Found problems: 196
2016 Purple Comet Problems, 4
One side of a rectangle has length 18. The area plus the perimeter of the rectangle is 2016. Find the perimeter of the rectangle.
2017 Purple Comet Problems, 15
Find the remainder when $7^{7^7}$ is divided by $1000$.
2021 Purple Comet Problems, 3
The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.
2015 Purple Comet Problems, 3
The repeating decimal $2.0151515\ldots$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2021 Purple Comet Problems, 30
For positive integer $k$, define $x_k=3k+\sqrt{k^2-1}-2(\sqrt{k^2-k}+\sqrt{k^2+k})$. Then $\sqrt{x_1}+\sqrt{x_2}+\cdots+\sqrt{x_{1681}}=\sqrt{m}-n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2022 Purple Comet Problems, 5
Let $A_1, A_2, A_3, \ldots , A_{12}$ be the vertices of a regular $12-$gon (dodecagon). Find the number of points in the plane that are equidistant to at least $3$ distinct vertices of this $12-$gon.
2015 Purple Comet Problems, 10
In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of different positive weights of chemicals that Gerome could measure.
2021 Purple Comet Problems, 15
Find the value of $x$ where the graph of $$y=\log_3(\sqrt{x^2+729}+x)-2\log_3(\sqrt{x^2+729}-x)$$ crosses the $x$-axis.
2016 Purple Comet Problems, 16
The figure below shows a barn in the shape of two congruent pentagonal prisms that intersect at right angles and have a common center. The ends of the prisms are made of a 12 foot by 7 foot rectangle surmounted by an isosceles triangle with sides 10 feet, 10 feet, and 12 feet. Each prism is 30 feet long. Find the volume of the barn in cubic feet.
[center][img]https://snag.gy/Ox9CUp.jpg[/img][/center]
2016 Purple Comet Problems, 5
A 2 meter long bookshelf is filled end-to-end with 46 books. Some of the books are 3 centimeters thick while all the others are 5 centimeters thick. Find the number of books on the shelf that are 3 centimeters thick.
2015 Purple Comet Problems, 12
Right triangle ABC with a right angle at A has AB = 20 and AC = 15. Point D is on AB with BD = 2. Points E and F are placed on ray CA and ray CB, respectively, such that CD is a median of $\triangle$ CEF. Find the area of $\triangle$CEF.
2015 Purple Comet Problems, 2
The diagram below is made up of a rectangle AGHB, an equilateral triangle AFG, a rectangle ADEF, and a parallelogram ABCD. Find the degree measure of ∠ABC. For diagram go to http://www.purplecomet.org/welcome/practice, the 2015 middle school contest, and go to #2
2015 Purple Comet Problems, 8
Gwendoline rolls a pair of six-sided dice and records the product of the two values rolled. Gwendoline
repeatedly rolls the two dice and records the product of the two values until one of the values she records
appears for a third time. What is the maximum number of times Gwendoline will need to roll the two dice?
2015 Purple Comet Problems, 27
A container is shaped like a right circular cone open at the top surmounted by a frustum which is open at
the top and bottom as shown below. The lower cone has a base with radius 2 centimeters and height 6
centimeters while the frustum has bases with radii 2 and 8 centimeters and height 6 centimeters. If there is
a rainfall measuring 2 centimeter of rain, the rain falling into the container will fill the container to a
height of $m + 3\sqrt{n}$ cm, where m and n are positive integers. Find m + n.
2017 Purple Comet Problems, 9
In $\triangle{ADE}$ points $B$ and $C$ are on side $AD$ and points $F$ and $G$ are on side $AE$ so that $BG \parallel CF \parallel DE$, as shown. The area of $\triangle{ABG}$ is $36$, the area of trapezoid $CFED$ is $144$, and $AB = CD$. Find the area of trapezoid $BGFC$.
[center][img]https://snag.gy/SIuOLB.jpg[/img][/center]
2015 Purple Comet Problems, 4
Janet played a song on her clarinet in 3 minutes and 20 seconds. Janet’s goal is to play the song at a 25%
faster rate. Find the number of seconds it will take Janet to play the song when she reaches her goal.
2015 Purple Comet Problems, 20
The diagram below shows an $8$x$7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two
triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenuse of each triangle. The diameter of the circle is $\frac{m}{n}$ , where m and n are relatively prime positive integers. Find m + n.
For diagram go to http://www.purplecomet.org/welcome/practice, go to the 2015 middle school contest questions, and then go to #20
2021 Purple Comet Problems, 9
Find $k$ such that $k\pi$ is the area of the region of points in the plane satisfying $$\frac{x^2+y^2+1}{11} \le x \le \frac{x^2+y^2+1}{7}.$$
2021 Purple Comet Problems, 13
Two infinite geometric series have the same sum. The first term of the first series is $1$, and the first term of the second series is $4$. The fifth terms of the two series are equal. The sum of each series can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
2022 Purple Comet Problems, 19
Given that $a_1, a_2, a_3, . . . , a_{99}$ is a permutation of $1, 2, 3, . . . , 99,$ find the maximum possible value of
$$|a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|.$$
2015 Purple Comet Problems, 25
You have a collection of small wooden blocks that are rectangular solids measuring $3$×$4$×$6$. Each of the six faces of each block is to be painted a solid color, and you have three colors of paint to use. 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.)
2015 Purple Comet Problems, 9
Find the sum of all positive integers n with the property that the digits of n add up to 2015−n.
2015 Purple Comet Problems, 16
Jamie, Linda, and Don bought bundles of roses at a flower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a fixed price which was higher than the price that the flower shop charged. At the end of the
fair, Jamie, Linda, and Don donated their unsold bundles of roses to the fair organizers. Jamie had bought
20 bundles of roses, sold 15 bundles of roses, and made $60$ profit. Linda had bought 34 bundles of roses,
sold 24 bundles of roses, and made $69 profit. Don had bought 40 bundles of roses and sold 36 bundles of
roses. How many dollars profit did Don make?
2021 Purple Comet Problems, 7
Among the $100$ constants $a_1,a_2,a_3,...,a_{100}$, there are $39$ equal to $-1$ and $61$ equal to $+1$. Find the sum of all the products $a_ia_j$, where $1\le i < j \le 100$.
2021 Purple Comet Problems, 11
There are nonzero real numbers $a$ and $b$ so that the roots of $x^2 + ax + b$ are $3a$ and $3b$. There are relatively prime positive integers $m$ and $n$ so that $a - b = \tfrac{m}{n}$. Find $m + n$.