instruction_seed stringlengths 15 8.79k | response_seed stringlengths 15 8.79k | response stringlengths 96 53.2k | __original_row_idx int64 0 50k | conversations listlengths 2 2 |
|---|---|---|---|---|
Define $a$ $\$$ $b$ to be $a(b + 1) + ab$. What is the value of $(-2)$ $\$$ $3$? | Define $a$ $\$$ $b$ to be $a(b + 1) + ab$. What is the value of $(-2)$ $\$$ $3$? | To solve for \( (-2) \) \( \$ \) \( 3 \), we first need to use the definition of the operation:
\[
a \; \$ \; b = a(b + 1) + ab
\]
Here, we set \( a = -2 \) and \( b = 3 \). Now we substitute these values into the operation:
\[
(-2) \; \$ \; 3 = (-2)(3 + 1) + (-2)(3)
\]
Now, we calculate each part of the expression... | 10,000 | [
{
"from": "human",
"value": "Define $a$ $\\$$ $b$ to be $a(b + 1) + ab$. What is the value of $(-2)$ $\\$$ $3$?"
},
{
"from": "gpt",
"value": "To solve for \\( (-2) \\) \\( \\$ \\) \\( 3 \\), we first need to use the definition of the operation:\n\n\\[\na \\; \\$ \\; b = a(b + 1) + ab\n\\]\n\nHe... |
Four non-negative real numbers $a,$ $b,$ $c,$ $d$ satisfy
$a^3\times b^3\times c^3\times d^3 = 1024.$
Find the smallest possible value of $a^2+b^2+c^2+d^2.$ | Four non-negative real numbers $a,$ $b,$ $c,$ $d$ satisfy
$a^3\times b^3\times c^3\times d^3 = 1024.$
Find the smallest possible value of $a^2+b^2+c^2+d^2.$ | To solve for the smallest possible value of \( a^2 + b^2 + c^2 + d^2 \) given the constraint \( a^3 b^3 c^3 d^3 = 1024 \), we start by expressing this constraint in a more manageable form.
First, we write \( a^3 b^3 c^3 d^3 = (abcd)^3 = 1024 \). Taking the cube root gives us:
\[
abcd = \sqrt[3]{1024} = 10.0794 \appr... | 10,001 | [
{
"from": "human",
"value": "Four non-negative real numbers $a,$ $b,$ $c,$ $d$ satisfy\n$a^3\\times b^3\\times c^3\\times d^3 = 1024.$\nFind the smallest possible value of $a^2+b^2+c^2+d^2.$"
},
{
"from": "gpt",
"value": "To solve for the smallest possible value of \\( a^2 + b^2 + c^2 + d^2 \\) ... |
What is the smallest possible value of the sum $\lvert x - 1\rvert + \lvert x + 1\rvert + \lvert x + 3\rvert$? | What is the smallest possible value of the sum $\lvert x - 1\rvert + \lvert x + 1\rvert + \lvert x + 3\rvert$? | We need to minimize the expression
\[
S(x) = \lvert x - 1 \rvert + \lvert x + 1 \rvert + \lvert x + 3 \rvert.
\]
To analyze this expression, we will consider the points where any of the absolute value terms change. These points occur where each term inside the absolute value is equal to zero:
- The term \(\lvert x ... | 10,002 | [
{
"from": "human",
"value": "What is the smallest possible value of the sum $\\lvert x - 1\\rvert + \\lvert x + 1\\rvert + \\lvert x + 3\\rvert$?"
},
{
"from": "gpt",
"value": "We need to minimize the expression \n\n\\[\nS(x) = \\lvert x - 1 \\rvert + \\lvert x + 1 \\rvert + \\lvert x + 3 \\rver... |
Xanthe is attempting to solve the equation $25x^2+30x-36=0$ by completing the square. She manages to rewrite it in the form $(px+q)^2=r$, where $p$, $q$, and $r$ are integers and $p>0$. Find the value of $p+q+r$. | Xanthe is attempting to solve the equation $25x^2+30x-36=0$ by completing the square. She manages to rewrite it in the form $(px+q)^2=r$, where $p$, $q$, and $r$ are integers and $p>0$. Find the value of $p+q+r$. | We start with the quadratic equation
\[
25x^2 + 30x - 36 = 0.
\]
First, we will move the constant term to the other side of the equation:
\[
25x^2 + 30x = 36.
\]
Next, we factor out the coefficient of \(x^2\) (which is 25) from the left side:
\[
25(x^2 + \frac{30}{25} x) = 36,
\]
which simplifies to
\[
25(x^2 + \... | 10,003 | [
{
"from": "human",
"value": "Xanthe is attempting to solve the equation $25x^2+30x-36=0$ by completing the square. She manages to rewrite it in the form $(px+q)^2=r$, where $p$, $q$, and $r$ are integers and $p>0$. Find the value of $p+q+r$."
},
{
"from": "gpt",
"value": "We start with the quadr... |
A bookstore sells a total of 300 books on a certain day. The number of fiction books sold was 40 more than the number of non-fiction books sold. If the average price of a fiction book is $15 and the average price of a non-fiction book is $20, and the total revenue from book sales that day was $5700, how many fiction bo... | A bookstore sells a total of 300 books on a certain day. The number of fiction books sold was 40 more than the number of non-fiction books sold. If the average price of a fiction book is $15 and the average price of a non-fiction book is $20, and the total revenue from book sales that day was $5700, how many fiction bo... | Let \( f \) be the number of fiction books sold and \( n \) be the number of non-fiction books sold. We have the following information from the problem:
1. The total number of books sold is:
\[
f + n = 300
\]
2. The number of fiction books sold is 40 more than the number of non-fiction books sold:
\[
f... | 10,004 | [
{
"from": "human",
"value": "A bookstore sells a total of 300 books on a certain day. The number of fiction books sold was 40 more than the number of non-fiction books sold. If the average price of a fiction book is $15 and the average price of a non-fiction book is $20, and the total revenue from book sale... |
The sum of the digits of a two-digit number is 12, and the difference between the digits is 4. What is the two-digit number? | The sum of the digits of a two-digit number is 12, and the difference between the digits is 4. What is the two-digit number? | Let the two-digit number be represented as \( 10a + b \), where \( a \) is the tens digit and \( b \) is the units digit.
According to the problem, we have two equations based on the information given:
1. The sum of the digits is 12:
\[
a + b = 12
\]
2. The difference between the digits is 4:
\[
a - b... | 10,005 | [
{
"from": "human",
"value": "The sum of the digits of a two-digit number is 12, and the difference between the digits is 4. What is the two-digit number?"
},
{
"from": "gpt",
"value": "Let the two-digit number be represented as \\( 10a + b \\), where \\( a \\) is the tens digit and \\( b \\) is ... |
What is $\frac{1234_{7}}{25_{5}}-3456_{8}+7890_{6}$? Express your answer in base 10. | What is $\frac{1234_{7}}{25_{5}}-3456_{8}+7890_{6}$? Express your answer in base 10. | To solve the expression \(\frac{1234_{7}}{25_{5}} - 3456_{8} + 7890_{6}\), we first need to convert each number from their respective bases to base 10.
1. **Convert \(1234_{7}\) to base 10**:
\[
1234_{7} = 1 \cdot 7^3 + 2 \cdot 7^2 + 3 \cdot 7^1 + 4 \cdot 7^0
\]
Calculating each term:
\[
1 \cdot 7^3 ... | 10,006 | [
{
"from": "human",
"value": "What is $\\frac{1234_{7}}{25_{5}}-3456_{8}+7890_{6}$? Express your answer in base 10."
},
{
"from": "gpt",
"value": "To solve the expression \\(\\frac{1234_{7}}{25_{5}} - 3456_{8} + 7890_{6}\\), we first need to convert each number from their respective bases to base... |
Find the integer $m,$ $-180 < m < 180,$ such that $\cos m^\circ = \cos 72^\circ.$ | Find the integer $m,$ $-180 < m < 180,$ such that $\cos m^\circ = \cos 72^\circ.$ | To find the integer \( m \) such that \( -180 < m < 180 \) and \( \cos m^\circ = \cos 72^\circ \), we start by using the property of the cosine function:
\[
\cos \theta = \cos(360^\circ - \theta) = \cos(-\theta)
\]
This means that if \( \cos m^\circ = \cos 72^\circ \), then \( m \) can take values of the form:
\[
m ... | 10,007 | [
{
"from": "human",
"value": "Find the integer $m,$ $-180 < m < 180,$ such that $\\cos m^\\circ = \\cos 72^\\circ.$"
},
{
"from": "gpt",
"value": "To find the integer \\( m \\) such that \\( -180 < m < 180 \\) and \\( \\cos m^\\circ = \\cos 72^\\circ \\), we start by using the property of the cos... |
Simplify $\sqrt{20} \div \sqrt{45}$. | Simplify $\sqrt{20} \div \sqrt{45}$. | To simplify \(\sqrt{20} \div \sqrt{45}\), we can first use the property of square roots that states \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\). Applying this, we have:
\[
\sqrt{20} \div \sqrt{45} = \frac{\sqrt{20}}{\sqrt{45}} = \sqrt{\frac{20}{45}}.
\]
Next, we simplify the fraction \(\frac{20}{45}\). We can ... | 10,008 | [
{
"from": "human",
"value": "Simplify $\\sqrt{20} \\div \\sqrt{45}$."
},
{
"from": "gpt",
"value": "To simplify \\(\\sqrt{20} \\div \\sqrt{45}\\), we can first use the property of square roots that states \\(\\frac{\\sqrt{a}}{\\sqrt{b}} = \\sqrt{\\frac{a}{b}}\\). Applying this, we have:\n\n\\[\n... |
The staff of the school math lab consists of a director, 4 deputy directors, 9 group leaders and 27 instructors. Today they have a meeting. If the director meets with 4 deputy directors every 5 days, with 3 group leaders every 4 days, and with 5 instructors every 7 days, how many days will pass before he next meets wit... | The staff of the school math lab consists of a director, 4 deputy directors, 9 group leaders and 27 instructors. Today they have a meeting. If the director meets with 4 deputy directors every 5 days, with 3 group leaders every 4 days, and with 5 instructors every 7 days, how many days will pass before he next meets wit... | To find the next day the director meets with all 4 deputy directors, 3 group leaders, and 5 instructors at the same time, we need to find the least common multiple (LCM) of the time intervals for each group.
1. **Deputy Directors**: The director meets with 4 deputy directors every 5 days.
- Interval: \( 5 \) days
... | 10,009 | [
{
"from": "human",
"value": "The staff of the school math lab consists of a director, 4 deputy directors, 9 group leaders and 27 instructors. Today they have a meeting. If the director meets with 4 deputy directors every 5 days, with 3 group leaders every 4 days, and with 5 instructors every 7 days, how man... |
The three-digit numbers 512, 522, and 532 form an arithmetic progression. If the sum of five consecutive numbers of an arithmetic progression with a common difference of 3 is equal to 1845, what is the middle number? | The three-digit numbers 512, 522, and 532 form an arithmetic progression. If the sum of five consecutive numbers of an arithmetic progression with a common difference of 3 is equal to 1845, what is the middle number? | To find the middle number of the five consecutive numbers in an arithmetic progression with a common difference of 3, we denote the five consecutive terms as follows:
Let the first term be \( a \). The five consecutive terms can then be expressed as:
\[
a, \quad a + 3, \quad a + 6, \quad a + 9, \quad a + 12.
\]
The s... | 10,010 | [
{
"from": "human",
"value": "The three-digit numbers 512, 522, and 532 form an arithmetic progression. If the sum of five consecutive numbers of an arithmetic progression with a common difference of 3 is equal to 1845, what is the middle number?"
},
{
"from": "gpt",
"value": "To find the middle ... |
Consider the sequence of positive integers which are not divisible by 7 but leave a remainder of 3 when divided by 4. What is the number of terms in the sequence less than 100? | Consider the sequence of positive integers which are not divisible by 7 but leave a remainder of 3 when divided by 4. What is the number of terms in the sequence less than 100? | To find the sequence of positive integers that are not divisible by 7 and leave a remainder of 3 when divided by 4, we start by describing the integers satisfying these conditions.
1. **Remainder when divided by 4**:
An integer \( n \) that leaves a remainder of 3 when divided by 4 can be expressed in the form:
... | 10,011 | [
{
"from": "human",
"value": "Consider the sequence of positive integers which are not divisible by 7 but leave a remainder of 3 when divided by 4. What is the number of terms in the sequence less than 100?"
},
{
"from": "gpt",
"value": "To find the sequence of positive integers that are not divi... |
Find the range of the function $$f(x)=\frac{x^2+4}{x^2+4x+4}$$Express your answer as an interval. | Find the range of the function $$f(x)=\frac{x^2+4}{x^2+4x+4}$$Express your answer as an interval. | Let's consider the function
\[
f(x) = \frac{x^2 + 4}{x^2 + 4x + 4}.
