problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$. | 1. Define the sequences \(a_k\) and \(b_k\) as follows:
\[
a_k = (k^2 + 1)k!
\]
\[
b_k = a_1 + a_2 + a_3 + \cdots + a_k
\]
2. We start by simplifying \(a_k\):
\[
a_k = (k^2 + 1)k! = k^2 k! + k! = k(k \cdot k!) + k! = k(k+1)! - k \cdot k! + k!
\]
\[
= k(k+1)! - k!(k-1)
\]
3. Next, w... | 99 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In the addition problem
\[ \setlength{\tabcolsep}{1mm}\begin{tabular}{cccccc}& W & H & I & T & E\\ + & W & A & T & E & R \\\hline P & I & C & N & I & C\end{tabular} \] each distinct letter represents a different digit. Find the number represented by the answer PICNIC. | 1. **Determine the value of \( P \):**
- Since \( P \) is the leading digit of the sum \( PICNIC \), and the sum of two 5-digit numbers can be at most \( 199,998 \), \( P \) must be 1.
- Therefore, \( P = 1 \).
2. **Determine the value of \( C \):**
- The last digit of the sum \( PICNIC \) is \( C \). Since \... | 169069 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Bill’s age is one third larger than Tracy’s age. In $30$ years Bill’s age will be one eighth larger than Tracy’s age. How many years old is Bill? | 1. Let \( b \) be Bill's age and \( t \) be Tracy's age. From the problem, we know that Bill’s age is one third larger than Tracy’s age. This can be written as:
\[
b = \frac{4t}{3}
\]
This equation comes from the fact that one third larger means \( t + \frac{t}{3} = \frac{4t}{3} \).
2. In 30 years, Bill’s ... | 24 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. S... | To find the 20th special number, we need to follow the given rule: a number \( n \) is special if it is relatively prime to the sum of all special numbers less than \( n \). We will list out the special numbers and their sums step-by-step.
1. \( 1 \) is special by definition.
- Sum of special numbers: \( 1 \)
2. \... | 28 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other? | 1. **Fixing the Coaches**: Since the two coaches must sit opposite each other, we can fix one coach in a position. There are 2 ways to place the second coach opposite the first one. This is because once we place the first coach, the second coach has only one position available directly opposite.
2. **Arranging the Mat... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Eight identical cubes with of size $1 \times 1 \times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$, number $3$ written on the face opposite number $5$, and number $4$ written on the face opposite number $6$. The eight cubes are stacked into ... | 1. **Understanding the Problem:**
- We have eight identical cubes, each with faces numbered from 1 to 6.
- The numbers on opposite faces are: 1 opposite 2, 3 opposite 5, and 4 opposite 6.
- These cubes are stacked to form a $2 \times 2 \times 2$ cube.
- We need to find the sum of the numbers on the outer su... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In January Jeff’s investment went up by three quarters. In February it went down by one quarter. In March it went up by one third. In April it went down by one fifth. In May it went up by one seventh. In June Jeff’s investment fell by $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. If Jeff’s inv... | 1. Let Jeff's initial investment be $I = 1$ (for simplicity).
2. In January, the investment went up by three quarters:
\[
I_{\text{Jan}} = 1 + \frac{3}{4} = \frac{7}{4}
\]
3. In February, the investment went down by one quarter:
\[
I_{\text{Feb}} = \frac{7}{4} \times \left(1 - \frac{1}{4}\right) = \frac{... | 11 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The graph of the equation $y = 5x + 24$ intersects the graph of the equation $y = x^2$ at two points. The two points are a distance $\sqrt{N}$ apart. Find $N$. | 1. **Find the points of intersection:**
To find the points where the graphs of \( y = 5x + 24 \) and \( y = x^2 \) intersect, we set the equations equal to each other:
\[
5x + 24 = x^2
\]
Rearrange the equation to form a standard quadratic equation:
\[
x^2 - 5x - 24 = 0
\]
Solve this quadrati... | 3146 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positi... | 1. **Simplify the problem by ignoring the blue candies**: Since the blue candies do not affect the order in which the yellow and red candies are eaten, we can ignore them. We are left with 2 yellow candies and 4 red candies.
2. **Calculate the total number of ways to arrange the yellow and red candies**: We need to fi... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The straight river is one and a half kilometers wide and has a current of $8$ kilometers per hour. A boat capable of traveling $10$ kilometers per hour in still water, sets out across the water. How many minutes will it take the boat to reach a point directly across from where it started? | 1. **Identify the given values:**
- Width of the river: \(1.5\) kilometers
- Speed of the current: \(8\) kilometers per hour
- Speed of the boat in still water: \(10\) kilometers per hour
2. **Determine the effective speed of the boat across the river:**
- The boat's speed in still water is \(10\) km/h.
... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise? | 1. Let \( a \) be the tailor's age, \( b \) be the tortoise's age, and \( c \) be the tree's age.
2. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. This gives us the equation:
\[
\frac{a}{4} = 2a - b
\]
Solving for \( a \):
\[
\frac{a}{4} = 2a - b \implies a... | 77 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Functions $f$ and $g$ are defined so that $f(1) = 4$, $g(1) = 9$, and for each integer $n \ge 1$, $f(n+1) = 2f(n) + 3g(n) + 2n $ and $g(n+1) = 2g(n) + 3 f(n) + 5$. Find $f(2005) - g(2005)$. | 1. Given the functions \( f \) and \( g \) with initial conditions \( f(1) = 4 \) and \( g(1) = 9 \), and the recurrence relations:
\[
f(n+1) = 2f(n) + 3g(n) + 2n
\]
\[
g(n+1) = 2g(n) + 3f(n) + 5
\]
