of non-negative integers such that \[C!+C!+A!+M!=B!\] [i]2019 CCA Math Bonanza Team Round #4[/i]

Let $AX$ and $BZ$ be altitudes of the triangle $ABC$. Let $AY$ and $BT$ be its angle bisectors. It is given that angles $XAY$ and $ZBT$ are equal. Does this necessarily imply that $ABC$ is isosceles? [i]The Jury[/i]

Find a necessary and sufficient condition on the positive integer $n$ that the equation \[x^n + (2 + x)^n + (2 - x)^n = 0\] have a rational root.

[hide=Note][i]The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.[/i][/hide] For each integer $n\ge2$, consider a regular polygon with $2n$ sides, all of length $1$. Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length $1$. Determine the limit inferior and the limit superior of the sequence defined by $$\frac1{n^2}\log_2C(n).$$

Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$. Find the largest value of $p$.

Let $ABC$ be a triangle with incenter $I$. Points $E$ and $F$ are on segments $AC$ and $BC$ respectively such that, $AE=AI$ and $BF=BI$. If $EF$ is the perpendicular bisector of $CI$, then $\angle{ACB}$ in degrees can be written as $\frac{m}{n}$ where $m$ and $n$ are co-prime positive integers. Find the value of $m+n$.

As a reward for working for NIMO, Evan divides $100$ indivisible marbles among three of his volunteers: David, Justin, and Michael. (Of course, each volunteer must get at least one marble!) However, Evan knows that, in the middle of the night, Lewis will select a positive integer $n > 1$ and, for each volunteer, steal exactly $\frac 1n$ of his marbles (if possible, i.e. if $n$ divides the number of marbles). In how many ways can Evan distribute the $100$ marbles so that Lewis is unable to steal marbles from every volunteer, regardless of which $n$ he selects? [i]Proposed by Jack Cornish[/i]

If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is: $\textbf{(A)}\ 0\qquad \textbf{(B)}\ \sqrt{50}\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ \sqrt{200}\qquad \textbf{(E)}\ \text{none of these}$

The value of $$\left\lfloor\frac1{3!}+\frac4{4!}+\frac9{5!}+\ldots\right\rfloor$$

There are $15$ lights on the ceiling of a room, numbered from $1$ to $15$. All lights are turned off. In another room, there are $15$ switches: a switch for lights $1$ and $2$, a switch for lights $2$ and $3$, a switch for lights $3$ en $4$, etcetera, including a sqitch for lights $15$ and $1$. When the switch for such a pair of lights is turned, both of the lights change their state (from on to off, or vice versa). The switches are put in a random order and all look identical. Raymond wants to find out which switch belongs which pair of lights. From the room with the switches, he cannot see the lights. He can, however, flip a number of switches, and then go to the other room to see which lights are turned on. He can do this multiple times. What is the minimum number of visits to the other room that he has to take to determine for each switch with certainty which pair of lights it corresponds to?

Several boys and girls are dancing a waltz at a ball. Is it possible that each girl can always get to dance the next dance with either a more handsome or more clever boy than for the previous dance, and that each time at least $80\%$ of the girls get to dance the next dance with a boy who is more handsome and more clever? (The numbers of boys and girls are equal and all are dancing.) (AY Belov)

Let $x$ and $y$ be positive real numbers. 1. Prove: if $x^3 - y^3 \ge 4x$, then $x^2 > 2y$. 2. Prove: if $x^5 - y^3 \ge 2x$, then $x^3 \ge 2y$.

Circles $ S$ and $ T$ intersect at $ P$ and $ Q$, with $ S$ passing through the centre of $ T$. Distinct points $ A$ and $ B$ lie on $ S$, inside $ T$, and are equidistant from the centre of $ T$. The line $ PA$ meets $ T$ again at $ D$. Prove that $ |AD| \equal{} |PB|$.

Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $. a) Prove that the lines $ EF $ and $ AB $ are perpendicular. b) Determine the locus of the point $ E $. c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.

