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 |
|---|---|---|---|---|
If the seven-digit number $854n526$ is divisible by $11$, what is $n$? | If the seven-digit number $854n526$ is divisible by $11$, what is $n$? | To determine the value of \( n \) that makes the seven-digit number \( 854n526 \) divisible by \( 11 \), we will apply the divisibility rule for \( 11 \). According to this rule, a number is divisible by \( 11 \) if the difference between the sum of its digits in odd positions and the sum of its digits in even position... | 10,300 | [
{
"from": "human",
"value": "If the seven-digit number $854n526$ is divisible by $11$, what is $n$?"
},
{
"from": "gpt",
"value": "To determine the value of \\( n \\) that makes the seven-digit number \\( 854n526 \\) divisible by \\( 11 \\), we will apply the divisibility rule for \\( 11 \\). Ac... |
In a music festival, there are 4 bands, each consisting of 3 members: a singer, a guitarist, and a drummer. The festival organizer wants to form a supergroup by selecting one member from each band. However, there are some compatibility issues between the members: the singer from Band A cannot work with the drummer from... | In a music festival, there are 4 bands, each consisting of 3 members: a singer, a guitarist, and a drummer. The festival organizer wants to form a supergroup by selecting one member from each band. However, there are some compatibility issues between the members: the singer from Band A cannot work with the drummer from... | To determine the number of ways to form a supergroup consisting of one member from each of the four bands while considering the compatibility issues, we can proceed as follows:
### Step 1: Calculate the total combinations without restrictions.
Each band consists of 3 members: a singer, a guitarist, and a drummer. The... | 10,301 | [
{
"from": "human",
"value": "In a music festival, there are 4 bands, each consisting of 3 members: a singer, a guitarist, and a drummer. The festival organizer wants to form a supergroup by selecting one member from each band. However, there are some compatibility issues between the members: the singer from... |
Let $\tau : \mathbb{N} \to \mathbb{N}$ be a bijection satisfying
$$\text{(i)}\,\,\tau(a+b)=\tau(a)+\tau(b)\quad\forall a,b \in \mathbb{N},$$
$$\text{(ii)}\,\,\tau(a\cdot b)=\tau(a)\cdot \tau(b)\quad\forall a,b \in \mathbb{N},$$
and
$$\text{(iii)}\,\, \gcd (\tau (a),\tau (b))=1 \Longrightarrow \gcd (a,b)=1\quad\forall ... | Let $\tau : \mathbb{N} \to \mathbb{N}$ be a bijection satisfying
$$\text{(i)}\,\,\tau(a+b)=\tau(a)+\tau(b)\quad\forall a,b \in \mathbb{N},$$
$$\text{(ii)}\,\,\tau(a\cdot b)=\tau(a)\cdot \tau(b)\quad\forall a,b \in \mathbb{N},$$
and
$$\text{(iii)}\,\, \gcd (\tau (a),\tau (b))=1 \Longrightarrow \gcd (a,b)=1\quad\forall ... | Let \( \tau : \mathbb{N} \to \mathbb{N} \) be a bijection that satisfies the properties:
1. \( \tau(a+b) = \tau(a) + \tau(b) \) for all \( a,b \in \mathbb{N} \).
2. \( \tau(a \cdot b) = \tau(a) \cdot \tau(b) \) for all \( a,b \in \mathbb{N} \).
3. If \( \gcd(\tau(a), \tau(b)) = 1 \), then \( \gcd(a,b) = 1 \) for all \... | 10,302 | [
{
"from": "human",
"value": "Let $\\tau : \\mathbb{N} \\to \\mathbb{N}$ be a bijection satisfying\n$$\\text{(i)}\\,\\,\\tau(a+b)=\\tau(a)+\\tau(b)\\quad\\forall a,b \\in \\mathbb{N},$$\n$$\\text{(ii)}\\,\\,\\tau(a\\cdot b)=\\tau(a)\\cdot \\tau(b)\\quad\\forall a,b \\in \\mathbb{N},$$\nand \n$$\\text{(iii)}\... |
Let $n$ be a positive number such that when $179$ is divided by $n$, the remainder is $7.$ Compute the sum of all possible two-digit values of $n$. | Let $n$ be a positive number such that when $179$ is divided by $n$, the remainder is $7.$ Compute the sum of all possible two-digit values of $n$. | To find values of \( n \) such that when \( 179 \) is divided by \( n \), the remainder is \( 7 \), we start with the condition given:
\[
179 \equiv 7 \pmod{n}.
\]
This can be rewritten as:
\[
179 - 7 = 172 \equiv 0 \pmod{n},
\]
which indicates that \( n \) must be a divisor of \( 172 \). Furthermore, since the rem... | 10,303 | [
{
"from": "human",
"value": "Let $n$ be a positive number such that when $179$ is divided by $n$, the remainder is $7.$ Compute the sum of all possible two-digit values of $n$."
},
{
"from": "gpt",
"value": "To find values of \\( n \\) such that when \\( 179 \\) is divided by \\( n \\), the rema... |
Determine the value of $a+b$ where the value of $\log_{10}{52641}$ falls between two consecutive integers $a$ and $b$. | Determine the value of $a+b$ where the value of $\log_{10}{52641}$ falls between two consecutive integers $a$ and $b$. | To determine the consecutive integers \( a \) and \( b \) such that
\[
a < \log_{10}{52641} < b,
\]
we start by estimating \( 52641 \) in terms of powers of \( 10 \).
Calculating powers of \( 10 \):
\[
10^4 = 10000,
\]
\[
10^5 = 100000.
\]
Since \( 52641 \) lies between \( 10^4 \) and \( 10^5 \), we conclude that... | 10,304 | [
{
"from": "human",
"value": "Determine the value of $a+b$ where the value of $\\log_{10}{52641}$ falls between two consecutive integers $a$ and $b$."
},
{
"from": "gpt",
"value": "To determine the consecutive integers \\( a \\) and \\( b \\) such that \n\n\\[\na < \\log_{10}{52641} < b,\n\\]\n\n... |
Suppose that \( \gamma \) is directly proportional to \( \delta \). If \( \gamma = 2 \) when \( \delta = 5 \), find \( \gamma \) when \( \delta = 3 \). Express your answer as a fraction. | Suppose that \( \gamma \) is directly proportional to \( \delta \). If \( \gamma = 2 \) when \( \delta = 5 \), find \( \gamma \) when \( \delta = 3 \). Express your answer as a fraction. | Since \( \gamma \) is directly proportional to \( \delta \), we can express this relationship as:
\[
\gamma = k \delta
\]
for some constant \( k \). To find the value of \( k \), we can use the values given: \( \gamma = 2 \) when \( \delta = 5 \).