\]
First, notice that the denominator can be simplified:
\[
x^2 + 4x + 4 = (x+2)^2.
\]
Thus, we can rewrite the function as:
\[
f(x) = \frac{x^2 + 4}{(x+2)^2}.
\]
Next, we will analyze the function to find its range. To do this, we will set
\[
y... | 10,012 | [
{
"from": "human",
"value": "Find the range of the function $$f(x)=\\frac{x^2+4}{x^2+4x+4}$$Express your answer as an interval."
},
{
"from": "gpt",
"value": "Let's consider the function\n\n\\[\nf(x) = \\frac{x^2 + 4}{x^2 + 4x + 4}.\n\\]\n\nFirst, notice that the denominator can be simplified:\n... |
What is the greatest common factor of 32 and 48? | What is the greatest common factor of 32 and 48? | To find the greatest common factor (GCF) of 32 and 48, we can use the prime factorization method.
First, let's find the prime factorization of each number:
- For 32:
- 32 is \(2 \times 16\)
- 16 is \(2 \times 8\)
- 8 is \(2 \times 4\)
- 4 is \(2 \times 2\)
- Thus, the prime factorization of 32 is \(2^5\).
... | 10,013 | [
{
"from": "human",
"value": "What is the greatest common factor of 32 and 48?"
},
{
"from": "gpt",
"value": "To find the greatest common factor (GCF) of 32 and 48, we can use the prime factorization method.\n\nFirst, let's find the prime factorization of each number:\n\n- For 32:\n - 32 is \\(2... |
What is the remainder of $8^6 + 7^7 + 6^8$ divided by 5? | What is the remainder of $8^6 + 7^7 + 6^8$ divided by 5? | To find the remainder of \( 8^6 + 7^7 + 6^8 \) when divided by 5, we can apply modular arithmetic to each term separately.
1. **Calculate \( 8^6 \mod 5 \)**:
First, reduce 8 modulo 5:
\[
8 \equiv 3 \mod 5
\]
Now we need to calculate \( 3^6 \mod 5 \). To do this efficiently, we can use the fact that \( 3... | 10,014 | [
{
"from": "human",
"value": "What is the remainder of $8^6 + 7^7 + 6^8$ divided by 5?"
},
{
"from": "gpt",
"value": "To find the remainder of \\( 8^6 + 7^7 + 6^8 \\) when divided by 5, we can apply modular arithmetic to each term separately.\n\n1. **Calculate \\( 8^6 \\mod 5 \\)**:\n First, re... |
Find the value of the expression: $\lceil 12.8 \rceil + \lceil -12.8 \rceil$ | Find the value of the expression: $\lceil 12.8 \rceil + \lceil -12.8 \rceil$ | To solve the expression \( \lceil 12.8 \rceil + \lceil -12.8 \rceil \), we need to compute the ceiling of \( 12.8 \) and the ceiling of \( -12.8 \) separately.
1. Compute \( \lceil 12.8 \rceil \):
The ceiling function \( \lceil x \rceil \) gives the smallest integer greater than or equal to \( x \). For \( 12.8 \):... | 10,015 | [
{
"from": "human",
"value": "Find the value of the expression: $\\lceil 12.8 \\rceil + \\lceil -12.8 \\rceil$"
},
{
"from": "gpt",
"value": "To solve the expression \\( \\lceil 12.8 \\rceil + \\lceil -12.8 \\rceil \\), we need to compute the ceiling of \\( 12.8 \\) and the ceiling of \\( -12.8 \... |
What is the length of the longest side of a right triangle if the hypotenuse is $\sqrt{29}$ and one of the other sides is 5? | What is the length of the longest side of a right triangle if the hypotenuse is $\sqrt{29}$ and one of the other sides is 5? | Let the right triangle have legs \( a \) and \( b \), where \( c \) is the hypotenuse. We are given that \( c = \sqrt{29} \) and one of the legs \( a = 5 \). We need to find the length of the other leg \( b \).
Using the Pythagorean theorem, we have:
\[
a^2 + b^2 = c^2
\]
Substituting the known values, we get:
\[
5... | 10,016 | [
{
"from": "human",
"value": "What is the length of the longest side of a right triangle if the hypotenuse is $\\sqrt{29}$ and one of the other sides is 5?"
},
{
"from": "gpt",
"value": "Let the right triangle have legs \\( a \\) and \\( b \\), where \\( c \\) is the hypotenuse. We are given that... |
Write $2.1+5.76$ as a decimal. | Write $2.1+5.76$ as a decimal. | To add \( 2.1 \) and \( 5.76 \), we can line up the numbers by their decimal points:
```
2.10
+ 5.76
-------
```
Now, we'll add them column by column, starting from the right:
1. In the hundredths place, we have \( 0 + 6 = 6 \).
2. In the tenths place, we have \( 1 + 7 = 8 \).
3. In the units place, we have \( 2 +... | 10,017 | [
{
"from": "human",
"value": "Write $2.1+5.76$ as a decimal."
},
{
"from": "gpt",
"value": "To add \\( 2.1 \\) and \\( 5.76 \\), we can line up the numbers by their decimal points:\n\n```\n 2.10\n+ 5.76\n-------\n```\n\nNow, we'll add them column by column, starting from the right:\n\n1. In the ... |
Find the number of distinct prime numbers $p$ for which there exists an integer solution to the congruence $x^2+5x+6\equiv 0\pmod p$. | Find the number of distinct prime numbers $p$ for which there exists an integer solution to the congruence $x^2+5x+6\equiv 0\pmod p$. | To solve the congruence \( x^2 + 5x + 6 \equiv 0 \pmod{p} \), we can first rewrite the quadratic equation:
\[
x^2 + 5x + 6 = 0
\]
The discriminant of this quadratic is given by:
\[
D = 5^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1
\]
For the quadratic to have integer solutions modulo \( p \), the discriminant \( D \) must ... | 10,018 | [
{
"from": "human",
"value": "Find the number of distinct prime numbers $p$ for which there exists an integer solution to the congruence $x^2+5x+6\\equiv 0\\pmod p$."
},
{
"from": "gpt",
"value": "To solve the congruence \\( x^2 + 5x + 6 \\equiv 0 \\pmod{p} \\), we can first rewrite the quadratic... |
A bag contains 8 blue marbles and 5 yellow marbles. A marble is drawn from the bag, its color noted, and then put back in the bag. A second marble is drawn and its color noted. What is the probability that the two marbles have different colors? | A bag contains 8 blue marbles and 5 yellow marbles. A marble is drawn from the bag, its color noted, and then put back in the bag. A second marble is drawn and its color noted. What is the probability that the two marbles have different colors? | To solve this problem, we first need to determine the total number of marbles in the bag.
There are 8 blue marbles and 5 yellow marbles, so the total number of marbles is:
\[
8 + 5 = 13 \text{ marbles.}
\]
Next, we can analyze the possible outcomes for drawing two marbles of different colors. The two different color... | 10,019 | [
{
"from": "human",
"value": "A bag contains 8 blue marbles and 5 yellow marbles. A marble is drawn from the bag, its color noted, and then put back in the bag. A second marble is drawn and its color noted. What is the probability that the two marbles have different colors?"
},
{
"from": "gpt",
"... |
A geometric sequence has the property that the sum of its first 4 terms is 40, and the sum of its first 8 terms is 60. What is the common ratio of this sequence? | A geometric sequence has the property that the sum of its first 4 terms is 40, and the sum of its first 8 terms is 60. What is the common ratio of this sequence? | We denote the first term of the geometric sequence as \( a \) and the common ratio as \( r \).
The sum of the first \( n \) terms of a geometric sequence can be calculated using the formula:
\[
S_n = a \frac{1 - r^n}{1 - r} \quad \text{for } r \neq 1
\]
Given the problem, we have two equations based on the sums of t... | 10,020 | [
{
"from": "human",
"value": "A geometric sequence has the property that the sum of its first 4 terms is 40, and the sum of its first 8 terms is 60. What is the common ratio of this sequence?"
},
{
"from": "gpt",
"value": "We denote the first term of the geometric sequence as \\( a \\) and the co... |
If only one third of the world's population lives in urban areas, and only one fifth of that urban population has access to high-speed internet, what fraction of the world's population has access to high-speed internet? | If only one third of the world's population lives in urban areas, and only one fifth of that urban population has access to high-speed internet, what fraction of the world's population has access to high-speed internet? | Let's break down the problem step by step:
1. **Fraction of the world's population that lives in urban areas**:
\[
\text{Urban population} = \frac{1}{3}
\]
2. **Fraction of the urban population that has access to high-speed internet**:
\[
\text{Population with access to high-speed internet} = \frac{1}{... | 10,021 | [
{
"from": "human",
"value": "If only one third of the world's population lives in urban areas, and only one fifth of that urban population has access to high-speed internet, what fraction of the world's population has access to high-speed internet?"
},
{
"from": "gpt",
"value": "Let's break down... |
There are 5 representatives from each of 3 companies at a conference. At the beginning of the conference, every person shakes hands once with every person except the other representatives from their company. How many handshakes are there? | There are 5 representatives from each of 3 companies at a conference. At the beginning of the conference, every person shakes hands once with every person except the other representatives from their company. How many handshakes are there? | To calculate the total number of handshakes at the conference, we begin by determining how many representatives are present in total. Each of the 3 companies has 5 representatives, leading to the following calculation for the total number of participants:
\[
\text{Total representatives} = 3 \times 5 = 15
\]
Next, we ... | 10,022 | [
{
"from": "human",
"value": "There are 5 representatives from each of 3 companies at a conference. At the beginning of the conference, every person shakes hands once with every person except the other representatives from their company. How many handshakes are there?"
},
{
"from": "gpt",
"value"... |
If $$1+6+11+16+21+26+\cdots+91+96+101 \equiv n\pmod{15},$$where $0\le n<15$, what is the value of $n$? | If $$1+6+11+16+21+26+\cdots+91+96+101 \equiv n\pmod{15},$$where $0\le n<15$, what is the value of $n$? | To solve the problem, we need to find the sum of the arithmetic series \(1 + 6 + 11 + 16 + 21 + 26 + \ldots + 101\).
First, we can recognize that this series is an arithmetic sequence where the first term \(a\) is \(1\) and the common difference \(d\) is \(5\).
Next, we need to identify the number of terms in the se... | 10,023 | [
{
"from": "human",
"value": "If $$1+6+11+16+21+26+\\cdots+91+96+101 \\equiv n\\pmod{15},$$where $0\\le n<15$, what is the value of $n$?"
},
{
"from": "gpt",
"value": "To solve the problem, we need to find the sum of the arithmetic series \\(1 + 6 + 11 + 16 + 21 + 26 + \\ldots + 101\\).\n\nFirst,... |
Find the 427th digit after the decimal point in the decimal representation of $\frac{13}{48}$. | Find the 427th digit after the decimal point in the decimal representation of $\frac{13}{48}$. | To find the 427th digit after the decimal point in the decimal representation of \(\frac{13}{48}\), we first need to perform the division.
Calculating \(\frac{13}{48}\):
1. Perform long division of 13 by 48.
- \( 48 \) goes into \( 130 \) (the decimal representation starts after adding a decimal point and a zero) \(... | 10,024 | [
{
"from": "human",
"value": "Find the 427th digit after the decimal point in the decimal representation of $\\frac{13}{48}$."
},
{
"from": "gpt",
"value": "To find the 427th digit after the decimal point in the decimal representation of \\(\\frac{13}{48}\\), we first need to perform the division... |
In a certain triangle, the measures of angles $A$, $B$, and $C$ are positive integer numbers of degrees. Angle $A$ is a multiple of angle $B$, and angle $B$ is a multiple of angle $C$. If the sum of the measures of angles $A$ and $C$ is 90 degrees, how many different measures are possible for angle $A$? | In a certain triangle, the measures of angles $A$, $B$, and $C$ are positive integer numbers of degrees. Angle $A$ is a multiple of angle $B$, and angle $B$ is a multiple of angle $C$. If the sum of the measures of angles $A$ and $C$ is 90 degrees, how many different measures are possible for angle $A$? | Let the measures of angles \( A \), \( B \), and \( C \) be represented as follows:
- \( A = kb \) for some integer \( k \)
- \( B = mc \) for some integer \( m \)
From the problem, we have the following relationships:
1. \( A + B + C = 180^\circ \)
2. \( A + C = 90^\circ \)
From equation 2, we can express \( B \) as... | 10,025 | [
{
"from": "human",
"value": "In a certain triangle, the measures of angles $A$, $B$, and $C$ are positive integer numbers of degrees. Angle $A$ is a multiple of angle $B$, and angle $B$ is a multiple of angle $C$. If the sum of the measures of angles $A$ and $C$ is 90 degrees, how many different measures ar... |
Find the $r$ that satisfies $\log_{16} (r+16) = \frac{5}{4}$. | Find the $r$ that satisfies $\log_{16} (r+16) = \frac{5}{4}$. | To solve for \( r \) in the equation
\[
\log_{16} (r+16) = \frac{5}{4},
\]
we can rewrite the logarithmic equation in its exponential form. The general form of the logarithmic equation \( \log_b(x) = y \) is converted to
\[
x = b^y.
