Found problems: 196
2015 Purple Comet Problems, 30
Cindy and Neil wanted to paint the side of a staircase in the six-square pattern shown below so that each
of the six squares is painted a solid color, and no two squares that share an edge are the same color. Cindy
draws all n patterns that can be colored using the four colors red, white, blue, and green. Neil looked at
these patterns and claimed that k of the patterns Cindy drew were incorrect because two adjacent squares
were colored with the same color. This is because Neil is color-blind and cannot distinguish red from
green. Find $n + k$. For picture go to http://www.purplecomet.org/welcome/practice
2015 Purple Comet Problems, 7
Talya went for a 6 kilometer run. She ran 2 kilometers at 12 kilometers per hour followed by 2 kilometers
at 10 kilometers per hour followed by 2 kilometers at 8 kilometers per hour. Talya’s average speed for the 6 kilometer run was $\frac{m}{n}$ kilometers per hour, where m and n are relatively prime positive integers. Find m + n.
2016 Purple Comet Problems, 1
Two integers have a sum of 2016 and a difference of 500. Find the larger of the two integers.
2021 Purple Comet Problems, 23
The sum $$\sum_{k=3}^{\infty} \frac{1}{k(k^4-5k^2+4)^2}$$ is equal to $\frac{m^2}{2n^2}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2016 Purple Comet Problems, 2
The figure below was formed by taking four squares, each with side length 5, and putting one on each side of a square with side length 20. Find the perimeter of the figure below.
[center][img]https://snag.gy/LGimC8.jpg[/img][/center]
2022 Purple Comet Problems, 4
A jar contains red, blue, and yellow candies. There are $14\%$ more yellow candies than blue candies, and $14\%$ fewer red candies than blue candies. Find the percent of candies in the jar that are yellow.
2003 Purple Comet Problems, 12
How many triangles appear in the diagram below:
[asy]
import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.6)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=0,xmax=8,ymin=0,ymax=8; draw((0,8)--(0,0)); draw((0,0)--(8,0)); draw((8,0)--(8,8)); draw((8,8)--(0,8)); draw((0,8)--(1,7)); draw((1,7)--(2,8)); draw((2,8)--(3,7)); draw((3,7)--(4,8)); draw((4,8)--(5,7)); draw((5,7)--(6,8)); draw((6,8)--(7,7)); draw((7,7)--(8,8)); draw((8,6)--(7,7)); draw((0,6)--(1,7)); draw((1,7)--(2,6)); draw((2,6)--(3,7)); draw((3,7)--(4,6)); draw((4,6)--(5,7)); draw((5,7)--(6,6)); draw((6,6)--(7,7)); draw((1,5)--(0,6)); draw((1,5)--(2,6)); draw((2,6)--(3,5)); draw((3,5)--(4,6)); draw((4,6)--(5,5)); draw((5,5)--(6,6)); draw((6,6)--(7,5)); draw((7,5)--(8,6)); draw((7,5)--(8,4)); draw((0,4)--(1,5)); draw((1,5)--(2,4)); draw((2,4)--(3,5)); draw((3,5)--(4,4)); draw((4,4)--(5,5)); draw((5,5)--(6,4)); draw((6,4)--(7,5)); draw((1,3)--(0,4)); draw((1,3)--(2,4)); draw((3,3)--(4,4)); draw((3,3)--(2,4)); draw((5,3)--(4,4)); draw((5,3)--(6,4)); draw((6,4)--(7,3)); draw((7,3)--(8,4)); draw((8,2)--(7,3)); draw((0,2)--(1,3)); draw((1,3)--(2,2)); draw((2,2)--(3,3)); draw((3,3)--(4,2)); draw((5,3)--(4,2)); draw((5,3)--(6,2)); draw((7,3)--(6,2)); draw((7,1)--(6,2)); draw((7,1)--(8,2)); draw((7,1)--(8,0)); draw((6,0)--(7,1)); draw((4,0)--(5,1)); draw((5,1)--(6,0)); draw((2,0)--(3,1)); draw((3,1)--(4,0)); draw((0,0)--(1,1)); draw((1,1)--(2,0)); draw((1,1)--(0,2)); draw((1,1)--(2,2)); draw((2,2)--(3,1)); draw((3,1)--(4,2)); draw((4,2)--(5,1)); draw((5,1)--(6,2));
dot((8,0),ds);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]
1991 AMC 8, 22
Each spinner is divided into $3$ equal parts. The results obtained from spinning the two spinners are multiplied. What is the probability that this product is an even number?