2. Define \( d(n) = f(n) - g(n) \). We need to find \( d(2005) \).
3. Using the recurrence relations,... | 1004 | Other | math-word-problem | Yes | Yes | aops_forum | false |
The summation $\sum_{k=1}^{360} \frac{1}{k \sqrt{k+1} + (k+1)\sqrt{k}}$ is the ratio of two relatively prime positive integers $m$ and $n$. Find $m + n$. | 1. We start with the given summation:
\[
\sum_{k=1}^{360} \frac{1}{k \sqrt{k+1} + (k+1)\sqrt{k}}
\]
2. To simplify the expression, we rationalize the denominator. Multiply the numerator and the denominator by the conjugate of the denominator:
\[
\frac{1}{k \sqrt{k+1} + (k+1)\sqrt{k}} \cdot \frac{k \sqrt... | 37 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $d_k$ be the greatest odd divisor of $k$ for $k = 1, 2, 3, \ldots$. Find $d_1 + d_2 + d_3 + \ldots + d_{1024}$. | 1. **Understanding the Problem:**
We need to find the sum of the greatest odd divisors of the numbers from 1 to 1024. Let \( d_k \) be the greatest odd divisor of \( k \).
2. **Odd Numbers:**
For any odd number \( k \), the greatest odd divisor is \( k \) itself. The odd numbers between 1 and 1024 are \( 1, 3, ... | 349526 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At the beginning of each hour from $1$ o’clock AM to $12$ NOON and from $1$ o’clock PM to $12$ MIDNIGHT a coo-coo clock’s coo-coo bird coo-coos the number of times equal to the number of the hour. In addition, the coo-coo clock’s coo-coo bird coo-coos a single time at $30$ minutes past each hour. How many times does th... | To solve this problem, we need to calculate the total number of coo-coos from 12:42 PM on Monday to 3:42 AM on Wednesday. We will break this down into three parts: the remaining time on Monday, the entire day of Tuesday, and the time on Wednesday up to 3:42 AM.
1. **Remaining time on Monday (from 12:42 PM to 12:00 AM)... | 289 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the student... | 1. Let the number of freshmen be \( f \), sophomores be \( o \), juniors be \( j \), and seniors be \( s \).
2. Given the ratios:
\[
\frac{f}{o} = \frac{5}{4} \implies f = \frac{5}{4}o
\]
\[
\frac{o}{j} = \frac{7}{8} \implies o = \frac{7}{8}j \implies j = \frac{8}{7}o
\]
\[
\frac{j}{s} = \frac{... | 630 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
We draw a radius of a circle. We draw a second radius $23$ degrees clockwise from the first radius. We draw a third radius $23$ degrees clockwise from the second. This continues until we have drawn $40$ radii each $23$ degrees clockwise from the one before it. What is the measure in degrees of the smallest angle betwee... | 1. **Understanding the Problem:**
We need to find the smallest angle between any two of the 40 radii drawn at 23-degree intervals on a circle.
2. **Listing the Angles:**
The angles where the radii are drawn can be listed as follows:
\[
0^\circ, 23^\circ, 46^\circ, 69^\circ, 92^\circ, 115^\circ, 138^\circ, ... | 23 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers divide the number $10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10$ ? | To determine the number of positive integers that divide \(10!\), we need to find the total number of divisors of \(10!\). This can be done by using the prime factorization of \(10!\).
1. **Prime Factorization of \(10!\)**:
\[
10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times... | 270 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before.... | 1. **Define the variables and initial conditions:**
- Let \( R_n \) be the number of rabbits caught on day \( n \).
- Let \( S_n \) be the number of squirrels caught on day \( n \).
- Initially, \( R_1 = 200 \) and \( S_1 = 18 \).
2. **Establish the recurrence relations:**
- Each day, the number of rabbits... | 5491 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers less than $2006$ which are both multiples of six and one more than a multiple of seven. | 1. We need to find the sum of all positive integers less than $2006$ which are both multiples of six and one more than a multiple of seven. Let these integers be denoted by $x$.
2. We start with the condition that $x$ is a multiple of six, so $x = 6n$ for some integer $n$.
3. Additionally, $x$ is one more than a multip... | 47094 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A rogue spaceship escapes. $54$ minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels $12\%$ faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues? | 1. Let's denote the speed of the rogue spaceship as \( v \) units per hour.
2. The police spaceship travels \( 12\% \) faster than the rogue spaceship, so its speed is \( v + 0.12v = 1.12v \) units per hour.
3. The rogue spaceship has a head start of 54 minutes. Converting this to hours, we get:
\[
\frac{54}{60}... | 450 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$. | 1. We start with the given equation:
\[
v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}
\]
Since \(v\) is a positive integer and \(1 < \frac{222}{155} < 2\), it follows that \(v = 1\).
2. Subtracting \(v = 1\) from both sides, we get:
\[
\frac{1}{w + \frac{1}{x + \frac{1}{... | 12354 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Heather and Kyle need to mow a lawn and paint a room. If Heather does both jobs by herself, it will take her a total of nine hours. If Heather mows the lawn and, after she finishes, Kyle paints the room, it will take them a total of eight hours. If Kyle mows the lawn and, after he finishes, Heather paints the room, it ... | 1. Let \( H_m \) and \( H_p \) be the hours Heather takes to mow the lawn and paint the room, respectively. Similarly, let \( K_m \) and \( K_p \) be the hours Kyle takes to mow the lawn and paint the room, respectively.
2. From the problem, we have the following information:
- Heather does both jobs in 9 hours: \(... | 41 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$. Find the number of zeros there are at the right end of the decimal representation for $k$. | To determine the number of zeros at the right end of the decimal representation of \( k \), we need to count the number of factors of 10 in \( k \). Since \( 10 = 2 \times 5 \), we need to count the number of pairs of factors of 2 and 5 in \( k \).
Given that \( k \) is the product of every third positive integer fro... | 168 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$12$ students need to form five study groups. They will form three study groups with $2$ students each and two study groups with $3$ students each. In how many ways can these groups be formed? | 1. **Choosing the groups of 3 students:**
- First, we choose 3 students out of 12. This can be done in $\binom{12}{3}$ ways.
