Let $s(n) = \frac16 n^3 - \frac12 n^2 + \frac13 n$. (a) Show that $s(n)$ is an integer whenever $n$ is an integer. (b) How many integers $n$ with $0 < n \le 2008$ are such that $s(n)$ is divisible by $4$?

Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$.

Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.

Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.

How many are there $10$-digit numbers composed from the digits $1, 2, 3$ only and in which, two neighbouring digits differ by $1$ : (A): $48$ (B): $64$ (C): $72$ (D): $128$ (E): None of the above.

Prove that for every odd positive integer $n$ the number $n^n-n$ is divisible by $24$.

Let $a_n$ be a sequence of natural numbers defined by $a_1 = m$ and for $n > 1$. We call apair$ (a_k, a_{\ell })$ [i]interesting [/i] if (i) $0 < \ell - k < 2016$, (ii) $a_k$ divides $a_{\ell }$. Show that there exists a $m$ such that the sequence $a_n$ contains no interesting pair.

A triple of positive integers $(a,b,c)$ is [i]brazilian[/i] if $$a|bc+1$$ $$b|ac+1$$ $$c|ab+1$$ Determine all the brazilian triples.

Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k,m \in N = \{0,1,2,...,\}$

Let $a\geq 4$ be a positive integer. Determine the smallest value of $n\geq 5$, $n\neq a$, such that $a$ can be represented in the form$$a=\frac{x_1^2+x_2^2+\cdots + x_n^ 2}{x_1x_2\cdots x_n}$$for a suitable choice of the $n$ positive integers $x_1,x_2,\ldots ,x_n$.

Let $S$ be the set of positive perfect squares that are of the form $\overline{AA}$, i.e. the concatenation of two equal integers $A$. (Integers are not allowed to start with zero.) (a) Prove that $S$ is infinite. (b) Does there exist a function $f:S\times S \rightarrow S$ such that if $a,b,c \in S$ and $a,b | c$, then $f(a,b) | c$? (If such a function $f$ exists, we call $f$ an LCM function)

Find all pairs $(m,p)$ of natural numbers , such that $p$ is a prime and \[2^mp^2+27\] is the third power of a natural numbers

Find the largest integer $x<1000$ such that $\left(\begin{array}{c}1515 \\ x\end{array}\right)$ and $\left(\begin{array}{c}1975 \\ x\end{array}\right)$ are both odd.

Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

Solve the following equation in integers with gcd (x, y) = 1 $x^2 + y^2 = 2 z^2$

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

The number $1000$ can be written as the sum of $16$ consecutive natural numbers: $$1000 = 55 + 56 + ... + 70.$$ Determines all natural numbers that cannot be written as the sum of two or more consecutive natural numbers .

[hide=Intro]The answers to each of the ten questions in this section are integers containing only the digits $ 1$ through $ 8$, inclusive. Each answer can be written into the grid on the answer sheet, starting from the cell with the problem number, and continuing across or down until the entire answer has been written. Answers may cross dark lines. If the answers are correctly filled in, it will be uniquely possible to write an integer from $ 1$ to $ 8$ in every cell of the grid, so that each number will appear exactly once in every row, every column, and every marked $2$ by $4$ box. You will get $7$ points for every correctly filled answer, and a $15$ point bonus for filling in every gridcell. It will help to work back and forth between the grid and the problems, although every problem is uniquely solvable on its own. Please write clearly within the boxes. No points will be given for a cell without a number, with multiple numbers, or with illegible handwriting.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f4db097a9e3c2602b8608be64f06498bd9d58c.png[/img] [b]1 ACROSS:[/b] Jack puts $ 10$ red marbles, $ 8$ green marbles and 4 blue marbles in a bag. If he takes out $11$ marbles, what is the expected number of green marbles taken out? [b]2 DOWN:[/b] What is the closest integer to $6\sqrt{35}$ ? [b]3 ACROSS: [/b]Alan writes the numbers $ 1$ to $64$ in binary on a piece of paper without leading zeroes. How many more times will he have written the digit $ 1$ than the digit $0$? [b]4 ACROSS:[/b] Integers a and b are chosen such that $-50 < a, b \le 50$. How many ordered pairs $(a, b)$ satisfy the below equation? $$(a + b + 2)(a + 2b + 1) = b$$ [b]5 DOWN: [/b]Zach writes the numbers $ 1$ through $64$ in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence “$10$”? [b]6 ACROSS:[/b] If you are in a car that travels at $60$ miles per hour, $\$1$ is worth $121$ yen, there are $8$ pints in a gallon, your car gets $10$ miles per gallon, a cup of coffee is worth $\$2$, there are 2 cups in a pint, a gallon of gas costs $\$1.50$, 1 mile is about $1.6$ kilometers, and you are going to a coffee shop 32 kilometers away for a gallon of coffee, how much, in yen, will it cost? [b]7 DOWN:[/b] Clive randomly orders the letters of “MIXING THE LETTERS, MAN”. If $\frac{p}{m^nq}$ is the probability that he gets “LMT IS AN EXTREME THING” where p and q are odd integers, and $m$ is a prime number, then what is $m + n$? [b]8 ACROSS:[/b] Joe is playing darts. A dartboard has scores of $10, 7$, and $4$ on it. If Joe can throw $12$ darts, how many possible scores can he end up with? [b]9 ACROSS:[/b] What is the maximum number of bounded regions that $6$ overlapping ellipses can cut the plane into? [b]10 DOWN:[/b] Let $ABC$ be an equilateral triangle, such that $A$ and $B$ both lie on a unit circle with center $O$. What is the maximum distance between $O$ and $C$? Write your answer be in the form $\frac{a\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime, and $a$ and $c$ share no common factor. What is $abc$ ? PS. You had better use hide for answers.