[asy]
draw(circle((0,0),2)); draw(circle((5,0),2));
draw((0,0)--(sqrt(3),1)); draw((0,0)--(-sqrt(3),1)); draw((0,0)--(0,-2));
draw((5,0)--(5+sqrt(3),1)); draw((5,0)--(5-sqrt(3),1)); draw((5,0)--(5,-2));
fill((0,5/3)--(2/3,7/3)--(1/3,7/3)--(1/3,3)--(-1/3,3)--(-1/3,7/3)--(-2/3,7/3)--cycle,black);
fill((5,5/3)--(17/3,7/3)--(16/3,7/3)--(16/3,3)--(14/3,3)--(14/3,7/3)--(13/3,7/3)--cycle,black);
label("$1$",(0,1/2),N); label("$2$",(sqrt(3)/4,-1/4),ESE); label("$3$",(-sqrt(3)/4,-1/4),WSW);
label("$4$",(5,1/2),N); label("$5$",(5+sqrt(3)/4,-1/4),ESE); label("$6$",(5-sqrt(3)/4,-1/4),WSW);
[/asy]
$\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ \frac{7}{9} \qquad \text{(E)}\ 1$
2022 Purple Comet Problems, 17
There are real numbers $x, y,$ and $z$ such that the value of $$x+y+z-\left(\frac{x^2}{5}+\frac{y^2}{6}+\frac{z^2}{7}\right)$$ reaches its maximum of $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n + x + y + z.$
2016 Purple Comet Problems, 17
The cubic polynomials $p(x)$ and $q(x)$ satisfy
• $p(1) = q(2)$
• $p(3) = q(4)$
• $p(5) = q(6)$
• $p(7) = q(8) + 13$.
Find $p(9)-q(10)$.
2022 Purple Comet Problems, 2
Cary made an investment of $\$1000$. During years $1, 2, 3, \text{and } 4$, his investment went up $20$ percent, down
$50$ percent, up $30$ percent, and up $40$ percent, respectively. Find the number of dollars Cary’s investment
was worth at the end of the fourth year.
2016 Purple Comet Problems, 5
Julius has a set of five positive integers whose mean is 100. If Julius removes the median of the set of five numbers, the mean of the set increases by 5, and the median of the set decreases by 5. Find the maximum possible value of the largest of the five numbers Julius has.
2021 Purple Comet Problems, 2
A furniture store set the sticker price of a table $40$ percent higher than the wholesale price that the store paid for the table. During a special sale, the table sold for $35$ percent less than this sticker price. Find the percent the final sale price was of the original wholesale price of the table.
2016 Purple Comet Problems, 25
For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$. Evaluate $$ 4\sum^{30}_{n=1} n \cdot T(n).$$
2009 Purple Comet Problems, 3
The [i]Purple Comet! Math Meet[/i] runs from April 27 through May 3, so the sum of the calendar dates for these seven days is $27 + 28 + 29 + 30 + 1 + 2 + 3 = 120.$ What is the largest sum of the calendar dates for seven consecutive Fridays occurring at any time in any year?
2021 Purple Comet Problems, 10
A semicircle has diameter $AB$ with $AB = 100$. Points $C$ and $D$ lie on the semicircle such that $AC = 28$ and $BD = 60$. Find $CD$.
2015 Purple Comet Problems, 20
For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.
2021 Purple Comet Problems, 28
Let $z_1$, $z_2$, $z_3$, $\cdots$, $z_{2021}$ be the roots of the polynomial $z^{2021}+z-1$. Evaluate $$\frac{z_1^3}{z_{1}+1}+\frac{z_2^3}{z_{2}+1}+\frac{z_3^3}{z_{3}+1}+\cdots+\frac{z_{2021}^3}{z_{2021}+1}.$$
2022 Purple Comet Problems, 8
Find the number of divisors of $20^{22}$ that are perfect squares.
2016 Purple Comet Problems, 3
Find the positive integer n such that $10^n$ cubic centimeters is the same as 1 cubic kilometer.
2017 Purple Comet Problems, 12
Let $x$, $y$, and $z$ be real numbers such that
$$12x - 9y^2 = 7$$
$$6y - 9z^2 = -2$$
$$12z - 9x^2 = 4$$
Find $6x^2 + 9y^2 + 12z^2$.
2015 Purple Comet Problems, 13
Given that x, y, and z are real numbers satisfying $x+y +z = 10$ and $x^2 +y^2 +z^2 = 50$, find the maximum possible value of $(x + 2y + 3z)^2 + (y + 2z + 3x)^2 + (z + 2x + 3y)^2$.
2016 Purple Comet Problems, 3
The sum of the numbers $3a - 4$, $3b - 4$, and $3c - 4$ is $2016$. Find the sum of the numbers $4a - 3$, $4b - 3$, and $4c - 3$.
2022 Purple Comet Problems, 10
Find the positive integer $n$ such that a convex polygon with $3n + 2$ sides has $61.5$ percent fewer diagonals than a convex polygon with $5n - 2$ sides.
2022 Purple Comet Problems, 16
A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three rectangular sides of the box meet at a corner of the box. The center points of those three rectangular sides are the vertices of a triangle with area $30$ square inches. Find $m + n.$