Prove that a necessary and sufficient for the existence of a set $ S \subset \{1,2,...,n \}$ with the property that the integers $ 0,1,...,n\minus{}1$ all have an odd number of representations in the form $ x\minus{}y, x,y \in S$, is that $ (2n\minus{}1)$ has a multiple of the form $ 2.4^k\minus{}1$ [i]L. Lovasz, J. Pelikan[/i]

For positive integers $k$ and $n$, we know $k \geq n!$. Prove that $ \phi (k) \geq (n-1)!$

For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers. Prove that: 1. There does not exist a solution $(a, b, c)$ for $n = 2017$. 2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$. 3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.

$10$ real numbers are given $a_1,a_2,\ldots ,a_{10} $, and the $45$ sums of two of these numbers are formed $a_i+a_j $, $1\leq i<j\leq 10$ . It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the $45$ sums there are $k$ that are not integers and $45-k$ that are integers.

A positive integer $n$ is good if $n$ can be written as the sum of $2004$ positive integers $a_1$, $a_2$, ..., $a_{2004}$ such that $1 \le a_1 < a_2<...<a_{2004}$ and $a_i$ divides $a_{i+1}$ for $i=1$, $2$, ..., $2003$. Show that there are only finitely many positive integers that are not good.

Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.) [i]David Yang.[/i]

If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0 + qy_0.$ Determine the maximum value of $b - a$, where $a$ and $b$ are positive integers with the following property: If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q - 1$ and $0 \leq y \leq p - 1$ such that $t = px + qy.$

A positive integer $d$ is called [i]nice[/i] iff for all positive integers $x,y$ hold: $d$ divides $(x+y)^{5}-x^{5}-y^{5}$ iff $d$ divides $(x+y)^{7}-x^{7}-y^{7}$ . a) Is 29 nice? b) Is 2006 nice? c) Prove that infinitely many nice numbers exist.

[b]p1.[/b] If $A = 0$, $B = 1$, $C = 2$, $...$, $Z = 25$, then what is the sum of $A + B + M+ C$? [b]p2.[/b] Eric is playing Tetris against Bryan. If Eric wins one-fifth of the games he plays and he plays $15$ games, find the expected number of games Eric will win. [b]p3.[/b] What is the sum of the measures of the exterior angles of a regular $2023$-gon in degrees? [b]p4.[/b] If $N$ is a base $10$ digit of $90N3$, what value of $N$ makes this number divisible by $477$? [b]p5.[/b] What is the rightmost non-zero digit of the decimal expansion of $\frac{1}{2^{2023}}$ ? [b]p6.[/b] if graphs of $y = \frac54 x + m$ and $y = \frac32 x + n$ intersect at $(16, 27)$, what is the value of $m + n$? [b]p7.[/b] Bryan is hitting the alphabet keys on his keyboard at random. If the probability he spells out ABMC at least once after hitting $6$ keys is $\frac{a}{b^c}$ , for positive integers $a$, $b$, $c$ where $b$, $c$ are both as small as possible, find $a+b+c$. Note that the letters ABMC must be adjacent for it to count: AEBMCC should not be considered as correctly spelling out ABMC. [b]p8.[/b] It takes a Daniel twenty minutes to change a light bulb. It takes a Raymond thirty minutes to change a light bulb. It takes a Bryan forty-five minutes to change a light bulb. In the time that it takes two Daniels, three Raymonds, and one and a half Bryans to change $42$ light bulbs, how many light bulbs could half a Raymond change? Assume half a person can work half as productively as a whole person. [b]p9.[/b] Find the value of $5a + 4b + 3c + 2d + e$ given $a, b, c, d, e$ are real numbers satisfying the following equations: $$a^2 = 2e + 23$$ $$b^2 = 10a - 34$$ $$c^2 = 8b - 23$$ $$d^2 = 6c - 14$$ $$e^2 = 4d - 7.$$ [b]p10.[/b] How many integers between $1$ and $1000$ contain exactly two $1$’s when written in base $2$? [b]p11.[/b] Joe has lost his $2$ sets of keys. However, he knows that he placed his keys in one of his $12$ mailboxes, each labeled with a different positive integer from $1$ to $12$. Joe plans on opening the $2$ mailbox labeled $1$ to see if any of his keys are there. However, a strong gust of wind blows by, opening mailboxes $11$ and $12$, revealing that they are empty. If Joe decides to open one of the mailboxes labeled $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ , or $10$, the probability that he finds at least one of his sets of keys can be expressed as $\frac{a}{b}$, where a and b are relatively prime positive integers. Find the sum $a + b$. Note that a single mailbox can contain $0$, $1$, or $2$ sets of keys, and the mailboxes his sets of keys were placed in are determined independently at random. [b]p12.[/b] As we all know, the top scientists have recently proved that the Earth is a flat disc. Bob is standing on Earth. If he takes the shortest path to the edge, he will fall off after walking $1$ meter. If he instead turns $90$ degrees away from the shortest path and walks towards the edge, he will fall off after $3$ meters. Compute the radius of the Earth. [b]p13.[/b] There are $999$ numbers that are repeating decimals of the form $0.abcabcabc...$ . The sum of all of the numbers of this form that do not have a $1$ or $2$ in their decimal representation can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Find $a + b$. [b]p14.[/b] An ant is crawling along the edges of a sugar cube. Every second, it travels along an edge to another adjacent vertex randomly, interested in the sugar it notices. Unfortunately, the cube is about to be added to some scalding coffee! In $10$ seconds, it must return to its initial vertex, so it can get off and escape. If the probability the ant will avoid a tragic doom can be expressed as $\frac{a}{3^{10}}$ , where $a$ is a positive integer, find $a$. Clarification: The ant needs to be on its initial vertex in exactly $10$ seconds, no more or less. [b]p15.[/b] Raymond’s new My Little Pony: Friendship is Magic Collector’s book arrived in the mail! The book’s pages measure $4\sqrt3$ inches by $12$ inches, and are bound on the longer side. If Raymond keeps one corner in the same plane as the book, what is the total area one of the corners can travel without ripping the page? If the desired area in square inches is $a\pi+b\sqrt{c}$ where $a$, $b$, and $c$ are integers and $c$ is squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A sequence of positive integers is defined by $a_0=1$ and $a_{n+1}=a_n^2+1$ for each $n\ge0$. Find $\text{gcd}(a_{999},a_{2004})$.

