Miscellaneous Number Theory: all problems

Suppose

What is the value of $x$?

Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome: it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period?

A palindrome, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x + 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?

Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top?

A set of numbers is called sum-free if whenever $x$ and $y$ are (not necessarily distinct) elements of the set, $x + y$ is not an element of the set. For example, $\{1, 4, 6\}$ and the empty set are sum-free, but $\{2, 4, 5\}$ is not. What is the greatest possible number of elements in a sum-free subset of $\{1, 2, 3, \ldots, 20\}$?

Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1) = 1$, $d(2) = 2$, and $d(12) = 6$. (This function is known as the divisor function.) Let

There is a unique positive integer $N$ such that $f(N) > f(n)$ for all positive integers $n \ne N$. What is the sum of the digits of $N$?

A palindrome between $1000$ and $10{,}000$ is chosen at random. What is the probability that it is divisible by $7$?

How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2, 2, 3, 3, 5, 5, 5$?

Alice and Bob are playing a game with $n$ coins in a pile. Every turn, a player can take $1$ or $4$ coins from the pile, and Alice goes first. The player that takes the last coin wins. How many values of $n$ less than or equal to $2024$ are there such that no matter what Alice does, Bob has a winning strategy?

A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base $10$ with no leading zeros. A $6$-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\dfrac{n}{11}$ is also a palindrome?

What is the ones digit of the sum

(Recall that $\lfloor x \rfloor$ represents the greatest integer less than or equal to $x$.)

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?

Ninety-four bricks, each measuring $4'' \times 10'' \times 19''$, are to be stacked one on top of another to form a tower $94$ bricks tall. Each brick can be oriented so it contributes $4''$ or $10''$ or $19''$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks?

Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r$th root of $N$. What is

An array of numbers is constructed beginning with the numbers $-1 \quad 3 \quad 1$ in the top row. Each adjacent pair of numbers is summed to produce a number in the next row. Each row begins and ends with $-1$ and $1$, respectively.

If the process continues, one of the rows will sum to $12{,}288$. In that row, what is the third number from the left?

Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5$, $n$, and $n + 1$ cents, $91$ cents is the greatest postage that cannot be formed.