Let $k>2$ be a positive integer such that the $k$-digit number $n_k=133\dots 3$, consisting of a digit $1$ followed by $k-1$ digits $3$ is prime. Show that $24 \mid k(k+2)$.

Let $m$ and $n$ be positive integers. Let $S$ be the set of all points $(x, y)$ with integer coordinates such that $1 \leq x,y \leq m+n-1$ and $m+1 \leq x +y \leq 2m+n-1.$ Let $L$ be the set of the $3m+3n-3$ lines parallel to one of $x = 0, y = 0,$ or $x + y = 0$ and passing through at least one point in $S$. For which pairs $(m, n)$ does there exist a subset $T$ of $S$ such that every line in $L$ intersects an odd number of elements of $T$?

Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$

Let $c$ be a fixed positive integer, and let ${a_n}^{\inf}_{n=1}$ be a sequence of positive integers such that $a_n < a_{n+1} < a_n+c$ for every positive integer $n$. Let $s$ denote the infinite string of digits obtained by writing the terms in the sequence consecutively from left to right, starting from the first term. For every positive integer $k$, let $s_k$ denote the number whose decimal representation is identical to the $k$ most left digits of $s$. Prove that for every positive integer $m$ there exists a positive integer $k$ such that $s_k$ is divisible by $m$.

[u]Round 1[/u] [b]1.1.[/b] Suppose a certain menu has $3$ sandwiches and $5$ drinks. How many ways are there to pick a meal so that you have exactly a drink and a sandwich? [b]1.2.[/b] If $a + b = 4$ and $a + 3b = 222222$, find $10a + b$. [b]1.3.[/b] Compute $$\left\lfloor \frac{2019 \cdot 2017}{2018} \right\rfloor $$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. [u]Round 2[/u] [b]2.1.[/b] Andrew has $10$ water bottles, each of which can hold at most $10$ cups of water. Three bottles are thirty percent filled, five are twenty-four percent filled, and the rest are empty. What is the average amount of water, in cups, contained in the ten water bottles? [b]2.2.[/b] How many positive integers divide $195$ evenly? [b]2.3.[/b] Square $A$ has side length $\ell$ and area $128$. Square $B$ has side length $\ell/2$. Find the length of the diagonal of Square $B$. [u]Round 3[/u] [b]3.1.[/b] A right triangle with area $96$ is inscribed in a circle. If all the side lengths are positive integers, what is the area of the circle? Express your answer in terms of $\pi$. [b]3.2.[/b] A circular spinner has four regions labeled $3, 5, 6, 10$. The region labeled $3$ is $1/3$ of the spinner, $5$ is $1/6$ of the spinner, $6$ is $1/10$ of the spinner, and the region labeled $10$ is $2/5$ of the spinner. If the spinner is spun once randomly, what is the expected value of the number on which it lands? [b]3.3.[/b] Find the integer k such that $k^3 = 8353070389$ [u]Round 4[/u] [b]4.1.[/b] How many ways are there to arrange the letters in the word [b]zugzwang [/b] such that the two z’s are not consecutive? [b]4.2.[/b] If $O$ is the circumcenter of $\vartriangle ABC$, $AD$ is the altitude from $A$ to $BC$, $\angle CAB = 66^o$ and $\angle ABC = 44^o$, then what is the measure of $\angle OAD$ ? [b]4.3.[/b] If $x > 0$ satisfies $x^3 +\frac{1}{x^3} = 18$, find $x^5 +\frac{1}{x^5}$ [u]Round 5[/u] [b]5.1.[/b] Let $C$ be the answer to Question $3$. Neethen decides to run for school president! To be entered onto the ballot, however, Neethen needs $C + 1$ signatures. Since no one else will support him, Neethen gets the remaining $C$ other signatures through bribery. The situation can be modeled by $k \cdot N = 495$, where $k$ is the number of dollars he gives each person, and $N$ is the number of signatures he will get. How many dollars does Neethen have to bribe each person with to get exactly C signatures? [b]5.2.[/b] Let $A$ be the answer to Question $1$. With $3A - 1$ total votes, Neethen still comes short in the election, losing to Serena by just $1$ vote. Darn! Neethen sneaks into the ballot room, knowing that if he destroys just two ballots that voted for Serena, he will win the election. How many ways can Neethen choose two ballots to destroy? [b]5.3.[/b] Let $B$ be the answer to Question $2$. Oh no! Neethen is caught rigging the election by the principal! For his punishment, Neethen needs to run the perimeter of his school three times. The school is modeled by a square of side length $k$ furlongs, where $k$ is an integer. If Neethen runs $B$ feet in total, what is $k + 1$? (Note: one furlong is $1/8$ of a mile). [u]Round 6[/u] [b]6.1.[/b] Find the unique real positive solution to the equation $x =\sqrt{6 + 2\sqrt6 + 2x}- \sqrt{6 - 2\sqrt6 - 2x} -\sqrt6$. [b]6.2.[/b] Consider triangle ABC with $AB = 13$ and $AC = 14$. Point $D$ lies on $BC$, and the lengths of the perpendiculars from $D$ to $AB$ and $AC$ are both $\frac{56}{9}$. Find the largest possible length of $BD$. [b]6.3.[/b] Let $f(x, y) = \frac{m}{n}$, where $m$ is the smallest positive integer such that $x$ and $y$ divide $m$, and $n$ is the largest positive integer such that $n$ divides both $x$ and $y$. If $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, what is the median of the distinct values that $f(a, b)$ can take, where $a, b \in S$? [u]Round 7[/u] [b]7.1.[/b] The polynomial $y = x^4 - 22x^2 - 48x - 23$ can be written in the form $$y = (x - \sqrt{a} - \sqrt{b} - \sqrt{c})(x - \sqrt{a} +\sqrt{b} +\sqrt{c})(x +\sqrt{a} -\sqrt{b} +\sqrt{c})(x +\sqrt{a} +\sqrt{b} -\sqrt{c})$$ for positive integers $a, b, c$ with $a \le b \le c$. Find $(a + b)\cdot c$. [b]7.2.[/b] Varun is grounded for getting an $F$ in every class. However, because his parents don’t like him, rather than making him stay at home they toss him onto a number line at the number $3$. A wall is placed at $0$ and a door to freedom is placed at $10$. To escape the number line, Varun must reach 10, at which point he walks through the door to freedom. Every $5$ minutes a bell rings, and Varun may walk to a different number, and he may not walk to a different number except when the bell rings. Being an $F$ student, rather than walking straight to the door to freedom, whenever the bell rings Varun just randomly chooses an adjacent integer with equal chance and walks towards it. Whenever he is at $0$ he walks to $ 1$ with a $100$ percent chance. What is the expected number of times Varun will visit $0$ before he escapes through the door to freedom? [b]7.3.[/b] Let $\{a_1, a_2, a_3, a_4, a_5, a_6\}$ be a set of positive integers such that every element divides $36$ under the condition that $a_1 < a_2 <... < a_6$. Find the probability that one of these chosen sets also satisfies the condition that every $a_i| a_j$ if $i|j$. [u]Round 8[/u] [b]8.[/b] How many numbers between $1$ and $100, 000$ can be expressed as the product of at most $3$ distinct primes? Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input. $$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.1 |I|}, 13 - \frac{|I-X|}{0.1 |I-2X|} \right\} \right\rceil \right\}$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy \[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]