Find all primes $p$ for which $p^2 - p + 1$ is a perfect cube.

Consider the system of congruences $$\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}$$ Find the number of triples $(x,y, z) $ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$.

A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$) $f (1992 + x) = f (1992 - x)$ for all integers $x$. Can all positive divisors of $92$ occur as values of f?

Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?

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

Call a number $n$ [i]good[/i] if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers. (a) Prove that there exist infinitely many sets of $4$ consecutive good numbers. (b) Find all sets of $5$ consecutive good numbers. [i]Proposed by Michael Ma[/i]

Let $a,b,c$ be three integers for which the sum \[ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}\] is integer. Prove that each of the three numbers \[ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}{a}\] is integer. (Proposed by Gerhard J. Woeginger)

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Somya has a football game $4$ days from today. If the day before yesterday was Wednesday, what day of the week is the game? [b]p2.[/b] Sammy writes the following equation: $$\frac{2 + 2}{8 + 8}=\frac{x}{8}.$$ What is the value of $x$ in Sammy's equation? [b]p3.[/b] On $\pi$ day, Peter buys $7$ pies. The pies costed $\$3$, $\$1$, $\$4$, $\$1$, $\$5$, $\$9$, and $\$2$. What was the median price of Peter's $7$ pies in dollars? [b]p4.[/b] Antonio draws a line on the coordinate plane. If the line passes through the points ($1, 3$) and ($-1,-1$), what is slope of the line? [b]p5.[/b] Professor Varun has $25$ students in his science class. He divides his students into the maximum possible number of groups of $4$, but $x$ students are left over. What is $x$? [b]p6.[/b] Evaluate the following: $$4 \times 5 \div 6 \times 3 \div \frac47$$ [b]p7.[/b] Jonny, a geometry expert, draws many rectangles with perimeter $16$. What is the area of the largest possible rectangle he can draw? [b]p8.[/b] David always drives at $60$ miles per hour. Today, he begins his trip to MIT by driving $60$ miles. He stops to take a $20$ minute lunch break and then drives for another $30$ miles to reach the campus. What is the total time in minutes he spends getting to MIT? [b]p9.[/b] Richard has $5$ hats: blue, green, orange, red, and purple. Richard also has 5 shirts of the same colors: blue, green, orange, red, and purple. If Richard needs a shirt and a hat of different colors, how many outts can he wear? [b]p10.[/b] Poonam has $9$ numbers in her bag: $1, 2, 3, 4, 5, 6, 7, 8, 9$. Eric runs by with the number $36$. How many of Poonam's numbers evenly divide Eric's number? [b]p11.[/b] Serena drives at $45$ miles per hour. If her car runs at $6$ miles per gallon, and each gallon of gas costs $2$ dollars, how many dollars does she spend on gas for a $135$ mile trip? [b]p12.[/b] Grace is thinking of two integers. Emmie observes that the sum of the two numbers is $56$ but the difference of the two numbers is $30$. What is the sum of the squares of Grace's two numbers? [b]p13.[/b] Chang stands at the point ($3,-3$). Fang stands at ($-3, 3$). Wang stands in-between Chang and Fang; Wang is twice as close to Fang as to Chang. What is the ordered pair that Wang stands at? [b]p14.[/b] Nithin has a right triangle. The longest side has length $37$ inches. If one of the shorter sides has length $12$ inches, what is the perimeter of the triangle in inches? [b]p15.[/b] Dora has $2$ red socks, $2$ blue socks, $2$ green socks, $2$ purple socks, $3$ black socks, and $4$ gray socks. After a long snowstorm, her family loses electricity. She picks socks one-by-one from the drawer in the dark. How many socks does she have to pick to guarantee a pair of socks that are the same color? [b]p16.