Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$. [i]Proposed by Ryan Alweiss[/i]

Let \(d(n)\) denote the number of positive divisors of \(n\). The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined as follows: \(a_0=1\), and for all integers \(n\ge1\), \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\] Show that for all integers \(n\ge1\), we have \(a_n\le3n\). [i]Proposed by Karthik Vedula[/i]

How many $8$-digit numbers in base $4$ formed of the digits $1,2, 3$ are divisible by $3$?

[b]p1.[/b] Fluffy the Dog is an extremely fluffy dog. Because of his extreme fluffiness, children always love petting Fluffy anywhere. Given that Fluffy likes being petted $1/4$ of the time, out of $120$ random people who each pet Fluffy once, what is the expected number of times Fluffy will enjoy being petted? [b]p2.[/b] Andy thinks of four numbers $27$, $81$, $36$, and $41$ and whispers the numbers to his classmate Cynthia. For each number she hears, Cynthia writes down every factor of that number on the whiteboard. What is the sum of all the different numbers that are on the whiteboard? (Don't include the same number in your sum more than once) [b]p3.[/b] Charles wants to increase the area his square garden in his backyard. He increases the length of his garden by $2$ and increases the width of his garden by $3$. If the new area of his garden is $182$, then what was the original area of his garden? [b]p4.[/b] Antonio is trying to arrange his flute ensemble into an array. However, when he arranges his players into rows of $6$, there are $2$ flute players left over. When he arranges his players into rows of $13$, there are $10$ flute players left over. What is the smallest possible number of flute players in his ensemble such that this number has three prime factors? [b]p5.[/b] On the AMC $9$ (Acton Math Competition $9$), $5$ points are given for a correct answer, $2$ points are given for a blank answer and $0$ points are given for an incorrect answer. How many possible scores are there on the AMC $9$, a $15$ problem contest? [b]p6.[/b] Charlie Puth produced three albums this year in the form of CD's. One CD was circular, the second CD was in the shape of a square, and the final one was in the shape of a regular hexagon. When his producer circumscribed a circle around each shape, he noticed that each time, the circumscribed circle had a radius of $10$. The total area occupied by $1$ of each of the different types of CDs can be expressed in the form $a + b\pi + c\sqrt{d}$ where $d$ is not divisible by the square of any prime. Find $a + b + c + d$. [b]p7.[/b] You are picking blueberries and strawberries to bring home. Each bushel of blueberries earns you $10$ dollars and each bushel of strawberries earns you $8$ dollars. However your cart can only fit $24$ bushels total and has a weight limit of $100$ lbs. If a bushel of blueberries weighs $8$ lbs and each bushel of strawberries weighs $6$ lbs, what is your maximum profit. (You can only pick an integer number of bushels) [b]p8.[/b] The number $$\sqrt{2218 + 144\sqrt{35} + 176\sqrt{55} + 198\sqrt{77}}$$ can be expressed in the form $a\sqrt5 + b\sqrt7 + c\sqrt{11}$ for positive integers $a, b, c$. Find $abc$. [b]p9.[/b] Let $(x, y)$ be a point such that no circle passes through the three points $(9,15)$, $(12, 20)$, $(x, y)$, and no circle passes through the points $(0, 17)$, $(16, 19)$, $(x, y)$. Given that $x - y = -\frac{p}{q}$ for relatively prime positive integers $p$, $q$, Find $p + q$. [b]p10.[/b] How many ways can Alfred, Betty, Catherine, David, Emily and Fred sit around a $6$ person table if no more than three consecutive people can be in alphabetical order (clockwise)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the integer $\sqrt[5]{55^5 + 3183^5 + 28969^5 + 85282^5}$.

A prime number $p$ and integers $x, y, z$ with $0 < x < y < z < p$ are given. Show that if the numbers $x^3, y^3, z^3$ give the same remainder when divided by $p$, then $x^2 + y^2 + z^2$ is divisible by $x + y + z.$

Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right)$. $\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 99$

Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

Arithmetical progression $a_1, a_2, a_3, a_4,...$ contains $a_1^2 , a_2^2$ and $a_3^2$ at some positions. Prove that all terms of this progression are integers.

