In the five-sided star shown, the letters $A$, $B$, $C$, $D$, and $E$ are replaced by the numbers $3$, $5$, $6$, $7$, and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $AB$, $BC$, $CD$, $DE$, and $EA$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?
Nine consecutive positive even integers are entered into the 3 $\times$ 3 grid shown so that the sums of the three numbers in each row, each column and each diagonal are the same. What is the average value of the five numbers that are missing? Express your answer as a decimal to the nearest tenth.
Abigail, Bartholomew and Cromwell play a game in which they take turns adding 1, 2, 3 or 4 to a sum in order to create an increasing sequence of primes. For example, Abigail must start with either 2 or 3. If she chooses 2, then Bartholomew can add 1 to make 3, or he can add 3 to make 5. If Bartholomew makes 3, then Cromwell can add 2 to make 5, or he can add 4 to make 7. Abigail, Bartholomew and Cromwell take turns, in that order, until no more primes can be made, and the game ends. The player who makes the last prime wins. If Bartholomew wins, how many primes were made?
Four nickels, one penny and one dime were divided among three piggy banks so that each bank received two coins. Labels indicating the amount in each bank were made (6 cents, 10 cents and 15 cents), but when the labels were put on the banks, no bank had the correct label attached. Soraya shook the piggy bank labeled as 15 cents, and out fell a penny. What was the actual combined value of the two coins contained in the piggy bank that was labeled 6 cents?
In some languages, every consonant must be followed by a vowel. How many seven-letter "words" can be made from the Hawaiian word MAKAALA if each consonant must be followed by a vowel?
The sum of five consecutive, positive even integers is a perfect square. What is the smallest possible integer that could be the least of these five integers?
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?
Call a positive integer squarish if it contains the digits of the squares of its digits in order but not necessarily contiguous. For example, $14263$ contains $1^2 = 1$, $4^2 = 16$ and $2^2 = 4$. However, it is not squarish because it does not contain $3^2 = 9$, and $6^2 = 36$ is not in order. What is the smallest squarish number that includes at least one digit greater than $1$?
Marti lives in New York and wishes to call her friend Kathy who lives in Honolulu. The chart below shows the times in several cities when it is 12:00 noon in New York. If Marti calls Kathy when the time is 6:30 p.m. in New York, what time is it in Honolulu?
The apples collected by Ms. Pinski's class are represented in the bar graph shown. How many more red apples than yellow apples were collected?
One witness to a crime said that the suspect was 25 years old and 69 inches tall. A second witness claimed that the suspect was 35 years old and 74 inches tall. The third witness reported that the suspect was 65 inches tall and 35 years old. Each witness correctly identified either the suspect's height or age, but not both. If $a$ is the suspect's age in years, and $b$ is the suspect's height in inches, what is the value of the sum $a + b$?
A 3-inch by 8-inch sheet of paper and a 2-inch by 12-inch sheet of paper have the same area. Using just one cut (not necessarily straight), the 3-inch by 8-inch sheet can be divided into two pieces that can be rearranged to completely cover the 2-inch by 12-inch sheet. What is the length of the cut?
Four consecutive integers are substituted in every possible way for distinct values $a$, $b$, $c$ and $d$. What is the positive difference between the smallest and largest possible values of $(ab + cd)$?
CDs sell for $3$ different amounts. Three customers bought $3$ CDs each but none bought three of the same price. The first customer spent $\$4$, the second spent $\$9$ and the third spent $\$12$. We must find the price of the most expensive CD.
There is more than one four-digit positive integer in which the thousands digit is the number of 0s in the four-digit number, the hundreds digit is the number of 1s, the tens digit is the number of 2s and the units digit is the number of 3s. What is the sum of all such integers?
A polyhedron has $60$ vertices, $150$ edges, and $92$ faces. If all of the faces are either regular pentagons or equilateral triangles, how many of the $92$ faces are pentagons?
Five red lines and three blue lines are drawn on a plane. Given that $x$ pairs of lines of the same color intersect and $y$ pairs of lines of different colors intersect, find the maximum possible value of $(y - x)$.
A solitaire game is played with $8$ red, $9$ green, and $10$ blue cards. Totoro plays each of the cards exactly once in some order, one at a time. When he plays a card of color $c$, he gains a number of points equal to the number of cards that are not of color $c$ in his hand. Find the maximum number of points that he can obtain by the end of the game.
Five checkers are on the squares of an $8 \times 8$ checkerboard such that no two checkers are in the same row or the same column. How many squares on the checkerboard share neither a row nor a column with any of the five checkers?
A positive integer $x$ is $sunny$ if $3x$ has more digits than $x$. If all sunny numbers are written in increasing order, what is the 50th number written?
There are $10$ monsters, each with 6 units of health. On turn $n$, you can attack one monster, reducing its health by $n$ units. If a monster's health drops to $0$ or below, the monster dies. What is the minimum number of turns necessary to kill all of the monsters?
Month $A$ has three Wednesdays, but neither its first day nor its last day is Wednesday. What day of the week is the first day of month $A$?
Joe puts $63$ cards, number from $1$ to $63$, on a regular chessboard in sequence. The last space on the chessboard is left empty. A card can be moved to a neighboring space if that space is empty. Joe wants to just switch the card $1$ and card $2$, but leave all other cards at their original spaces, after a series of moves. Is it possible?