\]
Thus, we have
\[
r + 16 = 16^{\frac{5}{4}}.
\]
Next, we need to compute \( ... | 10,026 | [
{
"from": "human",
"value": "Find the $r$ that satisfies $\\log_{16} (r+16) = \\frac{5}{4}$."
},
{
"from": "gpt",
"value": "To solve for \\( r \\) in the equation \n\n\\[\n\\log_{16} (r+16) = \\frac{5}{4},\n\\]\n\nwe can rewrite the logarithmic equation in its exponential form. The general form ... |
A bookshelf has 5 shelves, and each shelf can hold 8 rows of books. If the bookshelf is currently empty, how many books can be placed on it if each book takes up a space of 3 inches and the total length of the bookshelf is 48 inches? | A bookshelf has 5 shelves, and each shelf can hold 8 rows of books. If the bookshelf is currently empty, how many books can be placed on it if each book takes up a space of 3 inches and the total length of the bookshelf is 48 inches? | To determine how many books can be placed on the bookshelf, we need to consider the dimensions provided:
- The total length of the bookshelf is 48 inches, and each book takes up a space of 3 inches.
First, we calculate how many books can fit in one row of the shelf:
\[
\text{Number of books per row} = \frac{\text{To... | 10,027 | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold 8 rows of books. If the bookshelf is currently empty, how many books can be placed on it if each book takes up a space of 3 inches and the total length of the bookshelf is 48 inches?"
},
{
"from": "gpt",
"value": "To... |
Find the maximum value of $x^2 - 8x + 16$ subject to the constraint that $x$ is a positive integer.
Let $f(x) = x^2 - 8x + 16$. We can rewrite $f(x)$ as $f(x) = (x - 4)^2$, which is a quadratic function with a minimum value of $0$ at $x = 4$.
Since we want to find the maximum value of $f(x)$, we need to find the larg... | Find the maximum value of $x^2 - 8x + 16$ subject to the constraint that $x$ is a positive integer.
Let $f(x) = x^2 - 8x + 16$. We can rewrite $f(x)$ as $f(x) = (x - 4)^2$, which is a quadratic function with a minimum value of $0$ at $x = 4$.
Since we want to find the maximum value of $f(x)$, we need to find the larg... | Let's first analyze the given function \( f(x) = x^2 - 8x + 16 \).
We can rewrite the function as follows:
\[
f(x) = (x - 4)^2
\]
This expression indicates that \( f(x) \) is always non-negative and reaches its minimum value of 0 at \( x = 4 \).
The function \( f(x) = (x - 4)^2 \) is a parabola opening upwards. As \(... | 10,028 | [
{
"from": "human",
"value": "Find the maximum value of $x^2 - 8x + 16$ subject to the constraint that $x$ is a positive integer.\n\nLet $f(x) = x^2 - 8x + 16$. We can rewrite $f(x)$ as $f(x) = (x - 4)^2$, which is a quadratic function with a minimum value of $0$ at $x = 4$.\n\nSince we want to find the maxi... |
A carpenter is building a bookshelf using wooden planks of different lengths. The lengths of the planks are 24 cm, 32 cm, 40 cm, and 48 cm. If the total length of the bookshelf is exactly 240 cm, how many planks of each length are used? | A carpenter is building a bookshelf using wooden planks of different lengths. The lengths of the planks are 24 cm, 32 cm, 40 cm, and 48 cm. If the total length of the bookshelf is exactly 240 cm, how many planks of each length are used? | To solve the problem, we need to find non-negative integer values \(x\), \(y\), \(z\), and \(w\) such that:
\[
24x + 32y + 40z + 48w = 240
\]
where:
- \(x\) is the number of 24 cm planks,
- \(y\) is the number of 32 cm planks,
- \(z\) is the number of 40 cm planks,
- \(w\) is the number of 48 cm planks.
We can firs... | 10,029 | [
{
"from": "human",
"value": "A carpenter is building a bookshelf using wooden planks of different lengths. The lengths of the planks are 24 cm, 32 cm, 40 cm, and 48 cm. If the total length of the bookshelf is exactly 240 cm, how many planks of each length are used?"
},
{
"from": "gpt",
"value": ... |
What is the smallest positive integer $n$ such that the sum of the digits of $n$ is a divisor of $n$, and $n$ is divisible by the sum of the digits of $n$ plus one? | What is the smallest positive integer $n$ such that the sum of the digits of $n$ is a divisor of $n$, and $n$ is divisible by the sum of the digits of $n$ plus one? | Let \( S(n) \) denote the sum of the digits of the integer \( n \). We need to find the smallest positive integer \( n \) such that both \( S(n) \) divides \( n \) and \( n \) is divisible by \( S(n) + 1 \).
Let \( S(n) = s \). The conditions can be rewritten as:
1. \( s \mid n \)
2. \( n \equiv 0 \pmod{s + 1} \)
We ... | 10,030 | [
{
"from": "human",
"value": "What is the smallest positive integer $n$ such that the sum of the digits of $n$ is a divisor of $n$, and $n$ is divisible by the sum of the digits of $n$ plus one?"
},
{
"from": "gpt",
"value": "Let \\( S(n) \\) denote the sum of the digits of the integer \\( n \\).... |
In $\triangle DEF$, the sides have integer lengths and $DE=DF$. Circle $\omega_1$ has its center at the incenter of $\triangle DEF$. Suppose that the incircle of $\triangle DEF$ is externally tangent to $\omega_1$, and two of the excircles are both internally tangent to $\omega_1$. Find the minimum possible value of th... | In $\triangle DEF$, the sides have integer lengths and $DE=DF$. Circle $\omega_1$ has its center at the incenter of $\triangle DEF$. Suppose that the incircle of $\triangle DEF$ is externally tangent to $\omega_1$, and two of the excircles are both internally tangent to $\omega_1$. Find the minimum possible value of th... | In triangle \( DEF \), we know that \( DE = DF \) implies that \( \triangle DEF \) is isosceles. Let \( DE = DF = a \) and \( EF = b \). The perimeter of triangle \( DEF \) is given by \( P = 2a + b \).
Given that the incircle of \( \triangle DEF \) is externally tangent to the circle \( \omega_1 \) centered at the in... | 10,031 | [
{
"from": "human",
"value": "In $\\triangle DEF$, the sides have integer lengths and $DE=DF$. Circle $\\omega_1$ has its center at the incenter of $\\triangle DEF$. Suppose that the incircle of $\\triangle DEF$ is externally tangent to $\\omega_1$, and two of the excircles are both internally tangent to $\\... |
In a class of 20 students, 5 students are selected to participate in a science fair. How many ways can the teacher choose the group of 5 students such that it includes exactly 2 boys and 3 girls? | In a class of 20 students, 5 students are selected to participate in a science fair. How many ways can the teacher choose the group of 5 students such that it includes exactly 2 boys and 3 girls? | To solve the problem of selecting 5 students from a class of 20, consisting specifically of 2 boys and 3 girls, we will follow these steps:
1. **Determine the number of boys and girls in the class.**
We need to know how many of the 20 students are boys and how many are girls. For this example, let's assume there ar... | 10,032 | [
{
"from": "human",
"value": "In a class of 20 students, 5 students are selected to participate in a science fair. How many ways can the teacher choose the group of 5 students such that it includes exactly 2 boys and 3 girls?"
},
{
"from": "gpt",
"value": "To solve the problem of selecting 5 stud... |
Given a function $f(x) = x^3 - x$, determine the values of $x$ for which $f(x) = f(f(f(x)))$. Enter all the solutions, separated by commas. | Given a function $f(x) = x^3 - x$, determine the values of $x$ for which $f(x) = f(f(f(x)))$. Enter all the solutions, separated by commas. | Firstly, we simplify the problem by letting \( y = f(x) \). This gives us the equation:
\[
f(x) = x^3 - x
\]
Now, we seek \( f(f(f(x))) = f(y) \). To find \( f(y) \) where \( y = x^3 - x \), we need to compute \( f(y) \):
\[
f(y) = f(x^3 - x) = (x^3 - x)^3 - (x^3 - x)
\]
Next, we need to calculate \( (x^3 - x)^3 \):... | 10,033 | [
{
"from": "human",
"value": "Given a function $f(x) = x^3 - x$, determine the values of $x$ for which $f(x) = f(f(f(x)))$. Enter all the solutions, separated by commas."
},
{
"from": "gpt",
"value": "Firstly, we simplify the problem by letting \\( y = f(x) \\). This gives us the equation:\n\n\\[... |
Let $a,$ $b,$ $c,$ and $d$ be the roots of \[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.\]Find the value of \[\frac{1}{ab} + \frac{1}{ac} + \frac{1}{ad} + \frac{1}{bc} + \frac{1}{bd} + \frac{1}{cd}.\]
Let $a,$ $b,$ $c,$ and $d$ be the roots of \[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.\]Find the value of \[\frac{1}{ab} + \frac{1}{ac} + \f... | Let $a,$ $b,$ $c,$ and $d$ be the roots of \[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.\]Find the value of \[\frac{1}{ab} + \frac{1}{ac} + \frac{1}{ad} + \frac{1}{bc} + \frac{1}{bd} + \frac{1}{cd}.\]
Let $a,$ $b,$ $c,$ and $d$ be the roots of \[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.\]Find the value of \[\frac{1}{ab} + \frac{1}{ac} + \f... | To find the value of
\[
S = \frac{1}{ab} + \frac{1}{ac} + \frac{1}{ad} + \frac{1}{bc} + \frac{1}{bd} + \frac{1}{cd},
\]
we can rewrite \(S\) in terms of the elementary symmetric sums of the roots \(a, b, c, d\). Specifically, we have:
\[
S = \frac{cd + bd + bc + ad + ac + ab}{abcd}.
\]
Using Vieta's formulas for t... | 10,034 | [
{
"from": "human",
"value": "Let $a,$ $b,$ $c,$ and $d$ be the roots of \\[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.\\]Find the value of \\[\\frac{1}{ab} + \\frac{1}{ac} + \\frac{1}{ad} + \\frac{1}{bc} + \\frac{1}{bd} + \\frac{1}{cd}.\\]\n\nLet $a,$ $b,$ $c,$ and $d$ be the roots of \\[x^4 + 8x^3 + 9x^2 + 5x + 4 = 0.... |
Compute the number of ways to arrange 10 objects in a row, where 3 of the objects are identical red balls, 2 are identical blue balls, and the remaining 5 are distinct toys. | Compute the number of ways to arrange 10 objects in a row, where 3 of the objects are identical red balls, 2 are identical blue balls, and the remaining 5 are distinct toys. | To find the number of ways to arrange 10 objects, where we have 3 identical red balls, 2 identical blue balls, and 5 distinct toys, we can use the formula for permutations of a multiset. The formula to compute the number of distinct permutations of \( n \) objects, of which there are \( n_1 \) indistinguishable of one ... | 10,035 | [
{
"from": "human",
"value": "Compute the number of ways to arrange 10 objects in a row, where 3 of the objects are identical red balls, 2 are identical blue balls, and the remaining 5 are distinct toys."
},
{
"from": "gpt",
"value": "To find the number of ways to arrange 10 objects, where we hav... |
In $\triangle ABC$ the angle bisector from $A$ is given perpendicular to the angle bisector from $B$. If $AB=5$ and $AC=7$, find the length of $BC$. | In $\triangle ABC$ the angle bisector from $A$ is given perpendicular to the angle bisector from $B$. If $AB=5$ and $AC=7$, find the length of $BC$. | Let \( D \) be the point where the angle bisector from \( A \) meets \( BC \), and let \( E \) be the point where the angle bisector from \( B \) meets \( AC \). According to the problem, the angle bisector \( AD \) is perpendicular to the angle bisector \( BE \).
Applying the Angle Bisector Theorem, we have:
\[
\fr... | 10,036 | [
{
"from": "human",
"value": "In $\\triangle ABC$ the angle bisector from $A$ is given perpendicular to the angle bisector from $B$. If $AB=5$ and $AC=7$, find the length of $BC$."
