For a positive integer $n$, let $R(n)$ be the sum of the remainders when $n$ is divided by $1,2, \cdots , n$. For example, $R(4) = 0 + 0 + 1 + 0 = 1,$ $R(7) = 0 + 1 + 1 + 3 + 2 + 1 + 0 = 8$. Find all positive integers such that $R(n) = n-1$.

Find all the natural numbers that are $300$ times the sum of its digits.

[b]p1.[/b] How many real roots does the equation $2x^7 + x^5 + 4x^3 + x + 2 = 0$ have? [b]p2.[/b] Given that $f(n) = 1 +\sum^n_{j=1}(1 +\sum^j_{i=1}(2i + 1))$, find the value of $f(99)-\sum^{99}_{i=1} i^2$. [b]p3.[/b] A rectangular prism with dimensions $1\times a \times b$, where $1 < a < b < 2$, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute $ab$. Note: Two rectangular prisms of dimensions $p \times q\times r$ and$ x\times y\times z$ are similar if $\frac{p}{x} = \frac{q}{y} = \frac{r}{z}$ . [b]p4.[/b] For fixed real values of $p$, $q$, $r$ and $s$, the polynomial $x^4 + px^3 + qx^2 + rx + s$ has four non real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Compute $q$. [b]p5.[/b] There are $10$ seats in a row in a theater. Say we have an infinite supply of indistinguishable good kids and bad kids. How many ways can we seat $10$ kids such that no two bad kids are allowed to sit next to each other? [b]p6.[/b] There are an infinite number of people playing a game. They each pick a different positive integer $k$, and they each win the amount they chose with probability $\frac{1}{k^3}$ . What is the expected amount that all of the people win in total? [b]p7.[/b] There are $100$ donuts to be split among $4$ teams. Your team gets to propose a solution about how the donuts are divided amongst the teams. (Donuts may not be split.) After seeing the proposition, every team either votes in favor or against the propisition. The proposition is adopted with a majority vote or a tie. If the proposition is rejected, your team is eliminated and will never receive any donuts. Another remaining team is chosen at random to make a proposition, and the process is repeated until a proposition is adopted, or only one team is left. No promises or deals need to be kept among teams besides official propositions and votes. Given that all teams play optimally to maximize the expected value of the number of donuts they receive, are completely indifferent as to the success of the other teams, but they would rather not eliminate a team than eliminate one (if the number of donuts they receive is the same either way), then how much should your team propose to keep? [b]p8.[/b] Dominic, Mitchell, and Sitharthan are having an argument. Each of them is either credible or not credible – if they are credible then they are telling the truth. Otherwise, it is not known whether they are telling the truth. At least one of Dominic, Mitchell, and Sitharthan is credible. Tim knows whether Dominic is credible, and Ethan knows whether Sitharthan is credible. The following conversation occurs, and Tim and Ethan overhear: Dominic: “Sitharthan is not credible.” Mitchell: “Dominic is not credible.” Sitharthan: “At least one of Dominic or Mitchell is credible.” Then, at the same time, Tim and Ethan both simultaneously exclaim: “I can’t tell exactly who is credible!” They each then think for a moment, and they realize that they can. If Tim and Ethan always tell the truth, then write on your answer sheet exactly which of the other three are credible. [b]p9.[/b] Pick an integer $n$ between $1$ and $10$. If no other team picks the same number, we’ll give you $\frac{n}{10}$ points. [b]p10.[/b] Many quantities in high-school mathematics are left undefined. Propose a definition or value for the following expressions and justify your response for each. We’ll give you $\frac15$ points for each reasonable argument. $$(i) \,\,\,(.5)! \,\,\, \,\,\,(ii) \,\,\,\infty \cdot 0 \,\,\, \,\,\,(iii) \,\,\,0^0 \,\,\, \,\,\,(iv)\,\,\, \prod_{x\in \emptyset}x \,\,\, \,\,\,(v)\,\,\, 1 - 1 + 1 - 1 + ...$$ [b]p11.[/b] On the back of your answer sheet, write the “coolest” math question you know, and include the solution. If the graders like your question the most, then you’ll get a point. (With your permission, we might include your question on the Mixer next year!) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a set of $1997$ numbers such that if each number in the set, replace with the sum of the rest, you get the same set. Prove that the product of numbers in the set is equal to $0$.

