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Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$, $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers. | 1. We need to find a ten-digit number \(a_0a_1a_2\ldots a_9\) such that for each \(k = 0, 1, \ldots, 9\), \(a_k\) is equal to the number of times the digit \(k\) occurs in the number.
2. First, note that the sum of all digits \(a_0 + a_1 + a_2 + \ldots + a_9 = 10\) because the number has ten digits.
3. Let's denote \(a... | 6210001000 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$. | 1. **Understanding the problem**: We need to find the minimum degree \( n \) of a monic polynomial \( P(x) \) with integer coefficients such that \( 2013 \) divides \( P(r) \) for exactly \( 1000 \) values of \( r \) between \( 1 \) and \( 2013 \) inclusive.
2. **Prime factorization of 2013**:
\[
2013 = 3 \time... | 50 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$. | To solve the problem, we need to determine the number of prime numbers \( p \) between 100 and 200 for which the congruence \( x^{11} + y^{16} \equiv 2013 \pmod{p} \) has a solution in integers \( x \) and \( y \).
1. **Analyzing the Congruence:**
We start by analyzing the given congruence \( x^{11} + y^{16} \equiv... | 21 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The area of a circle centered at the origin, which is inscribed in the parabola $y=x^2-25$, can be expressed as $\tfrac ab\pi$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$? | 1. **Identify the points of intersection:**
The circle is inscribed in the parabola \( y = x^2 - 25 \). Let the points of intersection be \((x, x^2 - 25)\) and \((-x, x^2 - 25)\).
2. **Determine the slope of the line connecting the origin and the point of intersection:**
The slope between \((x, x^2 - 25)\) and t... | 103 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $m$ such that $2^m$ can be expressed as a sum of four factorials (of positive integers).
Note: The factorials do not have to be distinct. For example, $2^4=16$ counts, because it equals $3!+3!+2!+2!$. | To find the sum of all positive integers \( m \) such that \( 2^m \) can be expressed as a sum of four factorials of positive integers, we start by setting up the equation:
\[ 2^m = a! + b! + c! + d! \]
where \( a \ge b \ge c \ge d > 0 \).
### Case 1: \( d = 1 \)
If \( d = 1 \), then \( d! = 1 \), and the right-hand... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers $n$ less than $1000$ have the property that the number of positive integers less than $n$ which are coprime to $n$ is exactly $\tfrac n3$? | 1. We start by noting that the problem involves Euler's Totient Function, $\varphi(n)$, which counts the number of positive integers less than $n$ that are coprime to $n$. The given condition is:
\[
\varphi(n) = \frac{n}{3}
\]
2. Suppose $n$ has the prime factorization $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k... | 41 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$, where $n$ is either $2012$ or $2013$. | 1. **Clear the denominators:**
\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{n^2}
\]
Multiply through by \(xy \cdot n^2\):
\[
n^2 y + n^2 x = xy
\]
Rearrange the equation:
\[
xy - n^2 x - n^2 y = 0
\]
Add \(n^4\) to both sides:
\[
xy - n^2 x - n^2 y + n^4 = n^4
\]
Factor the le... | 338 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $k$ be a positive integer with the following property: For every subset $A$ of $\{1,2,\ldots, 25\}$ with $|A|=k$, we can find distinct elements $x$ and $y$ of $A$ such that $\tfrac23\leq\tfrac xy\leq\tfrac 32$. Find the smallest possible value of $k$. | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( k \) such that for every subset \( A \) of \(\{1, 2, \ldots, 25\}\) with \(|A| = k\), there exist distinct elements \( x \) and \( y \) in \( A \) satisfying \(\frac{2}{3} \leq \frac{x}{y} \leq \frac{3}{2} \).
2. **Analyzing the Cond... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Floyd looked at a standard $12$ hour analogue clock at $2\!:\!36$. When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\!:\!00$ was that? | 1. **Determine the movement of the hour and minute hands:**
- Let \( h \) be the degrees moved by the hour hand.
- The minute hand moves \( 12h \) degrees for every \( h \) degrees the hour hand moves (since the minute hand moves 12 times faster than the hour hand).
2. **Set up the equation for the total movemen... | 14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Asheville, Bakersfield, Charter, and Darlington are four small towns along a straight road in that order. The distance from Bakersfield to Charter is one-third the distance from Asheville to Charter and one-quarter the distance from Bakersfield to Darlington. If it is $12$ miles from Bakersfield to Charter, how many mi... | 1. Let the towns be denoted as follows:
- \( A \) for Asheville
- \( B \) for Bakersfield
- \( C \) for Charter
- \( D \) for Darlington
2. Given:
- The distance from Bakersfield to Charter (\( BC \)) is 12 miles.
- The distance from Bakersfield to Charter (\( BC \)) is one-third the distance from As... | 72 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all four-digit integers whose digits are a rearrangement of the digits $1$, $2$, $3$, $4$, such as $1234$, $1432$, or $3124$. | 1. **Calculate the sum of the digits:**
\[
1 + 2 + 3 + 4 = 10
\]
2. **Determine the number of four-digit integers that can be formed using the digits \(1, 2, 3, 4\):**
Since we are rearranging 4 unique digits, the number of permutations is:
\[
4! = 24
\]
3. **Calculate the contribution of each di... | 66660 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$, and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter,... | 1. **First Week:**
- The quarry sells $60\%$ of the gravel at full price.
- Full price of the gravel is $3200$ dollars.
- Amount sold in the first week: $0.6 \times 3200 = 1920$ dollars.
2. **Second Week:**
- The remaining gravel is $40\%$ of the original amount.
- The price is reduced by $10\%$, so the... | 3000 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For positive integer $n$ let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999,488$ and $a_7 = 9,999,999,488$. Find the value of $n$ so that an is divisible by the highest power of $2$. | 1. Let's start by expressing \( a_n \) in a more manageable form. Given \( a_n \) is the integer consisting of \( n \) digits of 9 followed by the digits 488, we can write:
\[
a_n = 10^n \cdot 999 + 488
\]
This is because \( 999 \ldots 999 \) (with \( n \) digits of 9) can be written as \( 10^n - 1 \).
2. ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, and $c$ be positive real numbers such that $a^2+b^2+c^2=989$ and $(a+b)^2+(b+c)^2+(c+a)^2=2013$. Find $a+b+c$. | 1. We start with the given equations:
\[
a^2 + b^2 + c^2 = 989
\]
and
\[
(a+b)^2 + (b+c)^2 + (c+a)^2 = 2013.