},
{
"from": "gpt",
"value": "Let \\( D \\) be the point where the angle bisector from \\( A \\) meets \\( BC \\), a... |
Find the matrix $\mathbf{M},$ with real entries, such that
\[\mathbf{M}^2 + \mathbf{M}^{-1} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}.\] | Find the matrix $\mathbf{M},$ with real entries, such that
\[\mathbf{M}^2 + \mathbf{M}^{-1} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}.\] | Let the matrix \(\mathbf{M}\) be represented as:
\[
\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]
The condition given in the problem is:
\[
\mathbf{M}^2 + \mathbf{M}^{-1} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}
\]
First, we need to compute \(\mathbf{M}^2\) and \(\mathbf{M}^{-1}\).
### Step 1:... | 10,037 | [
{
"from": "human",
"value": "Find the matrix $\\mathbf{M},$ with real entries, such that\n\\[\\mathbf{M}^2 + \\mathbf{M}^{-1} = \\begin{pmatrix} 2 & 0 \\\\ 0 & 2 \\end{pmatrix}.\\]"
},
{
"from": "gpt",
"value": "Let the matrix \\(\\mathbf{M}\\) be represented as:\n\n\\[\n\\mathbf{M} = \\begin{pm... |
A bacterial culture is known to triple every hour. If a petri dish contains 100 million bacteria after 5 hours, how many bacteria were initially present? | A bacterial culture is known to triple every hour. If a petri dish contains 100 million bacteria after 5 hours, how many bacteria were initially present? | To solve the problem of how many bacteria were initially present in the petri dish, we can use the information provided about the growth pattern of the bacteria.
We know the bacteria triple every hour. Therefore, after each hour, the number of bacteria multiplies by 3.
Let \( N_0 \) be the initial number of bacteria.... | 10,038 | [
{
"from": "human",
"value": "A bacterial culture is known to triple every hour. If a petri dish contains 100 million bacteria after 5 hours, how many bacteria were initially present?"
},
{
"from": "gpt",
"value": "To solve the problem of how many bacteria were initially present in the petri dish... |
Tom has been saving money for a new bike and has $120 in his piggy bank. He wants to buy a bike that costs $180. Tom's parents have agreed to give him an allowance of $5 per week for doing chores. However, every time Tom receives his allowance, he spends $2 on candy. How many weeks will it take Tom to save enough money... | Tom has been saving money for a new bike and has $120 in his piggy bank. He wants to buy a bike that costs $180. Tom's parents have agreed to give him an allowance of $5 per week for doing chores. However, every time Tom receives his allowance, he spends $2 on candy. How many weeks will it take Tom to save enough money... | Tom currently has $120 and needs to buy a bike that costs $180. To determine how much more he needs to save, we subtract what he has from the cost of the bike:
\[
180 - 120 = 60
\]
So, Tom needs an additional $60.
Tom receives an allowance of $5 per week, but he spends $2 on candy each week. Therefore, his net savin... | 10,039 | [
{
"from": "human",
"value": "Tom has been saving money for a new bike and has $120 in his piggy bank. He wants to buy a bike that costs $180. Tom's parents have agreed to give him an allowance of $5 per week for doing chores. However, every time Tom receives his allowance, he spends $2 on candy. How many we... |
Find $z$ so that the vectors $\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ -2 \\ z \end{pmatrix}$ are orthogonal. | Find $z$ so that the vectors $\begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 3 \\ -2 \\ z \end{pmatrix}$ are orthogonal. | To determine the value of \( z \) that makes the vectors \( \begin{pmatrix} 2 \\ 1 \\ -1 \end{pmatrix} \) and \( \begin{pmatrix} 3 \\ -2 \\ z \end{pmatrix} \) orthogonal, we need to ensure that their dot product equals zero.
The dot product of two vectors \( \begin{pmatrix} a \\ b \\ c \end{pmatrix} \) and \( \begin{p... | 10,040 | [
{
"from": "human",
"value": "Find $z$ so that the vectors $\\begin{pmatrix} 2 \\\\ 1 \\\\ -1 \\end{pmatrix}$ and $\\begin{pmatrix} 3 \\\\ -2 \\\\ z \\end{pmatrix}$ are orthogonal."
},
{
"from": "gpt",
"value": "To determine the value of \\( z \\) that makes the vectors \\( \\begin{pmatrix} 2 \\\... |
Simplify the expression $\frac{a^m \cdot a^{-n}}{a^n \cdot a^{-m}}$ when $a$, $m$, and $n$ are integers, $a\neq 0$, and $m>n$. | Simplify the expression $\frac{a^m \cdot a^{-n}}{a^n \cdot a^{-m}}$ when $a$, $m$, and $n$ are integers, $a\neq 0$, and $m>n$. | To simplify the expression \( \frac{a^m \cdot a^{-n}}{a^n \cdot a^{-m}} \), we start by applying the laws of exponents.
1. First, we simplify the numerator and denominator separately:
- The numerator is:
\[
a^m \cdot a^{-n} = a^{m - n}
\]
- The denominator is:
\[
a^n \cdot a^{-m} = a^{n... | 10,041 | [
{
"from": "human",
"value": "Simplify the expression $\\frac{a^m \\cdot a^{-n}}{a^n \\cdot a^{-m}}$ when $a$, $m$, and $n$ are integers, $a\\neq 0$, and $m>n$."
},
{
"from": "gpt",
"value": "To simplify the expression \\( \\frac{a^m \\cdot a^{-n}}{a^n \\cdot a^{-m}} \\), we start by applying the... |
In a triangular array of numbers, each row begins and ends with a 1, and the number in the kth position of the nth row is the sum of the numbers in the (k-1)th and kth positions of the (n-1)th row. How many ways are there to read the number 1 in the 7th row of this array? | In a triangular array of numbers, each row begins and ends with a 1, and the number in the kth position of the nth row is the sum of the numbers in the (k-1)th and kth positions of the (n-1)th row. How many ways are there to read the number 1 in the 7th row of this array? | In a triangular array of numbers known as Pascal's triangle, each row starts and ends with 1. The numbers in the triangle correspond to the binomial coefficients. The \(n\)th row consists of the numbers \(C(n, k)\), which represents the number of ways to choose \(k\) elements from \(n\) elements.
In Pascal's triangle,... | 10,042 | [
{
"from": "human",
"value": "In a triangular array of numbers, each row begins and ends with a 1, and the number in the kth position of the nth row is the sum of the numbers in the (k-1)th and kth positions of the (n-1)th row. How many ways are there to read the number 1 in the 7th row of this array?"
},
... |
Find the coefficient of the term containing $x^2y^3z^4$ in the expansion of $(x+y+z)^9$. | Find the coefficient of the term containing $x^2y^3z^4$ in the expansion of $(x+y+z)^9$. | To find the coefficient of the term \(x^2y^3z^4\) in the expansion of \((x+y+z)^9\), we can use the multinomial theorem, which states that:
\[
(x_1 + x_2 + \cdots + x_m)^n = \sum_{k_1+k_2+\cdots+k_m=n} \frac{n!}{k_1! k_2! \cdots k_m!} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}
\]
In this case, we have \(x_1 = x\), \(x_2 = ... | 10,043 | [
{
"from": "human",
"value": "Find the coefficient of the term containing $x^2y^3z^4$ in the expansion of $(x+y+z)^9$."
},
{
"from": "gpt",
"value": "To find the coefficient of the term \\(x^2y^3z^4\\) in the expansion of \\((x+y+z)^9\\), we can use the multinomial theorem, which states that:\n\n... |
A committee of 5 people is to be chosen from a club of 10 people, but with the constraint that at least 2 members must be from a specific subgroup of 4 people. How many ways can this committee be formed? | A committee of 5 people is to be chosen from a club of 10 people, but with the constraint that at least 2 members must be from a specific subgroup of 4 people. How many ways can this committee be formed? | To solve this problem, we need to form a committee of 5 people from a total of 10 people, ensuring that at least 2 members are from a specific subgroup of 4 people.
Let’s denote the specific subgroup of 4 people as \(A_1, A_2, A_3, A_4\) and the other 6 people as \(B_1, B_2, B_3, B_4, B_5, B_6\).
We will consider di... | 10,044 | [
{
"from": "human",
"value": "A committee of 5 people is to be chosen from a club of 10 people, but with the constraint that at least 2 members must be from a specific subgroup of 4 people. How many ways can this committee be formed?"
},
{
"from": "gpt",
"value": "To solve this problem, we need t... |
A coffee shop sells a combination of small and large coffee cups. On Monday, they sold 5 small cups and 2 large cups for a total of $\$$15. On Tuesday, they sold 3 small cups and 4 large cups for a total of $\$$24. If the shop offers a 10% discount for bulk orders of 10 or more cups, how much would it cost to buy 8 sma... | A coffee shop sells a combination of small and large coffee cups. On Monday, they sold 5 small cups and 2 large cups for a total of $\$$15. On Tuesday, they sold 3 small cups and 4 large cups for a total of $\$$24. If the shop offers a 10% discount for bulk orders of 10 or more cups, how much would it cost to buy 8 sma... | Let's denote the price of a small cup of coffee as \( x \) and the price of a large cup as \( y \).
From Monday's sales, we have the equation:
\[
5x + 2y = 15
\]
From Tuesday's sales, we have the equation:
\[
3x + 4y = 24
\]
We now have a system of linear equations. We can use substitution or elimination to solve fo... | 10,045 | [
{
"from": "human",
"value": "A coffee shop sells a combination of small and large coffee cups. On Monday, they sold 5 small cups and 2 large cups for a total of $\\$$15. On Tuesday, they sold 3 small cups and 4 large cups for a total of $\\$$24. If the shop offers a 10% discount for bulk orders of 10 or mor... |
The sum of a negative integer $N$ and its square is 6. What is the value of $N$? | The sum of a negative integer $N$ and its square is 6. What is the value of $N$? | Let \( N \) be the negative integer. According to the problem, we are given the equation:
\[
N + N^2 = 6
\]
We can rearrange this equation to form a standard quadratic equation:
\[
N^2 + N - 6 = 0
\]
Next, we will factor the quadratic expression. We need two numbers that multiply to \(-6\) (the constant term) and a... | 10,046 | [
{
"from": "human",
"value": "The sum of a negative integer $N$ and its square is 6. What is the value of $N$?"
},
{
"from": "gpt",
"value": "Let \\( N \\) be the negative integer. According to the problem, we are given the equation:\n\n\\[\nN + N^2 = 6\n\\]\n\nWe can rearrange this equation to f... |
In an isosceles triangle, one of the angles opposite an equal side is $40^{\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label("$\backslash$",(1.5,1)); label("{/}",(4.5,1));
label("$40^{\circ}$",(.5,0),dir(45));
[/asy] | In an isosceles triangle, one of the angles opposite an equal side is $40^{\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label("$\backslash$",(1.5,1)); label("{/}",(4.5,1));
label("$40^{\circ}$",(.5,0),dir(45));
[/asy] | In the problem, we have an isosceles triangle where one of the angles opposite an equal side is \( 40^\circ \). Let \( A \) and \( B \) be the two equal angles, both measuring \( 40^\circ \). The triangle's angles can be represented as follows:
- \( A = 40^\circ \)
- \( B = 40^\circ \)
- \( C \) is the third angle.
B... | 10,047 | [
{
"from": "human",
"value": "In an isosceles triangle, one of the angles opposite an equal side is $40^{\\circ}$. How many degrees are in the measure of the triangle's largest angle? [asy] draw((0,0)--(6,0)--(3,2)--(0,0)); label(\"$\\backslash$\",(1.5,1)); label(\"{/}\",(4.5,1));\nlabel(\"$40^{\\circ}$\",(.... |
Let $D$ be the circle with equation $x^2 + 8x + y^2 - 2y = 1$. If $(c, d)$ is the center of $D$ and $s$ is its radius, what is the value of $c + d - s$? | Let $D$ be the circle with equation $x^2 + 8x + y^2 - 2y = 1$. If $(c, d)$ is the center of $D$ and $s$ is its radius, what is the value of $c + d - s$? | To solve for the values of \(c\), \(d\), and \(s\) from the given circle equation \(x^2 + 8x + y^2 - 2y = 1\), we will start by rewriting the equation in standard form.
### Step 1: Complete the square for \(x\).
The terms involving \(x\) are:
\[
x^2 + 8x
\]
To complete the square:
\[
x^2 + 8x = (x + 4)^2 - 16
\]
### ... | 10,048 | [
{
"from": "human",
"value": "Let $D$ be the circle with equation $x^2 + 8x + y^2 - 2y = 1$. If $(c, d)$ is the center of $D$ and $s$ is its radius, what is the value of $c + d - s$?"