- Next, we choose 3 students out of the remaining 9. This can be done in $\binom{9}{3}$ ways.
- Since the order of choosing the groups does not matter, we divide by 2 to account for the ... | 138600 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$. Find the sum of all the possible values of $n$ in these ordered pairs. | 1. We start with the given equation:
\[
59m - 68n = mn
\]
Rearrange the equation to:
\[
59m - mn = 68n
\]
Factor out \(m\) on the left-hand side:
\[
m(59 - n) = 68n
\]
Rearrange to solve for \(m\):
\[
m = \frac{68n}{59 - n}
\]
2. Since \(m\) and \(n\) are positive integers,... | 237 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$f(x)$ and $g(x)$ are linear functions such that for all $x$, $f(g(x)) = g(f(x)) = x$. If $f(0) = 4$ and $g(5) = 17$, compute $f(2006)$. | 1. Given that $f(x)$ and $g(x)$ are linear functions such that $f(g(x)) = g(f(x)) = x$ for all $x$, it implies that $f(x)$ and $g(x)$ are inverses of each other. Therefore, we can write:
\[
f(g(x)) = x \quad \text{and} \quad g(f(x)) = x
\]
2. Let $f(x) = ax + b$. Since $f(x)$ and $g(x)$ are inverses, we can e... | 122 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can $100$ be written as the sum of three positive integers $x, y$, and $z$ satisfying $x < y < z$ ? | 1. We start by expressing the problem in terms of three positive integers \(x, y, z\) such that \(x < y < z\) and \(x + y + z = 100\).
2. To simplify, let \(y = x + a\) and \(z = y + b = x + a + b\), where \(a\) and \(b\) are positive integers. This transforms the equation into:
\[
x + (x + a) + (x + a + b) = 100... | 784 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There is a very popular race course where runners frequently go for a daily run. Assume that all runners randomly select a start time, a starting position on the course, and a direction to run. Also assume that all runners make exactly one complete circuit of the race course, all runners run at the same speed, and all ... | 1. **Understanding the Problem:**
- You are running on a race course for 1 hour.
- You pass 300 runners going in the opposite direction.
- Some of these runners are counted twice because you pass them twice.
- We need to find the expected number of different runners you pass, not counting duplicates.
2. **... | 225 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all the positive integers which have at most three not necessarily distinct prime factors where the primes come from the set $\{ 2, 3, 5, 7 \}$. | 1. **Generate all combinations:**
We will consider the following cases:
- Numbers with 0 prime factors: $1$
- Numbers with 1 prime factor: $2, 3, 5, 7$
- Numbers with 2 prime factors: $2^2, 2 \cdot 3, 2 \cdot 5, 2 \cdot 7, 3^2, 3 \cdot 5, 3 \cdot 7, 5^2, 5 \cdot 7, 7^2$
- Numbers with 3 prime factors: $2... | 1932 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $F_0 = 0, F_{1} = 1$, and for $n \ge 1, F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$ . Then there are rational numbers $A$ and $B$ such that $\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}$. Find $A + B$. | 1. We start with the given Fibonacci sequence \( F_0 = 0, F_1 = 1 \), and for \( n \ge 1 \), \( F_{n+1} = F_n + F_{n-1} \). We also have the sequence \( a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n \).
2. We need to find rational numbers \( A \) and \( B \) such that
\[
\frac{a_{30} + a_{29}}{a_{26} + ... | 188 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We have two positive integers both less than $1000$. The arithmetic mean and the geometric mean of these numbers are consecutive odd integers. Find the maximum possible value of the difference of the two integers. | 1. Let the two positive integers be \( a \) and \( b \) with \( a > b \).
2. The arithmetic mean (AM) of \( a \) and \( b \) is given by:
\[
\text{AM} = \frac{a + b}{2}
\]
3. The geometric mean (GM) of \( a \) and \( b \) is given by:
\[
\text{GM} = \sqrt{ab}
\]
4. According to the problem, the AM and... | 62 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be two real numbers such that $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$. Find $\tan^2 x + 2 \tan^2 y$. | 1. Start with the given equation:
\[
2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7
\]
Factor the expression:
\[
\sin y (2 \sin x + 6 \cos x) + 3 \cos y = 7
\]
2. Apply the Cauchy-Schwarz inequality:
\[
(\sin^2 y + \cos^2 y) \left( (2 \sin x + 6 \cos x)^2 + 9 \right) \geq 49
\]
Since... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Square $ABCD$ has side length $36$. Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$. Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$. | 1. **Identify the coordinates of the points:**
- Let the coordinates of the vertices of the square \(ABCD\) be:
\[
A = (0, 0), \quad B = (36, 0), \quad C = (36, 36), \quad D = (0, 36)
\]
- Point \(E\) is on side \(AB\) a distance \(12\) from \(B\), so:
\[
E = (24, 0)
\]
- Point \(... | 54 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Terry drove along a scenic road using $9$ gallons of gasoline. Then Terry went onto the freeway and used $17$ gallons of gasoline. Assuming that Terry gets $6.5$ miles per gallon better gas mileage on the freeway than on the scenic road, and Terry’s average gas mileage for the entire trip was $30$ miles per gallon, f... | 1. Let \( M \) be the total number of miles Terry drove.
2. Terry used 9 gallons of gasoline on the scenic road and 17 gallons on the freeway, so the total amount of gasoline used is:
\[
9 + 17 = 26 \text{ gallons}
\]
3. The average gas mileage for the entire trip is given as 30 miles per gallon. Therefore, we... | 780 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The product of two positive numbers is equal to $50$ times their sum and $75$ times their difference. Find their sum. | 1. Let the two positive numbers be \( x \) and \( y \).