\]
2. Expand the second equation:
\[
(a+b)^2 + (b+c)^2 + (c+a)^2 = a^2 + b^2 + 2ab + b^2 + c^2 + 2bc + c^2 + a^2 + 2ca.
\]
Simplify the expanded form:
\[
(a+b)^2 + (b+c)^2... | 32 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$. | 1. **Identify the problem**: We need to find the probability that a point randomly selected from inside a rectangle with side lengths 6 and 8 is closer to a side of the rectangle than to either diagonal.
2. **Calculate the area of the rectangle**:
\[
\text{Area of the rectangle} = 6 \times 8 = 48
\]
3. **De... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Two years ago Tom was $25\%$ shorter than Mary. Since then Tom has grown $20\%$ taller, and Mary has grown $4$ inches taller. Now Mary is $20\%$ taller than Tom. How many inches tall is Tom now? | 1. Let \( t \) be Tom's height two years ago and \( m \) be Mary's height two years ago. According to the problem, two years ago Tom was \( 25\% \) shorter than Mary. This can be written as:
\[
t = 0.75m
\]
2. Since then, Tom has grown \( 20\% \) taller. Therefore, Tom's current height is:
\[
1.2t
\]... | 45 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The greatest common divisor of $n$ and $180$ is $12$. The least common multiple of $n$ and $180$ is $720$. Find $n$. | 1. Given that the greatest common divisor (gcd) of \( n \) and \( 180 \) is \( 12 \), we write:
\[
\gcd(n, 180) = 12
\]
2. Also given that the least common multiple (lcm) of \( n \) and \( 180 \) is \( 720 \), we write:
\[
\text{lcm}(n, 180) = 720
\]
3. We use the relationship between gcd and lcm fo... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In four years Kay will be twice as old as Gordon. Four years after that Shaun will be twice as old as Kay. Four years after that Shaun will be three times as old as Gordon. How many years old is Shaun now? | 1. Let the current ages of Kay, Gordon, and Shaun be denoted by \( K \), \( G \), and \( S \) respectively.
2. According to the problem, in four years, Kay will be twice as old as Gordon:
\[
K + 4 = 2(G + 4)
\]
Simplifying this equation:
\[
K + 4 = 2G + 8 \implies K = 2G + 4
\]
3. Four years afte... | 48 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There is a pile of eggs. Joan counted the eggs, but her count was way off by $1$ in the $1$'s place. Tom counted in the eggs, but his count was off by $1$ in the $10$'s place. Raoul counted the eggs, but his count was off by $1$ in the $100$'s place. Sasha, Jose, Peter, and Morris all counted the eggs and got the corre... | 1. Let \( x \) be the correct number of eggs. The counts by Joan, Tom, and Raoul are off by \( \pm 1 \) in the 1's place, \( \pm 10 \) in the 10's place, and \( \pm 100 \) in the 100's place, respectively. The counts by Sasha, Jose, Peter, and Morris are correct.
2. The sum of the counts by all seven people is given b... | 439 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$|5x^2-\tfrac25|\le|x-8|$ if and only if $x$ is in the interval $[a, b]$. There are relatively prime positive integers $m$ and $n$ so that $b -a =\tfrac{m}{n}$ . Find $m + n$. | 1. We start with the inequality:
\[
|5x^2 - \frac{2}{5}| \le |x - 8|
\]
2. To solve this, we need to consider the cases where the absolute values can be removed. We will analyze the inequality by squaring both sides to eliminate the absolute values:
\[
(5x^2 - \frac{2}{5})^2 \le (x - 8)^2
\]
3. Expa... | 18 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The number $N$ is the product of two primes. The sum of the positive divisors of $N$ that are less than $N$ is $2014$. Find $N$. | 1. Let \( N = a \cdot b \), where \( a \) and \( b \) are prime numbers.
2. The positive divisors of \( N \) are \( 1, a, b, \) and \( N \).
3. The sum of all positive divisors of \( N \) is given by:
\[
1 + a + b + N
\]
4. According to the problem, the sum of the positive divisors of \( N \) that are less tha... | 4022 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits. | 1. Let the three-digit number be represented as \(100a + 10b + c\), where \(a, b, c\) are the digits of the number, and \(a \neq 0\) since it is a three-digit number.
2. According to the problem, the number is equal to the sum of its digits plus twice the product of its digits. Therefore, we have the equation:
\[
... | 397 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways. | To solve the problem of finding the number of ways to write \(12\) as an ordered sum of integers where the smallest of those integers is equal to \(2\), we can use a systematic approach.
1. **Identify the constraints**:
- The smallest integer in each sum must be \(2\).
- The sum of the integers must be \(12\).
... | 70 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x,y$ and $z$ are integers that satisfy the system of equations \[x^2y+y^2z+z^2x=2186\] \[xy^2+yz^2+zx^2=2188.\] Evaluate $x^2+y^2+z^2.$ | 1. We start with the given system of equations:
\[
x^2y + y^2z + z^2x = 2186
\]
\[
xy^2 + yz^2 + zx^2 = 2188
\]
2. Subtract the first equation from the second equation:
\[
(xy^2 + yz^2 + zx^2) - (x^2y + y^2z + z^2x) = 2188 - 2186
\]
Simplifying the right-hand side:
\[
xy^2 + yz^2 + ... | 245 | Algebra | other | Yes | Yes | aops_forum | false |
For positive integers $m$ and $n$, the decimal representation for the fraction $\tfrac{m}{n}$ begins $0.711$ followed by other digits. Find the least possible value for $n$. | 1. We are given that the decimal representation of the fraction $\frac{m}{n}$ begins with $0.711$ followed by other digits. This means that:
\[
0.711 \leq \frac{m}{n} < 0.712
\]
2. To find the least possible value for $n$, we need to find a fraction $\frac{m}{n}$ that lies within this interval and has the sma... | 45 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which:
(i) $f(x)\ge 0$ for all real $x,$ and
(ii) $a_n=0$ whenever $n$ is a multiple of $3.$
Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that t... | 1. **Define the set \( C \) and the cosine polynomials \( f(x) \):**
\[
C = \bigcup_{N=1}^{\infty} C_N,
\]
where \( C_N \) denotes the set of cosine polynomials of the form:
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2\pi nx),
\]
subject to the conditions:
- \( f(x) \geq 0 \) for all real \( x \),
... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be an integer larger than $1$ and let $S$ be the set of $n$-element subsets of the set $\{1,2,\ldots,2n\}$. Determine
\[\max_{A\in S}\left (\min_{x,y\in A, x \neq y} [x,y]\right )\] where $[x,y]$ is the least common multiple of the integers $x$, $y$. | 1. **Define the Set and Odd Values:**
Let \( T = \{1, 2, \ldots, 2n\} \). Define the *odd value* of an element \( t \in T \) as the largest odd divisor of \( t \). The odd values are in the set \( U = \{1, 3, \ldots, 2n-1\} \).