},
{
"from": "gpt",
"value": "To solve for the values of \\(c\\), \\(d\\), and \\(s\\) from the given circle equa... |
Find the angle between the plane of the equilateral triangle $ABC$ with side length $4$ and the plane of the isosceles triangle $DEF$ with base $DE$ of length $6$ and congruent legs $DF$ and $EF$ of length $5$, where $A$ and $D$ are the same point and $B$, $E$, $C$, and $F$ are distinct points in space. | Find the angle between the plane of the equilateral triangle $ABC$ with side length $4$ and the plane of the isosceles triangle $DEF$ with base $DE$ of length $6$ and congruent legs $DF$ and $EF$ of length $5$, where $A$ and $D$ are the same point and $B$, $E$, $C$, and $F$ are distinct points in space. | To find the angle between the planes of the two triangles, we first need to establish the equations for the planes based on the points of the triangles in 3D space.
For triangle \(ABC\):
- Place point \(A\) at the origin, \(A(0, 0, 0)\).
- Since triangle \(ABC\) is equilateral with side length 4, we can place points \... | 10,049 | [
{
"from": "human",
"value": "Find the angle between the plane of the equilateral triangle $ABC$ with side length $4$ and the plane of the isosceles triangle $DEF$ with base $DE$ of length $6$ and congruent legs $DF$ and $EF$ of length $5$, where $A$ and $D$ are the same point and $B$, $E$, $C$, and $F$ are ... |
A bookstore has five shelves, each containing books of a different genre (fiction, non-fiction, mystery, sci-fi, romance). The store also has five bookstands, each displaying a book from a different genre. If the store owner wants to display one book from each genre on the bookstands, but no book can be displayed on a ... | A bookstore has five shelves, each containing books of a different genre (fiction, non-fiction, mystery, sci-fi, romance). The store also has five bookstands, each displaying a book from a different genre. If the store owner wants to display one book from each genre on the bookstands, but no book can be displayed on a ... | To solve this problem, we need to consider the arrangement of books on the bookstands in relation to the shelves of different genres.
We have five genres: fiction, non-fiction, mystery, sci-fi, and romance. We'll denote the genres with the following labels:
1. F - Fiction
2. NF - Non-Fiction
3. M - Mystery
4. SF - Sc... | 10,050 | [
{
"from": "human",
"value": "A bookstore has five shelves, each containing books of a different genre (fiction, non-fiction, mystery, sci-fi, romance). The store also has five bookstands, each displaying a book from a different genre. If the store owner wants to display one book from each genre on the books... |
Five balls are numbered 1 through 5 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be ev... | Five balls are numbered 1 through 5 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that the product of the two numbers will be ev... | To find the probability that the product of the two chosen numbers is both even and greater than 10, we first explore the criteria needed for the product of two integers \(a\) and \(b\) to satisfy these conditions.
### Step 1: Determine when is the product even
The product \(ab\) is even if at least one of \(a\) or \(... | 10,051 | [
{
"from": "human",
"value": "Five balls are numbered 1 through 5 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that ... |
Given that $x+y=5$ and $x^2+y^2=17$, find $xy$. | Given that $x+y=5$ and $x^2+y^2=17$, find $xy$. | Given the equations \(x+y=5\) and \(x^2+y^2=17\), let us denote \(s = x+y\) and \(p = xy\).
We know from the identity for the sum of squares that:
\[
x^2 + y^2 = (x+y)^2 - 2xy.
\]
Substituting the expressions for \(s\) and \(p\) into this identity, we have:
\[
x^2 + y^2 = s^2 - 2p.
\]
Now, substituting the known v... | 10,052 | [
{
"from": "human",
"value": "Given that $x+y=5$ and $x^2+y^2=17$, find $xy$."
},
{
"from": "gpt",
"value": "Given the equations \\(x+y=5\\) and \\(x^2+y^2=17\\), let us denote \\(s = x+y\\) and \\(p = xy\\).\n\nWe know from the identity for the sum of squares that:\n\n\\[\nx^2 + y^2 = (x+y)^2 - ... |
Sandy's daughter has a playhouse in the back yard. She plans to cover the one shaded exterior wall and the two rectangular faces of the roof, also shaded, with a special siding to resist the elements. The siding is sold only in 8-foot by 12-foot sections that cost $\$27.30$ each. If Sandy can cut the siding when she ge... | Sandy's daughter has a playhouse in the back yard. She plans to cover the one shaded exterior wall and the two rectangular faces of the roof, also shaded, with a special siding to resist the elements. The siding is sold only in 8-foot by 12-foot sections that cost $\$27.30$ each. If Sandy can cut the siding when she ge... | To determine the total cost of the siding that Sandy needs to purchase, we first need to calculate the area of the shaded wall and the two rectangular faces of the roof.
1. **Calculate the area of the shaded wall**:
The dimensions of the shaded wall are given as:
- Width = 8 feet
- Height = 6 feet
The are... | 10,053 | [
{
"from": "human",
"value": "Sandy's daughter has a playhouse in the back yard. She plans to cover the one shaded exterior wall and the two rectangular faces of the roof, also shaded, with a special siding to resist the elements. The siding is sold only in 8-foot by 12-foot sections that cost $\\$27.30$ eac... |
The expression
$$\frac{3x}{2} + \frac{25}{x}$$
achieves its minimum value for what positive integer $x$? | The expression
$$\frac{3x}{2} + \frac{25}{x}$$
achieves its minimum value for what positive integer $x$? | Let \( f(x) = \frac{3x}{2} + \frac{25}{x} \). To find the minimum value of \( f(x) \) for positive integers \( x \), we will find the critical points of the function by differentiating \( f(x) \) and setting the derivative to zero.
First, we compute the derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx}\left(\frac{3x}{2... | 10,054 | [
{
"from": "human",
"value": "The expression \n$$\\frac{3x}{2} + \\frac{25}{x}$$ \nachieves its minimum value for what positive integer $x$?"
},
{
"from": "gpt",
"value": "Let \\( f(x) = \\frac{3x}{2} + \\frac{25}{x} \\). To find the minimum value of \\( f(x) \\) for positive integers \\( x \\), ... |
In a triangle, the length of the longest side is 15 cm and the perimeter is 35 cm. If the triangle is scalene and one of the other sides is exactly 10 cm longer than the shortest side, what is the length of the shortest side? | In a triangle, the length of the longest side is 15 cm and the perimeter is 35 cm. If the triangle is scalene and one of the other sides is exactly 10 cm longer than the shortest side, what is the length of the shortest side? | Let the lengths of the sides of the triangle be denoted as \( a \), \( b \), and \( c \), where \( c \) is the longest side. According to the information given:
- The longest side \( c = 15 \) cm.
- The perimeter of the triangle is \( a + b + c = 35 \) cm.
- One of the other sides is exactly 10 cm longer than the shor... | 10,055 | [
{
"from": "human",
"value": "In a triangle, the length of the longest side is 15 cm and the perimeter is 35 cm. If the triangle is scalene and one of the other sides is exactly 10 cm longer than the shortest side, what is the length of the shortest side?"
},
{
"from": "gpt",
"value": "Let the le... |
In the polynomial equation
$$x^4+ax^3+bx^2+cx+d=0$$
$a$, $b$, $c$, and $d$ are all integers, and the roots of the polynomial are $2$, $-3$, $5$, and $-\sqrt{3}$. If $d$ is a negative integer, what is the largest possible value of $d$? | In the polynomial equation
$$x^4+ax^3+bx^2+cx+d=0$$
$a$, $b$, $c$, and $d$ are all integers, and the roots of the polynomial are $2$, $-3$, $5$, and $-\sqrt{3}$. If $d$ is a negative integer, what is the largest possible value of $d$? | To solve for \( d \) in the polynomial equation given the roots \( 2 \), \( -3 \), \( 5 \), and \( -\sqrt{3} \), we can first express the polynomial in factored form:
\[
P(x) = (x - 2)(x + 3)(x - 5)(x + \sqrt{3})(x - (-\sqrt{3})) = (x - 2)(x + 3)(x - 5)(x^2 + 3)
\]
Next, we will expand the polynomial step-by-step.
S... | 10,056 | [
{
"from": "human",
"value": "In the polynomial equation\n$$x^4+ax^3+bx^2+cx+d=0$$\n$a$, $b$, $c$, and $d$ are all integers, and the roots of the polynomial are $2$, $-3$, $5$, and $-\\sqrt{3}$. If $d$ is a negative integer, what is the largest possible value of $d$?"
},
{
"from": "gpt",
"value":... |
The polynomial $x^3 + cx^2 + dx + 6$ has a factor of the form $x^2 + x - 2.$ Find $c+d$. | The polynomial $x^3 + cx^2 + dx + 6$ has a factor of the form $x^2 + x - 2.$ Find $c+d$. | Let \( p(x) = x^3 + cx^2 + dx + 6 \).
Given that \( p(x) \) has a factor \( x^2 + x - 2 \), we can express \( p(x) \) in the following form:
\[
p(x) = (x^2 + x - 2)(x + a)
\]
for some constant \( a \).
To find \( p(x) \), we first need to multiply \( x^2 + x - 2 \) by \( x + a \).
\[
(x^2 + x - 2)(x + a) = x^3 + ax... | 10,057 | [
{
"from": "human",
"value": "The polynomial $x^3 + cx^2 + dx + 6$ has a factor of the form $x^2 + x - 2.$ Find $c+d$."
},
{
"from": "gpt",
"value": "Let \\( p(x) = x^3 + cx^2 + dx + 6 \\).\nGiven that \\( p(x) \\) has a factor \\( x^2 + x - 2 \\), we can express \\( p(x) \\) in the following fo... |
What is the smallest positive integer that can be expressed as the sum of two perfect squares in exactly three different ways? | What is the smallest positive integer that can be expressed as the sum of two perfect squares in exactly three different ways? | To solve the problem, we need to find the smallest positive integer \( n \) that can be expressed as a sum of two perfect squares in exactly three different ways.
A number can be expressed as a sum of two squares if it satisfies certain conditions, particularly related to its prime factors of the form \( 4k+3 \). Spec... | 10,058 | [
{
"from": "human",
"value": "What is the smallest positive integer that can be expressed as the sum of two perfect squares in exactly three different ways?"
},
{
"from": "gpt",
"value": "To solve the problem, we need to find the smallest positive integer \\( n \\) that can be expressed as a sum ... |
What is the largest positive integer $n$ such that $2^n$ is less than $100$? | What is the largest positive integer $n$ such that $2^n$ is less than $100$? | To find the largest positive integer \( n \) such that \( 2^n < 100 \), we can start by calculating the powers of 2 and see how far we can go before we exceed 100.
Calculating the first few powers of 2:
\[
2^1 = 2
\]
\[
2^2 = 4
\]
\[
2^3 = 8
\]
\[
2^4 = 16
\]
\[
2^5 = 32
\]
\[
2^6 = 64
\]
\[
2^7 = 128
\]
Next, we ob... | 10,059 | [
{
"from": "human",
"value": "What is the largest positive integer $n$ such that $2^n$ is less than $100$?"
},
{
"from": "gpt",
"value": "To find the largest positive integer \\( n \\) such that \\( 2^n < 100 \\), we can start by calculating the powers of 2 and see how far we can go before we exc... |
Given three integers $a$, $b$, and $c$, where $a$, $b$, and $c$ are pairwise relatively prime (i.e., any two of them are coprime), find a formula for the product of the greatest common divisor (GCD) and the least common multiple (LCM) of the differences of these numbers, namely GCD($b-a$, $c-a$) and LCM($b-a$, $c-a$). | Given three integers $a$, $b$, and $c$, where $a$, $b$, and $c$ are pairwise relatively prime (i.e., any two of them are coprime), find a formula for the product of the greatest common divisor (GCD) and the least common multiple (LCM) of the differences of these numbers, namely GCD($b-a$, $c-a$) and LCM($b-a$, $c-a$). | To solve for the product of the greatest common divisor (GCD) and the least common multiple (LCM) of the differences \( b - a \) and \( c - a \), we denote \( x = b - a \) and \( y = c - a \). We will calculate \( \text{GCD}(x, y) \) and \( \text{LCM}(x, y) \), and then find their product.
### Step 1: Find GCD and LCM... | 10,060 | [
{
"from": "human",
"value": "Given three integers $a$, $b$, and $c$, where $a$, $b$, and $c$ are pairwise relatively prime (i.e., any two of them are coprime), find a formula for the product of the greatest common divisor (GCD) and the least common multiple (LCM) of the differences of these numbers, namely ... |
If the digit represented by $*$ satisfies the following addition problem $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & 5 & 3 & 2 & *_7\\ & & & * & 6 & 1_7\\ &+ & & & * & 3_7\\ \cline{2-6} & & 6 & 2 & * & 4_7\\ \end{array} $$what is the value of $*$? | If the digit represented by $*$ satisfies the following addition problem $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & 5 & 3 & 2 & *_7\\ & & & * & 6 & 1_7\\ &+ & & & * & 3_7\\ \cline{2-6} & & 6 & 2 & * & 4_7\\ \end{array} $$what is the value of $*$? | To solve for the digit represented by \( * \) in the given base \( 7 \) addition problem, we will start by converting the expressions from base \( 7 \) to base \( 10 \), process the addition step by step, and analyze the carries.