Substituting these values into the equation gives:
\[
2 = k \cdot 5
... | 10,305 | [
{
"from": "human",
"value": "Suppose that \\( \\gamma \\) is directly proportional to \\( \\delta \\). If \\( \\gamma = 2 \\) when \\( \\delta = 5 \\), find \\( \\gamma \\) when \\( \\delta = 3 \\). Express your answer as a fraction."
},
{
"from": "gpt",
"value": "Since \\( \\gamma \\) is direct... |
The hypotenuse of a particular right triangle is 5 inches. The radius of a particular sphere is also 5 inches. By how many cubic inches is the volume of the sphere greater than the volume of the right triangle? Express your answer as a decimal to the nearest tenth. | The hypotenuse of a particular right triangle is 5 inches. The radius of a particular sphere is also 5 inches. By how many cubic inches is the volume of the sphere greater than the volume of the right triangle? Express your answer as a decimal to the nearest tenth. | To find the difference in volumes between the sphere and the right triangle, we first need to calculate the volumes of both shapes separately.
1. **Volume of the Sphere**:
The formula for the volume \( V \) of a sphere with radius \( r \) is given by:
\[
V = \frac{4}{3} \pi r^3
\]
For a sphere with a ra... | 10,306 | [
{
"from": "human",
"value": "The hypotenuse of a particular right triangle is 5 inches. The radius of a particular sphere is also 5 inches. By how many cubic inches is the volume of the sphere greater than the volume of the right triangle? Express your answer as a decimal to the nearest tenth."
},
{
... |
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint. | To find the coordinates of the other endpoint of the segment, we can use the midpoint formula. The midpoint \((x_m, y_m)\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
x_m = \frac{x_1 + x_2}{2}, \quad y_m = \frac{y_1 + y_2}{2}
\]
We know one endpoint \((x_1, y_1) = (8, 0)\) and t... | 10,307 | [
{
"from": "human",
"value": "The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint."
},
{
"from": "gpt",
"value": "To find the coordinates of the other endpoint of the segment, we can use the midpoi... |
Find the number of terms in the sequence of positive integers formed by starting with 3 and repeatedly multiplying by 2 and adding 1, ending when the term exceeds 1000. | Find the number of terms in the sequence of positive integers formed by starting with 3 and repeatedly multiplying by 2 and adding 1, ending when the term exceeds 1000. | We start with the first term \( a_1 = 3 \). The sequence is generated by the following recurrence relation:
\[
a_n = 2a_{n-1} + 1
\]
Let's calculate the terms of the sequence step by step until we exceed 1000:
1. For \( n = 1 \):
\[
a_1 = 3
\]
2. For \( n = 2 \):
\[
a_2 = 2a_1 + 1 = 2 \times 3 + 1... | 10,308 | [
{
"from": "human",
"value": "Find the number of terms in the sequence of positive integers formed by starting with 3 and repeatedly multiplying by 2 and adding 1, ending when the term exceeds 1000."
},
{
"from": "gpt",
"value": "We start with the first term \\( a_1 = 3 \\). The sequence is gener... |
A certain lottery has 15 balls, numbered from 1 to 15. Each ticket randomly selects 6 distinct balls. What is the expected number of even-numbered balls selected by a randomly chosen ticket? Express your answer as a common fraction. | A certain lottery has 15 balls, numbered from 1 to 15. Each ticket randomly selects 6 distinct balls. What is the expected number of even-numbered balls selected by a randomly chosen ticket? Express your answer as a common fraction. | To find the expected number of even-numbered balls selected by a randomly chosen lottery ticket, we start by determining how many even-numbered balls are available from the 15 balls.
The even numbers from 1 to 15 are:
\[ 2, 4, 6, 8, 10, 12, 14 \]
Thus, there are a total of 7 even-numbered balls.
The total number of b... | 10,309 | [
{
"from": "human",
"value": "A certain lottery has 15 balls, numbered from 1 to 15. Each ticket randomly selects 6 distinct balls. What is the expected number of even-numbered balls selected by a randomly chosen ticket? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "To ... |
Line segment $\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 10:3.$ Then
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B}\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$
[asy]
unitsize(1 cm);
pair A, B, P;
A = (0,0);
B = (5,1);
P = interp(A,B,10/7);
draw(A--P);
d... | Line segment $\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 10:3.$ Then
\[\overrightarrow{P} = t \overrightarrow{A} + u \overrightarrow{B}\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$
[asy]
unitsize(1 cm);
pair A, B, P;
A = (0,0);
B = (5,1);
P = interp(A,B,10/7);
draw(A--P);
d... | To solve for the position of point \( P \) in relation to points \( A \) and \( B \), we will first express the condition given by the ratio \( AP:PB = 10:3 \).
Let the distance between points \( A \) and \( B \) be represented by \( d \). According to the problem, the segment \( AP \) can be represented as \( 10x \) ... | 10,310 | [
{
"from": "human",
"value": "Line segment $\\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 10:3.$ Then\n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrightarrow{B}\\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$\n\n[asy]\nunitsize(1 cm);\n\npair A, B, P;\n\nA = ... |
In the diagram below, we have $\sin \angle RPQ = \frac{7}{25}$. What is $\sin \angle RPS$?
[asy]
pair R,P,Q,SS;
SS = (-2,0);
P = (0,0);
Q = (2,0);
R = rotate(aSin(7/25))*(1.5,0);
dot("$S$",SS,S);
dot("$Q$",Q,S);
dot("$R$",R,N);
dot("$P$",P,S);
draw(Q--SS);
draw(P--R);
[/asy] | In the diagram below, we have $\sin \angle RPQ = \frac{7}{25}$. What is $\sin \angle RPS$?