2. **Analyze the Set \( A \):**
Consider any \( n \)-element subset \( A \) of \( T \... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given five fixed points in the space. It is known that these points are centers of five spheres, four of which are pairwise externally tangent, and all these point are internally tangent to the fifth one. It turns out that it is impossible to determine which of the marked points is the center of the largest sphere. Fin... | 1. **Identify the centers and radii of the spheres:**
Let the five fixed points be \( A, B, C, X, Y \). Assume \( X \) and \( Y \) are the centers of the largest spheres, and \( A, B, C \) are the centers of the smaller spheres. Denote the radii of the spheres centered at \( A, B, C \) as \( r_A, r_B, r_C \), and th... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
2013 is the first year since the Middle Ages that consists of four consecutive digits. How many such years are there still to come after 2013 (and before 10000)? | To solve this problem, we need to identify all the years between 2013 and 9999 that consist of four consecutive digits. Let's break down the solution step-by-step:
1. **Identify the range of years:**
- We are looking for years between 2013 and 9999.
2. **Identify the pattern of four consecutive digits:**
- A ye... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$f(x)=\sum\limits_{i=1}^{2013}\left[\dfrac{x}{i!}\right]$. A integer $n$ is called [i]good[/i] if $f(x)=n$ has real root. How many good numbers are in $\{1,3,5,\dotsc,2013\}$? | 1. We start with the function \( f(x) = \sum_{i=1}^{2013} \left\lfloor \frac{x}{i!} \right\rfloor \). We need to determine how many odd numbers in the set \(\{1, 3, 5, \dots, 2013\}\) are "good," meaning \( f(x) = n \) has a real root for these \( n \).
2. First, we note that \( f(x) \) is an increasing function becau... | 587 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B mak... | 1. **Determine the total sum of the numbers assigned to the acrobats:**
\[
\text{Total sum} = \frac{12 \times 13}{2} = 78
\]
Since there are 12 acrobats, each assigned a distinct number from 1 to 12, the sum of these numbers is 78.
2. **Calculate the sum of the numbers in circle A:**
Since the acrobats ... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the numbers $\{24,27,55,64,x\}$. Given that the mean of these five numbers is prime and the median is a multiple of $3$, compute the sum of all possible positive integral values of $x$. | 1. **Identify the conditions:**
- The mean of the numbers $\{24, 27, 55, 64, x\}$ is prime.
- The median of the numbers is a multiple of 3.
2. **Calculate the sum of the numbers:**
The sum of the numbers is $24 + 27 + 55 + 64 + x = 170 + x$.
3. **Determine the range for the mean:**
Since the mean is prime... | 60 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given the digits $1$ through $7$, one can form $7!=5040$ numbers by forming different permutations of the $7$ digits (for example, $1234567$ and $6321475$ are two such permutations). If the $5040$ numbers are then placed in ascending order, what is the $2013^{\text{th}}$ number? | To find the \(2013^{\text{th}}\) number in the ascending order of permutations of the digits \(1\) through \(7\), we can use the factorial number system and step-by-step elimination.
1. **Determine the first digit:**
- Each digit in the first position has \(6!\) permutations of the remaining 6 digits.
- \(6! = 7... | 3546127 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Ca... | 1. **Identify the problem**: We need to measure exactly 1 minute using candles that burn for specific times. Big candles burn for 16 minutes and cost 16 cents, while small candles burn for 7 minutes and cost 7 cents. We need to find the cheapest combination of these candles to measure exactly 1 minute.
2. **List multi... | 97 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Karl likes the number $17$ his favorite polynomials are monic quadratics with integer coefficients such that $17$ is a root of the quadratic and the roots differ by no more than $17$. Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose $x^2$ term... | 1. **Form of the Polynomial:**
Since \(17\) is a root of the monic quadratic polynomial, we can write the polynomial as:
\[
P(x) = (x - 17)(x - r)
\]
Expanding this, we get:
\[
P(x) = x^2 - (17 + r)x + 17r
\]
Therefore, the polynomial is:
\[
P(x) = x^2 - (17 + r)x + 17r
\]
2. **Cond... | 8960 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of positive integers $b$ where $b \le 2013$, $b \neq 17$, and $b \neq 18$, such that there exists some positive integer $N$ such that $\dfrac{N}{17}$ is a perfect $17$th power, $\dfrac{N}{18}$ is a perfect $18$th power, and $\dfrac{N}{b}$ is a perfect $b$th power. | 1. We start by analyzing the given conditions. We need to find the number of positive integers \( b \) such that \( b \leq 2013 \), \( b \neq 17 \), and \( b \neq 18 \), and there exists some positive integer \( N \) such that:
\[
\frac{N}{17} \text{ is a perfect } 17 \text{th power},
\]
\[
\frac{N}{18} ... | 690 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Robin goes birdwatching one day. he sees three types of birds: penguins, pigeons, and robins. $\frac23$ of the birds he sees are robins. $\frac18$ of the birds he sees are penguins. He sees exactly $5$ pigeons. How many robins does Robin see? | 1. Let \( N \) be the total number of birds Robin sees.
2. According to the problem, \(\frac{2}{3}\) of the birds are robins, \(\frac{1}{8}\) of the birds are penguins, and he sees exactly 5 pigeons.
