The coordinates of the center of the circle are (0, 593/16) = (0, 37.0625).  Solution Without Calculus:  The equation of the circle can be expressed as $x^2+(y-a)^2=9$.  We must determine the value of $a$ for which the circle and the parabola with equation $y=1+4x^2$ intersect in exactly two locations.  In other words, we want the system […]

The largest possible number in the collection is 18.  In order to have a mean of 10, the sum of all ten numbers must be 100.   Additionally, with the range equal to 10, the maximum number in the set is 10 more than the minimum.   The minimum number in the collection must be less than […]

The area of the region where their paths overlap is ${288\over 83}\text{ ft}^2 \approx 3.47 \text{ ft}^2$. In our solution, we will use the following notational convention $$XY = \text{ the length of the side joining } X \text{ and } Y.$$  We will take advantage of various proportions that result from similar triangles found […]

The area of the shaded region is $\pi$. In our solution, we will first obtain a more general result where the chord of length 2 is replaced with an unspecified length $h$ which must of course be no bigger than the radius of the largest semicircle. Let $a$, $b$, and $c$ be the radii of […]

Most of the numbers can be created in more than one way and all of them can be created using all four numbers.  Below, we provide a single solution for each of the numbers. 0 = 2 + 3 – 2 – 3 1 = (2 + 3) / (2 + 3) 2 = 2^3 […]

Both diagrams have multiple solutions.  Below we provide a single example of a solution for each.    

Below is one possible solution:

The answer is:  840 polar rectangles.   For the sake of the explanation, we will define the “bottom” side of a polar rectangle to be the side formed by the smaller of the two arcs and the “top” to be the side formed by the larger arc.  With the polar rectangle oriented such that the top is facing […]

Answer:  The last remaining coin will be a dime. To see this, we observe that no matter what two coins are selected from the jug, we will always end up with one less coin in the jug after each selection as one coin is always added to the jug after two are removed.  Additionally, in each […]

Answer:  The coins that are in a position that is a perfect square will be tails up, while all other coins will be heads up.   Since all coins were heads up at the start, any quarter that was turned over an even number of times will be heads up at the end of the procedure and […]