[asy]
pair R,P,Q,SS;
SS = (-2,0);
P = (0,0);
Q = (2,0);
R = rotate(aSin(7/25))*(1.5,0);
dot("$S$",SS,S);
dot("$Q$",Q,S);
dot("$R$",R,N);
dot("$P$",P,S);
draw(Q--SS);
draw(P--R);
[/asy] | To solve for \(\sin \angle RPS\), we observe the relationships involving the angles \(\angle RPQ\) and \(\angle RPS\).
Given that:
\[
\sin \angle RPQ = \frac{7}{25}
\]
we can use the sine function to find \(\cos \angle RPQ\). From the identity \(\sin^2 x + \cos^2 x = 1\), we calculate:
\[
\cos^2 \angle RPQ = 1 - \s... | 10,311 | [
{
"from": "human",
"value": "In the diagram below, we have $\\sin \\angle RPQ = \\frac{7}{25}$. What is $\\sin \\angle RPS$?\n\n[asy]\npair R,P,Q,SS;\nSS = (-2,0);\nP = (0,0);\nQ = (2,0);\nR = rotate(aSin(7/25))*(1.5,0);\ndot(\"$S$\",SS,S);\ndot(\"$Q$\",Q,S);\ndot(\"$R$\",R,N);\ndot(\"$P$\",P,S);\ndraw(Q--... |
If $\mathbf{u} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 5 \\ -1 \end{pmatrix}$, find the value of $a\mathbf{u} + b\mathbf{v}$, where $a$ and $b$ are the roots of the equation $x^2 + 3x - 2 = 0$. | If $\mathbf{u} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 5 \\ -1 \end{pmatrix}$, find the value of $a\mathbf{u} + b\mathbf{v}$, where $a$ and $b$ are the roots of the equation $x^2 + 3x - 2 = 0$. | To solve the problem, we need to first find the values of \( a \) and \( b \) which are the roots of the quadratic equation \( x^2 + 3x - 2 = 0 \). We can use the quadratic formula for this purpose:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( x^2 + 3x - 2 = 0 \), we identify \( a = 1 \), \( b = 3... | 10,312 | [
{
"from": "human",
"value": "If $\\mathbf{u} = \\begin{pmatrix} 1 \\\\ -2 \\\\ 3 \\end{pmatrix}$ and $\\mathbf{v} = \\begin{pmatrix} 4 \\\\ 5 \\\\ -1 \\end{pmatrix}$, find the value of $a\\mathbf{u} + b\\mathbf{v}$, where $a$ and $b$ are the roots of the equation $x^2 + 3x - 2 = 0$."
},
{
"from": "g... |
What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$ | What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$ | To determine what percent of the positive integers less than or equal to \( 100 \) have no remainder when divided by \( 5 \), we first need to identify all the integers in this range that are divisible by \( 5 \).
The integers that are divisible by \( 5 \) can be represented as:
\[
5, 10, 15, 20, 25, 30, 35, 40, 45, ... | 10,313 | [
{
"from": "human",
"value": "What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$"
},
{
"from": "gpt",
"value": "To determine what percent of the positive integers less than or equal to \\( 100 \\) have no remainder when divided by \\( 5 \\), ... |
Round 0.0006438107 to four decimal places. | Round 0.0006438107 to four decimal places. | To round the number 0.0006438107 to four decimal places, we look at the fifth decimal place to determine how to round the fourth decimal place.
The number is:
0.0006438107
The fourth decimal place is 3 (the number is 0.0006**4**3...). The fifth decimal place is 8. Since the fifth decimal place (8) is greater than 5, ... | 10,314 | [
{
"from": "human",
"value": "Round 0.0006438107 to four decimal places."
},
{
"from": "gpt",
"value": "To round the number 0.0006438107 to four decimal places, we look at the fifth decimal place to determine how to round the fourth decimal place.\n\nThe number is:\n0.0006438107\n\nThe fourth dec... |
If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 2,$ then find
\[\begin{vmatrix} a & 2b \\ c & 2d \end{vmatrix}.\] | If $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = 2,$ then find
\[\begin{vmatrix} a & 2b \\ c & 2d \end{vmatrix}.\] | To find the determinant
\[
\begin{vmatrix} a & 2b \\ c & 2d \end{vmatrix},
\]
we can use the properties of determinants. Specifically, we can factor constants out of the rows of the determinant.
Starting with the determinant,
\[
\begin{vmatrix} a & 2b \\ c & 2d \end{vmatrix},
\]
we can factor out 2 from the secon... | 10,315 | [
{
"from": "human",
"value": "If $\\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} = 2,$ then find\n\\[\\begin{vmatrix} a & 2b \\\\ c & 2d \\end{vmatrix}.\\]"
},
{
"from": "gpt",
"value": "To find the determinant \n\n\\[\n\\begin{vmatrix} a & 2b \\\\ c & 2d \\end{vmatrix},\n\\]\n\nwe can use the ... |
Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. | To solve the equation \( |4x| + 3 = 35 \), we first isolate the absolute value term:
\[
|4x| + 3 = 35
\]
Subtracting 3 from both sides gives us:
\[
|4x| = 35 - 3
\]
\[
|4x| = 32
\]
Next, we recognize that the equation \( |4x| = 32 \) can be broken down into two separate cases:
1. \( 4x = 32 \)
2. \( 4x = -32 \)
We... | 10,316 | [
{
"from": "human",
"value": "Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$."
},
{
"from": "gpt",
"value": "To solve the equation \\( |4x| + 3 = 35 \\), we first isolate the absolute value term:\n\n\\[\n|4x| + 3 = 35\n\\]\nSubtracting 3 from both sides gives us:\n\n\... |
Find the value of $b$ if the lines $x+y=5$ and $bx - y=1$ are perpendicular. | Find the value of $b$ if the lines $x+y=5$ and $bx - y=1$ are perpendicular. | To determine the value of \( b \) such that the lines \( x + y = 5 \) and \( bx - y = 1 \) are perpendicular, we first find the slopes of the two lines.
1. **Finding the slope of the first line**: The equation of the line is given by
\[
x + y = 5.
\]
We can rearrange this to slope-intercept form \( y = m... | 10,317 | [
{
"from": "human",
"value": "Find the value of $b$ if the lines $x+y=5$ and $bx - y=1$ are perpendicular."