[/b] Justin selects a random positive $2$-digit integer. What is the probability that the sum of the two digits of Justin's number equals $11$? [b]p17.[/b] Eddie correctly computes $1! + 2! + .. + 9! + 10!$. What is the remainder when Eddie's sum is divided by $80$? [b]p18.[/b] $\vartriangle PQR$ is drawn such that the distance from $P$ to $\overline{QR}$ is $3$, the distance from $Q$ to $\overline{PR}$ is $4$, and the distance from $R$ to $\overline{PQ}$ is $5$. The angle bisector of $\angle PQR$ and the angle bisector of $\angle PRQ$ intersect at $I$. What is the distance from $I$ to $\overline{PR}$? [b]p19.[/b] Maxwell graphs the quadrilateral $|x - 2| + |y + 2| = 6$. What is the area of the quadrilateral? [b]p20.[/b] Uncle Gowri hits a speed bump on his way to the hospital. At the hospital, patients who get a rare disease are given the option to choose treatment $A$ or treatment $B$. Treatment $A$ will cure the disease $\frac34$ of the time, but since the treatment is more expensive, only $\frac{8}{25}$ of the patients will choose this treatment. Treatment $B$ will only cure the disease $\frac{1}{2}$ of the time, but since it is much more aordable, $\frac{17}{25}$ of the patients will end up selecting this treatment. Given that a patient was cured, what is the probability that the patient selected treatment $A$? [b]p21.[/b] In convex quadrilateral $ABCD$, $AC = 28$ and $BD = 15$. Let $P, Q, R, S$ be the midpoints of $AB$, $BC$, $CD$ and $AD$ respectively. Compute $PR^2 + QS^2$. [b]p22.[/b] Charlotte writes the polynomial $p(x) = x^{24} - 6x + 5$. Let its roots be $r_1$, $r_2$, $...$, $r_{24}$. Compute $r^{24}_1 +r^{24}_2 + r^{24}_3 + ... + r^{24}_24$. [b]p23.[/b] In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Let $E$ be a point on $CD$, and let $F$ be the point on $AB$ which lies on the bisector of $\angle BED$. If $FD^2 + EF^2 = 52$, what is the length of $BE$? [b]p24.[/b] In $\vartriangle ABC$, the measure of $\angle A$ is $60^o$ and the measure of $\angle B$ is $45^o$. Let $O$ be the center of the circle that circumscribes $\vartriangle ABC$. Let $I$ be the center of the circle that is inscribed in $\vartriangle ABC$. Finally, let $H$ be the intersection of the $3$ altitudes of the triangle. What is the angle measure of $\angle OIH$ in degrees? [b]p25.[/b] Kaitlyn fully expands the polynomial $(x^2 + x + 1)^{2018}$. How many of the coecients are not divisible by $3$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$x$ and $y$ are integer $5$-digits numbers, such that in the decimal notation, all ten digits are used exactly once. Also $\tan{x}-\tan{y}=1+\tan{x}\tan{y}$, where $x,y$ are angles in degrees. Find maximum of $x$

The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$

$a$ and $b$ are given positive integers. Prove that there are infinitely many positive integers $n$ such that $n^b+1$ doesn't divide $a^n+1$.

An infinite sequence $x_1,x_2,\ldots$ of positive integers satisfies \[ x_{n+2}=\gcd(x_{n+1},x_n)+2006 \] for each positive integer $n$. Does there exist such a sequence which contains exactly $10^{2006}$ distinct numbers?

For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$

Show that there are at least $3$ and at most $4$ powers of $2$ with $m$ digits. For which $m$ are there $4$?

You are given a $2 \times 2$ array with a positive integer in each field. If we add the product of the numbers in the first column, the product of the numbers in the second column, the product of the numbers in the first row and the product of the numbers in the second row, we get $2021$. a) Find possible values for the sum of the four numbers in the table. b) Find the number of distinct arrays that satisfy the given conditions that contain four pairwise distinct numbers in arrays.