2. According to the problem, the product of the two numbers is equal to \( 50 \) times their sum:
\[
xy = 50(x + y)
\]
3. The product of the two numbers is also equal to \( 75 \) times their difference:
\[
xy = 75(x - y)
\]
4. Since both expressi... | 360 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There is an interval $[a, b]$ that is the solution to the inequality \[|3x-80|\le|2x-105|\] Find $a + b$. | 1. Start with the given inequality:
\[
|3x - 80| \leq |2x - 105|
\]
2. Square both sides to eliminate the absolute values:
\[
(3x - 80)^2 \leq (2x - 105)^2
\]
3. Expand both sides:
\[
(3x - 80)^2 = 9x^2 - 480x + 6400
\]
\[
(2x - 105)^2 = 4x^2 - 420x + 11025
\]
4. Set up the inequa... | 12 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Penelope plays a game where she adds $25$ points to her score each time she wins a game and deducts $13$ points from her score each time she loses a game. Starting with a score of zero, Penelope plays $m$ games and has a total score of $2007$ points. What is the smallest possible value for $m$? | 1. Let \( w \) be the number of games Penelope wins and \( l \) be the number of games she loses. We know that each win adds 25 points and each loss deducts 13 points. Therefore, the total score equation can be written as:
\[
25w - 13l = 2007
\]
2. We also know that the total number of games played \( m \) is... | 87 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class. Each year at the college one third of the freshman class drops out of school, $40$ students in the sophomore class drop out of school, an... | 1. **Determine the number of students needed in the junior class at the beginning of the year:**
- Given that one tenth of the junior class drops out, we need to start with enough students so that after the dropouts, there are 3400 students remaining.
- Let \( J \) be the number of students in the junior class at... | 5727 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Tom can run to Beth's house in $63$ minutes. Beth can run to Tom's house in $84$ minutes. At noon Tom starts running from his house toward Beth's house while at the same time Beth starts running from her house toward Tom's house. When they meet, they both run at Beth's speed back to Beth's house. At how many minute... | 1. **Determine the running rates of Tom and Beth:**
- Tom's running rate is $\frac{1}{63}$ of the distance between the houses per minute.
- Beth's running rate is $\frac{1}{84}$ of the distance between the houses per minute.
2. **Calculate the combined rate at which the distance between them decreases:**
\[
... | 78 | Other | math-word-problem | Yes | Yes | aops_forum | false |
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to eac... | To solve this problem, we need to determine the smallest positive integer \( n \) such that applying the given permutation \( n \) times to \( T_0 \) results in \( T_0 \) again. This means we need to find the order of the permutation.
1. **Identify the permutation:**
The permutation \( T_1 \) is given as \( \text{J... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If you alphabetize all of the distinguishable rearrangements of the letters in the word [b]PURPLE[/b], find the number $n$ such that the word [b]PURPLE [/b]is the $n$th item in the list. | To find the position of the word "PURPLE" in the list of all its distinguishable rearrangements when sorted alphabetically, we need to follow these steps:
1. **Calculate the total number of distinguishable permutations of "PURPLE":**
The word "PURPLE" has 6 letters where 'P' appears twice. The total number of disti... | 226 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We have some identical paper squares which are black on one side of the sheet and white on the other side. We can join nine squares together to make a $3$ by $3$ sheet of squares by placing each of the nine squares either white side up or black side up. Two of these $3$ by $3$ sheets are distinguishable if neither ca... | To solve this problem, we will use Burnside's Lemma, which helps in counting the number of distinct objects under group actions, such as rotations and reflections.
1. **Identify the group actions:**
The group actions here are the rotations of the $3 \times 3$ grid. The possible rotations are $0^\circ$, $90^\circ$, ... | 70 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A rectangular storage bin measures $10$ feet by $12$ feet, is $3$ feet tall, and sits on a flat plane. A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground $4$ feet away from the base of the storage bin as shown. The number o... | 1. **Calculate the volume of dirt on the sides of the storage bin:**
- The storage bin has two pairs of opposite sides: one pair with dimensions \(12 \text{ feet} \times 3 \text{ feet}\) and the other pair with dimensions \(10 \text{ feet} \times 3 \text{ feet}\).
- The dirt forms a triangular prism along each si... | 280 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A positive number $\dfrac{m}{n}$ has the property that it is equal to the ratio of $7$ plus the number’s reciprocal and $65$ minus the number’s reciprocal. Given that $m$ and $n$ are relatively prime positive integers, find $2m + n$. | 1. Let \( \frac{m}{n} = a \). We are given that \( a \) satisfies the equation:
\[
a = \frac{7 + \frac{1}{a}}{65 - \frac{1}{a}}
\]
2. Simplify the right-hand side of the equation:
\[
a = \frac{7a + 1}{65a - 1}
\]
3. Cross-multiplying to eliminate the fractions, we get:
\[
a(65a - 1) = 7a + 1
... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Allowing $x$ to be a real number, what is the largest value that can be obtained by the function $25\sin(4x)-60\cos(4x)?$ | 1. Consider the function \( f(x) = 25\sin(4x) - 60\cos(4x) \).
2. To find the maximum value, we can use calculus by taking the derivative and setting it to zero. First, compute the derivative:
\[
f'(x) = 100\cos(4x) + 240\sin(4x)
\]
3. Set the derivative equal to zero to find the critical points:
\[
100\... | 65 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
You know that the Jones family has five children, and the Smith family has three children. Of the eight children you know that there are five girls and three boys. Let $\dfrac{m}{n}$ be the probability that at least one of the families has only girls for children. Given that $m$ and $n$ are relatively prime positiv... | To find the probability that at least one of the families has only girls for children, we need to consider two cases: either the Jones family has all five girls, or the Smith family has all three girls.