},
{
"from": "gpt",
"value": "To determine the value of \\( b \\) such that the lines \\( x + y = 5 \\) and \\( bx - y = 1 \\) are perpendicular, we first find the slopes of the two lines.\... |
The number 3600 can be expressed as a sum of consecutive positive integers. What is the minimum number of consecutive positive integers that sum to 3600, if the sum starts with the number 12? | The number 3600 can be expressed as a sum of consecutive positive integers. What is the minimum number of consecutive positive integers that sum to 3600, if the sum starts with the number 12? | To express the number \( 3600 \) as a sum of consecutive positive integers starting from \( 12 \), we can denote the first \( n \) consecutive integers starting from \( 12 \) as follows:
\[
12, 13, 14, \ldots, (12 + (n-1)) = (11 + n)
\]
The sum of these \( n \) integers can be computed using the formula for the sum o... | 10,318 | [
{
"from": "human",
"value": "The number 3600 can be expressed as a sum of consecutive positive integers. What is the minimum number of consecutive positive integers that sum to 3600, if the sum starts with the number 12?"
},
{
"from": "gpt",
"value": "To express the number \\( 3600 \\) as a sum ... |
When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$? | When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \cdot b \cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such that $p_1 + p_2 + p_3 = a+b+c$? | To solve the problem, we follow these steps:
1. **Determine the prime factorization of 5002.**
2. **Identify distinct primes that equal \(a\), \(b\), and \(c\).**
3. **Calculate the sum \(a + b + c\).**
4. **Find how many other positive integers can be expressed as products of three distinct primes that sum up to \(a ... | 10,319 | [
{
"from": "human",
"value": "When the digits in the number $2005$ are reversed we obtain the number $5002,$ and $5002 = a \\cdot b \\cdot c$, such that $a$, $b$ and $c$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $p_1$, $p_2$ and $p_3$ such t... |
At the start of a basketball game, the ball is tossed upwards at an angle. If the ball reaches a height of 15 feet above the court and lands on the court 10 feet away from the point of release, what is the angle of elevation, to the nearest degree, of the initial trajectory of the ball?
Express your answer as a decima... | At the start of a basketball game, the ball is tossed upwards at an angle. If the ball reaches a height of 15 feet above the court and lands on the court 10 feet away from the point of release, what is the angle of elevation, to the nearest degree, of the initial trajectory of the ball?
Express your answer as a decima... | To find the angle of elevation of the initial trajectory of the basketball, we can use the properties of right triangles and trigonometric functions.
We know two key pieces of information:
1. The height of the ball when it reaches its peak is \( h = 15 \) feet.
2. The horizontal distance covered when it lands is \( d ... | 10,320 | [
{
"from": "human",
"value": "At the start of a basketball game, the ball is tossed upwards at an angle. If the ball reaches a height of 15 feet above the court and lands on the court 10 feet away from the point of release, what is the angle of elevation, to the nearest degree, of the initial trajectory of t... |
A point (x,y) is chosen at random inside the triangle with vertices (0,0), (0,1), and (1,0). Determine the probability that $x+y<\frac{1}{2}$ | A point (x,y) is chosen at random inside the triangle with vertices (0,0), (0,1), and (1,0). Determine the probability that $x+y<\frac{1}{2}$ | To find the probability that a randomly chosen point \((x,y)\) inside the triangle with vertices \((0,0)\), \((0,1)\), and \((1,0)\) satisfies the condition \(x + y < \frac{1}{2}\), we need to do the following steps:
1. **Identify the area of the triangle**:
The triangle is formed with the vertices \((0,0)\), \((0,... | 10,321 | [
{
"from": "human",
"value": "A point (x,y) is chosen at random inside the triangle with vertices (0,0), (0,1), and (1,0). Determine the probability that $x+y<\\frac{1}{2}$"
},
{
"from": "gpt",
"value": "To find the probability that a randomly chosen point \\((x,y)\\) inside the triangle with ve... |
Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it to Alice, he keeps it. Alice starts with th... | Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it to Alice, he keeps it. Alice starts with th... | To find the probability that Alice has the ball again after two turns, we can analyze the situation based on the transitions of the ball between Alice and Bob.
1. **Initial state**: Alice has the ball.
2. **Turn 1**:
- There are two possible actions:
1. Alice keeps the ball (with probability \( \frac{1}{2} \)... | 10,322 | [
{
"from": "human",
"value": "Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it to... |
Let $f(x) = x^4 + bx^3 + cx^2 + dx + e$ be a polynomial with integer coefficients. Define a function $g(n)$ as the number of integer roots of $f(x)$, counting multiplicity, that are greater than or equal to $n$. For example, if $f(x) = (x-2)^2(x-5)(x+3)$, then $g(4) = 2$ because $f(x)$ has two integer roots (5 and 2, w... | Let $f(x) = x^4 + bx^3 + cx^2 + dx + e$ be a polynomial with integer coefficients. Define a function $g(n)$ as the number of integer roots of $f(x)$, counting multiplicity, that are greater than or equal to $n$. For example, if $f(x) = (x-2)^2(x-5)(x+3)$, then $g(4) = 2$ because $f(x)$ has two integer roots (5 and 2, w... | To solve for the smallest possible value of \( e \) in the polynomial \( f(x) = x^4 + bx^3 + cx^2 + dx + e \) under the conditions \( g(0) = 2 \), \( g(10) = 1 \), and \( g(100) = 0 \), we first interpret these conditions in terms of the roots of the polynomial.
1. **Understanding \( g(n) \)**:
- \( g(0) = 2 \): Th... | 10,323 | [
{
"from": "human",
"value": "Let $f(x) = x^4 + bx^3 + cx^2 + dx + e$ be a polynomial with integer coefficients. Define a function $g(n)$ as the number of integer roots of $f(x)$, counting multiplicity, that are greater than or equal to $n$. For example, if $f(x) = (x-2)^2(x-5)(x+3)$, then $g(4) = 2$ because... |
Find the missing term of an arithmetic sequence whose sum is 39 and whose terms are all positive integers greater than 5. The last term of the sequence is 14. | Find the missing term of an arithmetic sequence whose sum is 39 and whose terms are all positive integers greater than 5. The last term of the sequence is 14. | Let the arithmetic sequence be defined by the first term \( a \) and the common difference \( d \). The terms of the sequence can be represented as:
\[
a, a + d, a + 2d, \ldots, a + (n-1)d
\]
where \( n \) is the number of terms in the sequence. The last term of the sequence is given as 14, which leads us to the equa... | 10,324 | [
{
"from": "human",
"value": "Find the missing term of an arithmetic sequence whose sum is 39 and whose terms are all positive integers greater than 5. The last term of the sequence is 14."
