Positive integer numbers a and b satisfy $(a^2- 9b^2)^2 - 33b = 1$. a) Prove $|a -3b|\ge 1$. b) Find all pairs of positive integers $(a, b)$ satisfying the equality.

Let $\Phi$ and $\Psi$ denote the Euler totient and Dedekind‘s totient respectively. Determine all $n$ such that $\Phi(n)$ divides $n +\Psi (n)$.

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Prove that for any integer $n$ the number $n (n^2 + 5)$ is divisible by $6$.

Find all possible non-negative integer solution $(x,y)$ of the following equation- $$x! + 2^y =(x+1)!$$ Note: $x!=x \cdot (x-1)!$ and $0!=1$. For example, $5! = 5\times 4\times 3\times 2\times 1 = 120$.

a) Show that there are five integers $A, B, C, D$, and $E$ such that $2018 = A^5 + B^5 + C^5 + D^5 + E^5$ b) Show that there are no four integers $A, B, C$ and $D$ such that $2018 = A^5 + B^5 + C^5 + D^5$

Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

Let $n$ be a positive integer. Prove that $a(n) = n^5 +5^n$ is divisible by $11$ if and only if $b(n) = n^5 · 5^n +1$ is divisible by $11$. [i](Walther Janous)[/i]

Professor Conway collects a total of $58$ midterms from the two sections of his introductory linear algebra course. He notices that the number of midterms from the smaller section is equal to the product of the digits of the number of midterms from his larger section. Assuming that every student handed in a midterm, how many students are there in the smaller section?