1. **Case 1: The Jones family has five girls**
The probability that the first girl is in the Jones family is $\f... | 67 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a particular value of the angle $\theta$ we can take the product of the two complex numbers $(8+i)\sin\theta+(7+4i)\cos\theta$ and $(1+8i)\sin\theta+(4+7i)\cos\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find the largest value for $a+b$. | 1. We start by multiplying the two given complex numbers:
\[
\left[(8+i)\sin\theta + (7+4i)\cos\theta\right] \left[(1+8i)\sin\theta + (4+7i)\cos\theta\right]
\]
2. We expand the product using the distributive property:
\[
\left[(8+i)\sin\theta + (7+4i)\cos\theta\right] \left[(1+8i)\sin\theta + (4+7i)\co... | 125 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A dart board looks like three concentric circles with radii of 4, 6, and 8. Three darts are thrown at the board so that they stick at three random locations on then board. The probability that one dart sticks in each of the three regions of the dart board is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime posit... | 1. **Calculate the total area of the dart board:**
The dart board consists of three concentric circles with radii 4, 6, and 8. The total area of the dart board is the area of the largest circle:
\[
\text{Total area} = \pi \times 8^2 = 64\pi
\]
2. **Calculate the area of each region:**
- The smallest reg... | 617 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Six chairs sit in a row. Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left. There is only a problem if two people sitting next to each other have the person on the right crossing t... | 1. **Define the problem and initial conditions:**
- We have 6 chairs and 6 people.
- Each person can choose one of three positions: feet on the floor (F), legs crossed to the right (R), or legs crossed to the left (L).
- A problem occurs if two adjacent people have the person on the right crossing their legs t... | 1106 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two circles with radius $2$ and radius $4$ have a common center at P. Points $A, B,$ and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$. | 1. **Identify the given information and the goal:**
- Two concentric circles with radii 2 and 4 centered at point \( P \).
- Points \( A, B, \) and \( C \) form an equilateral triangle on the larger circle.
- Point \( D \) is the intersection of the smaller circle and the line segment \( PB \).
- We need to... | 192 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Starting on April 15, 2008, you can go one day backward and one day forwards to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer k so that if you go k days backward and k days forward you get two calendar date... | 1. **Identify the problem constraints**: We need to find the smallest positive integer \( k \) such that going \( k \) days backward and \( k \) days forward from a given date results in the same calendar date.
2. **Consider the month with the fewest days**: February is the month with the fewest days, having 28 days i... | 14 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The student population at one high school consists of freshmen, sophomores, juniors, and seniors. There are 25 percent more freshmen than juniors, 10 percent fewer sophomores than freshmen, and 20 percent of the students are seniors. If there are 144 sophomores, how many students attend the school? | 1. Let \( F \) be the number of freshmen, \( S \) be the number of sophomores, \( J \) be the number of juniors, and \( R \) be the number of seniors.
2. We are given that there are 144 sophomores, so \( S = 144 \).
3. We are also given that there are 10 percent fewer sophomores than freshmen. Therefore, \( S = 0.9F \)... | 540 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all the digits in the decimal representations of all the positive integers less than $1000.$ | 1. **Counting from 000 to 999:**
- We include leading zeros and the number 0 itself, so we count from 000 to 999. This gives us a total of 1000 numbers.
2. **Sum of the units digits:**
- The units digit cycles through 0 to 9 repeatedly. Each complete cycle (0 through 9) sums to:
\[
0 + 1 + 2 + 3 + 4 + ... | 13500 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A line through the origin passes through the curve whose equation is $5y=2x^2-9x+10$ at two points whose $x-$coordinates add up to $77.$ Find the slope of the line. | 1. Let the equation of the line through the origin be \( y = kx \), where \( k \) is the slope of the line.
2. Substitute \( y = kx \) into the given equation of the parabola \( 5y = 2x^2 - 9x + 10 \):
\[
5(kx) = 2x^2 - 9x + 10
\]
Simplify this to:
\[
5kx = 2x^2 - 9x + 10
\]
3. Rearrange the equati... | 29 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A container is shaped like a square-based pyramid where the base has side length $23$ centimeters and the height is $120$ centimeters. The container is open at the base of the pyramid and stands in an open field with its vertex pointing down. One afternoon $5$ centimeters of rain falls in the open field partially filli... | 1. **Understanding the problem**: We need to find the depth of rainwater collected in a square-based pyramid container after 5 cm of rain falls. The base of the pyramid has a side length of 23 cm, and the height of the pyramid is 120 cm. The pyramid is open at the base and stands with its vertex pointing down.
2. **Vo... | 60 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all the integers $N > 1$ with the properties that the each prime factor of $N $ is either $2, 3,$ or $5,$ and $N$ is not divisible by any perfect cube greater than $1.$ | 1. We need to find the sum of all integers \( N > 1 \) such that each prime factor of \( N \) is either \( 2, 3, \) or \( 5 \), and \( N \) is not divisible by any perfect cube greater than \( 1 \).
2. The numbers \( N \) satisfying these conditions can be written in the form \( 2^a \times 3^b \times 5^c \), where \( ... | 2820 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A circular track with diameter $500$ is externally tangent at a point A to a second circular track with diameter $1700.$ Two runners start at point A at the same time and run at the same speed. The first runner runs clockwise along the smaller track while the second runner runs clockwise along the larger track. Ther... | 1. **Define the problem and setup the geometry:**
- Let the diameter of the smaller track be \(500\) units, so its radius \(r_1 = \frac{500}{2} = 250\) units.
- Let the diameter of the larger track be \(1700\) units, so its radius \(r_2 = \frac{1700}{2} = 850\) units.
- Let the centers of the smaller and large... | 501 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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 | 1. Let the sides of the triangle be \( a = 75 \), \( b = x \), and \( c = 2x \).