3. We need to find \( N \) such that the sum of the fractions of robins, penguins, and pigeons equals 1 (the total numbe... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Jimmy runs a successful pizza shop. In the middle of a busy day, he realizes that he is running low on ingredients. Each pizza must have 1 lb of dough, $\frac14$ lb of cheese, $\frac16$ lb of sauce, and $\frac13$ lb of toppings, which include pepperonis, mushrooms, olives, and sausages. Given that Jimmy currently ha... | 1. **Determine the limiting ingredient for each component:**
- Dough: Each pizza requires 1 lb of dough. Jimmy has 200 lbs of dough.
\[
\text{Number of pizzas from dough} = \frac{200 \text{ lbs}}{1 \text{ lb/pizza}} = 200 \text{ pizzas}
\]
- Cheese: Each pizza requires \(\frac{1}{4}\) lb of cheese.... | 80 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Queen Jack likes a 5-card hand if and only if the hand contains only queens and jacks. Considering all possible 5-card hands that can come from a standard 52-card deck, how many hands does Queen Jack like? | 1. First, we need to identify the total number of queens and jacks in a standard 52-card deck. There are 4 queens and 4 jacks, making a total of 8 cards that are either a queen or a jack.
2. We are asked to find the number of 5-card hands that can be formed using only these 8 cards.
3. The number of ways to choose 5 ca... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
According to Moor's Law, the number of shoes in Moor's room doubles every year. In 2013, Moor's room starts out having exactly one pair of shoes. If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes? Note that left and righ... | 1. According to Moor's Law, the number of pairs of shoes doubles every year. Starting in 2013 with 1 pair of shoes, the number of pairs of shoes in year \( n \) is given by:
\[
x_n = 2^{n-2013}
\]
where \( n \) is the year.
2. To find the earliest year when Moor can wear at least 500 mismatched pairs of sh... | 2018 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$\mathbb{R}^2$-tic-tac-toe is a game where two players take turns putting red and blue points anywhere on the $xy$ plane. The red player moves first. The first player to get $3$ of their points in a line without any of their opponent's points in between wins. What is the least number of moves in which Red can guaran... | 1. **First Move by Red:**
- Red places the first point anywhere on the $xy$ plane. This move does not affect the strategy significantly as it is the initial point.
2. **First Move by Blue:**
- Blue places the first point anywhere on the $xy$ plane. This move also does not affect the strategy significantly as it ... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not da... | 1. First, we need to determine the number of ways to split 9 people into three groups of three. We start by choosing the first group of 3 people from the 9 people. This can be done in $\binom{9}{3}$ ways.
\[
\binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 8... | 280 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x\le 29$, a subset of $A$ sums to $x$? | To solve this problem, we need to find all sets \( A \) of 5 distinct positive integers such that any positive integer \( x \leq 29 \) can be represented as the sum of some subset of \( A \).
1. **Identify the smallest elements:**
- The smallest elements must be \( 1 \) and \( 2 \). This is because we need to form... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $3\times 6$ grid is filled with the numbers in the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ according to the following rules: (1) Both the first three columns and the last three columns contain the integers 1 through 9. (2) No numbers appear more than once in a given row. Let $N$ be the number of ways to fi... | 1. **Understanding the problem**: We need to fill a $3 \times 6$ grid with the numbers from the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ such that:
- Both the first three columns and the last three columns contain the integers 1 through 9.
- No numbers appear more than once in a given row.
- We need to f... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$. What is the expected value of the number of games they will play? | To solve this problem, we will use a Markov chain to model the states of the game. We define the states based on the number of consecutive wins a player has. Let \( S_0 \) be the state where no player has won any consecutive games, \( S_1 \) be the state where one player has won one consecutive game, \( S_2 \) be the s... | 14 | Other | math-word-problem | Yes | Yes | aops_forum | false |
$x$ is a base-$10$ number such that when the digits of $x$ are interpreted as a base-$20$ number, the resulting number is twice the value as when they are interpreted as a base-$13$ number. Find the sum of all possible values of $x$. | 1. We start by interpreting the problem statement. We need to find a base-10 number \( x \) such that when its digits are interpreted as a base-20 number, the resulting number is twice the value of the number when its digits are interpreted as a base-13 number.
2. Let's denote \( x \) as a 3-digit number \( abc \) in ... | 198 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $f$ is a monic cubic polynomial with $f(0)=-64$, and all roots of $f$ are non-negative real numbers, what is the largest possible value of $f(-1)$? (A polynomial is monic if it has a leading coefficient of $1$.) | 1. Given that \( f \) is a monic cubic polynomial with \( f(0) = -64 \) and all roots are non-negative real numbers, we can express \( f(x) \) as:
\[
f(x) = (x-a)(x-b)(x-c)
\]
where \( a, b, c \geq 0 \) and \( abc = -f(0) = 64 \).
2. We need to find the largest possible value of \( f(-1) \). First, we subs... | -125 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In square $ABCD$ with side length $2$, let $P$ and $Q$ both be on side $AB$ such that $AP=BQ=\frac{1}{2}$. Let $E$ be a point on the edge of the square that maximizes the angle $PEQ$. Find the area of triangle $PEQ$. | 1. **Identify the coordinates of points \( P \) and \( Q \):**
- Since \( AP = BQ = \frac{1}{2} \) and \( AB = 2 \), we can place the square \( ABCD \) in the coordinate plane with \( A = (0, 0) \), \( B = (2, 0) \), \( C = (2, 2) \), and \( D = (0, 2) \).
- Point \( P \) is on \( AB \) such that \( AP = \frac{1}... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions.
(1) Find the cross-sectional area $... | 1. **Find the cross-sectional area \( S(x) \) at the height \( x \):**
The equation of the semicircle is given by:
\[
x^2 + y^2 = 1
\]
At height \( x \), the length of the base \( BC \) of the right-angled triangle is:
\[
BC = \sqrt{1 - x^2}
\]
The length \( AB \) is:
\[
AB = 1 - x
... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Player $A$ places an odd number of boxes around a circle and distributes $2013$ balls into some of these boxes. Then the player $B$ chooses one of these boxes and takes the balls in it. After that the player $A$ chooses half of the remaining boxes such that none of two are consecutive and take the balls in them. If pla... | 1. **Initial Setup:**
Player \( A \) places an odd number of boxes around a circle and distributes \( 2013 \) balls into some of these boxes. Let's denote the number of boxes by \( n \), where \( n \) is odd.