[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Euclid eats $\frac17$ of a pie in $7$ seconds. Euler eats $\frac15$ of an identical pie in $10$ seconds. Who eats faster? [b]p2.[/b] Given that $\pi = 3.1415926...$ , compute the circumference of a circle of radius 1. Express your answer as a decimal rounded to the nearest hundred thousandth (i.e. $1.234562$ and $1.234567$ would be rounded to $1.23456$ and $1.23457$, respectively). [b]p3.[/b] Alice bikes to Wonderland, which is $6$ miles from her house. Her bicycle has two wheels, and she also keeps a spare tire with her. If each of the three tires must be used for the same number of miles, for how many miles will each tire be used? [b]p4.[/b] Simplify $\frac{2010 \cdot 2010}{2011}$ to a mixed number. (For example, $2\frac12$ is a mixed number while $\frac52$ and $2.5$ are not.) [b]p5.[/b] There are currently $175$ problems submitted for $EMC^2$. Chris has submitted $51$ of them. If nobody else submits any more problems, how many more problems must Chris submit so that he has submitted $\frac13$ of the problems? [b]p6.[/b] As shown in the diagram below, points $D$ and $L$ are located on segment $AK$, with $D$ between $A$ and $L$, such that $\frac{AD}{DK}=\frac{1}{3}$ and $\frac{DL}{LK}=\frac{5}{9}$. What is $\frac{DL}{AK}$? [img]https://cdn.artofproblemsolving.com/attachments/9/a/3f92bd33ffbe52a735158f7ebca79c4c360d30.png[/img] [b]p7.[/b] Find the number of possible ways to order the letters $G, G, e, e, e$ such that two neighboring letters are never $G$ and $e$ in that order. [b]p8.[/b] Find the number of odd composite integers between $0$ and $50$. [b]p9.[/b] Bob tries to remember his $2$-digit extension number. He knows that the number is divisible by $5$ and that the first digit is odd. How many possibilities are there for this number? [b]p10.[/b] Al walks $1$ mile due north, then $2$ miles due east, then $3$ miles due south, and then $4$ miles due west. How far, in miles, is he from his starting position? (Assume that the Earth is flat.) [b]p11.[/b] When n is a positive integer, $n!$ denotes the product of the first $n$ positive integers; that is, $n! = 1 \cdot 2 \cdot 3 \cdot ... \cdot n$. Given that $7! = 5040$, compute $8! + 9! + 10!$. [b]p12.[/b] Sam's phone company charges him a per-minute charge as well as a connection fee (which is the same for every call) every time he makes a phone call. If Sam was charged $\$4.88$ for an $11$-minute call and $\$6.00$ for a $19$-minute call, how much would he be charged for a $15$-minute call? [b]p13.[/b] For a positive integer $n$, let $s_n$ be the sum of the n smallest primes. Find the least $n$ such that $s_n$ is a perfect square (the square of an integer). [b]p14.[/b] Find the remainder when $2011^{2011}$ is divided by $7$. [b]p15.[/b] Let $a, b, c$, and $d$ be $4$ positive integers, each of which is less than $10$, and let $e$ be their least common multiple. Find the maximum possible value of $e$. [b]p16.[/b] Evaluate $100 - 1 + 99 - 2 + 98 - 3 + ... + 52 - 49 + 51 - 50$. [b]p17.[/b] There are $30$ basketball teams in the Phillips Exeter Dorm Basketball League. In how ways can $4$ teams be chosen for a tournament if the two teams Soule Internationals and Abbot United cannot be chosen at the same time? [b]p18.[/b] The numbers $1, 2, 3, 4, 5, 6$ are randomly written around a circle. What is the probability that there are four neighboring numbers such that the sum of the middle two numbers is less than the sum of the other two? [b]p19.[/b] What is the largest positive $2$-digit factor of $3^{2^{2011}} - 2^{2^{2011}}$? [b]p20.[/b] Rhombus $ABCD$ has vertices $A = (-12,-4)$, $B = (6, b)$, $C = (c,-4)$ and $D = (d,-28)$, where $b$, $c$, and $d$ are integers. Find a constant $m$ such that the line y = $mx$ divides the rhombus into two regions of equal area. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 1[/u] [b]p1.[/b] Alec rated the movie Frozen $1$ out of $5$ stars. At least how many ratings of $5$ out of $5$ stars does Eric need to collect to make the average rating for Frozen greater than or equal to $4$ out of $5$ stars? [b]p2.[/b] Bessie shuffles a standard $52$-card deck and draws five cards without replacement. She notices that all five of the cards she drew are red. If she draws one more card from the remaining cards in the deck, what is the probability that she draws another red card? [b]p3.[/b] Find the value of $121 \cdot 1020304030201$. [u]Round 2[/u] [b]p4.[/b] Find the smallest positive integer $c$ for which there exist positive integers $a$ and $b$ such that $a \ne b$ and $a^2 + b^2 = c$ [b]p5.[/b] A semicircle with diameter $AB$ is constructed on the outside of rectangle $ABCD$ and has an arc length equal to the length of $BC$. Compute the ratio of the area of the rectangle to the area of the semicircle. [b]p6.[/b] There are $10$ monsters, each with $6$ units of health. On turn $n$, you can attack one monster, reducing its health by $n$ units. If a monster's health drops to $0$ or below, the monster dies. What is the minimum number of turns necessary to kill all of the monsters? [u]Round 3[/u] [b]p7.[/b] It is known that $2$ students make up $5\%$ of a class, when rounded to the nearest percent. Determine the number of possible class sizes. [b]p8.[/b] At $17:10$, Totoro hopped onto a train traveling from Tianjin to Urumuqi. At $14:10$ that same day, a train departed Urumuqi for Tianjin, traveling at the same speed as the $17:10$ train. If the duration of a one-way trip is $13$ hours, then how many hours after the two trains pass each other would Totoro reach Urumuqi? [b]p9.[/b] Chad has $100$ cookies that he wants to distribute among four friends. Two of them, Jeff and Qiao, are rivals; neither wants the other to receive more cookies than they do. The other two, Jim and Townley, don't care about how many cookies they receive. In how many ways can Chad distribute all $100$ cookies to his four friends so that everyone is satisfied? (Some of his four friends may receive zero cookies.) [u]Round 4[/u] [b]p10.[/b] Compute the smallest positive integer with at least four two-digit positive divisors. [b]p11.[/b] Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, $BC = 10$ and $AD = 18$. Given that the two circles with diameters $BC$ and $AD$ are tangent, find the perimeter of $ABCD$. [b]p12.[/b] How many length ten strings consisting of only $A$s and Bs contain neither "$BAB$" nor "$BBB$" as a substring? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2934037p26256063]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If $p$ is a prime number, what is the product of elements like $g$ such that $1\le g\le p^2$ and $g$ is a primitive root modulo $p$ but it's not a primitive root modulo $p^2$, modulo $p^2$?($\frac{100}{6}$ points)