The addition in base \( 7 \) is given as:
\[
\begin{array}{cccccc}
& & 5 & 3 & 2 & *_7\\
... | 10,061 | [
{
"from": "human",
"value": "If the digit represented by $*$ satisfies the following addition problem $$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c} & & 5 & 3 & 2 & *_7\\\\ & & & * & 6 & 1_7\\\\ &+ & & & * & 3_7\\\\ \\cline{2-6} & & 6 & 2 & * & 4_7\\\\ \\end{array} $$what is the value of $*$?"
},
{
"fr... |
Consider the two points $(3,-1)$ and $(5,6)$. What is the $y$-coordinate of the point on the $y$-axis that is equidistant from these two points? | Consider the two points $(3,-1)$ and $(5,6)$. What is the $y$-coordinate of the point on the $y$-axis that is equidistant from these two points? | To find the point on the \( y \)-axis that is equidistant from the points \((3, -1)\) and \((5, 6)\), we first denote a point on the \( y \)-axis as \((0, y)\).
Next, we calculate the distances from \((0, y)\) to the points \((3, -1)\) and \((5, 6)\).
1. **Distance from \((0, y)\) to \((3, -1)\)**:
\[
d_1 = \s... | 10,062 | [
{
"from": "human",
"value": "Consider the two points $(3,-1)$ and $(5,6)$. What is the $y$-coordinate of the point on the $y$-axis that is equidistant from these two points?"
},
{
"from": "gpt",
"value": "To find the point on the \\( y \\)-axis that is equidistant from the points \\((3, -1)\\) a... |
In a company, there are several teams, and each team has a different number of employees. The number of teams is equal to the average number of employees per team. How many employees are in the company? | In a company, there are several teams, and each team has a different number of employees. The number of teams is equal to the average number of employees per team. How many employees are in the company? | Let \( n \) be the number of teams and let \( k \) be the average number of employees per team. According to the problem, we have:
\[ n = k \]
The total number of employees \( E \) in the company can be calculated as the product of the number of teams and the average number of employees per team:
\[
E = n \cdot k
\]... | 10,063 | [
{
"from": "human",
"value": "In a company, there are several teams, and each team has a different number of employees. The number of teams is equal to the average number of employees per team. How many employees are in the company?"
},
{
"from": "gpt",
"value": "Let \\( n \\) be the number of te... |
Rachel has some 2 cent stamps and some 5 cent stamps. What is the least number of stamps she can combine so the value of the stamps is 27 cents? | Rachel has some 2 cent stamps and some 5 cent stamps. What is the least number of stamps she can combine so the value of the stamps is 27 cents? | To determine the least number of stamps Rachel can use to total 27 cents with 2 cent and 5 cent stamps, we can set up a mathematical approach.
Let:
- \( x \) = number of 2 cent stamps
- \( y \) = number of 5 cent stamps
The total value of the stamps can be expressed by the equation:
\[
2x + 5y = 27
\]
We want to min... | 10,064 | [
{
"from": "human",
"value": "Rachel has some 2 cent stamps and some 5 cent stamps. What is the least number of stamps she can combine so the value of the stamps is 27 cents?"
},
{
"from": "gpt",
"value": "To determine the least number of stamps Rachel can use to total 27 cents with 2 cent and 5 ... |
In a three-dimensional space, five identical rectangular prisms are arranged to form a larger rectangular prism. The larger prism has dimensions $x$, $2x$, and $3x$, and its volume is 16000. What is the length of the diagonal of the base of one of the smaller prisms, rounded off to the nearest integer? | In a three-dimensional space, five identical rectangular prisms are arranged to form a larger rectangular prism. The larger prism has dimensions $x$, $2x$, and $3x$, and its volume is 16000. What is the length of the diagonal of the base of one of the smaller prisms, rounded off to the nearest integer? | To begin solving the problem, we first calculate the volume of the larger rectangular prism, which is given by the formula for the volume of a rectangular prism:
\[
V = \text{length} \times \text{width} \times \text{height}
\]
For our larger prism, the dimensions are \(x\), \(2x\), and \(3x\), leading to its volume b... | 10,065 | [
{
"from": "human",
"value": "In a three-dimensional space, five identical rectangular prisms are arranged to form a larger rectangular prism. The larger prism has dimensions $x$, $2x$, and $3x$, and its volume is 16000. What is the length of the diagonal of the base of one of the smaller prisms, rounded o... |
What is $7-3\cdot2+12\div4$? | What is $7-3\cdot2+12\div4$? | To solve the expression \( 7 - 3 \cdot 2 + 12 \div 4 \), we need to follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
1. First, we perform the multiplication and division from left to right.
... | 10,066 | [
{
"from": "human",
"value": "What is $7-3\\cdot2+12\\div4$?"
},
{
"from": "gpt",
"value": "To solve the expression \\( 7 - 3 \\cdot 2 + 12 \\div 4 \\), we need to follow the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addit... |
Problem:
In a tetrahedron $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of edges $AB$, $BC$, $CD$, and $DA$, respectively. If $X$ is an arbitrary point, show that there exists a constant $m$ such that
\[XA^2 + XB^2 + XC^2 + XD^2 = m \cdot XM^2 + MN^2 + NP^2 + PQ^2 + MQ^2 + XN^2 + XP^2 + XQ^2.\]
Find $m$. | Problem:
In a tetrahedron $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of edges $AB$, $BC$, $CD$, and $DA$, respectively. If $X$ is an arbitrary point, show that there exists a constant $m$ such that
\[XA^2 + XB^2 + XC^2 + XD^2 = m \cdot XM^2 + MN^2 + NP^2 + PQ^2 + MQ^2 + XN^2 + XP^2 + XQ^2.\]
Find $m$. | To solve this problem, we first denote the position vectors of the points \( A, B, C, D \) in three-dimensional space as \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \). The midpoints can then be expressed as follows:
- \( \vec{m} = \frac{\vec{a} + \vec{b}}{2} \)
- \( \vec{n} = \frac{\vec{b} + \vec{c}}{2} \)
- \( \vec{p} = \... | 10,067 | [
{
"from": "human",
"value": "Problem:\nIn a tetrahedron $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of edges $AB$, $BC$, $CD$, and $DA$, respectively. If $X$ is an arbitrary point, show that there exists a constant $m$ such that\n\\[XA^2 + XB^2 + XC^2 + XD^2 = m \\cdot XM^2 + MN^2 + NP^2 + PQ^2 + MQ... |
Problem:
A library has 10 books on a specific subject, and a student needs to borrow exactly 4 books. However, there are some restrictions: the student must borrow at least one book from the 3 books written by author A, and at least one book from the 2 books written by author B. Additionally, the student cannot borrow ... | Problem:
A library has 10 books on a specific subject, and a student needs to borrow exactly 4 books. However, there are some restrictions: the student must borrow at least one book from the 3 books written by author A, and at least one book from the 2 books written by author B. Additionally, the student cannot borrow ... | To solve this problem, we need to consider the restrictions placed on the borrowing of books. We have a total of 10 books, categorized by their authors as follows:
- 3 books by author A (let's denote them as A1, A2, A3)
- 2 books by author B (denote them as B1, B2)
- 5 books by author C (denote them as C1, C2, C3, C4,... | 10,068 | [
{
"from": "human",
"value": "Problem:\nA library has 10 books on a specific subject, and a student needs to borrow exactly 4 books. However, there are some restrictions: the student must borrow at least one book from the 3 books written by author A, and at least one book from the 2 books written by author B... |
In a certain ellipse
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\]
where $a > b,$ the focal length is equal to $\frac{a}{3}.$ Find $\frac{b}{a}.$ | In a certain ellipse
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\]
where $a > b,$ the focal length is equal to $\frac{a}{3}.$ Find $\frac{b}{a}.$ | To solve for \(\frac{b}{a}\) in the given ellipse, we start with the standard form of the ellipse equation given by:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,
\]
where \(a > b\).
The distance \(c\) from the center of the ellipse to each focus is determined by the relationship:
\[
c = \sqrt{a^2 - b^2}.
\]
According... | 10,069 | [
{
"from": "human",
"value": "In a certain ellipse\n\\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1,\\]\nwhere $a > b,$ the focal length is equal to $\\frac{a}{3}.$ Find $\\frac{b}{a}.$"
},
{
"from": "gpt",
"value": "To solve for \\(\\frac{b}{a}\\) in the given ellipse, we start with the standard for... |
The perimeter of a rectangular room is 80 feet. If the length of the room is three times the width, what is the area of the room, in square feet? | The perimeter of a rectangular room is 80 feet. If the length of the room is three times the width, what is the area of the room, in square feet? | Let the width of the rectangular room be denoted as \( w \), and the length as \( l \). According to the problem, the relationship between the length and width is given by:
\[
l = 3w
\]
The formula for the perimeter \( P \) of a rectangle is:
\[
P = 2l + 2w
\]
We know the perimeter is 80 feet, so we can set up the ... | 10,070 | [
{
"from": "human",
"value": "The perimeter of a rectangular room is 80 feet. If the length of the room is three times the width, what is the area of the room, in square feet?"
},
{
"from": "gpt",
"value": "Let the width of the rectangular room be denoted as \\( w \\), and the length as \\( l \\)... |
What is the sum of $\left(\dfrac{1}{2}\right) + \left(\dfrac{1}{2}\right)^2 + \left(\dfrac{1}{2}\right)^3 + \left(\dfrac{1}{2}\right)^4$? | What is the sum of $\left(\dfrac{1}{2}\right) + \left(\dfrac{1}{2}\right)^2 + \left(\dfrac{1}{2}\right)^3 + \left(\dfrac{1}{2}\right)^4$? | We need to calculate the sum
\[
S = \left(\frac{1}{2}\right) + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 + \left(\frac{1}{2}\right)^4.
\]
We can rewrite this expression:
\[
S = \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}.
\]
To ... | 10,071 | [
{
"from": "human",
"value": "What is the sum of $\\left(\\dfrac{1}{2}\\right) + \\left(\\dfrac{1}{2}\\right)^2 + \\left(\\dfrac{1}{2}\\right)^3 + \\left(\\dfrac{1}{2}\\right)^4$?"
},
{
"from": "gpt",
"value": "We need to calculate the sum \n\n\\[\nS = \\left(\\frac{1}{2}\\right) + \\left(\\frac{... |
Determine the slope of a line that passes through the point (2,1) and has a $y$-intercept of 3. | Determine the slope of a line that passes through the point (2,1) and has a $y$-intercept of 3. | To determine the slope of a line given a point and the y-intercept, we can use the equation of a line in slope-intercept form, which is:
\[
y = mx + b
\]
where \( m \) is the slope and \( b \) is the y-intercept.
We are given the following information:
- The y-intercept \( b = 3 \).
- The line passes through the poi... | 10,072 | [
{
"from": "human",
"value": "Determine the slope of a line that passes through the point (2,1) and has a $y$-intercept of 3."
},
{
"from": "gpt",
"value": "To determine the slope of a line given a point and the y-intercept, we can use the equation of a line in slope-intercept form, which is:\n\n... |
The school's boys basketball team has 15 players, including a set of 4 brothers: John, Mike, Tom, and Alex. In how many ways can we choose 5 starters if the only restriction is that exactly two of the brothers must be in the starting lineup? | The school's boys basketball team has 15 players, including a set of 4 brothers: John, Mike, Tom, and Alex. In how many ways can we choose 5 starters if the only restriction is that exactly two of the brothers must be in the starting lineup? | To determine the number of ways to choose 5 starters from the basketball team with the restriction that exactly two of the four brothers (John, Mike, Tom, Alex) must be included, we can break the problem down into several steps.
1. **Select 2 Brothers:**
We start by selecting 2 brothers from the 4 available. We ca... | 10,073 | [
{
"from": "human",
"value": "The school's boys basketball team has 15 players, including a set of 4 brothers: John, Mike, Tom, and Alex. In how many ways can we choose 5 starters if the only restriction is that exactly two of the brothers must be in the starting lineup?"
