Algebraic Number Theory: all problems

There is a positive integer $n$ such that $(n + 1)! + (n + 2)! = n! \cdot 440$. What is the sum of the digits of $n$?

For all integers $n \ge 9$, the value of

is always which of the following?

Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n + 1)$?

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers?

When a student multiplied the number $66$ by the repeating decimal

where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $1.\underline{a}\,\underline{b}$. Later he found that his answer is $0.5$ less than the correct answer. What is the 2-digit number $\underline{a}\,\underline{b}$?

What is the remainder when $2^{202} + 202$ is divided by $2^{101} + 2^{51} + 1$?

Compute

A sequence of numbers is defined by $D_0 = 0$, $D_1 = 0$, $D_2 = 1$ and $D_n = D_{n-1} + D_{n-3}$ for $n \ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021}, D_{2022}, D_{2023})$, where $E$ denotes even and $O$ denotes odd?

Which of the following is equivalent to

The ages of Jonie's four cousins are distinct single-digit positive integers. Two of the cousins' ages multiplied together give $24$, while the other two multiply to $30$. What is the sum of the ages of Jonie's four cousins?

Let $f$ be a function defined on the set of positive rational numbers with the property that $f(a \cdot b) = f(a) + f(b)$ for all positive rational numbers $a$ and $b$. Suppose that $f$ also has the property that $f(p) = p$ for every prime number $p$. For which of the following numbers $x$ is $f(x) < 0$?

Let $f$ be the unique function defined on the positive integers such that

for all positive integers $n$, where the sum is taken over all positive divisors of $n$. What is $f(2023)$?

Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?

In the state of Coinland, coins have values $6$, $10$, and $15$ cents. Suppose $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of $x$?

When a standard 6-sided die is rolled $n$ times, the product of the $n$ numbers rolled can be any of $936$ possible values. What is $n$?

Square $ABCD$ has sides of length $4$. Points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{CD}$, respectively, with $AP = \frac{8}{5}$ and $DQ = \frac{10}{3}$. A path begins along the segment from $P$ to $Q$ and continues by reflecting against the sides of $ABCD$ (with congruent incoming and outgoing angles). If the path hits a vertex of the square, it terminates there; otherwise it continues forever. At which vertex does the path terminate?

Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$?

Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3 : 2 : 1$, what is the least possible total for the number of bananas?

There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \cdots < a_k$ such that

What is $k$?

For how many integers $n$ is $\dfrac{n}{20 - n}$ the square of an integer?

The number $2013$ is expressed in the form

where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?

For $k > 0$, let $I_k = 10\ldots064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?

Two fair dice, each with at least $6$ faces, are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10$, and the probability of rolling a sum of $12$ is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?

How many ordered pairs $(m, n)$ of positive integers, with $m \ge n$, have the property that their squares differ by $96$?

For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \cdots + n$?

Let $\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\mathcal{T}$ be the set of all numbers of the form $\frac{x - 256}{1000}$, where $x$ is in $\mathcal{S}$. In other words, $\mathcal{T}$ is the set of numbers that result when the last three digits of each number in $\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\mathcal{T}$ is divided by $1000$.

For each integer $n \ge 2$, let $S_n$ be the sum of all products $jk$, where $j$ and $k$ are integers and $1 \le j < k \le n$. What is the sum of the $10$ least values of $n$ such that $S_n$ is divisible by $3$?

Find the largest integer $n$ satisfying the following conditions:

(i) $n^{2}$ can be expressed as the difference of two consecutive cubes;

(ii) $2n + 79$ is a perfect square.