},
{
"from": "gpt",
"value": "Let the arithmetic sequence be defined by the first term \\( a \\) and the co... |
In right $\Delta ABC$, $\angle CAB$ is a right angle. Point $M$ is the midpoint of $\overline{BC}$. What is the number of centimeters in the length of median $\overline{AM}$? Express your answer as a decimal to the nearest tenth. [asy] pair A,B,C,M;
A = (0,0); B = (4,0); C = (0,3); M = (B+C)/2;
draw(M--A--B--C--A);
lab... | In right $\Delta ABC$, $\angle CAB$ is a right angle. Point $M$ is the midpoint of $\overline{BC}$. What is the number of centimeters in the length of median $\overline{AM}$? Express your answer as a decimal to the nearest tenth. [asy] pair A,B,C,M;
A = (0,0); B = (4,0); C = (0,3); M = (B+C)/2;
draw(M--A--B--C--A);
lab... | Given a right triangle \( \Delta ABC \) with \( \angle CAB \) as the right angle, we have the lengths \( AB = 4 \) cm and \( AC = 3 \) cm. We will find the length of median \( \overline{AM} \), where \( M \) is the midpoint of \( \overline{BC} \).
1. **Determine the coordinates of points \( A \), \( B \), and \( C \)*... | 10,325 | [
{
"from": "human",
"value": "In right $\\Delta ABC$, $\\angle CAB$ is a right angle. Point $M$ is the midpoint of $\\overline{BC}$. What is the number of centimeters in the length of median $\\overline{AM}$? Express your answer as a decimal to the nearest tenth. [asy] pair A,B,C,M;\nA = (0,0); B = (4,0); C ... |
Let $\triangle DEF$ be an equilateral triangle with side length 20, and let $G$ be its incenter. If the length of $DG$ is 12, what is the length of the inradius of $\triangle DEF$? | Let $\triangle DEF$ be an equilateral triangle with side length 20, and let $G$ be its incenter. If the length of $DG$ is 12, what is the length of the inradius of $\triangle DEF$? | To solve for the inradius \( r \) of triangle \( DEF \), we start by recalling some properties of equilateral triangles. In an equilateral triangle, the incenter is also the centroid and circumcenter. This fact simplifies our calculations.
### Step 1: Basic Properties of Equilateral Triangle
For an equilateral triang... | 10,326 | [
{
"from": "human",
"value": "Let $\\triangle DEF$ be an equilateral triangle with side length 20, and let $G$ be its incenter. If the length of $DG$ is 12, what is the length of the inradius of $\\triangle DEF$?"
},
{
"from": "gpt",
"value": "To solve for the inradius \\( r \\) of triangle \\( D... |
What is $\log_{7}{2400}$ rounded to the nearest integer? | What is $\log_{7}{2400}$ rounded to the nearest integer? | To find \( \log_{7}{2400} \) and round it to the nearest integer, we can use the change of base formula for logarithms:
\[
\log_{7}{2400} = \frac{\log_{10}{2400}}{\log_{10}{7}}
\]
We'll compute \( \log_{10}{2400} \) first. Since \( 2400 = 24 \times 100 \), we can express this as:
\[
\log_{10}{2400} = \log_{10}{(24 \... | 10,327 | [
{
"from": "human",
"value": "What is $\\log_{7}{2400}$ rounded to the nearest integer?"
},
{
"from": "gpt",
"value": "To find \\( \\log_{7}{2400} \\) and round it to the nearest integer, we can use the change of base formula for logarithms:\n\n\\[\n\\log_{7}{2400} = \\frac{\\log_{10}{2400}}{\\lo... |
Determine the smallest positive integer $n$ such that the decimal representation of $3^n$ has 3 as its units digit. | Determine the smallest positive integer $n$ such that the decimal representation of $3^n$ has 3 as its units digit. | To determine the smallest positive integer \( n \) such that the decimal representation of \( 3^n \) has 3 as its units digit, we can analyze the pattern of the units digits of powers of 3.
Calculating the first few powers of \( 3 \):
- \( 3^1 = 3 \) (units digit is 3)
- \( 3^2 = 9 \) (units digit is 9)
- \( 3^3 = 27... | 10,328 | [
{
"from": "human",
"value": "Determine the smallest positive integer $n$ such that the decimal representation of $3^n$ has 3 as its units digit."
},
{
"from": "gpt",
"value": "To determine the smallest positive integer \\( n \\) such that the decimal representation of \\( 3^n \\) has 3 as its un... |
For a positive integer $n$ and nonzero digits $d$, $e$, and $f$, let $D_n$ be the $n$-digit integer each of whose digits is equal to $d$; let $E_n$ be the $n$-digit integer each of whose digits is equal to $e$, and let $F_n$ be the $3n$-digit (not $n$-digit) integer each of whose digits is equal to $f$. What is the gre... | For a positive integer $n$ and nonzero digits $d$, $e$, and $f$, let $D_n$ be the $n$-digit integer each of whose digits is equal to $d$; let $E_n$ be the $n$-digit integer each of whose digits is equal to $e$, and let $F_n$ be the $3n$-digit (not $n$-digit) integer each of whose digits is equal to $f$. What is the gre... | To analyze the equation \( F_n - E_n = D_n^3 \) where:
- \( D_n = d \cdots d \) (n digits, each equals \( d \)),
- \( E_n = e \cdots e \) (n digits, each equals \( e \)),
- \( F_n = f \cdots f \) (3n digits, each equals \( f \)),
we must first express these integers in a mathematical form.
1. **Calculate \( D_n \):*... | 10,329 | [
{
"from": "human",
"value": "For a positive integer $n$ and nonzero digits $d$, $e$, and $f$, let $D_n$ be the $n$-digit integer each of whose digits is equal to $d$; let $E_n$ be the $n$-digit integer each of whose digits is equal to $e$, and let $F_n$ be the $3n$-digit (not $n$-digit) integer each of whos... |
It takes Rachel 15 minutes to complete the first puzzle in a book, and each subsequent puzzle takes 2 minutes longer than the previous one. How many total minutes does it take to complete the first four puzzles in the book? | It takes Rachel 15 minutes to complete the first puzzle in a book, and each subsequent puzzle takes 2 minutes longer than the previous one. How many total minutes does it take to complete the first four puzzles in the book? | To determine the total time it takes Rachel to complete the first four puzzles, we'll calculate the time for each individual puzzle based on her starting time and the increment for each subsequent puzzle.