},
{
"from": "gpt",
"va... |
In the diagram, $AOB$ is a sector of a circle with $\angle AOB=60^\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY
?$ [asy]
draw((0,0)--(12,0),black+linewidth(1));
draw((0,0)--(10.3923,-6)..(12,0)..(10.3923,6)--(0,0),black+linewidth(1));
draw((10.3923,-6)--(10.3923,6),b... | In the diagram, $AOB$ is a sector of a circle with $\angle AOB=60^\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY
?$ [asy]
draw((0,0)--(12,0),black+linewidth(1));
draw((0,0)--(10.3923,-6)..(12,0)..(10.3923,6)--(0,0),black+linewidth(1));
draw((10.3923,-6)--(10.3923,6),b... | To find the length of \(XY\) in the diagram, we begin by considering the geometrical properties of the circle sector \(AOB\) with \(\angle AOB = 60^\circ\), and where \(OA\) and \(OB\) are both radii of length \(12\).
### Step 1: Determine the coordinates for points \(A\) and \(B\)
We can place point \(O\) at the orig... | 10,074 | [
{
"from": "human",
"value": "In the diagram, $AOB$ is a sector of a circle with $\\angle AOB=60^\\circ.$ $OY$ is drawn perpendicular to $AB$ and intersects $AB$ at $X.$ What is the length of $XY\n?$ [asy]\ndraw((0,0)--(12,0),black+linewidth(1));\ndraw((0,0)--(10.3923,-6)..(12,0)..(10.3923,6)--(0,0),black+l... |
Find the range of the function
\[f(x) = \frac{x^2 - 4x + 3}{\sqrt{x^2 + 2x + 2} - \sqrt{x^2 - 2x + 2}}.\] | Find the range of the function
\[f(x) = \frac{x^2 - 4x + 3}{\sqrt{x^2 + 2x + 2} - \sqrt{x^2 - 2x + 2}}.\] | To find the range of the function
\[
f(x) = \frac{x^2 - 4x + 3}{\sqrt{x^2 + 2x + 2} - \sqrt{x^2 - 2x + 2}},
\]
we'll start by simplifying the function.
### Step 1: Rewrite the numerator
The numerator \(x^2 - 4x + 3\) can be factored:
\[
x^2 - 4x + 3 = (x-1)(x-3).
\]
### Step 2: Rewrite the denominator
Now, let's s... | 10,075 | [
{
"from": "human",
"value": "Find the range of the function\n\\[f(x) = \\frac{x^2 - 4x + 3}{\\sqrt{x^2 + 2x + 2} - \\sqrt{x^2 - 2x + 2}}.\\]"
},
{
"from": "gpt",
"value": "To find the range of the function\n\n\\[\nf(x) = \\frac{x^2 - 4x + 3}{\\sqrt{x^2 + 2x + 2} - \\sqrt{x^2 - 2x + 2}},\n\\]\n\n... |
Find the largest digit (in the usual base 10) that appears in the base 12 representation of $123456789_{10}$. | Find the largest digit (in the usual base 10) that appears in the base 12 representation of $123456789_{10}$. | To find the largest digit that appears in the base 12 representation of the decimal number \(123456789\), we need to first convert the decimal number into base 12.
The conversion process involves repeatedly dividing the number by 12 and keeping track of the remainders. The remainders will give us the digits of the num... | 10,076 | [
{
"from": "human",
"value": "Find the largest digit (in the usual base 10) that appears in the base 12 representation of $123456789_{10}$."
},
{
"from": "gpt",
"value": "To find the largest digit that appears in the base 12 representation of the decimal number \\(123456789\\), we need to first c... |
Determine the domain of the function
$$g(x) = \frac{1}{1-\sqrt{x}+\sqrt{1-x}}.$$
Express your answer in interval notation. | Determine the domain of the function
$$g(x) = \frac{1}{1-\sqrt{x}+\sqrt{1-x}}.$$
Express your answer in interval notation. | To determine the domain of the function
\[
g(x) = \frac{1}{1 - \sqrt{x} + \sqrt{1 - x}},
\]
we need to ensure that the denominator is not equal to zero and we also need to consider the restrictions imposed by the square root functions.
1. **Square Root Conditions**: The terms \(\sqrt{x}\) and \(\sqrt{1 - x}\) are d... | 10,077 | [
{
"from": "human",
"value": "Determine the domain of the function\n$$g(x) = \\frac{1}{1-\\sqrt{x}+\\sqrt{1-x}}.$$\nExpress your answer in interval notation."
},
{
"from": "gpt",
"value": "To determine the domain of the function \n\n\\[\ng(x) = \\frac{1}{1 - \\sqrt{x} + \\sqrt{1 - x}},\n\\]\n\nwe... |
Find the largest number of terms of a geometric sequence whose product is 30 and all terms are positive integers. | Find the largest number of terms of a geometric sequence whose product is 30 and all terms are positive integers. | To find the largest number of terms of a geometric sequence whose product is \(30\) and all terms are positive integers, we denote the terms of the geometric sequence as \(a, ar, ar^2, \ldots, ar^{n-1}\), where \(a\) is the first term and \(r\) is the common ratio. The product of the terms in an \(n\)-term geometric se... | 10,078 | [
{
"from": "human",
"value": "Find the largest number of terms of a geometric sequence whose product is 30 and all terms are positive integers."
},
{
"from": "gpt",
"value": "To find the largest number of terms of a geometric sequence whose product is \\(30\\) and all terms are positive integers,... |
In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $AD$.
Write another problem inspired by this one:
In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $A... | In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $AD$.
Write another problem inspired by this one:
In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $A... | To find \( AD \) in triangle \( ABC \), we start with the information given:
- \( AB = 10 \)
- \( AC = 17 \)
- The ratio \( BD : CD = 2 : 5 \)
Let \( BD = 2x \) and \( CD = 5x \). Therefore, the length of \( BC \) is:
\[
BC = BD + CD = 2x + 5x = 7x
\]
Next, we'll apply the Pythagorean theorem in conjunction with th... | 10,079 | [
{
"from": "human",
"value": "In triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from $A$ to $BC$. If $BD:CD = 2:5$, then find $AD$.\n\nWrite another problem inspired by this one:\n\nIn triangle $ABC$, $AB = 10$ and $AC = 17$. Let $D$ be the foot of the perpendicular from... |
What is the volume of the tetrahedron bounded by the planes $x=0,$ $y=0,$ $z=0,$ and $2x+3y+4z=1$? | What is the volume of the tetrahedron bounded by the planes $x=0,$ $y=0,$ $z=0,$ and $2x+3y+4z=1$? | To find the volume of the tetrahedron bounded by the planes \( x=0 \), \( y=0 \), \( z=0 \), and \( 2x + 3y + 4z = 1 \), we start by determining the vertices of the tetrahedron.
The planes \( x=0 \), \( y=0 \), and \( z=0 \) correspond to the coordinate axes in three-dimensional space. The plane \( 2x + 3y + 4z = 1 \)... | 10,080 | [
{
"from": "human",
"value": "What is the volume of the tetrahedron bounded by the planes $x=0,$ $y=0,$ $z=0,$ and $2x+3y+4z=1$?"
},
{
"from": "gpt",
"value": "To find the volume of the tetrahedron bounded by the planes \\( x=0 \\), \\( y=0 \\), \\( z=0 \\), and \\( 2x + 3y + 4z = 1 \\), we start... |
A jar contains a mixture of red and green jelly beans, with a total of 20 jelly beans in the jar. The probability of drawing a red jelly bean is 3/5. If a game is played where a player draws a jelly bean at random and wins $2 if it's red and $5 if it's green, what is the expected value of the player's winnings? | A jar contains a mixture of red and green jelly beans, with a total of 20 jelly beans in the jar. The probability of drawing a red jelly bean is 3/5. If a game is played where a player draws a jelly bean at random and wins $2 if it's red and $5 if it's green, what is the expected value of the player's winnings? | To find the expected value of the player's winnings in the jelly bean game, we first need to determine the number of red and green jelly beans in the jar. We know that:
1. The total number of jelly beans is 20.
2. The probability of drawing a red jelly bean is \( \frac{3}{5} \).
Let's denote:
- The number of red jell... | 10,081 | [
{
"from": "human",
"value": "A jar contains a mixture of red and green jelly beans, with a total of 20 jelly beans in the jar. The probability of drawing a red jelly bean is 3/5. If a game is played where a player draws a jelly bean at random and wins $2 if it's red and $5 if it's green, what is the expecte... |
Consider the four numbers:
\[2 + \sqrt{3}, \; 2 - \sqrt{3}, \; -1 + \sqrt{5}, \; -1 - \sqrt{5}\]
as roots of the same nonzero polynomial with rational coefficients.
What is the smallest possible degree of the polynomial? | Consider the four numbers:
\[2 + \sqrt{3}, \; 2 - \sqrt{3}, \; -1 + \sqrt{5}, \; -1 - \sqrt{5}\]
as roots of the same nonzero polynomial with rational coefficients.
What is the smallest possible degree of the polynomial? | To determine the smallest possible degree of a polynomial with rational coefficients that has the given numbers as roots, we need to identify any irrational and complex roots and account for their conjugates in our polynomial.
The given roots are:
1. \( 2 + \sqrt{3} \)
2. \( 2 - \sqrt{3} \)
3. \( -1 + \sqrt{5} \)
4. ... | 10,082 | [
{
"from": "human",
"value": "Consider the four numbers:\n\\[2 + \\sqrt{3}, \\; 2 - \\sqrt{3}, \\; -1 + \\sqrt{5}, \\; -1 - \\sqrt{5}\\]\nas roots of the same nonzero polynomial with rational coefficients.\nWhat is the smallest possible degree of the polynomial?"
},
{
"from": "gpt",
"value": "To ... |
The number of blue pencils and green pencils in a box is in the ratio of $5:4$. If there are 15 blue pencils, how many green pencils are there? | The number of blue pencils and green pencils in a box is in the ratio of $5:4$. If there are 15 blue pencils, how many green pencils are there? | To solve this problem, we start with the ratio of blue pencils to green pencils, which is given as 5:4. This means that for every 5 blue pencils, there are 4 green pencils.
Given that there are 15 blue pencils, we can denote the number of blue pencils as \( B \) and the number of green pencils as \( G \). Thus, we hav... | 10,083 | [
{
"from": "human",
"value": "The number of blue pencils and green pencils in a box is in the ratio of $5:4$. If there are 15 blue pencils, how many green pencils are there?"
},
{
"from": "gpt",
"value": "To solve this problem, we start with the ratio of blue pencils to green pencils, which is gi... |
A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5? | A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5? | To find the percentage of palindromes between 100 and 500 that contain at least one 5, we first need to identify all the palindromic numbers in that range.
### Step 1: Identify the range of palindromes
Palindromes between 100 and 500 are three-digit numbers where the first and last digits are the same. These numbers c... | 10,084 | [
{
"from": "human",
"value": "A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5?"
},
{
"from": "gpt",
"value": "To find the percentage of palindromes between 100 and 500 that contain at lea... |
Let
\[\mathbf{B} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} 0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \renewcommand{\arraystretch}{1}.\]Compute $\mathbf{B}^{2019}\mathbf{B}^{2020}$. | Let
\[\mathbf{B} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} 0 & 0 & -1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix} \renewcommand{\arraystretch}{1}.\]Compute $\mathbf{B}^{2019}\mathbf{B}^{2020}$. | To solve for \(\mathbf{B}^{2019} \mathbf{B}^{2020}\), we start by simplifying the expression using the properties of exponents:
\[
\mathbf{B}^{2019} \mathbf{B}^{2020} = \mathbf{B}^{2019 + 2020} = \mathbf{B}^{4039}.
\]
Next, we need to compute \(\mathbf{B}^{4039}\). To do this effectively, we first explore the powers ... | 10,085 | [
{
"from": "human",
"value": "Let\n\\[\\mathbf{B} = \\renewcommand{\\arraystretch}{1.5} \\begin{pmatrix} 0 & 0 & -1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} \\renewcommand{\\arraystretch}{1}.\\]Compute $\\mathbf{B}^{2019}\\mathbf{B}^{2020}$."
},
{
"from": "gpt",
"value": "To solve for \\(\\ma... |
Find the maximum possible value of $x + y + z$ for positive integers $x$, $y$, and $z$ such that the sum of their reciprocals is equal to $\frac{3}{2004}$. | Find the maximum possible value of $x + y + z$ for positive integers $x$, $y$, and $z$ such that the sum of their reciprocals is equal to $\frac{3}{2004}$. | To maximize \(x + y + z\) for positive integers \(x\), \(y\), and \(z\) such that
\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{3}{2004},
\]
we first express the equation in a clearer form:
\[
\frac{yz + xz + xy}{xyz} = \frac{3}{2004}.
\]
From this, we can cross-multiply to get:
\[
2004(yz + xz + xy) = 3xyz.
... | 10,086 | [
{
"from": "human",
"value": "Find the maximum possible value of $x + y + z$ for positive integers $x$, $y$, and $z$ such that the sum of their reciprocals is equal to $\\frac{3}{2004}$."