$(GBR 2)$ Let $a, b, x, y$ be positive integers such that $a$ and $b$ have no common divisor greater than $1$. Prove that the largest number not expressible in the form $ax + by$ is $ab - a - b$. If $N(k)$ is the largest number not expressible in the form $ax + by$ in only $k$ ways, find $N(k).$

Find all polynomials $ p$ of one variable with integer coefficients such that if $ a$ and $ b$ are natural numbers such that $ a \plus{} b$ is a perfect square, then $ p\left(a\right) \plus{} p\left(b\right)$ is also a perfect square.

Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$. Determine the largest mediocre number.

Let $(a_n)$ be defined by: $$ a_1 = 2, \qquad a_{n+1} = a_n^3 - a_n + 1 $$ Consider positive integers $n,p$, where $p$ is an odd prime. Prove that if $p | a_n$, then $p > n$.

For positive integers $a>b>1$, define \[x_n = \frac {a^n-1}{b^n-1}\] Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers. [i]V. Senderov[/i]

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

[b]p1.[/b] What is the slope of the line perpendicular to the the graph $\frac{x}{4}+\frac{y}{9}= 1$ at $(0, 9)$? [b]p2.[/b] A boy is standing in the middle of a very very long staircase and he has two pogo sticks. One pogo stick allows him to jump $220$ steps up the staircase. The second pogo stick allows him to jump $125$ steps down the staircase. What is the smallest positive number of steps that he can reach from his original position by a series of jumps? [b]p3.[/b] If you roll three fair six-sided dice, what is the probability that the product of the results will be a multiple of $3$? [b]p4.[/b] Right triangle $ABC$ has squares $ABXY$ and $ACWZ$ drawn externally to its legs and a semicircle drawn externally to its hypotenuse $BC$. If the area of the semicircle is $18\pi$ and the area of triangle $ABC$ is $30$, what is the sum of the areas of squares $ABXY$ and $ACWZ$? [img]https://cdn.artofproblemsolving.com/attachments/5/1/c9717e7731af84e5286335420b73b974da2643.png[/img] [b]p5.[/b] You have a bag containing $3$ types of pens: red, green, and blue. $30\%$ of the pens are red pens, and $20\%$ are green pens. If, after you add $10$ blue pens, $60\%$ of the pens are blue pens, how many green pens did you start with? [b]p6.[/b] Canada gained partial independence from the United Kingdom in $1867$, beginning its long role as the headgear of the United States. It gained its full independence in $1982$. What is the last digit of $1867^{1982}$? [b]p7.[/b] Bacon, Meat, and Tomato are dealing with paperwork. Bacon can fill out $5$ forms in $3$ minutes, Meat can fill out $7$ forms in $5$ minutes, and Tomato can staple $3$ forms in $1$ minute. A form must be filled out and stapled together (in either order) to complete it. How long will it take them to complete $105$ forms? [b]p8.[/b] Nice numbers are defined to be $7$-digit palindromes that have no $3$ identical digits (e.g., $1234321$ or $5610165$ but not $7427247$). A pretty number is a nice number with a $7$ in its decimal representation (e.g., $3781873$). What is the $7^{th}$ pretty number? [b]p9.[/b] Let $O$ be the center of a semicircle with diameter $AD$ and area $2\pi$. Given square $ABCD$ drawn externally to the semicircle, construct a new circle with center $B$ and radius $BO$. If we extend $BC$, this new circle intersects $BC$ at $P$. What is the length of $CP$? [img]https://cdn.artofproblemsolving.com/attachments/b/1/be15e45cd6465c7d9b45529b6547c0010b8029.png[/img] [b]p10.[/b] Derek has $10$ American coins in his pocket, summing to a total of $53$ cents. If he randomly grabs $3$ coins from his pocket, what is the probability that they're all different? [b]p11.[/b] What is the sum of the whole numbers between $6\sqrt{10}$ and $7\pi$ ? [b]p12.