2. **Distribution Strategy:**
Player \( A \) places the balls in the boxes such that the distribution fo... | 2012 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $G$ be a simple, undirected, connected graph with $100$ vertices and $2013$ edges. It is given that there exist two vertices $A$ and $B$ such that it is not possible to reach $A$ from $B$ using one or two edges. We color all edges using $n$ colors, such that for all pairs of vertices, there exists a way connecting ... | 1. **Understanding the Problem:**
We are given a simple, undirected, connected graph \( G \) with 100 vertices and 2013 edges. There exist two vertices \( A \) and \( B \) such that it is not possible to reach \( A \) from \( B \) using one or two edges. We need to color all edges using \( n \) colors such that for ... | 1915 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We put pebbles on some unit squares of a $2013 \times 2013$ chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each $19\times 19$ square formed by unit squares contains at least $21$ pebbles. | To solve this problem, we need to determine the minimum number of pebbles on a $2013 \times 2013$ chessboard such that every $19 \times 19$ square contains at least $21$ pebbles. We will use a combinatorial approach to find the lower bound on the number of pebbles.
1. **Understanding the Problem:**
- We have a $201... | 160175889 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point?
[i]A. Chukhnov[/i] | 1. **Subdivide each unit square**: Each face of the $7 \times 7 \times 7$ cube is divided into $1 \times 1$ unit squares. We further subdivide each $1 \times 1$ unit square into four smaller $1/2 \times 1/2$ cells. This results in each face having $7 \times 7 \times 4 = 196$ cells, and the entire cube having $6 \times ... | 74 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A finite sequence of integers $a_1, a_2, \dots, a_n$ is called [i]regular[/i] if there exists a real number $x$ satisfying \[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \] Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that the term $a_k$ is [i]forced[/i] if the fo... | 1. **Understanding the Problem:**
We need to find the maximum number of forced terms in a regular sequence of 1000 terms. A sequence is regular if there exists a real number \( x \) such that \( \left\lfloor kx \right\rfloor = a_k \) for \( 1 \le k \le n \). A term \( a_k \) is forced if the sequence \( a_1, a_2, \l... | 985 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Alex is trying to open a lock whose code is a sequence that is three letters long, with each of the letters being one of $\text A$, $\text B$ or $\text C$, possibly repeated. The lock has three buttons, labeled $\text A$, $\text B$ and $\text C$. When the most recent $3$ button-presses form the code, the lock opens. Wh... | 1. **Determine the total number of possible codes:**
Each code is a sequence of three letters, and each letter can be one of $\text{A}$, $\text{B}$, or $\text{C}$. Therefore, the total number of possible codes is:
\[
3^3 = 27
\]
2. **Understand the requirement for testing all codes:**
To guarantee that ... | 29 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$. | 1. Let the circle have center \( O \) and \( W, X, Y, Z \) be the points of tangency of the circle with \( AB, BC, CD, DA \) respectively. Notice that \(\angle OWB = \angle OYC = 90^\circ\), so \( WOY \) is a straight line and the altitude of the trapezoid.
2. Since the circle is tangent to all four sides of the quad... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,a_3,\dots$ be a sequence of positive real numbers such that $a_ka_{k+2}=a_{k+1}+1$ for all positive integers $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$. | 1. We start with the given recurrence relation:
\[
a_k a_{k+2} = a_{k+1} + 1
\]
Rearranging this, we get:
\[
a_{k+2} = \frac{a_{k+1} + 1}{a_k}
\]
2. Let \( a_1 = a \) and \( a_2 = b \), where \( a \) and \( b \) are positive integers. We will compute the next few terms in the sequence to identify ... | 4 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$. | 1. Start with the given equation:
\[
\frac{3}{x-3} + \frac{5}{x-5} + \frac{17}{x-17} + \frac{19}{x-19} = x^2 - 11x - 4.
\]
2. Add 4 to both sides and divide by \(x\):
\[
\frac{3}{x-3} + \frac{5}{x-5} + \frac{17}{x-17} + \frac{19}{x-19} + 4 = x^2 - 11x.
\]
\[
\frac{3}{x-3} + \frac{5}{x-5} + \fra... | 73 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A token starts at the point $(0,0)$ of an $xy$-coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends at a p... | 1. **Define the problem and notation:**
- The token starts at \((0,0)\) and makes 6 moves.
- Each move is 1 unit in one of the four directions: up (\(U\)), down (\(D\)), left (\(L\)), or right (\(R\)).
- We need to find the probability that the token ends on the graph of \(|y| = |x|\).
2. **Calculate the tota... | 1245 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$. | 1. Given that both \( N \) and \( N^2 \) end in the same sequence of four digits \( abcd \), we can express this condition mathematically as:
\[
N \equiv N^2 \pmod{10000}
\]
This implies:
\[
N^2 - N \equiv 0 \pmod{10000}
\]
Therefore:
\[
N(N-1) \equiv 0 \pmod{10000}
\]
2. To satisfy \(... | 937 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. within a population of men. For each of the three factors, the probability that a randomly selected man in the population as only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability th... | 1. **Define the problem and given probabilities:**
- Let \( P(A) \) be the probability that a randomly selected man has risk factor \( A \).
- Similarly, define \( P(B) \) and \( P(C) \) for risk factors \( B \) and \( C \).
- Given:
- \( P(\text{only } A) = 0.1 \)
- \( P(\text{only } B) = 0.1 \)
... | 76 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The repeating decimals $0.abab\overline{ab}$ and $0.abcabc\overline{abc}$ satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where $a,b$, and $c$ are (not necessarily distinct) digits. Find the three-digit number $abc$. | 1. **Convert the repeating decimals to fractions:**
Let \( x = 0.abab\overline{ab} \) and \( y = 0.abcabc\overline{abc} \).
For \( x \):
\[
x = 0.abab\overline{ab} = \frac{abab}{9999}
\]
where \( abab \) is the four-digit number formed by repeating \( ab \).
For \( y \):
\[
y = 0.abcabc\ov... | 447 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$. | 1. **Identify the roots and their properties:**
- Given that \( r \) and \( s \) are roots of \( p(x) = x^3 + ax + b \), the third root must be \( -r - s \) because the sum of the roots of a cubic polynomial \( x^3 + ax + b \) is zero (by Vieta's formulas).
