Found problems: 196
2015 Purple Comet Problems, 19
Let a,b,c, and d be real numbers such that \[a^2 + 3b^2 + \frac{c^2+3d^2}{2} = a + b + c+d-1.\] Find $1000a + 100b + 10c + d$.
2017 Purple Comet Problems, 4
The following diagram includes six circles with radius 4, one circle with radius 8, and one circle with radius 16. The area of the shaded region is $k\pi$. Find $k$.
[center][img]https://snag.gy/zNPBx0.jpg[/img][/center]
2022 Purple Comet Problems, 11
For positive integer $n,$ let $s(n)$ be the sum of the digits of n when n is expressed in base ten. For example, $s(2022) = 2 + 0 + 2 + 2 = 6.$ Find the sum of the two solutions to the equation $n - 3s(n) = 2022.$
2016 Purple Comet Problems, 3
Find the positive integer $n$ such that $10^n$ cubic centimeters is the same as 1 cubic kilometer.
2016 Purple Comet Problems, 30
Some identically sized spheres are piled in $n$ layers in the shape of a square pyramid with one sphere in the top layer, 4 spheres in the second layer, 9 spheres in the third layer, and so forth so that the bottom layer has a square array of $n^2$ spheres. In each layer the centers of the spheres form a square grid so that each sphere is tangent to any sphere adjacent to it on the grid. Each sphere in an upper level is tangent to the four spheres directly below it. The diagram shows how the first three layers of spheres are stacked. A square pyramid is built around the pile of spheres so that the sides of the pyramid are tangent to the spheres on the outside of the pile. There is a positive integer $m$ such that as $n$ gets large, the ratio of the volume of the pyramid to the total volume inside all of the spheres approaches $\frac{\sqrt{m}}{\pi}$. Find $m$.
[center][img]https://snag.gy/bIwyl6.jpg[/img][/center]
2015 Purple Comet Problems, 6
There are digits a and b so that the 15-digit number 7a7ba7ab7ba7b77 is divisible by 99.
Find 10a + b.
2015 Purple Comet Problems, 16
\[\left(1 + \frac{1}{1+2^1}\right)\left(1+\frac{1}{1+2^2}\right)\left(1 + \frac{1}{1+2^3}\right)\cdots\left(1 + \frac{1}{1+2^{10}}\right)= \frac{m}{n},\] where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2022 Purple Comet Problems, 2
Call a date mm/dd/yy $\textit{multiplicative}$ if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, $01/21/21$, $03/07/21$, and $07/03/21$ are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.
2017 Purple Comet Problems, 13
Find the number of positive integer divisors of $20^{17}$ that are either perfect squares or perfect cubes.
2018 Purple Comet Problems, 2
The following figure is made up of many $2$ × $4$ tiles such that adjacent tiles always share an edge of length $2$. Find the perimeter of this figure.
2016 Purple Comet Problems, 9
Find the sum of all perfect squares that divide 2016.
2021 Purple Comet Problems, 29
Two cubes with edge length $3$ and two cubes with edge length $4$ sit on plane $P$ so that the four cubes share a vertex, and the two larger cubes share no faces with each other as shown below. The cube vertices that do not touch $P$ or any of the other cubes are labeled $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$. The four cubes lie inside a right rectangular pyramid whose base is on $P$ and whose triangular sides touch the labeled vertices with one side containing vertices $A$, $B$, and $C$, another side containing vertices $D$, $E$, and $F$, and the two other sides each contain one of $G$ and $H$. Find the volume of the pyramid.
2015 Purple Comet Problems, 24
The complex number w has positive imaginary part and satisfies $|w| = 5$. The triangle in the complex plane with vertices at $w, w^2,$ and $w^3$ has a right angle at $w$. Find the real part of $w^3$.
2016 Purple Comet Problems, 29
Ten square tiles are placed in a row, and each can be painted with one of the four colors red (R), yellow (Y), blue (B), and white (W). Find the number of ways this can be done so that each block of five adjacent tiles contains at least one tile of each color. That is, count the patterns RWBWYRRBWY and WWBYRWYBWR but not RWBYYBWWRY because the five adjacent tiles colored BYYBW does not include the color red.
2015 Purple Comet Problems, 14
Evaluate
$\frac{\log_{10}20^2 \cdot \log_{20}30^2 \cdot \log_{30}40^2 \cdot \cdot \cdot \log_{990}1000^2}{\log_{10}11^2 \cdot \log_{11}12^2 \cdot \log_{12}13^2 \cdot \cdot \cdot \log_{99}100^2}$
.
2022 Purple Comet Problems, 7
In a room there are $144$ people. They are joined by $n$ other people who are each carrying $k$ coins. When these coins are shared among all $n + 144$ people, each person has $2$ of these coins. Find the minimum possible value of $2n + k$.
