[u]Round 1[/u] [b]p1.[/b] This year, the Mathathon consists of $7$ rounds, each with $3$ problems. Another math test, Aspartaime, consists of $3$ rounds, each with $5$ problems. How many more problems are on the Mathathon than on Aspartaime? [b]p2.[/b] Let the solutions to $x^3 + 7x^2 - 242x - 2016 = 0 $be $a, b$, and $c$. Find $a^2 + b^2 + c^2$. (You might find it helpful to know that the roots are all rational.) [b]p3.[/b] For triangle $ABC$, you are given $AB = 8$ and $\angle A = 30^o$ . You are told that $BC$ will be chosen from amongst the integers from $1$ to $10$, inclusive, each with equal probability. What is the probability that once the side length $BC$ is chosen there is exactly one possible triangle $ABC$? [u]Round 2 [/u] [b]p4.[/b] It’s raining! You want to keep your cat warm and dry, so you want to put socks, rain boots, and plastic bags on your cat’s four paws. Note that for each paw, you must put the sock on before the boot, and the boot before the plastic bag. Also, the items on one paw do not affect the items you can put on another paw. How many different orders are there for you to put all twelve items of rain footwear on your cat? [b]p5.[/b] Let $a$ be the square root of the least positive multiple of $2016$ that is a square. Let $b$ be the cube root of the least positive multiple of $2016$ that is a cube. What is $ a - b$? [b]p6.[/b] Hypersomnia Cookies sells cookies in boxes of $6, 9$ or $10$. You can only buy cookies in whole boxes. What is the largest number of cookies you cannot exactly buy? (For example, you couldn’t buy $8$ cookies.) [u]Round 3 [/u] [b]p7.[/b] There is a store that sells each of the $26$ letters. All letters of the same type cost the same amount (i.e. any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost of using a ‘t’ may be different from the cost of an ‘r.’ If the letters to spell out $1$ cost $\$1001$, the letters to spell out $2$ cost $\$1010$, and the letters to spell out $11$ cost $\$2015$, how much do the letters to spell out $12$ cost? [b]p8.[/b] There is a square $ABCD$ with a point $P$ inside. Given that $PA = 6$, $PB = 9$, $PC = 8$. Calculate $PD$. [b]p9.[/b] How many ordered pairs of positive integers $(x, y)$ are solutions to $x^2 - y^2 = 2016$? [u]Round 4 [/u] [b]p10.[/b] Given a triangle with side lengths $5, 6$ and $7$, calculate the sum of the three heights of the triangle. [b]p11. [/b]There are $6$ people in a room. Each person simultaneously points at a random person in the room that is not him/herself. What is the probability that each person is pointing at someone who is pointing back to them? [b]p12.[/b] Find all $x$ such that $\sum_{i=0}^{\infty} ix^i =\frac34$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782837p24446063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for any positive integer $n$, the number \[ S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n \] is the sum of two consecutive perfect squares. [i]Dorin Andrica[/i]

The sequence $f_0, f_1, \dots$ of polynomials in $\mathbb{F}_{11}[x]$ is defined by $f_0(x) = x$ and $f_{n+1}(x) = f_n(x)^{11} - f_n(x)$ for all $n \ge 0$. Compute the remainder when the number of nonconstant monic irreducible divisors of $f_{1000}(x)$ is divided by $1000$. [i]Proposed by Ankan Bhattacharya[/i]

During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^2/2$ miles on the $n^{\text{th}}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\text{th}}$ day?

Consider a triangle $ABC$. The midpoints of the sides $BC, CA$, and $AB$ are denoted by $D, E$, and $F$, respectively. Assume that the median $AD$ is perpendicular to the median $BE$ and that their lengths are given by $AD = 18$ and $BE = 13.5$. Compute the length of the third median $CF$. (K. Czakler, Vienna)

If $x, y, z$ are real numbers satisfying $x^2 +y^2 +z^2 = 2$, prove the inequality \[ x + y + z \leq 2 + xyz \] When does equality occur?

Let $(a_{n})_{n\geq 1}$ and $(b_{n})_{n\geq 1}$ be the sequence of positive integers defined by $a_{n+1}=na_{n}+1$ and $b_{n+1}=nb_{n}-1$ for $n\geq 1$. Show that the two sequence cannot have infinitely many common terms. [i]Laurentiu Panaitopol[/i]

Do there exist $10$ consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next $9$ integers? (Inspired by a diagram in an old text book)

Let $ABC$ be a triangle with $\angle C=90^{\circ}$, and $K, L $ be the midpoints of the minor arcs AC and BC of its circumcircle. Segment $KL$ meets $AC$,at point $N$. Find angle $NIC$ where $I$is the incenter of $ABC$.

Two different squares are randomly chosen from an $8\times8$ chessboard. What is the probability that two queens placed on the two squares can attack each other? Recall that queens in chess can attack any square in a straight line vertically, horizontally, or diagonally from their current position.

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities $$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$ Prove that the degrees of the three polynomials are all even.

Andy chooses not necessarily distinct digits $G$, $U$, $N$, and $A$ such that the $5$ digit number $GUNGA$ is divisible by $44$. Find the least possible value of $G+U+N+G+A$. [i]Proposed by Andy Xu[/i]

At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining $\frac{1}{2}$ points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.

Let $a,b,c$ be three postive real numbers such that $a+b+c=1$. Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.

Let $I$ be the incenter of $\triangle ABC$ with $AB>AC$. Let $\Gamma$ be the circle with diameter $AI$. The circumcircle of $\triangle ABC$ intersects $\Gamma$ at points $A,D$, with point $D$ lying on $\overarc{AC}$ (not containing $B$). Let the line passing through $A$ and parallel to $BC$ intersect $\Gamma$ at points $A,E$. If $DI$ is the angle bisector of $\angle CDE$, and $\angle ABC = 33^{\circ}$, find the value of $\angle BAC$.