- Therefore, \( p(x) \) can be written as:
\[
... | 62 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\tfrac23,$ and each of the other five sides has probability $\tfrac{1}{15}.$ Charles chooses one of the two dice at random and rolls it three times. Given that the first two rolls are both sixe... | 1. **Determine the probability of rolling two sixes with each die:**
- For the fair die, the probability of rolling a six on any single roll is \(\frac{1}{6}\). Therefore, the probability of rolling two sixes in a row is:
\[
\left(\frac{1}{6}\right)^2 = \frac{1}{36}
\]
- For the biased die, the pro... | 167 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Circle $C$ with radius $2$ has diameter $\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C,$ externally tangent to circle $D,$ and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$ and can be written in the for... | 1. **Define the radii of the circles:**
- Let the radius of circle \( D \) be \( 3x \).
- Let the radius of circle \( E \) be \( x \).
2. **Use the given information:**
- Circle \( C \) has a radius of 2.
- Circle \( D \) is internally tangent to circle \( C \) at point \( A \).
- Circle \( E \) is inte... | 254 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that the angles of $\triangle ABC$ satisfy $\cos(3A) + \cos(3B) + \cos(3C) = 1$. Two sides of the triangle have lengths $10$ and $13$. There is a positive integer $m$ so that the maximum possible length for the remaining side of $\triangle ABC$ is $\sqrt{m}$. Find $m$. | 1. Given the equation for the angles of $\triangle ABC$:
\[
\cos(3A) + \cos(3B) + \cos(3C) = 1
\]
We start by using the sum-to-product identity for the first two terms:
\[
\cos(3A) + \cos(3B) = 2 \cos\left(\frac{3(A+B)}{2}\right) \cos\left(\frac{3(A-B)}{2}\right)
\]
Note that since $A + B + C =... | 399 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5,$ no collection of $k$ pairs made by the child contains the shoes from exactly... | 1. **Labeling and Permutation**:
- Label the left shoes with numbers \(1, 2, \ldots, 10\) based on whose shoes they are.
- The child places a right shoe next to each left shoe, forming a permutation of \(1, 2, \ldots, 10\).
2. **Cycle Condition**:
- We need to ensure that no collection of \(k\) pairs (for \(k... | 57 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Call a natural number $n$ [i]good[/i] if for any natural divisor $a$ of $n$, we have that $a+1$ is also divisor of $n+1$. Find all good natural numbers.
[i]S. Berlov[/i] | 1. **Base Case: \( n = 1 \)**
- The divisors of \( n = 1 \) are \( 1 \).
- For \( a = 1 \), \( a + 1 = 2 \).
- We need to check if \( 2 \) is a divisor of \( n + 1 = 2 \). Indeed, \( 2 \) is a divisor of \( 2 \).
- Therefore, \( n = 1 \) is a good number.
2. **Case: \( n \) is an odd prime**
- Let \( n ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive rational numbers $a$ and $b$ are written as decimal fractions and each consists of a minimum period of 30 digits. In the decimal representation of $a-b$, the period is at least $15$. Find the minimum value of $k\in\mathbb{N}$ such that, in the decimal representation of $a+kb$, the length of period is at least ... | 1. **Understanding the Problem:**
We are given two positive rational numbers \(a\) and \(b\) with decimal representations having a minimum period of 30 digits. We need to find the smallest \(k \in \mathbb{N}\) such that the decimal representation of \(a + kb\) has a period of at least 15 digits.
2. **Rewriting \(a\... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$. Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$.
[i]N. Agakhanov[/i] | 1. Define \( m(n) \) to be the greatest proper natural divisor of \( n \in \mathbb{N} \). We need to find all \( n \in \mathbb{N} \) such that \( n + m(n) \) is a power of 10, i.e., \( n + m(n) = 10^k \) for some \( k \in \mathbb{N} \).
2. Note that \( n \geq 2 \) because the smallest natural number \( n \) is 1, and ... | 75 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$ and $b+2$. He then writes all six pairwise products of the numbers of notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$.
[i]S. Berlov[/i] | To solve the problem, we need to analyze the pairwise products of the numbers \(a\), \(a+2\), \(b\), and \(b+2\) and determine how many of these products can be perfect squares.
1. **List the pairwise products:**
The six pairwise products are:
\[
a(a+2), \quad a \cdot b, \quad a(b+2), \quad (a+2)b, \quad (a+... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$?
$\textbf{(A) }299\qquad
\textbf{(B) }300\qquad
\textbf{(C) }301\qquad
\textbf{(D) }302\qquad
\textbf{(E) }303\qquad$ | 1. **Define the function \( f_0(x) \):**
\[
f_0(x) = x + |x - 100| - |x + 100|
\]
We need to analyze this function in different intervals of \( x \).
2. **Analyze \( f_0(x) \) in different intervals:**
- For \( x < -100 \):
\[
f_0(x) = x + (100 - x) - (-100 - x) = x + 100 - x + 100 + x = x + 2... | 301 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $ABCDE$ be a pentagon inscribed in a circle such that $AB=CD=3$, $BC=DE=10$, and $AE=14$. The sum of the lengths of all diagonals of $ABCDE$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A) }129\qquad
\textbf{(B) }247\qquad
\textbf{(C) }353\qquad
\text... | 1. **Identify the given lengths and the diagonals to be found:**
- Given: \(AB = CD = 3\), \(BC = DE = 10\), \(AE = 14\).
- We need to find the sum of the lengths of all diagonals of pentagon \(ABCDE\).
2. **Label the unknown diagonal lengths:**
- Let \(AC = BD = CE = d\).
- Let \(AD = e\) and \(BE = f\).
... | 391 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The number $2017$ is prime. Let $S=\sum_{k=0}^{62}\binom{2014}{k}$. What is the remainder when $S$ is divided by $2017$?
$\textbf{(A) }32\qquad
\textbf{(B) }684\qquad
\textbf{(C) }1024\qquad
\textbf{(D) }1576\qquad
\textbf{(E) }2016\qquad$ | 1. We start with the given sum \( S = \sum_{k=0}^{62} \binom{2014}{k} \). We need to find the remainder when \( S \) is divided by 2017.
2. By Fermat's Little Theorem, for a prime \( p \), \( a^p \equiv a \pmod{p} \). This implies that \( (1+x)^{2017} \equiv 1 + x^{2017} \pmod{2017} \).
3. Consider the binomial expansi... | 1024 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$
$ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textb... | 1. Let the line through \( Q = (20, 14) \) be denoted by \(\ell\). The equation of the line \(\ell\) can be written in the slope-intercept form as \( y = mx + b \).
