A circle has radius 4 and perpendicular diameters `bar (AC)` and `bar (BD)`. Find the area of the shaded region.

Point `M` is the midpoint of `OA`, where `O` is the origin and `A(2, 5)`. Point `P (p, q)` is in the first quadrant and on the perpendicular bisector of `OA`. If `PM = AM`, find `p + q`.

The sum of the prime factors of `2025` is `3+3+3+3+5+5=22`.

`22` is not prime, so we continue. The sum of the prime factors of `22` is `2+11=13`.

`13` is prime, so we can say `13` is the rootprime of `2025`.

How many of the years since `2010` inclusive until now (`2025`) have a rootprime of `5`?

A small country has exactly 15 cities, with no 3 cities on the same straight line. Straight roads joining each pair of cities are constructed. Find the number of roads.

The first term of an arithmetic sequence is 1 and the fourth term is 13. Find the tenth term.

Suppose `F(n)` represents the number of positive factors of the integer `n`. For example, `F(8) = 4`, since the factors of `8` are `1, 2, 4,` and `8`. Find `F(168)`.

Consider the following rule to determine the terms of a sequence:

1. The first term is 20.

2. If a term, `x`, is even, then the next term is `1/2x`.

3. If a term, `x`, is odd, then the next term is `3x + 1`.

Find the value of the 900 ^th term.

Let `X` be a 2-digit number and let `Y` be the number with the digits reversed. Find the 2-digit number that satisfies `3X − 4Y = 8`.

Evaluate `(−1)^(1!) + (−1)^(2!) + (−1)^(3!) + ⋯ + (−1)^(100!)`.

Note: `7!` (called seven factorial) is equivalent to `7xx6xx5xx4xx3xx2xx1`, similarly `4! =4xx3xx2xx1`.

A Mersenne prime number is one of the form `m=2^p − 1` where `m` is prime. The number `2^44497 − 1` is a Mersenne Prime. What is its units digit?

Alysha spends five weeks training for a distance race. After keeping track of the number of hours for the five weeks, Alysha notices the following:
a. The first week she trained 2 more hours than the average for all five weeks.
b. On the second week, she trained for 36 hours.
c. On the third week, she trained 7 more hours than the average of the first two weeks.
d. On the fourth week, she trained 8 hours less than the average of weeks two and three.
e. On the fifth week, she trained 2 hours less than the fourth week.
How many hours did Alysha train for in total over the five weeks?