},
{
"from": "gpt",
"value": "To maximize \\(x + y + z\\) for positive integers \\(x\\), \\(y\\), and \\(z\\)... |
A publishing company has 20 different books, and it wants to package groups of 3 books together to sell at a discounted price. How many different sets of 3 books can the company create if it has 500 copies of each book? | A publishing company has 20 different books, and it wants to package groups of 3 books together to sell at a discounted price. How many different sets of 3 books can the company create if it has 500 copies of each book? | To determine how many different sets of 3 books can be created from 20 different books, we need to choose 3 books from the available 20. Since the order of selection does not matter, we will use the combination formula.
The formula for the number of combinations of choosing \( r \) items from \( n \) items is given by... | 10,087 | [
{
"from": "human",
"value": "A publishing company has 20 different books, and it wants to package groups of 3 books together to sell at a discounted price. How many different sets of 3 books can the company create if it has 500 copies of each book?"
},
{
"from": "gpt",
"value": "To determine how... |
An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction. | An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction. | To find the areas of the triangles and the trapezoid, we start with the equilateral triangles.
### Step 1: Calculate the area of the larger equilateral triangle.
The formula for the area \( A \) of an equilateral triangle with side length \( s \) is given by:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
For the larger triangle ... | 10,088 | [
{
"from": "human",
"value": "An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction."
... |
Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}.\]Compute $\operatorname{proj}_{\mathbf{w}} (5 \mathbf{v}).$ | Let $\mathbf{v}$ and $\mathbf{w}$ be vectors such that
\[\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}.\]Compute $\operatorname{proj}_{\mathbf{w}} (5 \mathbf{v}).$ | To begin, we will use the formula for the projection of a vector \(\mathbf{v}\) onto another vector \(\mathbf{w}\):
\[
\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w}
\]
Given that \(\operatorname{proj}_{\mathbf{w}} \mathbf{v} = \begin{pmatrix} ... | 10,089 | [
{
"from": "human",
"value": "Let $\\mathbf{v}$ and $\\mathbf{w}$ be vectors such that\n\\[\\operatorname{proj}_{\\mathbf{w}} \\mathbf{v} = \\begin{pmatrix} 3 \\\\ 2 \\end{pmatrix}.\\]Compute $\\operatorname{proj}_{\\mathbf{w}} (5 \\mathbf{v}).$"
},
{
"from": "gpt",
"value": "To begin, we will us... |
In the two concentric circles shown, the radius of the outer circle is twice the radius of the inner circle. What is the area of the gray region, in square feet, if the width of the gray region is 2 feet? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),4),gray);
filldraw(circle((0,0),2),white);
dra... | In the two concentric circles shown, the radius of the outer circle is twice the radius of the inner circle. What is the area of the gray region, in square feet, if the width of the gray region is 2 feet? Express your answer in terms of $\pi$.
[asy]
filldraw(circle((0,0),4),gray);
filldraw(circle((0,0),2),white);
dra... | Let the radius of the inner circle be denoted as \( r \). According to the problem, the radius of the outer circle is twice that of the inner circle, thus it can be expressed as \( 2r \).
We are also given that the width of the gray region (the area between the two circles) is 2 feet. This width is represented as the ... | 10,090 | [
{
"from": "human",
"value": "In the two concentric circles shown, the radius of the outer circle is twice the radius of the inner circle. What is the area of the gray region, in square feet, if the width of the gray region is 2 feet? Express your answer in terms of $\\pi$.\n\n[asy]\nfilldraw(circle((0,0),4)... |
Find $\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} + \begin{pmatrix} -3 & -8 \\ 2 & -1 \end{pmatrix}.$ | Find $\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} + \begin{pmatrix} -3 & -8 \\ 2 & -1 \end{pmatrix}.$ | To find the sum of the matrices \(\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}\) and \(\begin{pmatrix} -3 & -8 \\ 2 & -1 \end{pmatrix}\), we perform element-wise addition.
Let’s denote the first matrix as \( A = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix} \) and the second matrix as \( B = \begin{pmatrix} -3 & -8 \\... | 10,091 | [
{
"from": "human",
"value": "Find $\\begin{pmatrix} 2 & 1 \\\\ 4 & 3 \\end{pmatrix} + \\begin{pmatrix} -3 & -8 \\\\ 2 & -1 \\end{pmatrix}.$"
},
{
"from": "gpt",
"value": "To find the sum of the matrices \\(\\begin{pmatrix} 2 & 1 \\\\ 4 & 3 \\end{pmatrix}\\) and \\(\\begin{pmatrix} -3 & -8 \\\\ 2... |
A rectangular garden measures 12 feet by 15 feet. If it is scaled down proportionally to fit a 6 feet wide area, what would be the length of the scaled-down garden? | A rectangular garden measures 12 feet by 15 feet. If it is scaled down proportionally to fit a 6 feet wide area, what would be the length of the scaled-down garden? | To scale down the rectangular garden proportionally, we need to determine the scaling factor based on the width of the area we want it to fit into, which is 6 feet.
The original garden dimensions are 12 feet by 15 feet. The longer side of the garden is 15 feet.
We need to calculate the scaling factor based on the wi... | 10,092 | [
{
"from": "human",
"value": "A rectangular garden measures 12 feet by 15 feet. If it is scaled down proportionally to fit a 6 feet wide area, what would be the length of the scaled-down garden?"
},
{
"from": "gpt",
"value": "To scale down the rectangular garden proportionally, we need to determi... |
Consider the sequence defined recursively by $u_1 = a > 0$ and
\[u_{n + 1} = -\frac{1}{u_n + 1}\]for $n \ge 1.$ Express $u_{16}$ in terms of $a.$ | Consider the sequence defined recursively by $u_1 = a > 0$ and
\[u_{n + 1} = -\frac{1}{u_n + 1}\]for $n \ge 1.$ Express $u_{16}$ in terms of $a.$ | To express \( u_{16} \) in terms of \( a \), we will first analyze the recurrence relation given by
\[
u_{n + 1} = -\frac{1}{u_n + 1}.
\]
Starting with the initial value \( u_1 = a \), we will compute several terms in the sequence to identify a pattern.
1. Compute \( u_2 \):
\[
u_2 = -\frac{1}{u_1 + 1} = -\fra... | 10,093 | [
{
"from": "human",
"value": "Consider the sequence defined recursively by $u_1 = a > 0$ and\n\\[u_{n + 1} = -\\frac{1}{u_n + 1}\\]for $n \\ge 1.$ Express $u_{16}$ in terms of $a.$"
},
{
"from": "gpt",
"value": "To express \\( u_{16} \\) in terms of \\( a \\), we will first analyze the recurrenc... |
For how many integer values of $n$ does the inequality $\left(\frac{n-1}{n+1}\right)^n<\left(\frac{n+1}{n-1}\right)^{n-1}$ hold? | For how many integer values of $n$ does the inequality $\left(\frac{n-1}{n+1}\right)^n<\left(\frac{n+1}{n-1}\right)^{n-1}$ hold? | To analyze the given inequality
\[
\left(\frac{n-1}{n+1}\right)^n < \left(\frac{n+1}{n-1}\right)^{n-1},
\]
we start by manipulating both sides. The expression on the right can be rewritten as follows:
\[
\left(\frac{n+1}{n-1}\right)^{n-1} = \left(\frac{n-1}{n+1}\right)^{-(n-1)}.
\]
Thus, we can rewrite the inequal... | 10,094 | [
{
"from": "human",
"value": "For how many integer values of $n$ does the inequality $\\left(\\frac{n-1}{n+1}\\right)^n<\\left(\\frac{n+1}{n-1}\\right)^{n-1}$ hold?"
},
{
"from": "gpt",
"value": "To analyze the given inequality \n\n\\[\n\\left(\\frac{n-1}{n+1}\\right)^n < \\left(\\frac{n+1}{n-1}\... |
On a particular day in Salt Lake, UT, the temperature was given by $-t^2 +12t+50$ where $t$ is the time in hours past noon. What is the largest $t$ value at which the temperature was exactly 77 degrees? | On a particular day in Salt Lake, UT, the temperature was given by $-t^2 +12t+50$ where $t$ is the time in hours past noon. What is the largest $t$ value at which the temperature was exactly 77 degrees? | To find the largest \( t \) value at which the temperature is exactly 77 degrees, we first set the equation for temperature equal to 77:
\[
-t^2 + 12t + 50 = 77
\]
Next, we rearrange this equation to set it to zero:
\[
-t^2 + 12t + 50 - 77 = 0
\]
\[
-t^2 + 12t - 27 = 0
\]
To simplify, we can multiply the entire equ... | 10,095 | [
{
"from": "human",
"value": "On a particular day in Salt Lake, UT, the temperature was given by $-t^2 +12t+50$ where $t$ is the time in hours past noon. What is the largest $t$ value at which the temperature was exactly 77 degrees?"
},
{
"from": "gpt",
"value": "To find the largest \\( t \\) val... |
If $x + \frac{1}{x} = \sqrt{2}$, then find $x^{24}$. | If $x + \frac{1}{x} = \sqrt{2}$, then find $x^{24}$. | Given the equation \( x + \frac{1}{x} = \sqrt{2} \), we denote \( a = x + \frac{1}{x} \). From the problem, we have \( a = \sqrt{2} \).
We can derive powers of \( x \) and \( \frac{1}{x} \) using the relationship of \( a \):
\[
x^2 + \frac{1}{x^2} = a^2 - 2
\]
Calculating \( a^2 \):
\[
a^2 = (\sqrt{2})^2 = 2
\]
Th... | 10,096 | [
{
"from": "human",
"value": "If $x + \\frac{1}{x} = \\sqrt{2}$, then find $x^{24}$."
},
{
"from": "gpt",
"value": "Given the equation \\( x + \\frac{1}{x} = \\sqrt{2} \\), we denote \\( a = x + \\frac{1}{x} \\). From the problem, we have \\( a = \\sqrt{2} \\).\n\nWe can derive powers of \\( x \\... |
If 7 pings are equivalent to 9 zings, and 3 zings can be exchanged for 5 oranges, how many pings are required to buy 20 oranges? | If 7 pings are equivalent to 9 zings, and 3 zings can be exchanged for 5 oranges, how many pings are required to buy 20 oranges? | To determine how many pings are required to buy 20 oranges, we will first need to establish the relationship between pings, zings, and oranges.
1. **Relationship between pings and zings**:
We know that:
\[
7 \text{ pings} = 9 \text{ zings}
\]
Thus, we can express zings in terms of pings:
\[
1 \tex... | 10,097 | [
{
"from": "human",
"value": "If 7 pings are equivalent to 9 zings, and 3 zings can be exchanged for 5 oranges, how many pings are required to buy 20 oranges?"
},
{
"from": "gpt",
"value": "To determine how many pings are required to buy 20 oranges, we will first need to establish the relationshi... |
What is the minimum number of hours that must pass before the time on a 12-hour clock is the same as it is now, if it is currently 3 o'clock? | What is the minimum number of hours that must pass before the time on a 12-hour clock is the same as it is now, if it is currently 3 o'clock? | To determine the minimum number of hours that must pass before the time on a 12-hour clock shows the same time as it is currently (3 o'clock), we consider the following:
On a 12-hour clock, the time resets every 12 hours. Therefore, if it is currently 3:00, we need to find the next instance when the clock again reads ... | 10,098 | [
{
"from": "human",
"value": "What is the minimum number of hours that must pass before the time on a 12-hour clock is the same as it is now, if it is currently 3 o'clock?"
},
{
"from": "gpt",
"value": "To determine the minimum number of hours that must pass before the time on a 12-hour clock sho... |
If $60^a = 3$ and $60^b = 5,$ then find $12^{(1 - a - b)/(2(1 - b))}.$ | If $60^a = 3$ and $60^b = 5,$ then find $12^{(1 - a - b)/(2(1 - b))}.$ | We start with the equations given:
\[
60^a = 3
\]
\[
60^b = 5
\]
To express \(a\) and \(b\) in terms of logarithms, we take the logarithm of both sides of each equation.
For \(60^a = 3\):
\[
a = \frac{\log 3}{\log 60}
\]
For \(60^b = 5\):
\[
b = \frac{\log 5}{\log 60}
\]
Next, we need to find the expression \(1 ... | 10,099 | [
{
"from": "human",
"value": "If $60^a = 3$ and $60^b = 5,$ then find $12^{(1 - a - b)/(2(1 - b))}.$"
},
{
"from": "gpt",
"value": "We start with the equations given:\n\n\\[\n60^a = 3\n\\]\n\\[\n60^b = 5\n\\]\n\nTo express \\(a\\) and \\(b\\) in terms of logarithms, we take the logarithm of both ... |
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