2. Since the line passes through the point \( Q = (20, 14) \), we can substitute these coordinates into the line equation to find \( b \):
\[
14 = 2... | 80 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd? | 1. **Define the problem in terms of sequences:**
Let \( a_i \) be the number of ways to paint \( i \) seats such that the number of consecutive seats painted in the same color is always odd, with the rightmost seat being one particular color (either red or green).
2. **Establish the recurrence relation:**
To det... | 1686 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer is correct. | 1. **Color the bottom face**: There are \(4!\) ways to color the four bottom corners of the cube. This is because there are 4 colors and 4 corners, and each corner must be a different color. Therefore, the number of ways to color the bottom face is:
\[
4! = 24
\]
2. **Determine the top face**: Once the bottom... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that for two real numbers $x$ and $y$ the following equality is true:
$$(x+ \sqrt{1+ x^2})(y+\sqrt{1+y^2})=1$$
Find (with proof) the value of $x+y$. | 1. Given the equation:
\[
(x + \sqrt{1 + x^2})(y + \sqrt{1 + y^2}) = 1
\]
We will use the hyperbolic sine and cosine functions to simplify the expression. Recall the definitions:
\[
\sinh(a) = \frac{e^a - e^{-a}}{2}, \quad \cosh(a) = \frac{e^a + e^{-a}}{2}
\]
and the identity:
\[
\cosh^2(a... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Using squares of side 1, a stair-like figure is formed in stages following the pattern of the drawing.
For example, the first stage uses 1 square, the second uses 5, etc. Determine the last stage for which the corresponding figure uses less than 2014 squares.
[img]http://www.artofproblemsolving.com/Forum/download/fil... | 1. **Identify the pattern of the number of squares used at each stage:**
- At the 1st stage, 1 square is used.
- At the 2nd stage, 5 squares are used.
- At the 3rd stage, 13 squares are used.
- We need to find a general formula for the number of squares used at the \(k\)-th stage.
2. **Derive the formula f... | 32 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.
Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i... | To find the smallest positive constant \( c \) such that for any simple graph \( G = G(V, E) \), if \( |E| \geq c|V| \), then \( G \) contains 2 cycles with no common vertex, and one of them contains a chord, we proceed as follows:
1. **Claim**: The answer is \( c = 4 \).
2. **Construction**: Consider the bipartite g... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.
(Day 1, 3rd problem
author: Michal Rolínek) | 1. **Identify the total number of edges on the chessboard:**
- The $8 \times 8$ chessboard has $8$ rows and $8$ columns, creating $9$ horizontal and $9$ vertical lines of edges.
- Each row has $8$ horizontal edges, and there are $9$ rows, so there are $8 \times 9 = 72$ horizontal edges.
- Each column has $8$ v... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sets $A$ and $B$ are subsets of the positive integers. The sum of any two distinct elements of $A$ is an element of $B$. The quotient of any two distinct elements of $B$ (where we divide the largest by the smallest of the two) is an element of $A$. Determine the maximum number of elements in $A\cup B$. | 1. **Assume \( |A| \geq 3 \)**:
- Let \( x < y < z \) be the three smallest elements of \( A \).
- Since the sum of any two distinct elements of \( A \) is in \( B \), we have:
\[
x + y, \quad x + z, \quad y + z \in B
\]
2. **Consider the quotients**:
- The quotient of any two distinct elements... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block fa... | ### Part (a)
1. **Initial Setup and Definitions**:
- Let \( r \) and \( b \) be odd positive integers.
- A red block falls every \( r \) years, and a blue block falls every \( b \) years.
- The cycles are offset so that no two blocks fall at the same time.
- We consider a period of \( rb \) years, starting... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the value of the expression $x^2 + y^2 + z^2$,
if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$. | 1. Given the equations:
\[
x + y + z = 13,
\]
\[
xyz = 72,
\]
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{3}{4},
\]
we need to determine the value of \( x^2 + y^2 + z^2 \).
2. First, we use the identity for the sum of the reciprocals:
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z... | 61 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$. | 1. Given that the radius \( r \) of a circle with center at the origin is an odd integer, and there is a point \((p^m, q^n)\) on the circle, where \( p \) and \( q \) are prime numbers and \( m \) and \( n \) are positive integers, we need to determine \( r \).
2. Since the point \((p^m, q^n)\) lies on the circle, it ... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Six distinguishable players are participating in a tennis tournament. Each player plays one match of tennis against every other player. The outcome of each tennis match is a win for one player and a loss for the other players; there are no ties. Suppose that whenever $A$ and $B$ are players in the tournament for which ... | 1. **Identify the player with the most wins:**
- Consider the player who won the most matches and call him \( Q \). According to the problem statement, if \( Q \) won more matches than any other player, then \( Q \) must have won against every other player. Therefore, \( Q \) has 5 wins.
2. **Determine the number o... | 720 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An [i]up-right path[/i] from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_z), \dots, (x_k, y_k)$ of points in $ \mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$, and for each $1 \le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{... | 1. **Define the problem and paths:**
- We need to find the number of pairs \((A, B)\) where \(A\) is an up-right path from \((0, 0)\) to \((4, 4)\) and \(B\) is an up-right path from \((2, 0)\) to \((6, 4)\), such that \(A\) and \(B\) do not intersect.
2. **Claim and bijection:**
- We claim that pairs of up-righ... | 1750 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f:\mathbb{N}\to\mathbb{N}$ be a function satisfying the following conditions:
(a) $f(1)=1$.
(b) $f(a)\leq f(b)$ whenever $a$ and $b$ are positive integers with $a\leq b$.
(c) $f(2a)=f(a)+1$ for all positive integers $a$.
How many possible values can the $2014$-tuple $(f(1),f(2),\ldots,f(2014))$ take? | To solve this problem, we need to analyze the function \( f \) based on the given conditions and determine the number of possible values for the 2014-tuple \((f(1), f(2), \ldots, f(2014))\).
1. **Initial Condition**:
\[
f(1) = 1
\]
2. **Monotonicity**:
\[
f(a) \leq f(b) \quad \text{whenever} \quad a \l... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $a,b,c,$ and $d$ are all (not necessarily distinct) factors of $30$ and $abcd>900$. | 1. **Identify the factors of 30**: The factors of 30 are \(1, 2, 3, 5, 6, 10, 15, 30\).