2. Using Heron's formula, the area \( A \) of a triangle with sides \( a, b, c \) is given by:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( s \) is the semi-perimeter:
\[
s = \frac{a + b + c}{2} = \frac{75 + x + 2x}{2} = \f... | 1125 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
At Mallard High School there are three intermural sports leagues: football, basketball, and baseball. There are 427 students participating in these sports: 128 play on football teams, 291 play on basketball teams, and 318 play on baseball teams. If exactly 36 students participate in all three of the sports, how many ... | 1. Let \( F \) be the set of students who play football, \( B \) be the set of students who play basketball, and \( S \) be the set of students who play baseball.
2. We are given:
\[
|F| = 128, \quad |B| = 291, \quad |S| = 318, \quad |F \cap B \cap S| = 36
\]
3. Let \( |F \cap B| \) be the number of students w... | 274 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits. | To find the least positive integer \( n \) such that the decimal representation of the binomial coefficient \( \binom{2n}{n} \) ends in four zero digits, we need to determine when \( \binom{2n}{n} \) is divisible by \( 10^4 = 2^4 \times 5^4 \). Since the number of factors of 2 in \( \binom{2n}{n} \) will always be grea... | 313 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For natural number $n$, define the function $f(n)$ to be the number you get by $f(n)$ adding the digits of the number $n$. For example, $f(16)=7$, $f(f(78))=6$, and $f(f(f(5978)))=2$.
Find the least natural number $n$ such that $f(f(f(n)))$ is not a one-digit number. | 1. We need to find the smallest natural number \( n \) such that \( f(f(f(n))) \) is not a one-digit number. This means \( f(f(f(n))) \geq 10 \).
2. Let's start by understanding the function \( f(n) \). The function \( f(n) \) is the sum of the digits of \( n \). For example:
\[
f(16) = 1 + 6 = 7
\]
\[
... | 19999999999999999999999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The trapezoid below has bases with lengths 7 and 17 and area 120. Find the difference of the areas of the two triangles.
[center]
[img]https://i.snag.gy/BlqcSQ.jpg[/img]
[/center] | 1. **Identify the given information and the goal:**
- The trapezoid has bases of lengths 7 and 17.
- The area of the trapezoid is 120.
- We need to find the difference in the areas of the two triangles formed by the diagonals.
2. **Use the formula for the area of a trapezoid:**
\[
\text{Area} = \frac{1}... | 50 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the positive integer $n$ such that $10^n$ cubic centimeters is the same as 1 cubic kilometer.
| 1. We start by noting that there are \(10^5\) centimeters in a kilometer. This is because:
\[
1 \text{ kilometer} = 1000 \text{ meters} \quad \text{and} \quad 1 \text{ meter} = 100 \text{ centimeters}
\]
Therefore:
\[
1 \text{ kilometer} = 1000 \times 100 = 10^5 \text{ centimeters}
\]
2. To find t... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
One side of a rectangle has length 18. The area plus the perimeter of the rectangle is 2016. Find the perimeter of the rectangle.
| 1. Let the length of one side of the rectangle be \(18\) and the other side be \(a\).
2. The perimeter \(P\) of the rectangle is given by:
\[
P = 2 \times (18 + a) = 2a + 36
\]
3. The area \(A\) of the rectangle is given by:
\[
A = 18 \times a = 18a
\]
4. According to the problem, the sum of the area ... | 234 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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. | 1. We start by noting that the mean of the five integers is 100. Therefore, the sum of the five integers is:
\[
5 \times 100 = 500
\]
2. When the median is removed, the mean of the remaining four integers increases by 5. Thus, the new mean is:
\[
100 + 5 = 105
\]
Therefore, the sum of the remainin... | 269 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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 $45^{\circ}$ 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.
[center... | 1. Let \( x \) be the length of the segment between two adjacent dots on any side of the original square. Since each side of the square is divided into 6 equal segments, the side length of the square is \( 6x \).
2. Each shaded square is formed by connecting dots that are \( x \) units apart along the side of the squa... | 360 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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.
| 1. We start with the given information:
- Least Common Multiple (LCM) of \( m \) and \( n \) is 480.
- Greatest Common Divisor (GCD) of \( m \) and \( n \) is 12.
- Both \( m \) and \( n \) are greater than 50.
2. We use the relationship between LCM and GCD:
\[
\text{LCM}(m, n) \times \text{GCD}(m, n) =... | 156 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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 ... | 1. **Determine the meeting point of Aaron and Michael:**
- Aaron starts at Acorn and bicycles east at 17 km/h.
- Michael starts at Midland and bicycles west at 7 km/h.
- The distance between Acorn and Midland is \(60 + 60 = 120\) km.
- The combined speed of Aaron and Michael is \(17 + 7 = 24\) km/h.
- Th... | 65 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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 t... | 1. **Count the total number of digits Jeremy wrote:**
Jeremy wrote all three-digit integers from 100 to 999. Each number has 3 digits, and there are \(999 - 100 + 1 = 900\) such numbers.
\[
\text{Total digits} = 900 \times 3 = 2700
\]
2. **Count the occurrences of each digit from 0 to 9:**
- Each digit ... | 275 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest possible value of $pq + r$, where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.
| To find the greatest possible value of \( pq + r \), where \( p \), \( q \), and \( r \) are prime numbers satisfying \( pq + qr + rp = 2016 \), we can follow these steps:
1. **Minimize \( r \) and make \( p \) and \( q \) as close as possible**:
- Start by setting \( r = 2 \) (the smallest prime number).
2. **Sub... | 1008 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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 | 1. We need to find the least positive integer of the form \( abaaba \), where \( a \) and \( b \) are distinct digits, such that the integer can be written as a product of six distinct primes.
2. Notice that \( abaaba \) can be factored as:
\[
abaaba = aba \cdot 1001
\]
where \( 1001 = 7 \cdot 11 \cdot 13 \... | 282282 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to each teacher in her sc... | 1. Let the number of pieces of candy in each one-pound bag be \( x \).