For each real number $x$, $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example $\lfloor 2.8 \rfloor = 2$. Let $r \ge 0$ be a real number such that for all integers $m, n, m|n$ implies $\lfloor mr \rfloor| \lfloor nr \rfloor$. Prove that $r$ is an integer.

The smallest positive integer $x$ for which $1260x=N^3$, where $N$ is an integer, is: $\textbf{(A)}\ 1050 \qquad \textbf{(B)}\ 1260 \qquad \textbf{(C)}\ 1260^2 \qquad \textbf{(D)}\ 7350 \qquad \textbf{(E)}\ 44100$

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 all triples $(p, q, r)$ of prime numbers for which \[(p+1)(q+2)(r+3)=4pqr. \]

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 \}.$

Find all non-negative integers $n$ so that $\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} $ is an integer.

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

How many integers are there in $(b,2008b]$, where $b$ ($b > 0$) is given.

For each positive integer $n$ whose canonical decomposition is $n = p_1^{a_1} \cdot p_2^{a_2} \cdot\cdot\cdot p_k^{a_k}$, we define $t(n) = (p_1 + 1) \cdot (p_2 + 1) \cdot\cdot\cdot (p_k + 1)$. For example, $t(20) = t(2^2\cdot 5^1) = (2 + 1) (5 + 1) = 18$, $t(30) = t(2^1\cdot 3^1\cdot 5^1) = (2 + 1) (3 + 1) (5 + 1) = 72$ and $t(125) = t(5^3) = (5 + 1) = 6$ . We say that a positive integer $n$ is [i]special [/i]if $t(n)$ is a divisor of $n$. How many positive divisors of the number $54610$ are special?

A pair of integers $ (m,n)$ is called [i]good[/i] if \[ m\mid n^2 \plus{} n \ \text{and} \ n\mid m^2 \plus{} m\] Given 2 positive integers $ a,b > 1$ which are relatively prime, prove that there exists a [i]good[/i] pair $ (m,n)$ with $ a\mid m$ and $ b\mid n$, but $ a\nmid n$ and $ b\nmid m$.

A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot? $ \textbf{(A)}\ 52 \qquad \textbf{(B)}\ 58 \qquad \textbf{(C)}\ 62 \qquad \textbf{(D)}\ 68 \qquad \textbf{(E)}\ 70 $

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

If rationals $x, y$ satisfy $x^5 + y^5 = 2 x^2 y^2$, show that $1-x y$ is the square of a rational.

Find all integers $n$ such that $2^n + n^2$ is a square of an integer. [i](Tomas Jurik )[/i]

Determine all positive integers in the form $a<b<c<d$ with the property that each of them divides the sum of the other three.

Find all three-digit numbers whose remainders after division by $11$ give quotient, equal to the sum of the squares of its digits.

Given a positive integer $a,$ prove that $n!$ is divisible by $n^2 + n + a$ for infinitely many positive integers $n.{}$ [i]Proposed by Andrei Bâra[/i]

Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

Find all positive integers $n$ for which there are distinct integers $a_1$, ..., $a_n$ such that $\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1+a_2+...+a_n}{2}$.