[/b] What is the volume of a cylinder whose radius is equal to its height and whose surface area is numerically equal to its volume? [b]p13.[/b] $15$ people, including Luke and Matt, attend a Berkeley Math meeting. If Luke and Matt sit next to each other, a fight will break out. If they sit around a circular table, all positions equally likely, what is the probability that a fight doesn't break out? [b]p14.[/b] A non-degenerate square has sides of length $s$, and a circle has radius $r$. Let the area of the square be equal to that of the circle. If we have a rectangle with sides of lengths $r$, $s$, and its area has an integer value, what is the smallest possible value for $s$? [b]p15.[/b] How many ways can you arrange the letters of the word "$BERKELEY$" such that no two $E$'s are next to each other? [b]p16.[/b] Kim, who has a tragic allergy to cake, is having a birthday party. She invites $12$ people but isn't sure if $11$ or $12$ will show up. However, she needs to cut the cake before the party starts. What is the least number of slices (not necessarily of equal size) that she will need to cut to ensure that the cake can be equally split among either $11$ or $12$ guests with no excess? [b]p17.[/b] Tom has $2012$ blue cards, $2012$ red cards, and $2012$ boxes. He distributes the cards in such a way such that each box has at least $1$ card. Sam chooses a box randomly, then chooses a card randomly. Suppose that Tom arranges the cards such that the probability of Sam choosing a blue card is maximized. What is this maximum probability? [b]p18.[/b] Allison wants to bake a pie, so she goes to the discount market with a handful of dollar bills. The discount market sells different fruit each for some integer number of cents and does not add tax to purchases. She buys $22$ apples and $7$ boxes of blueberries, using all of her dollar bills. When she arrives back home, she realizes she needs more fruit, though, so she grabs her checkbook and heads back to the market. This time, she buys $31$ apples and $4$ boxes of blueberries, for a total of $60$ cents more than her last visit. Given she spent less than $100$ dollars over the two trips, how much (in dollars) did she spend on her first trip to the market? [b]p19.[/b] Consider a parallelogram $ABCD$. Let $k$ be the line passing through A and parallel to the bisector of $\angle ABC$, and let $\ell$ be the bisector of $\angle BAD$. Let $k$ intersect line $CD$ at $E$ and $\ell$ intersect line $CD$ at $F$. If $AB = 13$ and $BC = 37$, find the length $EF$. [b]p20.[/b] Given for some real $a, b, c, d,$ $$P(x) = ax^4 + bx^3 + cx^2 + dx$$ $$P(-5) = P(-2) = P(2) = P(5) = 1$$ Find $P(10).$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P(x)=x^{2016}+2x^{2015}+...+2017,Q(x)=1399x^{1398}+...+2x+1$. Prove that there are strictly increasing sequances $a_i,b_i, i=1,...$ of positive integers such that $gcd(a_i,a_{i+1})=1$ for each $i$. Moreover, for each even $i$, $P(b_i) \nmid a_i, Q(b_i) | a_i$ and for each odd $i$, $P(b_i)|a_i,Q(b_i) \nmid a_i$ Proposed by [i]Shayan Talaei[/i]

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$ \[a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n\] Show that $2003$ is not divisor of any of the terms in these two sequences.

For every natural number $n$ we define $f(n)$ by the following rule: $f(1) = 1$ and for $n>1$ then $f(n) = 1 + a_1 \cdot p_1 + \ldots + a_k \cdot p_k$, where $n = p_1^{a_1} \cdots p_k^{a_k}$ is the canonical prime factorisation of $n$ ($p_1, \ldots, p_k$ are distinct primes and $a_1, \ldots, a_k$ are positive integers). For every positive integer $s$, let $f_s(n) = f(f(\ldots f(n))\ldots)$, where on the right hand side there are exactly $s$ symbols $f$. Show that for every given natural number $a$, there is a natural number $s_0$ such that for all $s > s_0$, the sum $f_s(a) + f_{s-1}(a)$ does not depend on $s$.