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... |
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