1. **Time for the first puzzle**:
- Rachel takes 15 minutes.
2. **Time for the second puzzle**:
- She ... | 10,330 | [
{
"from": "human",
"value": "It takes Rachel 15 minutes to complete the first puzzle in a book, and each subsequent puzzle takes 2 minutes longer than the previous one. How many total minutes does it take to complete the first four puzzles in the book?"
},
{
"from": "gpt",
"value": "To determine... |
A shirt was initially priced at $x$. During a sale, the store offered a discount of $15\%$. However, due to an error, the discount was mistakenly applied twice, resulting in a final price of $\$20$. What is the value of $x$? | A shirt was initially priced at $x$. During a sale, the store offered a discount of $15\%$. However, due to an error, the discount was mistakenly applied twice, resulting in a final price of $\$20$. What is the value of $x$? | To determine the initial price \( x \) of the shirt when a 15% discount is mistakenly applied twice leading to a final price of \$20, we can set up our calculations as follows:
1. **Calculate the price after a single 15% discount:**
The first discount of 15% reduces the price to 85% of its original price. The price... | 10,331 | [
{
"from": "human",
"value": "A shirt was initially priced at $x$. During a sale, the store offered a discount of $15\\%$. However, due to an error, the discount was mistakenly applied twice, resulting in a final price of $\\$20$. What is the value of $x$?"
},
{
"from": "gpt",
"value": "To determ... |
Find the number of positive integers less than 100 that have a remainder of 2 when divided by 7. | Find the number of positive integers less than 100 that have a remainder of 2 when divided by 7. | We are tasked with finding the positive integers less than 100 that leave a remainder of 2 when divided by 7.
A number \( n \) can be expressed in terms of its remainder when divided by 7 as:
\[
n \equiv 2 \mod 7
\]
This implies that \( n \) can be written in the form:
\[
n = 7k + 2
\]
for some integer \( k \). Ou... | 10,332 | [
{
"from": "human",
"value": "Find the number of positive integers less than 100 that have a remainder of 2 when divided by 7."
},
{
"from": "gpt",
"value": "We are tasked with finding the positive integers less than 100 that leave a remainder of 2 when divided by 7.\n\nA number \\( n \\) can be ... |
Find the value of $\sin 75^\circ$. | Find the value of $\sin 75^\circ$. | To find the value of \( \sin 75^\circ \), we can use the sine addition formula. Specifically, we can express \( 75^\circ \) as the sum of \( 45^\circ \) and \( 30^\circ \):
\[
75^\circ = 45^\circ + 30^\circ
\]
The sine addition formula states:
\[
\sin(a + b) = \sin a \cos b + \cos a \sin b
\]
Applying this formula,... | 10,333 | [
{
"from": "human",
"value": "Find the value of $\\sin 75^\\circ$."
},
{
"from": "gpt",
"value": "To find the value of \\( \\sin 75^\\circ \\), we can use the sine addition formula. Specifically, we can express \\( 75^\\circ \\) as the sum of \\( 45^\\circ \\) and \\( 30^\\circ \\):\n\n\\[\n75^\\... |
Let $N$ be the sum of the divisors of $200$. What is the largest prime factor of $N$? | Let $N$ be the sum of the divisors of $200$. What is the largest prime factor of $N$? | To find the sum of the divisors of \(200\), we first determine its prime factorization:
\[
200 = 2^3 \times 5^2
\]
The formula for the sum of the divisors \(d(n)\) of a number \(n\) given its prime factorization \(n = p_1^{k_1} p_2^{k_2} \cdots p_m^{k_m}\) is:
\[
\sigma(n) = (1 + p_1 + p_1^2 + \cdots + p_1^{k_1})(1 ... | 10,334 | [
{
"from": "human",
"value": "Let $N$ be the sum of the divisors of $200$. What is the largest prime factor of $N$?"
},
{
"from": "gpt",
"value": "To find the sum of the divisors of \\(200\\), we first determine its prime factorization:\n\n\\[\n200 = 2^3 \\times 5^2\n\\]\n\nThe formula for the su... |
What is the remainder when $2007$ is divided by $81$?
Another problem:
Find the greatest common divisor of $2007$ and $81$. | What is the remainder when $2007$ is divided by $81$?
Another problem:
Find the greatest common divisor of $2007$ and $81$. | To find the remainder when \(2007\) is divided by \(81\), we can perform the division directly:
1. First, we calculate how many times \(81\) goes into \(2007\):
\[
\frac{2007}{81} \approx 24.74
\]
This means \(81\) fits into \(2007\) a total of \(24\) times (since we need the integer part).
2. Now we mult... | 10,335 | [
{
"from": "human",
"value": "What is the remainder when $2007$ is divided by $81$?\n\nAnother problem:\n\nFind the greatest common divisor of $2007$ and $81$."
},
{
"from": "gpt",
"value": "To find the remainder when \\(2007\\) is divided by \\(81\\), we can perform the division directly:\n\n1. ... |
Compute $\tan (-3645^\circ)$. | Compute $\tan (-3645^\circ)$. | To compute \(\tan (-3645^\circ)\), we start by finding an equivalent angle that lies within the standard range for tangent functions, which is typically between \(0^\circ\) and \(360^\circ\).
First, we calculate the positive equivalent angle by adding \(360^\circ\) repeatedly to \(-3645^\circ\):
\[
-3645^\circ + 360^... | 10,336 | [
{
"from": "human",
"value": "Compute $\\tan (-3645^\\circ)$."
},
{
"from": "gpt",
"value": "To compute \\(\\tan (-3645^\\circ)\\), we start by finding an equivalent angle that lies within the standard range for tangent functions, which is typically between \\(0^\\circ\\) and \\(360^\\circ\\).\n\... |
Let $a\circ b = \sqrt{a^2+b^2}$. What is the value of $(3\circ5)\circ(2\circ7)$? | Let $a\circ b = \sqrt{a^2+b^2}$. What is the value of $(3\circ5)\circ(2\circ7)$? | To find the value of \((3 \circ 5) \circ (2 \circ 7)\), we start by calculating each operation one by one using the definition of \(a \circ b = \sqrt{a^2 + b^2}\).
