Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$

Samuel Tsui and Jason Yang each chose a different integer between $1$ and $60$, inclusive. They don’t know each others’ numbers, but they both know that the other person’s number is between $1$ and $60$ and distinct from their own. They have the following conversation: Samuel Tsui: Do our numbers have any common factors greater than $1$? Jason Yang: Definitely not. However their least common multiple must be less than$ 2023$. Samuel Tsui: Ok, thismeans that the sumof the factors of our two numbers are equal. What is the sumof Samuel Tsui’s and Jason Yang’s numbers? [i]Proposed by Samuel Tsui[/i]

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

Find all prime numbers $p$ such that the number $$3^p+4^p+5^p+9^p-98$$ has at most $6$ positive divisors.

Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that $$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$ (NB Vassiliev)

There are relatively prime positive integers $p$ and $q$ such that $\dfrac{p}{q}=\displaystyle\sum_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$. Find $p+q$.

After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$

[u]Round 1 [/u] [b]p1.[/b] A small pizza costs $\$4$ and has $6$ slices. A large pizza costs $\$9$ and has $14$ slices. If the MMATHS organizers got at least $400$ slices of pizza (having extra is okay) as cheaply as possible, how many large pizzas did they buy? [b]p2.[/b] Rachel flips a fair coin until she gets a tails. What is the probability that she gets an even number of heads before the tails? [b]p3.[/b] Find the unique positive integer $n$ that satisfies $n! \cdot (n + 1)! = (n + 4)!$. [u]Round 2 [/u] [b]p4.[/b] The Portland Malt Shoppe stocks $10$ ice cream flavors and $8$ mix-ins. A milkshake consists of exactly $1$ flavor of ice cream and between $1$ and $3$ mix-ins. (Mix-ins can be repeated, the number of each mix-in matters, and the order of the mix-ins doesn’t matter.) How many different milkshakes can be ordered? [b]p5.[/b] Find the minimum possible value of the expression $(x)^2 + (x + 3)^4 + (x + 4)^4 + (x + 7)^2$, where $x$ is a real number. [b]p6.[/b] Ralph has a cylinder with height $15$ and volume $\frac{960}{\pi}$ . What is the longest distance (staying on the surface) between two points of the cylinder? [u]Round 3 [/u] [b]p7.[/b] If there are exactly $3$ pairs $(x, y)$ satisfying $x^2 + y^2 = 8$ and $x + y = (x - y)^2 + a$, what is the value of $a$? [b]p8.[/b] If $n$ is an integer between $4$ and $1000$, what is the largest possible power of $2$ that $n^4 - 13n^2 + 36$ could be divisible by? (Your answer should be this power of $2$, not just the exponent.) [b]p9.[/b] Find the sum of all positive integers $n \ge 2$ for which the following statement is true: “for any arrangement of $n$ points in three-dimensional space where the points are not all collinear, you can always find one of the points such that the $n - 1$ rays from this point through the other points are all distinct.” [u]Round 4 [/u] [b]p10.[/b] Donald writes the number $12121213131415$ on a piece of paper. How many ways can he rearrange these fourteen digits to make another number where the digit in every place value is different from what was there before? [b]p11.[/b] A question on Joe’s math test asked him to compute $\frac{a}{b} +\frac34$ , where $a$ and $b$ were both integers. Because he didn’t know how to add fractions, he submitted $\frac{a+3}{b+4}$ as his answer. But it turns out that he was right for these particular values of $a$ and $b$! What is the largest possible value that a could have been? [b]p12.[/b] Christopher has a globe with radius $r$ inches. He puts his finger on a point on the equator. He moves his finger $5\pi$ inches North, then $\pi$ inches East, then $5\pi$ inches South, then $2\pi$ inches West. If he ended where he started, what is the largest possible value of $r$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2789002p24519497]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds: For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$. (Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$) Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.

[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].

Find all pairs of natural numbers $(k, m)$ such that for any natural $n{}$ the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!{}$. [i]Proposed by P. Kozhevnikov[/i]

Find the first digit after the decimal point of the number $\displaystyle \frac1{1009}+\frac1{1010}+\cdots + \frac1{2016}$

Solve the equation,$$ \sin (\pi \log x) + \cos (\pi \log x) = 1$$

Let $ n$ be a natural number, and $ n \equal{} 2^{2007}k\plus{}1$, such that $ k$ is an odd number. Prove that \[ n\not|2^{n\minus{}1}\plus{}1\]

Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.

Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.

Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that \[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$

Let $a>1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct.

Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$

Let $a,b$ be positive integers and $p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}$ be a prime number. Find the maximum value of $p$ and justify your answer.

Prove that for every natural number $n > 1$ there exists a suquence $a_1$,$a_2$, $...$, $a_n$ of the numbers $1,2,...,n$ such that for each $k \in \{1,2,...,n-1\}$ the number $a_{k+1}$ divides $a_1+a_2+...+a_k$.

Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$

Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$.

For which pairs of positive integers $(m,n)$ there exists a set $A$ such that for all positive integers $x,y$, if $|x-y|=m$, then at least one of the numbers $x,y$ belongs to the set $A$, and if $|x-y|=n$, then at least one of the numbers $x,y$ does not belong to the set $A$? [i]Adapted by Dan Schwarz from A.M.M.[/i]