Problems

In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is $k$ , where $0 < k < 1.$ The spaces between squares are alternately shaded as …

The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of 4, 4, and 5 is $$\frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}$$ What is the harmonic mean of all the real roots of …

Triangle $\triangle ABC$ has side lengths $AB = 80$ , $BC = 45$ , and $AC = 75$ . The bisector of $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P.$ What is $BP$ ?

Call a positive integer fair if no digit is used more than once, it has no 0s, and no digit is adjacent to two greater digits. For example, $196$ , $23$ , and $12463$ are fair, but $1546$ , $320$ , and $34321$ are not fair. How many fair positive integers are there?

Let $f(n)=n^3-5n^2+2n+8$ and $g(n)=n^3-6n^2+5n+12.$ What is the sum of all integers $n$ such that $\tfrac{f(n)}{g(n)}$ is an integer?

The sum $$\sum_{k=1}^{\infty} \frac{1}{k^3 + 6k^2 + 8k}$$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. What is $a + b$ ?

What is the value of $9901\cdot101-99\cdot10101?$

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 $M$ ?

The first three terms of a geometric sequence are the integers $a, 720$ and $b$ , where $a < 720 < b$ . What is the sum of the digits of the least possible value of $b$ ?

The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5$ . The numbers in positions $(5, 5)$ , $(2, 4)$ , $(4, 3)$ and $(3, 1)$ are $0$ , $48$ , $16$ , and $12$ , respectively. What number is in position $(1, 2)$ ? …

Integers $a$ , $b$ , and $c$ satisfy $ab + c = 100$ , $bc + a = 87$ , and $ca + b = 60$ . What is $ab + bc + ca?$

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 $N$ ?

Real numbers $a, b,$ and $c$ have arithmetic mean $0$ . The arithmetic mean of $a^2, b^2,$ and $c^2$ is $10$ . What is the arithmetic mean of $ab, ac,$ and $bc$ ?

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

A group of $16$ people will be partitioned into $4$ indistinguishable $4$ -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$ , where $r$ and $M$ are positive integers and $M$ is not divisible by $3$ . What is …

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 digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$ ?

A square of area $2$ is inscribed in a square of area $3$ , creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?

An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1.$ An example showing $8$ circles is displayed …

A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?

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 …

The numbers $16$ and $25$ are a pair of consecutive positive squares whose difference is $9$ . How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$ ?