First, we compute \(3 \circ 5\):
\[
3 \circ 5 = \sqrt{3^2 + 5^2}
\]
Calculating \(3^2\) and \(5^2\):
\[
3^2 = 9 \quad \text{and} \quad 5^2 = 25
\]
Now s... | 10,337 | [
{
"from": "human",
"value": "Let $a\\circ b = \\sqrt{a^2+b^2}$. What is the value of $(3\\circ5)\\circ(2\\circ7)$?"
},
{
"from": "gpt",
"value": "To find the value of \\((3 \\circ 5) \\circ (2 \\circ 7)\\), we start by calculating each operation one by one using the definition of \\(a \\circ b =... |
Let $m$ and $n$ be the two distinct solutions to the equation $$(x-2)(x+4) = 9x - 36.$$ If $m > n$, what is the value of $m - n$? | Let $m$ and $n$ be the two distinct solutions to the equation $$(x-2)(x+4) = 9x - 36.$$ If $m > n$, what is the value of $m - n$? | To find the distinct solutions \( m \) and \( n \) of the equation \( (x-2)(x+4) = 9x - 36 \), we start by expanding both sides.
First, we expand the left-hand side:
\[
(x - 2)(x + 4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8.
\]
Next, we simplify the right-hand side:
\[
9x - 36 = 9(x - 4).
\]
Now we set the two sides eq... | 10,338 | [
{
"from": "human",
"value": "Let $m$ and $n$ be the two distinct solutions to the equation $$(x-2)(x+4) = 9x - 36.$$ If $m > n$, what is the value of $m - n$?"
},
{
"from": "gpt",
"value": "To find the distinct solutions \\( m \\) and \\( n \\) of the equation \\( (x-2)(x+4) = 9x - 36 \\), we st... |
Determine a polynomial function of degree 3, with leading coefficient 1 and rational coefficients, that has the given roots: 2 + √5 and 3 - √7. Express the terms in descending order of degree. | Determine a polynomial function of degree 3, with leading coefficient 1 and rational coefficients, that has the given roots: 2 + √5 and 3 - √7. Express the terms in descending order of degree. | To determine a polynomial function of degree 3 with the given roots \(2 + \sqrt{5}\) and \(3 - \sqrt{7}\), we also need to include the conjugate roots due to the requirement for rational coefficients. Thus, we include the conjugate roots \(2 - \sqrt{5}\) and \(3 + \sqrt{7}\).
The roots of the polynomial are:
1. \(2 + ... | 10,339 | [
{
"from": "human",
"value": "Determine a polynomial function of degree 3, with leading coefficient 1 and rational coefficients, that has the given roots: 2 + √5 and 3 - √7. Express the terms in descending order of degree."
},
{
"from": "gpt",
"value": "To determine a polynomial function of degre... |
In a regular hexagon, a circle is inscribed in each of the six triangles obtained by connecting every other vertex. The radii of the circles are all equal to 2. Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be points on the circles such that $P_iP_{i+1}$ is tangent to the $i$th circle for each $i=1,2,3,4,5,6$, where... | In a regular hexagon, a circle is inscribed in each of the six triangles obtained by connecting every other vertex. The radii of the circles are all equal to 2. Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be points on the circles such that $P_iP_{i+1}$ is tangent to the $i$th circle for each $i=1,2,3,4,5,6$, where... | Let the vertices of the regular hexagon be denoted as \( A_1, A_2, A_3, A_4, A_5, A_6 \). Connecting every other vertex produces the six triangles \( A_1 A_3 A_5 \), \( A_2 A_4 A_6 \), \( A_1 A_2 A_4 \), \( A_2 A_3 A_5 \), \( A_1 A_6 A_4 \), and \( A_3 A_6 A_5 \).
Given that the radius of the inscribed circles (inradi... | 10,340 | [
{
"from": "human",
"value": "In a regular hexagon, a circle is inscribed in each of the six triangles obtained by connecting every other vertex. The radii of the circles are all equal to 2. Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be points on the circles such that $P_iP_{i+1}$ is tangent to the $i$... |
Tom has been saving money for a new bike and has $\$120$ in his savings account. 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, Tom also spends $\$2$ per week on snacks and other treats. How many weeks will it take Tom to have e... | Tom has been saving money for a new bike and has $\$120$ in his savings account. 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, Tom also spends $\$2$ per week on snacks and other treats. How many weeks will it take Tom to have e... | To determine how many weeks it will take Tom to save enough money to buy the bike, we first need to find out how much more money Tom needs.
Tom currently has:
\[
120 \text{ dollars}
\]
The cost of the bike is:
\[
180 \text{ dollars}
\]
The amount Tom still needs is:
\[
180 - 120 = 60 \text{ dollars}
\]
Next, we calcu... | 10,341 | [
{
"from": "human",
"value": "Tom has been saving money for a new bike and has $\\$120$ in his savings account. 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, Tom also spends $\\$2$ per week on snacks and other trea... |
We roll a fair 6-sided die 5 times. What is the probability that we get an odd number in exactly 4 of the 5 rolls? | We roll a fair 6-sided die 5 times. What is the probability that we get an odd number in exactly 4 of the 5 rolls? | To determine the probability of rolling an odd number exactly 4 times out of 5 rolls of a fair 6-sided die, we start by recognizing the outcomes of a single roll. The numbers on a 6-sided die are {1, 2, 3, 4, 5, 6}, where the odd numbers are {1, 3, 5}. Therefore, the probability of rolling an odd number in a single rol... | 10,342 | [
{
"from": "human",
"value": "We roll a fair 6-sided die 5 times. What is the probability that we get an odd number in exactly 4 of the 5 rolls?"
},
{
"from": "gpt",
"value": "To determine the probability of rolling an odd number exactly 4 times out of 5 rolls of a fair 6-sided die, we start by ... |
A particle is moving in a 3D space along a trajectory given by the equations:
x = 2sin(t)
y = 2cos(t)
z = t^2
where t is time and x, y, and z are the coordinates of the particle in a 3D space.
