Problems

Let $N$ be the unique positive integer such that dividing $273436$ by $N$ leaves a remainder of $16$ and dividing $272760$ by $N$ leaves a remainder of $15$ . What is the tens digit of $N$ ?

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 $\{1,4,5\}$ is not. What is the greatest possible number of elements in a sum-free subset of …

Jerry wrote down the ones digit of each of the first $2025$ positive squares: $1,4,9,6,5,6,\dots$ . What is the sum of all the numbers Jerry wrote down?

The value of the two-digit number $\underline{a}~\underline{b}$ in base seven equals the value of the two-digit number $\underline{b}~\underline{a}$ in base nine. What is $a+b?$

Emmy says to Max, "I ordered $36$ math club sweatshirts today." Max asks, "How much did each shirt cost?" Emmy responds, "I'll give you a hint. The total cost was $\$ \underline A~\underline B~\underline B.\underline B~\underline A$ , where $A$ and $B$ are digits and $A \neq 0 …

A rectangular grid of squares has $141$ rows and $91$ columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from $1$ through $141 \times 91 = 12{,}831$ into the squares. Horace fills the grid horizontally: he puts $1$ through $91$ in order from left to right into …

What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes?

The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?

What is the least value of $n$ such that $n!$ is a multiple of $2024$ ?

The product of three integers is $60$ . What is the least possible positive sum of the three integers?

How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$ ?

Let $M$ be the greatest integer such that both $M+1213$ and $M+3773$ are perfect squares. What is the units digit of $M$ ?

There are exactly $K$ positive integers $5 \leq b \leq 2024$ such that the base- $b$ integer $2024_{b}$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$ ?

Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold: - If $x$ and $y$ are distinct elements of $S$ , then $|x-y| > 2.$ - If $x$ and $y$ are distinct odd elements of $S$ , then $|x-y| > 6.$ What is the maximum possible number of elements in $S$ ?

A rectangle has integer length sides and an area of 2024. What is the least possible perimeter of the rectangle?

What is the remainder when $7^{2024}+7^{2025}+7^{2026}$ is divided by $19$ ?

Let $N$ be the product of all the positive integer divisors of $42$ . What is the units digit of $N$ ?

How many different remainders can result when the $100$ th power of an integer is divided by $125$ ?

Let $$P(m)=\frac{m}{2}+\frac{m^2}{4}+\frac{m^4}{8}+\frac{m^8}{8}$$ How many of the values $P(2022)$ , $P(2023)$ , $P(2024)$ , and $P(2025)$ are integers?

How many positive perfect squares less than $2023$ are divisible by $5$ ?

A digital display shows the current date as an $8$ -digit integer consisting of a $4$ -digit year, followed by a $2$ -digit month, followed by a $2$ -digit date within the month. For example, Arbor Day this year is displayed as $20230428.$ For how many dates in $2023$ does each digit appear an even number of times in …

How many three-digit positive integers $N$ satisfy the following properties? - The number $N$ is divisible by $7$ . - The number formed by reversing the digits of $N$ is divisible by $5$ .

If the positive integer $c$ has positive integer divisors $a$ and $b$ with $c = ab$ , then $a$ and $b$ are said to be $\textit{complementary}$ divisors of $c$ . Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $20$ and another pair of complementary divisors that differ …

Let $L_1 = 1$ , $L_2 = 3$ , and $L_{n+2} = L_{n+1}+L_n$ for $n \geq 1$ . How many terms in the sequence $L_1, L_2, L_3, \cdots, L_{2023}$ are even?

What is the units digit of $2022^{2023} + 2023^{2022}$ ?