2. Let the number of people in Mike's class be \( s \).
3. Let the number of teachers in Betsy's school be \( t \).
From the problem, we can set up the following equations based on the given information:
\[ 4x - 15s = 23 \]
\[ 5x - 23t = 15 \]
4. ... | 302 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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... | 1. **Identify the number of ways to select the desired combination:**
- We need to select a red marble from Jar #1, a blue marble from Jar #2, and a green marble from Jar #3.
- The number of ways to select a red marble from Jar #1 is 5 (since there are 5 red marbles in Jar #1).
- The number of ways to select a... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers a, b, c, d, and e satisfy the equations
$$(a + 1)(3bc + 1) = d + 3e + 1$$
$$(b + 1)(3ca + 1) = 3d + e + 13$$
$$(c + 1)(3ab + 1) = 4(26-d- e) - 1$$
Find $d^2+e^2$.
| 1. We start with the given equations:
\[
(a + 1)(3bc + 1) = d + 3e + 1
\]
\[
(b + 1)(3ca + 1) = 3d + e + 13
\]
\[
(c + 1)(3ab + 1) = 4(26 - d - e) - 1
\]
2. Let's sum all three equations:
\[
(a + 1)(3bc + 1) + (b + 1)(3ca + 1) + (c + 1)(3ab + 1) = (d + 3e + 1) + (3d + e + 13) + [4(26 -... | 146 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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 a... | 1. **Using Routh's Theorem**: Routh's Theorem is a useful tool for finding the ratio of areas of triangles formed by cevians intersecting at a point. For an equilateral triangle, if the cevians divide the sides in the ratio \( x : 1 \), \( y : 1 \), and \( z : 1 \), the area of the inner triangle formed by the cevians ... | 30 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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 g... | To find the mean distance between two distinct dots in a 4x4 grid, we need to calculate the total distance for all pairs of dots and then divide by the number of pairs. We will use casework to count the number of pairs for each possible distance.
1. **Calculate the number of pairs for each distance:**
- **Distance... | 36 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest prime $p$ such that $p$ divides $2^{p+1} + 3^{p+1} + 5^{p+1} + 7^{p+1}$. | 1. **Apply Fermat's Little Theorem**: Fermat's Little Theorem states that if \( p \) is a prime number and \( a \) is an integer not divisible by \( p \), then \( a^p \equiv a \pmod{p} \). For our problem, we need to consider \( a^{p+1} \).
2. **Transform the Exponents**: Using Fermat's Little Theorem, we have:
\[
... | 29 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all values of $a$ such that there are positive integers $a$ and $b$ satisfying $(a - b)\sqrt{ab} = 2016$. | 1. Let \( \gcd(a, b) = d \). Then, we can write \( a = dx^2 \) and \( b = dy^2 \) for some positive integers \( x \) and \( y \) such that \( \gcd(x, y) = 1 \). This is because \( a \) and \( b \) must be divisible by \( d \), and the remaining parts must be squares to ensure \( a \) and \( b \) are integers.
2. Subst... | 209 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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 $$ 4\sum^{30}_{n=1} n \cdot T(n).$$ | 1. First, we need to simplify the expression for \( T(n) \):
\[
T(n) = \cos^2(30^\circ - n) - \cos(30^\circ - n)\cos(30^\circ + n) + \cos^2(30^\circ + n)
\]
2. Notice that \( T(n) \) can be rewritten in a form that suggests a perfect square. Let's explore this:
\[
T(n) = (\cos(30^\circ - n) - \cos(30^\c... | 1395 | Calculus | other | Yes | Yes | aops_forum | false |
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] | 1. **Identify the structure of the figure:**
- The figure consists of a central square with side length 20.
- Four smaller squares, each with side length 5, are attached to each side of the central square.
2. **Calculate the perimeter of the central square:**
- The perimeter of a square is given by \(4 \times... | 140 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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] | 1. Let the side length of the original square be \(3x\). Then, the area of the original square is:
\[
(3x)^2 = 9x^2
\]
2. The shaded region is an octagon formed by cutting off four right triangles from the corners of the square. Each of these triangles has legs of length \(x\), so the area of each triangle is... | 135 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A 2 meter long bookshelf is filled end-to-end with 46 books. Some of the books are 3 centimeters thick while all the others are 5 centimeters thick. Find the number of books on the shelf that are 3 centimeters thick. | 1. Define the variables:
- Let \( x \) be the number of 3 cm thick books.
- Let \( y \) be the number of 5 cm thick books.
2. Set up the system of equations based on the given information:
- The total number of books is 46:
\[
x + y = 46
\]
- The total thickness of the books is 200 cm (since... | 15 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of three-digit positive integers where the digits are three different prime numbers. For example, count 235 but not 553. | 1. Identify the single-digit prime numbers. The single-digit prime numbers are \(2, 3, 5,\) and \(7\).
2. Choose 3 out of these 4 prime numbers to form a three-digit number. The number of ways to choose 3 out of 4 is given by the binomial coefficient:
\[
\binom{4}{3} = \frac{4!}{3!(4-3)!} = 4
\]
3. For each se... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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 p... | 1. **Understanding the Problem:**
- Mildred is tied to a point on the side of a square shed.
- The shed has a side length of 10 meters.
- The rope is attached 2 meters from one corner of the shed.
- The rope is 14 meters long.
- We need to find the area of grass that Mildred can reach, expressed as \( n\... | 155 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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 altoge... | 1. Let \( x \) be the number of \$20 bills.
2. According to the problem, the number of \$10 bills is twice the number of \$20 bills, so the number of \$10 bills is \( 2x \).
3. The number of \$5 bills is 20 more than the number of \$10 bills, so the number of \$5 bills is \( 2x + 20 \).
4. The total amount of money co... | 1210 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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] | 1. **Case 1: $1 \times 1$ square**
- By inspection, there is only $1$ square that works. This is the square that contains the dot directly in its center.