The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$. What is the fewest number of digits he could have erased? [i]Ray Li[/i]

[hide=D stands for Dedekind, Z stands for Zermelo]they had two problem sets under those two names[/hide] [u]Set 4[/u] [b]D16.[/b] The cooking club at Blair creates $14$ croissants and $21$ danishes. Daniel chooses pastries randomly, stopping when he gets at least one croissant and at least two danishes. How many pastries must he choose to guarantee that he has one croissant and two danishes? [b]D17.[/b] Each digit in a $3$ digit integer is either $1, 2$, or $4$ with equal probability. What is the probability that the hundreds digit is greater than the sum of the tens digit and the ones digit? [b]D18 / Z11.[/b] How many two digit numbers are there such that the product of their digits is prime? [b]D19 / Z9.[/b] In the coordinate plane, a point is selected in the rectangle defined by $-6 \le x \le 4$ and $-2 \le y \le 8$. What is the largest possible distance between the point and the origin, $(0, 0)$? [b]D20 / Z10.[/b] The sum of two numbers is $6$ and the sum of their squares is $32$. Find the product of the two numbers. [u]Set 5[/u] [b]D21 / Z12.[/b] Triangle $ABC$ has area $4$ and $\overline{AB} = 4$. What is the maximum possible value of $\angle ACB$? [b]D22 / Z13.[/b] Let $ABCD$ be an iscoceles trapezoid with $AB = CD$ and M be the midpoint of $AD$. If $\vartriangle ABM$ and $\vartriangle MCD$ are equilateral, and $BC = 4$, find the area of trapezoid $ABCD$. [b]D23 / Z14.[/b] Let $x$ and $y$ be positive real numbers that satisfy $(x^2 + y^2)^2 = y^2$. Find the maximum possible value of $x$. [b]D24 / Z17.[/b] In parallelogram $ABCD$, $\angle A \cdot \angle C - \angle B \cdot \angle D = 720^o$ where all angles are in degrees. Find the value of $\angle C$. [b]D25.[/b] The number $12ab9876543$ is divisible by $101$, where $a, b$ represent digits between $0$ and $9$. What is $10a + b$? [u]Set 6[/u] [b]D26 / Z26.[/b] For every person who wrote a problem that appeared on the final MBMT tests, take the number of problems they wrote, and then take that number’s factorial, and finally multiply all these together to get $n$. Estimate the greatest integer $a$ such that $2^a$ evenly divides $n$. [b]D27 / Z27.[/b] Circles of radius $5$ are centered at each corner of a square with side length $6$. If a random point $P$ is chosen randomly inside the square, what is the probability that $P$ lies within all four circles? [b]D28 / Z28.[/b] Mr. Rose’s evil cousin, Mr. Caulem, has teaches a class of three hundred bees. Every week, he tries to disrupt Mr. Rose’s $4$th period by sending three of his bee students to fly around and make human students panic. Unfortunately, no pair of bees can fly together twice, as then Mr. Rose will become suspicious and trace them back to Mr. Caulem. What’s the largest number of weeks Mr. Caulem can disrupt Mr. Rose’s class? [b]D29 / Z29. [/b]Two blind brothers Beard and Bored are driving their tractors in the middle of a field facing north, and both are $10$ meters west from a roast turkey. Beard, can turn exactly $0.7^o$ and Bored can turn exactly $0.2^o$ degrees. Driving at a consistent $2$ meters per second, they drive straight until they notice the smell of the turkey getting farther away, and then turn right and repeat until they get to the turkey. Suppose Beard gets to the Turkey in about $818.5$ seconds. Estimate the amount of time it will take Bored. [b]D30 / Z30.[/b] Let a be the probability that $4$ randomly chosen positive integers have no common divisor except for $1$. Estimate $300a$. Note that the integers $1, 2, 3, 4$ have no common divisor except for $1$. Remark. This problem is asking you to find $300 \lim_{n\to \infty} a_n$, if $a_n$ is defined to be the probability that $4$ randomly chosen integers from $\{1, 2, ..., n\}$ have greatest common divisor $1$. PS. You should use hide for answers. D.1-15 / Z.1-8 problems have been collected [url=https://artofproblemsolving.com/community/c3h2916240p26045561]here [/url]and Z.15-25 [url=https://artofproblemsolving.com/community/c3h2916258p26045774]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be the set of the points $(x_1, x_2, . . . , x_{2012})$ in $2012$-dimensional space such that $|x_1|+|x_2|+...+|x_{2012}| \le 1$. Let $T$ be the set of points in $2012$-dimensional space such that $\max^{2012}_{i=1}|x_i| = 2$. Let $p$ be a randomly chosen point on $T$. What is the probability that the closest point in $S$ to $p$ is a vertex of $S$?