Prove that for a natural number $n$ greater than 1, the following conditions are equivalent: a) $ n $ is an even number, b) there is a permutation $ (a_0, a_1, a_2, \ldots, a_{n-1}) $ of the set $ \{0,1,2,\ldots,n—1\} $ with the property that the sequence of residues from dividing by $ n $ the numbers $ a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots, a_0 + a_1 + a_2 + \ldots a_{n-1} $ is also a permutation of this set.

Sequence $\{ a_n \}$ satisfies: $a_1=3$, $a_2=7$, $a_n^2+5=a_{n-1}a_{n+1}$, $n \geq 2$. If $a_n+(-1)^n$ is prime, prove that there exists a nonnegative integer $m$ such that $n=3^m$.

Compute the number of positive integers $n$ less than $10^8$ such that at least two of the last five digits of $$ \lfloor 1000\sqrt{25n^2 + \frac{50}{9}n + 2022}\rfloor$$ are $6$. If your submitted estimate is a positive number $E$ and the true value is $A$, then your score is given by $\max \left(0, \left\lfloor 25 \min \left( \frac{E}{A}, \frac{A}{E}\right)^7\right\rfloor \right)$.

Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions: (i) $a,b\in\{1,2,\ldots,2013\}$; (ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.

Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite.

A coin that comes up heads with probability $ p > 0$ and tails with probability $ 1\minus{}p > 0$ independently on each flip is flipped eight times. Suppose the probability of three heads and five tails is equal to $ \frac{1}{25}$ of the probability of five heads and three tails. Let $ p \equal{} \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$

A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0. a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared". b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously : 1) $e-2$ , $e$ , $e+4$ are all prime numbers. 2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.

Let $p$, $q$, and $r$ be distinct positive prime numbers. Show that if \[pqr\mid (pq)^r+(qr)^p+(rp)^q-1,\] then \[(pqr)^3\mid 3((pq)^r+(qr)^p+(rp)^q-1).\]

a) Let $a,n,m$ be natural numbers, $a > 1$. Prove that if $(a^m + 1)$ is divisible by $(a^n + 1)$ than $m$ is divisible by $n$. b) Let $a,b,n,m$ be natural numbers, $a>1, a$ and $b$ are relatively prime. Prove that if $(a^m+b^m)$ is divisible by $(a^n+b^n)$ than $m$ is divisible by $n$.

Find the remainder when $19^{92}$ is divided by 92.

Find all non-negative integers $x, y$ and $z$ satisfying the equation: \[7^{x}+1=3^{y}+5^z\]

An integer $N > 1$ is written on the board. Alex writes a sequence of positive integers, obtaining new integers in the following manner: he takes any divisor greater than $1$ of the last number and either adds it to, or subtracts it from the number itself. Is it always (for all $N > 1$) possible for Alex to write the number $2011$ at some point?

Find all primes $p, q$ such that $p + q = (p-q)^3$.

For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let \[\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}\] For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$. Find the maximal number of inequivalent elements in $\mathbb{E}_p$.

The positive integers $a$ and $b$ are such that $a+k$ is divisible by $b+k$ for all positive integers numbers $k<b$. Prove that $a-k$ is divisible by $b-k$ for all positive integers $k<b$.

Find all pairs $a, b$ of positive integers, for which $$(a, b) + 3[a, b] = a^3 - b^3$$ Here $(a, b)$ denotes the greatest common divisor of $a, b$, and $[a, b]$ denotes the least common multiple of $a, b$. [i]Proposed by Oleksiy Masalitin[/i]

Let $ a $ and $ b $ be integer numbers. Let $ a = a^{2}+b^{2}-8b-2ab+16$. Prove that $ a $ is a perfect square.

A pair $(a,b)$ of positive integers is good if $\gcd(a,b)=1$ and for each pair of sets $A,B$ of positive integers such that $A,B$ are, respectively, complete residues system modulo $a,b$, there are $x \in A, y \in B$ such that $\gcd(x+y,ab)=1$. For each pair of positive integers $a,k$, let $f(N)$ the number of $b \leq N$ such $b$ has $k$ distinct prime factors and $(a,b)$ is good. Prove that \[\liminf_{n \to \infty} f(n)/\frac{n}{(\log n)^k}\ge e^{k}\]

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

For any positive integer $n$, let $a_n$ be the number of pairs $(x,y)$ of integers satisfying $|x^2-y^2| = n$. (a) Find $a_{1432}$ and $a_{1433}$. (b) Find $a_n$ .