2016 Purple Comet Problems, 11
One evening a theater sold 300 tickets for a concert. Each ticket sold for \$40, and all tickets were purchased using \$5, \$10, and \$20 bills. At the end of the evening the theater had received twice as many \$10 bills as \$20 bills, and 20 more \$5 bills than \$10 bills. How many bills did the theater receive altogether?
2021 Purple Comet Problems, 24
Let $x$ be a real number such that $$4^{2x}+2^{-x}+1=(129+8\sqrt2)(4^{x}+2^{-x}-2^{x}).$$ Find $10x$.
2012 Purple Comet Problems, 30
The diagram below shows four regular hexagons each with side length $1$ meter attached to the sides of a square. This figure is drawn onto a thin sheet of metal and cut out. The hexagons are then bent upward along the sides of the square so that $A_1$ meets $A_2$, $B_1$ meets $B_2$, $C_1$ meets $C_2$, and $D_1$ meets $D_2$. If the resulting dish is filled with water, the water will rise to the height of the corner where the $A_1$ and $A_2$ meet. there are relatively prime positive integers $m$ and $n$ so that the number of cubic meters of water the dish will hold is $\sqrt{\frac{m}{n}}$. Find $m+n$.
[asy]
/* File unicodetex not found. */
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
import graph; size(7cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.3, xmax = 14.52, ymin = -8.3, ymax = 6.3; /* image dimensions */
draw((0,1)--(0,0)--(1,0)--(1,1)--cycle);
draw((1,1)--(1,0)--(1.87,-0.5)--(2.73,0)--(2.73,1)--(1.87,1.5)--cycle);
draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle);
draw((0,0)--(1,0)--(1.5,-0.87)--(1,-1.73)--(0,-1.73)--(-0.5,-0.87)--cycle);
draw((0,1)--(0,0)--(-0.87,-0.5)--(-1.73,0)--(-1.73,1)--(-0.87,1.5)--cycle);
/* draw figures */
draw((0,1)--(0,0));
draw((0,0)--(1,0));
draw((1,0)--(1,1));
draw((1,1)--(0,1));
draw((1,1)--(1,0));
draw((1,0)--(1.87,-0.5));
draw((1.87,-0.5)--(2.73,0));
draw((2.73,0)--(2.73,1));
draw((2.73,1)--(1.87,1.5));
draw((1.87,1.5)--(1,1));
draw((0,1)--(1,1));
draw((1,1)--(1.5,1.87));
draw((1.5,1.87)--(1,2.73));
draw((1,2.73)--(0,2.73));
draw((0,2.73)--(-0.5,1.87));
draw((-0.5,1.87)--(0,1));
/* dots and labels */
dot((1.87,-0.5),dotstyle);
label("$C_1$", (1.72,-0.1), NE * labelscalefactor);
dot((1.87,1.5),dotstyle);
label("$B_2$", (1.76,1.04), NE * labelscalefactor);
dot((1.5,1.87),dotstyle);
label("$B_1$", (0.96,1.8), NE * labelscalefactor);
dot((-0.5,1.87),dotstyle);
label("$A_2$", (-0.26,1.78), NE * labelscalefactor);
dot((-0.87,1.5),dotstyle);
label("$A_1$", (-0.96,1.08), NE * labelscalefactor);
dot((-0.87,-0.5),dotstyle);
label("$D_2$", (-1.02,-0.18), NE * labelscalefactor);
dot((-0.5,-0.87),dotstyle);
label("$D_1$", (-0.22,-0.96), NE * labelscalefactor);
dot((1.5,-0.87),dotstyle);
label("$C_2$", (0.9,-0.94), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
2016 Purple Comet Problems, 7
Positive integers m and n are both greater 50, have a least common multiple equal to 480, and have a
greatest common divisor equal to 12. Find $m + n$.
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 degree 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.
For diagram go to http://www.purplecomet.org/welcome/practice
2017 Purple Comet Problems, 19
Find the sum of all values of $a + b$, where $(a, b)$ is an ordered pair of positive integers and $a^2+\sqrt{2017-b^2}$ is a perfect square.
2022 Purple Comet Problems, 6
At Ignus School there are $425$ students. Of these students $351$ study mathematics, $71$ study Latin, and $203$ study chemistry. There are $199$ students who study more than one of these subjects, and $8$ students who do not study any of these subjects. Find the number of students who study all three of these subjects.
2016 Purple Comet Problems, 9
Find the sum of all perfect squares that divide 2016.
2021 Purple Comet Problems, 8
Pam lists the four smallest positive prime numbers in increasing order. When she divides the positive integer $N$ by the first prime, the remainder is $1$. When she divides $N$ by the second prime, the remainder is $2$. When she divides $N$ by the third prime, the remainder is $3$. When she divides $N$ by the fourth prime, the remainder is $4$. Find the least possible value for $N$.