Found problems: 196
2016 Purple Comet Problems, 18
Find the least positive integer $N$ that is 50 times the number of positive integer divisors that $N$ has.
2022 Purple Comet Problems, 9
Let $a$ and $b$ be positive integers satisfying $3a < b$ and $a^2 + ab + b^2 = (b + 3)^2 + 27.$ Find the minimum possible value of $a + b.$
2016 Purple Comet Problems, 12
Find the number whose reciprocal is the sum of the reciprocal of $9 + 15i$ and the reciprocal of $9-15i$ .
2021 Purple Comet Problems, 6
A rectangular wooden block has a square top and bottom, its volume is $576$, and the surface area of its vertical sides is $384$. Find the sum of the lengths of all twelve of the edges of the block.
2016 Purple Comet Problems, 12
Find the number of squares such that the sides of the square are segments in the following diagram and where the dot is inside the square.
[center][img]https://snag.gy/qXBIY4.jpg[/img][/center]
2016 Purple Comet Problems, 21
On equilateral $\triangle{ABC}$ point D lies on BC a distance 1 from B, point E lies on AC a distance 1 from C, and point F lies on AB a distance 1 from A. Segments AD, BE, and CF intersect in pairs at points G, H, and J which are the vertices of another equilateral triangle. The area of $\triangle{ABC}$ is twice the area of $\triangle{GHJ}$. The side length of $\triangle{ABC}$ can be written $\frac{r+\sqrt{s}}{t}$, where r, s, and t are relatively prime positive integers. Find $r + s + t$.
[center][img]https://i.snag.gy/TKU5Fc.jpg[/img][/center]
2022 Purple Comet Problems, 14
Starting at $12:00:00$ AM on January $1,$ $2022,$ after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365.$ Find $y + d.$
2021 Purple Comet Problems, 18
The side lengths of a scalene triangle are roots of the polynomial $$x^3-20x^2+131x-281.3.$$ Find the square of the area of the triangle.
2022 Purple Comet Problems, 4
Of $450$ students assembled for a concert, $40$ percent were boys. After a bus containing an equal number of boys and girls brought more students to the concert, $41$ percent of the students at the concert were boys. Find the number of students on the bus.
2022 Purple Comet Problems, 7
The value of
$$\left(1-\frac{1}{2^2-1}\right)\left(1-\frac{1}{2^3-1}\right)\left(1-\frac{1}{2^4-1}\right)\dots\left(1-\frac{1}{2^{29}-1}\right)$$
can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $2m - n.$
2015 Purple Comet Problems, 29
Ten spherical balls are stacked in a pyramid. The bottom level of the stack has six balls each with radius 6 arranged
in a triangular formation with adjacent balls tangent to each other. The middle level of the stack has three
balls each with radius 5 arranged in a triangular formation each tangent to three balls in the bottom level.
The top level of the stack has one ball with radius 6 tangent to the three balls in the middle level. The
diagram shows the stack of ten balls with the balls in the middle shaded. The height of this stack of balls is m +$\sqrt{n}$, where m and n are positive integers. Find $m + n.$
2017 Purple Comet Problems, 1
Caden, Zoe, Noah, and Sophia shared a pizza. Caden ate 20 percent of the pizza. Zoe ate 50 percent more of the pizza than Caden ate. Noah ate 50 percent more of the pizza than Zoe ate, and Sophia ate the rest of the pizza. Find the percentage of the pizza that Sophia ate.
2016 Purple Comet Problems, 7
Positive integers m and n are both greater than 50, have a least common multiple equal to 480, and have a
greatest common divisor equal to 12. Find m + n.
2013 Purple Comet Problems, 4
One of the two Purple Comet! question writers is an adult whose age is the same as the last two digits of the year he was born. His birthday is in August. What is his age today?
2021 Purple Comet Problems, 25
The area of the triangle whose altitudes have lengths $36.4$, $39$, and $42$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2018 Purple Comet Problems, 1
Find $n$ such that the mean of $\frac74$, $\frac65$, and $\frac1n$ is $1$.
2015 Purple Comet Problems, 21
Find the remainder when $8^{2014}$ + $6^{2014}$ is divided by 100.
2015 Purple Comet Problems, 22
Let $x$ be a real number between 0 and $\tfrac{\pi}{2}$ for which the function $3\sin^2 x + 8\sin x \cos x + 9\cos^2 x$ obtains its maximum value, $M$. Find the value of $M + 100\cos^2x$.
2016 Purple Comet Problems, 14
Find the number of positive integers $n$ such that a regular polygon with $n$ sides has internal angles with measures equal to an integer number of degrees.
2016 Purple Comet Problems, 2
The trapezoid ABCD has bases with lengths 7 and 17 and area 120. Find the difference of the areas of $\triangle$ACD and $\triangle$CDB.
[asy]
pair A, B, C, D;
A = (0, 0);
B = (17, 0);
C = (1, 10);
D = (10, 10);
draw(A--B--D--C--cycle);
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, W);
label("$D$", D, E);
draw(B--C);
draw(A--D);
[/asy]
2022 Purple Comet Problems, 1
Find the maximum possible value obtainable by inserting a single set of parentheses into the expression $1 + 2 \times 3 + 4 \times 5 + 6$.
2022 Purple Comet Problems, 12
A rectangle with width $30$ inches has the property that all points in the rectangle are within $12$ inches of at least one of the diagonals of the rectangle. Find the maximum possible length for the rectangle in inches.
2017 Purple Comet Problems, 7
Find the number of positive integers less than 100 that are divisors of 300.
2022 Purple Comet Problems, 3
Find the least odd positive integer that is the middle number of five consecutive integers that are all composite.
2016 Purple Comet Problems, 1
Two integers have a sum of 2016 and a difference of 500. Find the larger of the two integers.