Purple Comet 2017 – Middle School
1. Caden, Zoe, Noah, and Sophia shared a pizza. Caden ate percent of the pizza. Zoe ate percent more of the pizza
than Caden ate. Noah ate percent more of the pizza than Zoe ate, and Sophia ate the rest of the pizza. Find the
percentage of the pizza that Sophia ate.
2. The figure below was made by gluing together non-overlapping congruent squares. The figure has area . Find the
perimeter of the figure.
3. The Stromquist Comet is visible every years. If the comet is visible in , what is the next leap year when the
comet will be visible?
4. The following diagram includes six circles with radius , one circle with radius , and one circle with radius . The
area of the shaded region is . Find .
5. Find the greatest odd divisor of .
6. On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the
bus stop to work arriving at work minutes after getting out of bed. One morning Aiden got out of bed late, so he
rushed through his morning preparation getting onto the bus in half the usual time, the bus ride took longer than
usual, and he ran from the bus stop to work in half the usual time it takes him to walk arriving at work minutes
after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning
preparation taking longer than usual to get onto the bus, his bus ride took less time than usual, and he
walked slowly from the bus stop to work taking longer than usual. How many minutes after Aiden got out of bed
did he arrive at work that day?
7. Find the number of positive integers less than that are divisors of .
8. The positive integer is a multiple of , and the positive integer is a multiple of . Their sum is .
Find .
9. In points and are on side and points and are on side so that , as shown. The area
of is , the area of trapezoid CFED is , and . Find the area of trapezoid .
Purple Comet 2017 – Middle School
10. Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter
such as TAMEMHET.
11. Find the greatest prime divisor of .
12. Let and be real numbers such that
Find .
13. Find the number of positive integer divisors of that are either perfect squares or perfect cubes.
14. Let and be positive integers such that and . Find .
15. Find the remainder when is divided by .
16. The set of positive real numbers that satisfy is an interval . Find .
17. Let be real numbers such that and
The value of is , where and are relatively prime positive integers. Find .
18. Omar has four fair standard six-sided dice. Omar invented a game where he rolls his four dice over and over again
until the number does not appear on the top face of any of the dice. Omar wins the game if on that roll the top faces
of his dice show at least one and at least one . The probability that Omar wins the game is , where and are
relatively prime positive integers. Find .
19. Find the sum of all values of , where is an ordered pair of positive integers and is a
perfect square.
20. A right circular cone has a height equal to three times its base radius and has volume 1. The cone is inscribed inside a
sphere as shown. The volume of the sphere is , where and are relatively prime positive integers. Find .