[b]p1.[/b] Aidan walks into a skyscraper’s first floor lobby and takes the elevator up $50$ floors. After exiting the elevator, he takes the stairs up another $10$ floors, then takes the elevator down $30$ floors. Find the floor number Aidan is currently on. [b]p2.[/b] Jeff flips a fair coin twice and Kaylee rolls a standard $6$-sided die. The probability that Jeff flips $2$ heads and Kaylee rolls a $4$ is $P$. Find $\frac{1}{P}$ . [b]p3.[/b] Given that $a\odot b = a + \frac{a}{b}$ , find $(4\odot 2)\odot 3$. [b]p4.[/b] The following star is created by gluing together twelve equilateral triangles each of side length $3$. Find the outer perimeter of the star. [img]https://cdn.artofproblemsolving.com/attachments/e/6/ad63edbf93c5b7d4c7e5d68da2b4632099d180.png[/img] [b]p5.[/b] In Lexington High School’sMath Team, there are $40$ students: $20$ of whom do science bowl and $22$ of whom who do LexMACS. What is the least possible number of students who do both science bowl and LexMACS? [b]p6.[/b] What is the least positive integer multiple of $3$ whose digits consist of only $0$s and $1$s? The number does not need to have both digits. [b]p7.[/b] Two fair $6$-sided dice are rolled. The probability that the product of the numbers rolled is at least $30$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p8.[/b] At the LHSMath Team Store, $5$ hoodies and $1$ jacket cost $\$13$, and $5$ jackets and $1$ hoodie cost $\$17$. Find how much $15$ jackets and $15$ hoodies cost, in dollars. [b]p9.[/b] Eric wants to eat ice cream. He can choose between $3$ options of spherical ice cream scoops. The first option consists of $4$ scoops each with a radius of $3$ inches, which costs a total of $\$3$. The second option consists of a scoop with radius $4$ inches, which costs a total of $\$2$. The third option consists of $5$ scoops each with diameter $2$ inches, which costs a total of $\$1$. The greatest possible ratio of volume to cost of ice cream Eric can buy is nπ cubic inches per dollar. Find $3n$. [b]p10.[/b] Jen claims that she has lived during at least part of each of five decades. What is the least possible age that Jen could be? (Assume that age is always rounded down to the nearest integer.) [b]p11.[/b] A positive integer $n$ is called Maisylike if and only if $n$ has fewer factors than $n -1$. Find the sum of the values of $n$ that are Maisylike, between $2$ and $10$, inclusive. [b]p12.[/b] When Ginny goes to the nearby boba shop, there is a $30\%$ chance that the employee gets her drink order wrong. If the drink she receives is not the one she ordered, there is a $60\%$ chance that she will drink it anyways. Given that Ginny drank a milk tea today, the probability she ordered it can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the value of $a +b$. [b]p13.[/b] Alex selects an integer $m$ between $1$ and $100$, inclusive. He notices there are the same number of multiples of $5$ as multiples of $7$ betweenm and $m+9$, inclusive. Find how many numbers Alex could have picked. [b]p14.[/b] In LMTown there are only rainy and sunny days. If it rains one day there’s a $30\%$ chance that it will rain the next day. If it’s sunny one day there’s a $90\%$ chance it will be sunny the next day. Over n days, as n approaches infinity, the percentage of rainy days approaches $k\%$. Find $10k$. [b]p15.[/b] A bag of letters contains $3$ L’s, $3$ M’s, and $3$ T’s. Aidan picks three letters at random from the bag with replacement, and Andrew picks three letters at random fromthe bag without replacement. Given that the probability that both Aidan and Andrew pick one each of L, M, and T can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p16.