[u]Round 1[/u] [b]p1.[/b] What is $2 + 22 + 1 + 3 - 31 - 3$? [b]p2.[/b] Let $ABCD$ be a rhombus. Given $AB = 5$, $AC = 8$, and $BD = 6$, what is the perimeter of the rhombus? [b]p3.[/b] There are $2$ hats on a table. The first hat has $3$ red marbles and 1 blue marble. The second hat has $2$ red marbles and $4$ blue marbles. Jordan picks one of the hats randomly, and then randomly chooses a marble from that hat. What is the probability that she chooses a blue marble? [u]Round 2[/u] [b]p4.[/b] There are twelve students seated around a circular table. Each of them has a slip of paper that they may choose to pass to either their clockwise or counterclockwise neighbor. After each person has transferred their slip of paper once, the teacher observes that no two students exchanged papers. In how many ways could the students have transferred their slips of paper? [b]p5.[/b] Chad wants to test David's mathematical ability by having him perform a series of arithmetic operations at lightning-speed. He starts with the number of cubic centimeters of silicon in his 3D printer, which is $109$. He has David perform all of the following operations in series each second: $\bullet$ Double the number $\bullet$ Subtract $4$ from the number $\bullet$ Divide the number by $4$ $\bullet$ Subtract $5$ from the number $\bullet$ Double the number $\bullet$ Subtract $4$ from the number Chad instructs David to shout out after three seconds the result of three rounds of calculations. However, David computes too slowly and fails to give an answer in three seconds. What number should David have said to Chad? [b]p6.[/b] Points $D, E$, and $F$ lie on sides $BC$, $CA$, and $AB$ of triangle $ABC$, respectively, such that the following length conditions are true: $CD = AE = BF = 2$ and $BD = CE = AF = 4$. What is the area of triangle $ABC$? [u]Round 3[/u] [b]p7.[/b] In the $2, 3, 5, 7$ game, players count the positive integers, starting with $1$ and increasing, which do not contain the digits $2, 3, 5$, and $7$, and also are not divisible by the numbers $2, 3, 5$, and $7$. What is the fifth number counted? [b]p8.[/b] If A is a real number for which $19 \cdot A = \frac{2014!}{1! \cdot 2! \cdot 2013!}$ , what is $A$? Note: The expression $k!$ denotes the product $k \cdot (k - 1) \cdot ...\cdot 2 \cdot 1$. [b]p9.[/b] What is the smallest number that can be written as both $x^3 + y^2$ and $z^3 + w^2$ for positive integers $x, y, z,$ and $w$ with $x \ne z$? [u]Round 4[/u] [i]Each of the three problems in this round depends on the answer to one of the other problems. There is only one set of correct answers to these problems; however, each problem will be scored independently, regardless of whether the answers to the other problems are correct. In addition, it is given that the answer to each of the following problems is a positive integer less than or equal to the problem number. [/i] [b]p10.[/b] Let $B$ be the answer to problem $11$ and let $C$ be the answer to problem $12$. What is the sum of a side length of a square with perimeter $B$ and a side length of a square with area $C$? [b]p11.[/b] Let $A$ be the answer to problem $10$ and let $C$ be the answer to problem $12$. What is $(C - 1)(A + 1) - (C + 1)(A - 1)$? [b]p12.[/b] Let $A$ be the answer to problem $10$ and let $B$ be the answer to problem $11$. Let $x$ denote the positive difference between $A$ and $B$. What is the sum of the digits of the positive integer $9x$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2915810p26040675]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that if $ \left| ax^2+bx+c\right|\le 1, $ for all $ x\in [-1,1], $ then $ |a|+|b|+|c|\le 4. $

Find all functions $f: \mathbb{Z^+} \to \mathbb{Z^+}$ such that for all integers $a, b, c$ we have $$ af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac). $$ [i]Note. The set $\mathbb{Z^+}$ refers to the set of positive integers.[/i] [i]Proposed by Mojtaba Zare, Iran[/i]

For which positive integer values of $n$ can the set $\{ 1, 2, 3, \ldots, 4n \}$ be split into $n$ disjoint $4$-element subsets $\{ a,b,c,d \}$ such that in each of these sets $a = \dfrac{b +c +d} {3}$.