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1.

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?

2.

Suppose that the symbol $x \ @ \ y$ means:

$\frac{1}{x} + \frac{1}{y}.$

If $2 \ @ \ (3 \ @ \ x) = 4,$

what is $x?$

3.

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

4.

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?

5.

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 $a + b + c?$

6.

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?

7.

Find the value of the positive number:

$\sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}}$

8.

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

9.

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

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

10.

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 $y?$