Found problems: 196
2015 Purple Comet Problems, 9
The figure below has only two sizes for its internal angles. The larger angles are three times the size of the
smaller angles. Find the degree measure of one of the larger angles. For figure here: http://www.purplecomet.org/welcome/practice
P.S The figure has 9 sides.
2017 Purple Comet Problems, 5
Find the greatest odd divisor of $160^3$.
2016 Purple Comet Problems, 2
The trapezoid below has bases with lengths 7 and 17 and area 120. Find the difference of the areas of the two triangles.
[center]
[img]https://i.snag.gy/BlqcSQ.jpg[/img]
[/center]
2021 Purple Comet Problems, 4
A building contractor needs to pay his $108$ workers $\$200$ each. He is carrying $122$ one hundred dollar bills and $188$ fifty dollar bills. Only $45$ workers get paid with two $\$100$ bills. Find the number of workers who get paid with four $\$50$ bills.
2015 Purple Comet Problems, 13
The diagram below shows a parallelogram ABCD with $AB = 36$ and $AD = 60$. Diagonal BD is
perpendicular to side AB. Points E and F bisect sides AD and BC, respectively. Points G and H are the
intersections of BD with AF and CE, respectively. Find the area of quadrilateral EGFH The diagram below shows a parallelogram ABCD with AB = 36 and AD = 60. Diagonal BD is
perpendicular to side AB. Points E and F bisect sides AD and BC, respectively. Points G and H are the
intersections of BD with AF and CE, respectively. Find the area of quadrilateral EGFH.
2017 Purple Comet Problems, 3
The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible?
2015 Purple Comet Problems, 23
Larry and Diane start $100$ miles apart along a straight road. Starting at the same time, Larry and Diane
drive their cars toward each other. Diane drives at a constant rate of 30 miles per hour. To make it
interesting, at the beginning of each 10 mile stretch, if the two drivers have not met, Larry flips a fair coin.
If the coin comes up heads, Larry drives the next 10 miles at 20 miles per hour. If the coin comes up tails,
Larry drives the next 10 miles at 60 miles per hour. Larry and Diane stop driving when they meet. The expected number of times that Larry flips the coin is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n.$
2016 Purple Comet Problems, 18
The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to each teacher in her school. Betsy had 15 pieces of candy left over. Find the least number of pieces of candy the Tasty Candy Company could have placed in each one-pound bag.
2017 Purple Comet Problems, 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 $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[center][img]https://snag.gy/92ikv3.jpg[/img][/center]
2016 Purple Comet Problems, 10
Jeremy wrote all the three-digit integers from $100$ to $999$ on a blackboard. Then Allison erased each of the
$2700$ digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every
1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n.$
2015 Purple Comet Problems, 17
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
2016 Purple Comet Problems, 9
Find the value of $x$ such that $2^{x+3} - 2^{x-3} = 2016$.
2012 Purple Comet Problems, 27
You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle?
[asy]
import graph; size(5cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((12,0)--(12,1)--(11,1)--(11,0)--cycle);
fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black);
draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle);
draw((15,0)--(15,1)--(16,1)--(16,0)--cycle);
fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black);
[/asy]
2021 Purple Comet Problems, 27
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
2015 Purple Comet Problems, 18
You have many identical cube-shaped wooden blocks. You have four colors of paint to use, and you paint
each face of each block a solid color so that each block has at least one face painted with each of the four
colors. Find the number of distinguishable ways you could paint the blocks. (Two blocks are
distinguishable if you cannot rotate one block so that it looks identical to the other block.)
2017 Purple Comet Problems, 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 120 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 25 percent 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 96 minutes after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning preparation taking 25 percent longer than usual to get onto the bus, his bus ride took 25 percent less time than usual, and he walked slowly from the bus stop to work taking 25 percent longer than usual. How many minutes after Aiden got out of bed did he arrive at work that day?
2021 Purple Comet Problems, 5
There were three times as many red candies as blue candies on a table. After Darrel took the same number of red candies and blue candies, there were four times as many red candies as blue candies left on the table. Then after Cloe took $12$ red candies and $12$ blue candies, there were five times as many red candies as blue candies left on the table. Find the total number of candies that Darrel took.
2010 Purple Comet Problems, 1
Let $x$ satisfy $(6x + 7) + (8x + 9) = (10 + 11x) + (12 + 13x).$ There are relatively prime positive integers so that $x = -\tfrac{m}{n}$. Find $m + n.$
2015 Purple Comet Problems, 2
How many sets of two positive prime numbers $\{p,q\}$ have the property that $p + q = 100$?
2016 Purple Comet Problems, 6
The following diagram shows a square where each side has seven dots that divide the side into six equal segments. All the line segments that connect these dots that form a $45^{\circ}$ angle with a side of the square are drawn as shown. The area of the shaded region is 75. Find the area of the original square.
[center][img]https://i.snag.gy/Jzx9Fn.jpg[/img][/center]
2016 Purple Comet Problems, 20
The 24 unshaded squares in the 5 × 5 grid below can be tiled with twelve 1 × 2 tiles. One such tiling is shown. Find the number of ways the grid can be tiled.
[center][img]https://snag.gy/KMoPrF.jpg[/img][/center]
2017 Purple Comet Problems, 16
The set of positive real numbers $x$ that satisfy $2 | x^2 - 9 | \le 9 | x | $ is an interval $[m, M]$. Find $10m + M$.
2021 Purple Comet Problems, 21
Let $a$, $b$, and $c$ be real numbers satisfying the equations $$a^3+abc=26$$ $$b^3+abc=78$$ $$c^3-abc=104.$$ Find $a^3+b^3+c^3$.
2017 Purple Comet Problems, 10
Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET.
2022 Purple Comet Problems, 1
The $12$-sided polygon below was created by placing three $3$ × $3$ squares with their sides parallel so that
vertices of two of the squares are at the center of the third square. Find the perimeter of this $12$-sided
polygon.