This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 196

2009 Purple Comet Problems, 9
Tags: Purple Comet
Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers?

2022 Purple Comet Problems, 9
Tags: Purple Comet , HS
For positive integer $n$ let $z_n=\sqrt{\frac{3}{n}}+i$, where $i=\sqrt{-1}$. Find $|z_1 \cdot z_2 \cdot z_3 \cdots z_{47}|$.

2016 Purple Comet Problems, 8
Tags: Purple Comet
The map below shows an east/west road connecting the towns of Acorn, Centerville, and Midland, and a north/south road from Centerville to Drake. The distances from Acorn to Centerville, from Centerville to Midland, and from Centerville to Drake are each 60 kilometers. At noon Aaron starts at Acorn and bicycles east at 17 kilometers per hour, Michael starts at Midland and bicycles west at 7 kilometers per hour, and David starts at Drake and bicycles at a constant rate in a straight line across an open field. All three bicyclists arrive at exactly the same time at a point along the road from Centerville to Midland. Find the number of kilometers that David bicycles. For the map go to http://www.purplecomet.org/welcome/practice

2022 Purple Comet Problems, 5
Tags: Purple Comet , HS
Below is a diagram showing a $6 \times 8$ rectangle divided into four $6 \times 2$ rectangles and one diagonal line. Find the total perimeter of the four shaded trapezoids.

2016 Purple Comet Problems, 8
Tags: Purple Comet
The figure below has a 1 × 1 square, a 2 × 2 square, a 3 × 3 square, a 4 × 4 square, and a 5 × 5 square. Each of the larger squares shares a corner with the 1 × 1 square. Find the area of the region covered by these five squares. [center][img]https://snag.gy/1AfJWt.jpg[/img][/center]

2016 Purple Comet Problems, 16
Tags: Purple Comet
Henry rolls a fair die. If the die shows the number $k$, Henry will then roll the die $k$ more times. The probability that Henry will never roll a 3 or a 6 either on his first roll or on one of the $k$ subsequent rolls is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2021 Purple Comet Problems, 19
Tags: Purple Comet
Let $a, b, c, d$ be an increasing arithmetic sequence of positive real numbers with common difference $\sqrt2$. Given that the product $abcd = 2021$, $d$ can be written as $\frac{m+\sqrt{n}}{\sqrt{p}}$ , where $m, n,$ and $p$ are positive integers not divisible by the square of any prime. Find $m + n + p$.

2021 Purple Comet Problems, 16
Tags: Purple Comet
Paula rolls three standard fair dice. The probability that the three numbers rolled on the dice are the side lengths of a triangle with positive area is $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2022 Purple Comet Problems, 30
Tags: Purple Comet
There is a positive integer s such that there are s solutions to the equation $64sin^2(2x)+tan^2(x)+cot^2(x)=46$ in the interval $(0,\frac{\pi}{2})$ all of the form $\frac{m_k}{n_k}\pi$ where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, . . . , s$. Find $(m_1 + n_1) + (m_2 + n_2) + (m_3 + n_3) + · · · + (m_s + n_s)$.

2016 Purple Comet Problems, 19
Tags: Purple Comet
Jar #1 contains five red marbles, three blue marbles, and one green marble. Jar #2 contains five blue marbles, three green marbles, and one red marble. Jar #3 contains five green marbles, three red marbles, and one blue marble. You randomly select one marble from each jar. Given that you select one marble of each color, the probability that the red marble came from jar #1, the blue marble came from jar #2, and the green marble came from jar #3 can be expressed as $\frac{m}{n}$, where m and n are relatively prime positive integers. Find m + n.

2008 Purple Comet Problems, 19
Tags: geometry , calculus , derivative , Purple Comet
One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle

2015 Purple Comet Problems, 8
Tags: Purple Comet
In the figure below $\angle$LAM = $\angle$LBM = $\angle$LCM = $\angle$LDM, and $\angle$AEB = $\angle$BFC = $\angle$CGD = 34 degrees. Given that $\angle$KLM = $\angle$KML, find the degree measure of $\angle$AEF. This is #8 on the 2015 Purple comet High School. For diagram go to http://www.purplecomet.org/welcome/practice

2022 Purple Comet Problems, 18
Tags: Purple Comet
In $\triangle ABC$ let point $D$ be the foot of the altitude from $A$ to $BC.$ Suppose that $\angle A = 90^{\circ}, AB - AC = 5,$ and $BD - CD = 7.$ Find the area of $\triangle ABC.$

2016 Purple Comet Problems, 24
Tags: Purple Comet
Find the largest prime $p$ such that $p$ divides $2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1}$.

2016 Purple Comet Problems, 10
Tags: Purple Comet
Mildred the cow is tied with a rope to the side of a square shed with side length 10 meters. The rope is attached to the shed at a point two meters from one corner of the shed. The rope is 14 meters long. The area of grass growing around the shed that Mildred can reach is given by $n\pi$ square meters, where $n$ is a positive integer. Find $n$.

2016 Purple Comet Problems, 13
Tags: Purple Comet
One afternoon Elizabeth noticed that twice as many cars on the expressway carried only a driver as compared to the number of cars that carried a driver and one passenger. She also noted that twice as many cars carried a driver and one passenger as those that carried a driver and two passengers. Only 10% of the cars carried a driver and three passengers, and no car carried more than four people. Any car containing at least three people was allowed to use the fast lane. Elizabeth calculated that $\frac{m}{n}$ of the people in cars on the expressway were allowed to ride in the fast lane, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2015 Purple Comet Problems, 15
Tags: Purple Comet , geometry
On the inside of a square with side length 60, construct four congruent isosceles triangles each with base 60 and height 50, and each having one side coinciding with a different side of the square. Find the area of the octagonal region common to the interiors of all four triangles.

2016 Purple Comet Problems, 23
Tags: Purple Comet
Sixteen dots are arranged in a four by four grid as shown. The distance between any two dots in the grid is the minimum number of horizontal and vertical steps along the grid lines it takes to get from one dot to the other. For example, two adjacent dots are a distance 1 apart, and two dots at opposite corners of the grid are a distance 6 apart. The mean distance between two distinct dots in the grid is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$. [center][img]https://i.snag.gy/c1tB7z.jpg[/img][/center]

2017 Purple Comet Problems, 8
Tags: Purple Comet
The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$.

2021 Purple Comet Problems, 1
Tags: Purple Comet
The diagram shows two intersecting line segments that form some of the sides of two squares with side lengths $3$ and $6$. Two line segments join vertices of these squares. Find the area of the region enclosed by the squares and segments.

2011 Purple Comet Problems, 2
Tags: geometry , perimeter , Purple Comet
The diagram below shows a $12$-sided figure made up of three congruent squares. The figure has total perimeter $60$. Find its area. [asy] size(150); defaultpen(linewidth(0.8)); path square=unitsquare; draw(rotate(360-135)*square^^rotate(345)*square^^rotate(105)*square); [/asy]