What value of $a$ satisfies $27x^3 - 16\sqrt{2}=(3x-2\sqrt{2})(9x^2 + 12x\sqrt{2}+a)$?

If the first $25$ positive integers are multiplied together, in how many zeros does the product terminate?

What is the smallest positive number $x$ for which $\left(16^\sqrt{2}\right)^x$ represents a positive integer?

Of the pairs of positive integers $(x, y)$ that satisfies $3x+7y=188$, which ordered pair has the least positive difference $x-y$?

What is the smallest positive integer greater than $5$ which leaves a remainder of $5$ when divided by each of $6$, $7$, $8$, and $9$?

What are all the ordered pairs of positive numbers $(x, y)$ for which $x=\sqrt{2y}$ and $y=\sqrt{x}$?

How many minutes past 4 o'clock are the hands of a standard 12-hour clock first perpendicular to each other?

I first drove 16 km at 48 km/hr, then I drove 20 km at 40 km/hr, and finally I drove 24 km at 36 km/hr. What was my average speed in km/hr, for the entire trip?

How many different triangles have vertices selected from the seven points (-4, 0), (-2, 0), (0,0), (2,0), (4,0), (0,2), and (0,4)?

Three circular cylinders are strapped together as shown. The cross-section of each cylinder is a circle of radius 1. Presuming that the strap used to bind the cylinders together has no thickness and no extra length, how long is the binding strap?

What are the coordinates of the reflection of (6,0) across the graph of $y=3x$?

If the area of a circle's inscribed square is 60, what is the area of its circumscribed square?

What are all values of $x$ for which $log_x\sqrt{x+12}>1$?

Determine the units digit of the sum $0!+1!+2!+\cdots+n!+\cdots+20!$?

What is the area of a trapezoid the lengths of whose bases are 10 and 16, and the lengths of whose legs are 8 and 10?

A diagonal of a square intersects a segment that connects one vertex of the square to the midpoint of an opposite side, as shown. If the length of the shorter section of the diagonal is 2, what is the area of the square?

What real value of $x$ satisfies $\sqrt{5x} - \sqrt{2x} = 5-2$?