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

Three times Dick's age plus Tom's age equals twice Harry's age. Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. Their respective ages are relatively prime to each other. The sum of the squares of their ages is $\textbf{(A) }42\qquad\textbf{(B) }46\qquad\textbf{(C) }122\qquad\textbf{(D) }290\qquad \textbf{(E) }326$

Let $p$ be a prime number and $k$ be a positive integer. Let \[t=\sum_{i=0}^\infty\bigg\lfloor\frac{k}{p^i}\bigg\rfloor.\]a) Let $f(x)$ be a polynomial of degree $k$ with integer coefficients such that its leading coefficient is $1$ and its constant is divisible by $p.$ prove that there exists $n\in\mathbb{N}$ for which $p\mid f(n),$ but $p^{t+1}\nmid f(n).$ b) Prove that the statement above is sharp, i.e. there exists a polynomial $g(x)$ of degree $k,$ integer coefficients, leading coefficient $1$ and constant divisible by $p$ such that if $p\mid g(n)$ is true for a certain $n\in\mathbb{N},$ then $p^t\mid g(n)$ also holds. [i]Proposed by Kristóf Szabó, Budapest[/i]

Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.

Let $a_1=1$, $a_2=2^{a_1}$, $a_3=3^{a_2}$, $a_4=4^{a_3}$, $\dots$, $a_9 = 9^{a_8}$. Find the last two digits of $a_9$.

Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.) What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)

Determine all natural numbers $n \geq 2$ with at most four natural divisors, which have the property that for any two distinct proper divisors $d_1$ and $d_2$ of $n$, the positive integer $d_1-d_2$ divides $n$.

Find the smallest positive integer $n$ such that both $n^3-n$ and $(n+1)^3-(n+1)$ are divisible by $2025$. [i]V. Kamianetski[/i]

[b]p1[/b] How many two-digit positive integers with distinct digits satisfy the conditions that 1) neither digit is $0$, and 2) the units digit is a multiple of the tens digit? [b]p2[/b] Mirabel has $47$ candies to pass out to a class with $n$ students, where $10\le n < 20$. After distributing the candy as evenly as possible, she has some candies left over. Find the smallest integer $k$ such that Mirabel could have had $k$ leftover candies. [b]p3[/b] Callie picks two distinct numbers from $\{1, 2, 3, 4, 5\}$ at random. The probability that the sum of the numbers she picked is greater than the sum of the numbers she didn’t pick is $p$. $p$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$. Find $a + b$.

Let $n > 1$ be a natural number. We write out the fractions $\frac{1}{n}$, $\frac{2}{n}$, $\dots$ , $\dfrac{n-1}{n}$ such that they are all in their simplest form. Let the sum of the numerators be $f(n)$. For what $n>1$ is one of $f(n)$ and $f(2015n)$ odd, but the other is even?

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square. Proof: there exists $c_1, c_2 \in \mathbb{R}^{+}$ s.t. for any "CGMO sequence" $(a_n)_{n \ge 1}$ , there is a positive integer $N$ that satisfies any $n \ge N$, $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$.

Let $k$ be a positive integer. Prove that there exists a positive integer $\ell$ with the following property: if $m$ and $n$ are positive integers relatively prime to $\ell$ such that $m^m\equiv n^n \pmod{\ell}$, then $m\equiv n \pmod k$.

Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121.

Find all pairs of positive integers $(m,n)$ such that $$lcm(1,2,\dots,n)=m!$$ where $lcm(1,2,\dots,n)$ is the smallest positive integer multiple of all $1,2,\dots n-1$ and $n$.

Find all positive integers $k<202$ for which there exist a positive integers $n$ such that $$\bigg {\{}\frac{n}{202}\bigg {\}}+\bigg {\{}\frac{2n}{202}\bigg {\}}+\cdots +\bigg {\{}\frac{kn}{202}\bigg {\}}=\frac{k}{2}$$