[/b] Circle $\omega$ is inscribed in a square with side length $2$. In each corner tangent to $2$ of the square’s sides and externally tangent to $\omega$ is another circle. The radius of each of the smaller $4$ circles can be written as $(a -\sqrt{b})$ where $a$ and $b$ are positive integers. Find $a +b$. [img]https://cdn.artofproblemsolving.com/attachments/d/a/c76a780ac857f745067a8d6c4433efdace2dbb.png[/img] [b]p17.[/b] In the nonexistent land of Lexingtopia, there are $10$ days in the year, and the Lexingtopian Math Society has $5$ members. The probability that no two of the LexingtopianMath Society’s members share the same birthday can be written as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p18.[/b] Let $D(n)$ be the number of diagonals in a regular $n$-gon. Find $$\sum^{26}_{n=3} D(n).$$ [b]p19.[/b] Given a square $A_0B_0C_0D_0$ as shown below with side length $1$, for all nonnegative integers $n$, construct points $A_{n+1}$, $B_{n+1}$, $C_{n+1}$, and $D_{n+1}$ on $A_nB_n$, $B_nC_n$, $C_nD_n$, and $D_nA_n$, respectively, such that $$\frac{A_n A_{n+1}}{A_{n+1}B_n}=\frac{B_nB_{n+1}}{B_{n+1}C_n} =\frac{C_nC_{n+1}}{C_{n+1}D_n}=\frac{D_nD_{n+1}}{D_{n+1}A_n} =\frac34.$$ [img]https://cdn.artofproblemsolving.com/attachments/6/a/56a435787db0efba7ab38e8401cf7b06cd059a.png[/img] The sum of the series $$\sum^{\infty}_{i=0} [A_iB_iC_iD_i ] = [A_0B_0C_0D_0]+[A_1B_1C_1D_1]+[A_2B_2C_2D_2]...$$ where $[P]$ denotes the area of polygon $P$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$. [b]p20.[/b] Let $m$ and $n$ be two real numbers such that $$\frac{2}{n}+m = 9$$ $$\frac{2}{m}+n = 1$$ Find the sum of all possible values of $m$ plus the sumof all possible values of $n$. [b]p21.[/b] Let $\sigma (x)$ denote the sum of the positive divisors of $x$. Find the smallest prime $p$ such that $$\sigma (p!) \ge 20 \cdot \sigma ([p -1]!).$$ [b]p22.[/b] Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. Let $M$ be the midpoint of side $\overline{AB}$. Suppose there exists a point X on the circle passing through points $A$, $M$, and $C$ such that $BMCX$ is a parallelogram and $M$ and $X$ are on opposite sides of line $BC$. Let segments $\overline{AX}$ and $\overline{BC}$ intersect at a point $Y$ . Given that $BY = 8$, find $AY^2$. [b]p23.[/b] Kevin chooses $2$ integers between $1$ and $100$, inclusive. Every minute, Corey can choose a set of numbers and Kevin will tell him how many of the $2$ chosen integers are in the set. How many minutes does Corey need until he is certain of Kevin’s $2$ chosen numbers? [b]p24.[/b] Evaluate $$1^{-1} \cdot 2^{-1} +2^{-1} \cdot 3^{-1} +3^{-1} \cdot 4^{-1} +...+(2015)^{-1} \cdot (2016)^{-1} \,\,\, (mod \,\,\,2017).$$ [b]p25.[/b] In scalene $\vartriangle ABC$, construct point $D$ on the opposite side of $AC$ as $B$ such that $\angle ABD = \angle DBC = \angle BC A$ and $AD =DC$. Let $I$ be the incenter of $\vartriangle ABC$. Given that $BC = 64$ and $AD = 225$, find$ BI$ . [img]https://cdn.artofproblemsolving.com/attachments/b/1/5852dd3eaace79c9da0fd518cfdcd5dc13aecf.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that numbers $1,2,...,16$ can be divided in sequence such that sum of any two neighboring numbers is perfect square

A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

Find the smallest natural number $x> 0$ so that all following fractions are simplified $\frac{3x+9}{8},\frac{3x+10}{9},\frac{3x+11}{10},...,\frac{3x+49}{48}$ , i.e. numerators and denominators are relatively prime.

Determine all pairs $(x, y)$ of positive integers such that \[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\] [i]Proposed by Titu Andreescu, USA[/i]

Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.

Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.