2. **Express the condition \(abcd > 900\)**: We need to find the number of ordered quadruples \((a, b, c, d)\) such that \(a, b, c, d\) are factors of 30 and \(abcd > 900\).
3. **Calculate the product of the maximum possible value... | 1940 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$ and $5$ divides $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$. | 1. **Identify the problem constraints and modular properties:**
We need to find the number of ordered quintuples \((a_1, a_2, a_3, a_4, a_5)\) such that \(0 \leq a_i \leq 7\) for \(i = 1, 2, 3, 4, 5\) and \(5\) divides \(2^{a_1} + 2^{a_2} + 2^{a_3} + 2^{a_4} + 2^{a_5}\).
2. **Determine the values of \(2^x \mod 5\) ... | 6528 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For an integer $n$, let $f_9(n)$ denote the number of positive integers $d\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_1,b_2,\ldots,b_m$ are real numbers such that $f_9(n)=\textstyle\sum_{j=1}^mb_jf_9(n-j)$ for all $n>m$. Find the smallest possible value of $m$. | 1. **Define the function \( f_9(n) \)**:
- \( f_9(n) \) is the number of positive integers \( d \leq 9 \) that divide \( n \).
- The divisors of \( n \) that are less than or equal to 9 are \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \).
2. **Generating function for \( f_9(n) \)**:
- We define the generating function \( F(x... | 28 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $(a_1,\ldots,a_{20})$ and $(b_1,\ldots,b_{20})$ are two sequences of integers such that the sequence $(a_1,\ldots,a_{20},b_1,\ldots,b_{20})$ contains each of the numbers $1,\ldots,40$ exactly once. What is the maximum possible value of the sum \[\sum_{i=1}^{20}\sum_{j=1}^{20}\min(a_i,b_j)?\] | 1. We are given two sequences of integers $(a_1, a_2, \ldots, a_{20})$ and $(b_1, b_2, \ldots, b_{20})$ such that the combined sequence $(a_1, a_2, \ldots, a_{20}, b_1, b_2, \ldots, b_{20})$ contains each of the numbers $1, 2, \ldots, 40$ exactly once.
2. We need to find the maximum possible value of the sum:
\[
... | 400 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that
\[s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k).\] | To find the smallest positive integer \( k \) such that
\[ s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k), \]
we need to analyze the sum of the digits function \( s(n) \) and how it behaves under multiplication.
1. **Initial Observations**:
- If \( k \) is a single-digit number, then \( s(11k) = 2s(k) \), whi... | 9999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The positive real numbers $a,b,c$ are such that $21ab+2bc+8ca\leq 12$. Find the smallest value of $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}$. | 1. Given the inequality \(21ab + 2bc + 8ca \leq 12\), we need to find the smallest value of \(\frac{1}{a} + \frac{2}{b} + \frac{3}{c}\).
2. To approach this problem, we can use the method of Lagrange multipliers or try to find a suitable substitution that simplifies the inequality. However, a more straightforward appr... | 11 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$. | 1. **Restate the problem and notation:**
Let \( G \) be a graph with \( n = 2014 \) vertices that does not contain any 3-cliques (triangles). The set of degrees of the vertices of \( G \) is \( \{1, 2, \ldots, k\} \). We need to find the greatest possible value of \( k \).
2. **Initial claim and reasoning:**
We ... | 1342 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The real function $f$ is defined for $\forall$ $x\in \mathbb{R}$ and $f(0)=0$. Also $f(9+x)=f(9-x)$ and $f(x-10)=f(-x-10)$ for $\forall$ $x\in \mathbb{R}$. What’s the least number of zeros $f$ can have in the interval $[0;2014]$? Does this change, if $f$ is also continuous? | 1. Given the function \( f \) defined for all \( x \in \mathbb{R} \) with \( f(0) = 0 \), and the properties:
\[
f(9 + x) = f(9 - x) \quad \text{and} \quad f(x - 10) = f(-x - 10) \quad \forall x \in \mathbb{R}
\]
we need to determine the least number of zeros \( f \) can have in the interval \([0, 2014]\).
... | 107 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Some number of coins is firstly separated into 200 groups and then to 300 groups. One coin is [i]special[/i], if on the second grouping it is in a group that has less coins than the previous one, in the first grouping, that it was in. Find the least amount of [i]special[/i] coins we can have. | 1. **Initial Setup and Definitions:**
- Let \( n \) be the total number of coins.
- The coins are first divided into 200 groups, and then into 300 groups.
- A coin is considered *special* if, in the second grouping, it is in a group with fewer coins than the group it was in during the first grouping.
2. **Exa... | 101 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$ | 1. Given the function \( f: \mathbb{N} \to \mathbb{N}_0 \) with the properties:
- \( f(2) = 0 \)
- \( f(3) > 0 \)
- \( f(6042) = 2014 \)
- \( f(m + n) - f(m) - f(n) \in \{0, 1\} \) for all \( m, n \in \mathbb{N} \)
2. We need to determine \( f(2014) \).
3. Consider the equation \( f(6042) - f(2014) - f(40... | 671 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Starting with the triple $(1007\sqrt{2},2014\sqrt{2},1007\sqrt{14})$, define a sequence of triples $(x_{n},y_{n},z_{n})$ by
$x_{n+1}=\sqrt{x_{n}(y_{n}+z_{n}-x_{n})}$
$y_{n+1}=\sqrt{y_{n}(z_{n}+x_{n}-y_{n})}$
$ z_{n+1}=\sqrt{z_{n}(x_{n}+y_{n}-z_{n})}$
for $n\geq 0$.Show that each of the sequences $\langle x_n\rangle ... | To show that each of the sequences $\langle x_n \rangle_{n \geq 0}$, $\langle y_n \rangle_{n \geq 0}$, $\langle z_n \rangle_{n \geq 0}$ converges to a limit and to find these limits, we will proceed as follows:
1. **Initial Setup and Area Calculation:**
We start with the initial triple $(x_0, y_0, z_0) = (1007\sqrt... | 2014 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence? | 1. **Identify the first term of the sequence:**
The first term is given as \(2014\).
2. **Calculate the second term:**
The second term is the sum of the cubes of the digits of the first term.
\[
2^3 + 0^3 + 1^3 + 4^3 = 8 + 0 + 1 + 64 = 73
\]
3. **Calculate the third term:**
The third term is the sum... | 370 | Other | math-word-problem | Yes | Yes | aops_forum | false |
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