In isosceles trapezoid $ABCD$, shown here, $AB = 4$ units and $CD = 10$ units. Points $E$ and $F$ are on $\overline{CD}$ with $\overline{BE}$ parallel to $\overline{AD}$ and $\overline{AF}$ parallel to $\overline{BC}$. $\overline{AF}$ and $\overline{BE}$ intersect at point $G$. What is the ratio of the area of triangle $EFG$ to the area of trapezoid $ABCD$? Express your answer as a common fraction.

A box contains $r$ red balls and $g$ green balls. When $r$ more red balls are added to the box, the probability of drawing a red ball at random from the box increases by $25\%$. What was the probability of randomly drawing a red ball from the box originally? Express your answer as a common fraction.

The game of Connex contains one 4-unit piece, two identical 3-unit pieces, three identical 2-unit pieces and four identical 1-unit pieces. How many different arrangements of pieces will make a 10-unit segment? The 10-unit segments consisting of the pieces 4-3-2-1 and 1-2-3-4 are two such arrangements to include.

Let $\triangle ABC$ be a right triangle whose three sides' lengths are all integers. Prove among its three sides' lengths, at lease one is a multiple of $3$, one is a multiple of $4$, and one is a multiple of $5$. (Note: they can be the same side. For example, in the $5-12-13$, $12$ is both a multiple of $3$ and $4$.)

The dart board shown here contains 20 uniquely numbered sectors. When Malaika aims for a particular number, she hits it half the time. The other half of the time, she randomly hits an adjacent number on either side with equal probability. The number in the sector that her dart hits is the number of points scored. Trying to earn the highest possible score, Malaika decides to aim for the same number for each of her next 20 throws. Based on the given information, for which number should Malaika aim?

In rectangle ABCD, BC = 2AB. Points O and M are the midpoints of $\overline{AD}$ and $\overline{BC}$ , respectively. Point P bisects $\overline{AO}$ . If OB = $6\sqrt{2}$ units, what is the area of $\triangle{NOP}$?

In right $\triangle{ABC}$, shown here, AC = 24 units and BC = 7 units. Point D lies on $\overline{AB}$ so that $\overline{CD} \perp \overline{AB}$. The bisector of the smallest angle of $\triangle{ABC}$ intersects $\overline{CD}$ at point E. What is the length of $\overline{ED}$ ? Express your answer as a common fraction.

The single-digit prime numbers 2, 3, 5 and 7 are used to replace $a$, $b$, $c$ and $d$ in the multiplication table shown here. The four products are found and then added together. What is the greatest possible value of this sum?

The circular pizza, shown here, is cut 5 times with straight line cuts before being removed from the pan. What is the maximum number of pieces that can be made which contain none of the pizza's outer crust, located around its circumference?

After tossing a red, then a green and, finally, a white standard six-faced die, Patrick used the numbers showing on the upper faces of each die, in order, to create the incorrect equation below, such that red - green = white. By rotating each die a quarter turn in some direction so that the number on the top face moves to a lateral face, he finds that he can make a correct equation. Given that the opposite faces of a die have a sum of 7, how many correct equations are possible?

A square prism has dimensions $5' \times 5' \times 10'$, where ABCD is a square. AP = ER = 2 ft and QC = SG = 1 ft. The plane containing $\overline{PQ}$ and $\overline{RS}$ slices the original prism into two new prisms. What is the volume of the larger of these two prisms?

How many collections of six positive, odd integers have a sum of $18$? Note that $1 + 1 + 1 + 3 + 3 + 9$ and $9 + 1 + 3 + 1 + 3 + 1$ are considered to be the same collection.