2. **Case 2: $2 \times 2$ square**
- By inspection, there are $4$ possible $2 \times 2$ squares that work. These squares are formed by choosing any $2 \times 2... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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 car... | 1. Let \( x \) be the fraction of cars that have only a driver (1 person).
2. According to the problem, twice as many cars have a driver and one passenger (2 people) as those with only a driver. Therefore, the fraction of cars with 2 people is \( 2x \).
3. Similarly, twice as many cars have a driver and two passengers ... | 113 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
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. | 1. The measure of an internal angle of a regular polygon with \( n \) sides is given by:
\[
\text{Internal angle} = 180^\circ - \frac{360^\circ}{n}
\]
For this angle to be an integer, \(\frac{360^\circ}{n}\) must also be an integer.
2. This implies that \( n \) must be a divisor of \( 360 \). We need to fi... | 22 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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 $m + n$. | 1. **Identify the total number of possible outcomes:**
Since Suzie flips a fair coin 6 times, each flip has 2 possible outcomes (heads or tails). Therefore, the total number of possible outcomes is:
\[
2^6 = 64
\]
2. **Identify the favorable outcomes:**
We need to count the number of sequences where Suz... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$. | 1. Let \( n \) be the positive integer such that the least common multiple (LCM) of \( n \) and \( n - 30 \) is \( n + 1320 \). We can write this as:
\[
\text{LCM}(n, n - 30) = n + 1320
\]
2. Recall the relationship between the LCM and the greatest common divisor (GCD):
\[
\text{LCM}(a, b) = \frac{a \cd... | 165 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The figure below was made by gluing together 5 non-overlapping congruent squares. The figure has area 45. Find the perimeter of the figure.
[center][img]https://snag.gy/ZeKf4q.jpg[/center][/img] | 1. **Determine the area of each square:**
The total area of the figure is given as \(45\) square units. Since the figure is composed of 5 congruent squares, the area of each square is:
\[
\text{Area of each square} = \frac{45}{5} = 9 \text{ square units}
\]
2. **Find the side length of each square:**
Th... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible? | 1. The Stromquist Comet is visible every 61 years. We need to find the next leap year after 2017 when the comet will be visible.
2. A leap year is defined as a year that is divisible by 4, except for end-of-century years, which must be divisible by 400. For simplicity, we will first check for divisibility by 4.
3. We n... | 2444 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the... | 1. Let \( a \) be the time it takes Aiden for his morning preparation.
2. Let \( b \) be the time it takes Aiden to ride the bus.
3. Let \( c \) be the time it takes Aiden to walk from the bus stop to work.
From the problem, we have the following information:
- On a typical morning, Aiden arrives at work 120 minutes a... | 126 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$. | 1. Let \( m = 111x \) and \( n = 31y \), where \( x \) and \( y \) are positive integers.
2. Given that \( m + n = 2017 \), we substitute \( m \) and \( n \) into the equation:
\[
111x + 31y = 2017
\]
3. We need to find integer solutions for \( x \) and \( y \). To do this, we can use the method of solving Dio... | 463 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle{ADE}$ points $B$ and $C$ are on side $AD$ and points $F$ and $G$ are on side $AE$ so that $BG \parallel CF \parallel DE$, as shown. The area of $\triangle{ABG}$ is $36$, the area of trapezoid $CFED$ is $144$, and $AB = CD$. Find the area of trapezoid $BGFC$.
[center][img]https://snag.gy/SIuOLB.jpg[/img][... | 1. Let \( AB = CD = x \) and \( BC = y \).
2. Given that the area of \(\triangle ABG\) is 36, we can express this area in terms of the area of \(\triangle ADE\). Since \(BG \parallel CF \parallel DE\), the triangles \(\triangle ABG\) and \(\triangle ADE\) are similar. The ratio of their areas is the square of the rati... | 45 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of rearrangements of the letters in the word MATHMEET that begin and end with the same letter such as TAMEMHET. | To find the number of rearrangements of the letters in the word "MATHMEET" that begin and end with the same letter, we can break the problem into three cases based on which letter is at the ends.
1. **Case 1: The word begins and ends with the letter T.**
- The word "MATHMEET" contains 8 letters: M, A, T, H, M, E, E... | 540 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$, $y$, and $z$ be real numbers such that
$$12x - 9y^2 = 7$$
$$6y - 9z^2 = -2$$
$$12z - 9x^2 = 4$$
Find $6x^2 + 9y^2 + 12z^2$. | 1. We start with the given equations:
\[
12x - 9y^2 = 7
\]
\[
6y - 9z^2 = -2
\]
\[
12z - 9x^2 = 4
\]
2. Add all three equations:
\[
(12x - 9y^2) + (6y - 9z^2) + (12z - 9x^2) = 7 - 2 + 4
\]
Simplifying the right-hand side:
\[
12x + 6y + 12z - 9x^2 - 9y^2 - 9z^2 = 9
\]
3.... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The set of positive real numbers $x$ that satisfy $2 | x^2 - 9 | \le 9 | x | $ is an interval $[m, M]$. Find $10m + M$. | To solve the problem, we need to find the set of positive real numbers \( x \) that satisfy the inequality \( 2 | x^2 - 9 | \le 9 | x | \). We will consider different cases based on the value of \( x \).
1. **Case 1: \( x > 3 \)**
- Here, \( | x^2 - 9 | = x^2 - 9 \) and \( | x | = x \).
- The inequality becomes:... | 21 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
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. | 1. We start with the equation \(a^2 + \sqrt{2017 - b^2} = k^2\) where \(k\) is an integer. For this to be true, \(\sqrt{2017 - b^2}\) must also be an integer. Let \(\sqrt{2017 - b^2} = m\) where \(m\) is an integer. Therefore, we have:
\[
2017 - b^2 = m^2
\]
This can be rewritten as:
\[
2017 = b^2 + m... | 67 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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