[u]Round 1[/u] [b]p1.[/b] Goldilocks enters the home of the three bears – Papa Bear, Mama Bear, and Baby Bear. Each bear is wearing a different-colored shirt – red, green, or blue. All the bears look the same to Goldilocks, so she cannot otherwise tell them apart. The bears in the red and blue shirts each make one true statement and one false statement. The bear in the red shirt says: “I'm Blue's dad. I'm Green's daughter.” The bear in the blue shirt says: “Red and Green are of opposite gender. Red and Green are my parents.” Help Goldilocks find out which bear is wearing which shirt. [b]p2.[/b] The University of Washington is holding a talent competition. The competition has five contests: math, physics, chemistry, biology, and ballroom dancing. Any student can enter into any number of the contests but only once for each one. For example, a student may participate in math, biology, and ballroom. It turned out that each student participated in an odd number of contests. Also, each contest had an odd number of participants. Was the total number of contestants odd or even? [b]p3.[/b] The $99$ greatest scientists of Mars and Venus are seated evenly around a circular table. If any scientist sees two colleagues from her own planet sitting an equal number of seats to her left and right, she waves to them. For example, if you are from Mars and the scientists sitting two seats to your left and right are also from Mars, you will wave to them. Prove that at least one of the $99$ scientists will be waving, no matter how they are seated around the table. [b]p4.[/b] One hundred boys participated in a tennis tournament in which every player played each other player exactly once and there were no ties. Prove that after the tournament, it is possible for the boys to line up for pizza so that each boy defeated the boy standing right behind him in line. [b]p5.[/b] To celebrate space exploration, the Science Fiction Museum is going to read Star Wars and Star Trek stories for $24$ hours straight. A different story will be read each hour for a total of $12$ Star Wars stories and $12$ Star Trek stories. George and Gene want to listen to exactly $6$ Star Wars and $6$ Star Trek stories. Show that no matter how the readings are scheduled, the friends can find a block of $12$ consecutive hours to listen to the stories together. [u]Round 2[/u] [b]p6.[/b] $2013$ people attended Cinderella's ball. Some of the guests were friends with each other. At midnight, the guests started turning into mice. After the first minute, everyone who had no friends at the ball turned into a mouse. After the second minute, everyone who had exactly one friend among the remaining people turned into a mouse. After the third minute, everyone who had two human friends left in the room turned into a mouse, and so on. What is the maximal number of people that could have been left at the ball after $2013$ minutes? [b]p7.[/b] Bill and Charlie are playing a game on an infinite strip of graph paper. On Bill’s turn, he marks two empty squares of his choice (not necessarily adjacent) with crosses. Charlie, on his turn, can erase any number of crosses, as long as they are all adjacent to each other. Bill wants to create a line of $2013$ crosses in a row. Can Charlie stop him? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares.

Show that there exists a real constant $C>1$ with the following property: For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.) [i]Proposed by Jean-Marie De Koninck and Florian Luca[/i]

Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]

Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)

A positive six-digit integer begins and ends in $8$, and is also the product of three consecutive even numbers. What is the sum of the three even numbers?

Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?

Find an arithmetic progression of $2016$ natural numbers such that neither is a perfect power but its multiplication is a perfect power. Clarification: A perfect power is a number of the form $n^k$ where $n$ and $k$ are both natural numbers greater than or equal to $2$.

If $\phi$ is the Golden Ratio, we know that $\frac1\phi = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac1{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, $a, b, c$ positive integers, and the greatest common divisor of $a$ and $c$ is 1, find $a + b + c$.

Let $u,v$ be positive integers. Define sequence $\{a_n\}$ as follows: $a_1=u+v$, and for integers $m\ge 1$, \[\begin{array}{lll} \begin{cases} a_{2m}=a_m+u, \\ a_{2m+1}=a_m+v, \end{cases} \end{array}\] Let $S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )$. Prove that there are infinitely many perfect squares in the sequence $\{S_n\}$.

Find all positive integers $n$ such that the number \[A=\sqrt{\frac{9n-1}{n+7}}\] is rational.

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

Aladdin has several gold ingots, and sometimes he asks the Genie to give him more. The Genie first adds a thousand ingots, but then takes half of the total number. Could it be possible that after asking the Genie for gold ten times, the number of Aladdin’s gold ingots increased, assuming that each time the Genie took half, he took an integer number of ingots? (5 points) Alexandr Perepechko

Let $n,k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there is an integer $m$ divisible by $n$ whose sum of digits in base $10$ equals $k$.

If the number $3n +1$, where n is integer, is multiple of $7$, find the possible remainders of the following divisions: (a) of $n$ with divisor $7$, (b) of $n^{m}$ with divisor $7$, for all values of the positive integer $m, m >1$.

Let $S$ be the set of positive integers $x$ for which there exist positive integers $y$ and $m$ such that $y^2-2^m=x^2$. (a) Find all of the elements of $S$. (b) Find all $x$ such that both $x$ and $x+1$ are in $S$.

A time is called [i]reflexive [/i] if its representation on an analog clock would still be permissible if the hour and minute hand were switched. In a given non-leap day ($12:00:00.00$ a.m. to $11:59:59.99$ p.m.), how many times are reflexive?

Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

Prove that if the sum of two irreducible fractions is an integer then the two fractions have the same denominator.

Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties: $(1)T$ consists of finitely many prime numbers; $(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$

Prove that there exist infinitely many even positive integers $k$ such that for every prime $p$ the number $p^2+k$ is composite.

We say that a four-digit number $\overline{abcd}$ , which starts at $a$ and ends at $d$, is [i]interchangeable [/i] if there is an integer $n >1$ such that $n \times \overline{abcd}$ is a four-digit number that begins with $d$ and ends with $a$. For example, $1009$ is interchangeable since $1009\times 9=9081$. Find the largest interchangeable number.

$n$ is a positive integer. The number of solutions of $x^2+2016y^2=2017^n$ is $k$. Write $k$ with $n$.

A positive integer $n$ is called $special$ if there exist integers $a > 1$ and $b > 1$ such that $n=a^b + b$. Is there a set of $2014$ consecutive positive integers that contains exactly $2012$ $special$ numbers?