Found problems: 196
2017 Purple Comet Problems, 17
Let $a_0$, $a_1$, ..., $a_6$ be real numbers such that $a_0 + a_1 + ... + a_6 = 1$ and
$$a_0 + a_2 + a_3 + a_4 + a_5 + a_6 =\frac{1}{2}$$
$$a_0 + a_1 + a_3 + a_4 + a_5 + a_6 = \frac{2}{3}$$
$$a_0 + a_1 + a_2 + a_4 + a_5 + a_6 =\frac{7}{8}$$
$$a_0 + a_1 + a_2 + a_3 + a_5 + a_6 =\frac{29}{30}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_6 =\frac{143}{144}$$
$$a_0 + a_1 + a_2 + a_3 + a_4 + a_5 =\frac{839}{840}$$
The value of $a_0$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2015 Purple Comet Problems, 11
Suppose that the vertices of a polygon all lie on a rectangular lattice of points where adjacent points on
the lattice are a distance 1 apart. Then the area of the polygon can be found using Pick’s Formula: $I + \frac{B}{2}$ −1, where I is the number of lattice points inside the polygon, and B is the number of lattice points on the boundary of the polygon. Pat applied Pick’s Formula to find the area of a polygon but mistakenly interchanged the values of I and B. As a result, Pat’s calculation of the area was too small by 35. Using the correct values for I and B, the ratio n = $\frac{I}{B}$ is an integer. Find the greatest possible value of n.
2016 Purple Comet Problems, 28
Find the sum of all the possible values of xy such that x and y are positive integers satisfying $(x^2 + 1)(y^2 + 1) + 2(x -y)(1 - xy) = 4(1 + xy) + 140$.
2016 Purple Comet Problems, 15
Find the least positive integer of the form [u]a[/u] [u]b[/u] [u]a[/u] [u]a[/u] [u]b[/u] [u]a[/u], where a and b are distinct digits, such that the integer can be written as a product of six distinct primes
2015 Purple Comet Problems, 4
Six boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. Find the least $N$ for which it is guaranteed that for at least one $k$, box number $k$ contains at least $k^2$ balls.
2015 Purple Comet Problems, 3
The Fahrenheit temperature ($F$) is related to the Celsius temperature ($C$) by $F = \tfrac{9}{5} \cdot C + 32$. What is the temperature in Fahrenheit degrees that is one-fifth as large if measured in Celsius degrees?
2022 Purple Comet Problems, 6
Let $a_1=2021$ and for $n \ge 1$ let $a_{n+1}=\sqrt{4+a_n}$. Then $a_5$ can be written as $$\sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}},$$ where $m$ and $n$ are positive integers. Find $10m+n$.
2015 Purple Comet Problems, 12
The product $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$ can be expressed in the form $m$ $\cdot$ $n^3$, where m and n are positive integers, and m is not divisible by the cube of any prime. Find m.
2008 Purple Comet Problems, 15
Each of the distinct letters in the following subtraction problem represents a different digit. Find the number represented by the word [b]TEAM[/b]
[size=150][b]
PURPLE
- COMET
________
[color=#FFFFFF].....[/color]TEAM [/b][/size]
2021 Purple Comet Problems, 12
Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$. There are two distinct points, $P$ and $Q$, that are each equidistant from $F$, from line $L_1$, and from line $L_2$. Find the area of $\triangle{FPQ}$.
2015 Purple Comet Problems, 1
Arvin ate 11 halves of tarts, Bernice ate 12 quarters of tarts, Chrisandra ate 13 eighths of tarts, and Drake ate 14 sixteenths of tarts. How many tarts were eaten?
2015 Purple Comet Problems, 28
Let $A = {1,2,3,4,5}$ and $B = {0,1,2}$. Find the number of pairs of functions ${{f,g}}$ where both f and g map the set A into the set B and there are exactly two elements $x \in A$ where $f(x) = g(x)$. For example, the function f that maps $1 \rightarrow 0,2 \rightarrow 1,3 \rightarrow 0,4 \rightarrow 2,5 \rightarrow 1$ and the constant function g which maps each element of A to 0 form such a pair of functions.
2017 Purple Comet Problems, 14
Let a and b be positive integers such that $a + ab = 1443$ and $ab + b = 1444$. Find $10a + b$.
2016 Purple Comet Problems, 7
Melanie has $4\frac{2}{5}$ cups of flour. The recipe for one batch of cookies calls for $1\frac{1}{2}$ cups of flour. Melanie plans to make $2\frac{1}{2}$ batches of cookies. When she is done, she will have $\frac{m}{n}$ cups of flour remaining, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2015 Purple Comet Problems, 7
How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side
lengths and perimeter less than 20?
2016 Purple Comet Problems, 10
Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where m and n are relatively prime positive integers. Find $m + n$.
2016 Purple Comet Problems, 17
Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2021 Purple Comet Problems, 14
Each of the cells of a $7 \times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the expected number of hidden edges is less than $3$.
2015 Purple Comet Problems, 11
The Purple Plant Garden Store sells grass seed in ten-pound bags and fifteen-pound bags. Yesterday half
of the grass seed they had was in ten-pound bags. This morning the store received a shipment of 27 more
ten-pound bags, and now they have twice as many ten-pound bags as fifteen-pound bags. Find the total
weight in pounds of grass seed the store now has.
2017 Purple Comet Problems, 11
Find the greatest prime divisor of $29! + 33!$.
2015 Purple Comet Problems, 19
Problem 19 The diagram below shows a 24×24 square ABCD. Points E and F lie on sides AD and CD, respectively, so that DE = DF = 8. Set X consists of the shaded triangle ABC with its interior, while set Y consists of
the shaded triangle DEF with its interior. Set Z consists of all the points that are midpoints of segments
connecting a point in set X with a point in set Y . That is, Z = {z | z is the midpoint of xy for x ∈ X and y ∈ Y}. Find the area of the set Z.
For diagram to http://www.purplecomet.org/welcome/practice
2012 Purple Comet Problems, 1
Evaluate $5^4-4^3-3^2-2^1-1^0$.
2022 Purple Comet Problems, 8
The product
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positive integers. Find $s$.
2016 Purple Comet Problems, 1
Mike has 12 books, Sean has 9 books, and little Sherry has only 4 books. Find the percentage of these books that Sean has.
2016 Purple Comet Problems, 4
The following diagram shows a square where each side has four dots that divide the side into three equal segments. The shaded region has area 105. Find the area of the original square.
[center][img]https://snag.gy/r60Y7k.jpg[/img][/center]