What 2-digit number has the property that if you subtract 5 times its tens digit from it, the result is the 2-digit number with the digits swapped?

Suppose that the symbol \(x \ @ \ y\) means:

\[ \frac{1}{x} + \frac{1}{y}. \]

If \[2 \ @ \ (3 \ @ \ x) = 4,\]

what is \(x?\)

What is the sum of all odd 2-digit numbers?

Starting from a point on the ground, a ball is thrown straight up to a height of \(1\) meter. It bounces up and down vertically, each time reaching a maximum height that is \(\frac{5}{6}\) its previous maximum height. How many total meters will the ball travel, if it keeps bouncing forever?

Simplify

\[ \sqrt{15 + 4 \sqrt{14}} \]

and rewrite it in simplest radical form as

\[a \sqrt{b} + \sqrt{c}, \]

where \(a,\) \(b,\) and \(c\) are integers. What is the value of \(a + b + c?\)

Elijah counts \(31\) creepy-crawlers, all of which are spiders (which have \(8\) legs) or ants (which have \(6\)) legs. The creepy-crawlers have \(214\) legs altogether. How many spiders are there?

Find the value of the positive number:

\[ \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}} \]

A square's area plus its perimeter equals \(221.\) What is the side length of the square?

A straight line passes through the points

\[ \left( -10, \frac{1}{3} \right) \]

and

\[ \left( -2, \frac{17}{3} \right). \]

What is the \(y\)-intercept of another line perpendicular to this first line which passes through the midpoint between

\[ \left( -10, \frac{1}{3} \right) \]

and

\[ \left( -2, \frac{17}{3} \right)? \]

A triangle has corners \(A(0, 0),\) \(B(12, 0),\) and \(C(0, 6).\) If the midpoint of \(AB\) is connected to \(C\) and the midpoint of \(AC\) is connected to \(B,\) they intersect at a point with coordinates \((x, y).\) What is \(y?\)