Find the velocity vector of the particle at time t = 0. | A particle is moving in a 3D space along a trajectory given by the equations:
x = 2sin(t)
y = 2cos(t)
z = t^2
where t is time and x, y, and z are the coordinates of the particle in a 3D space.
Find the velocity vector of the particle at time t = 0. | To find the velocity vector of the particle at time \( t = 0 \), we first need to differentiate the position vector with respect to time. The position of the particle is given by the parametric equations:
\[
x(t) = 2\sin(t)
\]
\[
y(t) = 2\cos(t)
\]
\[
z(t) = t^2
\]
The velocity vector \( \mathbf{v}(t) \) is the deriv... | 10,343 | [
{
"from": "human",
"value": "A particle is moving in a 3D space along a trajectory given by the equations:\nx = 2sin(t)\ny = 2cos(t)\nz = t^2\n\nwhere t is time and x, y, and z are the coordinates of the particle in a 3D space.\n\nFind the velocity vector of the particle at time t = 0."
},
{
"from":... |
Consider the following system of equations:
\begin{align*}
x+4y&=b,\\
6x+7y&=3b,
\end{align*} If the system has a solution $(x,y)$ when $y=1$, find the value of $b$. | Consider the following system of equations:
\begin{align*}
x+4y&=b,\\
6x+7y&=3b,
\end{align*} If the system has a solution $(x,y)$ when $y=1$, find the value of $b$. | We are given the system of equations:
1. \( x + 4y = b \)
2. \( 6x + 7y = 3b \)
We need to find the value of \( b \) when \( y = 1 \).
Starting with the first equation, we substitute \( y = 1 \):
\[
x + 4(1) = b
\]
\[
x + 4 = b \quad \text{(1)}
\]
Now substituting \( y = 1 \) into the second equation:
\[
6x + 7(1... | 10,344 | [
{
"from": "human",
"value": "Consider the following system of equations:\n\\begin{align*}\nx+4y&=b,\\\\\n6x+7y&=3b,\n\\end{align*} If the system has a solution $(x,y)$ when $y=1$, find the value of $b$."
},
{
"from": "gpt",
"value": "We are given the system of equations:\n\n1. \\( x + 4y = b \\)... |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ be vectors, and let $E$ be the determinant of the matrix whose column vectors are $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}.$ Then the determinant of the matrix whose column vectors are $\mathbf{a} \times \mathbf{b},$ $\mathbf{b} \times \ma... | Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}$ be vectors, and let $E$ be the determinant of the matrix whose column vectors are $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ and $\mathbf{d}.$ Then the determinant of the matrix whose column vectors are $\mathbf{a} \times \mathbf{b},$ $\mathbf{b} \times \ma... | To find the determinant of the matrix formed by the columns \(\mathbf{a} \times \mathbf{b}\), \(\mathbf{b} \times \mathbf{c}\), \(\mathbf{c} \times \mathbf{d}\), and \(\mathbf{d} \times \mathbf{a}\), we need to express this determinant in terms of the determinant \(E\) of the original columns \(\mathbf{a}\), \(\mathbf{... | 10,345 | [
{
"from": "human",
"value": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c},$ and $\\mathbf{d}$ be vectors, and let $E$ be the determinant of the matrix whose column vectors are $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c},$ and $\\mathbf{d}.$ Then the determinant of the matrix whose column vectors are $\\ma... |
Suppose that $\{b_n\}$ is an arithmetic sequence with $$
b_1 + b_2 + \cdots + b_{50} = 150 \quad \text{and} \quad
b_{51} + b_{52} + \cdots + b_{100} = 300.
$$
What is the value of $b_2 - b_1$? Express your answer as a common fraction. | Suppose that $\{b_n\}$ is an arithmetic sequence with $$
b_1 + b_2 + \cdots + b_{50} = 150 \quad \text{and} \quad
b_{51} + b_{52} + \cdots + b_{100} = 300.
$$
What is the value of $b_2 - b_1$? Express your answer as a common fraction. | Let the first term of the arithmetic sequence \(\{b_n\}\) be \(b_1 = a\) and the common difference be \(d\). Thus, the terms can be expressed as:
\[
b_n = a + (n-1)d.
\]
First, we calculate the sum \(b_1 + b_2 + \cdots + b_{50}\). Using the formula for the sum of an arithmetic series, we have:
\[
b_1 + b_2 + \cdots + ... | 10,346 | [
{
"from": "human",
"value": "Suppose that $\\{b_n\\}$ is an arithmetic sequence with $$\nb_1 + b_2 + \\cdots + b_{50} = 150 \\quad \\text{and} \\quad\nb_{51} + b_{52} + \\cdots + b_{100} = 300.\n$$\nWhat is the value of $b_2 - b_1$? Express your answer as a common fraction."
},
{
"from": "gpt",
... |
For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\frac{n}{3150}$ terminate? | For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\frac{n}{3150}$ terminate? | To determine how many integer values of \( n \) between 1 and 2000 inclusive make the decimal representation of \( \frac{n}{3150} \) terminate, we first analyze when a fraction \( \frac{a}{b} \) has a terminating decimal form. This occurs if and only if the denominator \( b \), in its simplest form, contains no prime f... | 10,347 | [
{
"from": "human",
"value": "For how many integer values of $n$ between 1 and 2000 inclusive does the decimal representation of $\\frac{n}{3150}$ terminate?"
},
{
"from": "gpt",
"value": "To determine how many integer values of \\( n \\) between 1 and 2000 inclusive make the decimal representati... |
Let $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}.$ Find the value of $t$ such that the vector
\[t \begin{pmatrix} 1 \\ 2 \end{pmatrix}\]lies on the line passing through $\mathbf{a}$ and $\mathbf{b}.$ | Let $\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}.$ Find the value of $t$ such that the vector
\[t \begin{pmatrix} 1 \\ 2 \end{pmatrix}\]lies on the line passing through $\mathbf{a}$ and $\mathbf{b}.$ | To determine the value of \( t \) such that the vector \( t \begin{pmatrix} 1 \\ 2 \end{pmatrix} \) lies on the line passing through the points \( \mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \) and \( \mathbf{b} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} \), we first recognize that \( \mathbf{b} \) can be expressed a... | 10,348 | [
{
"from": "human",
"value": "Let $\\mathbf{a} = \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 4 \\\\ 6 \\end{pmatrix}.$ Find the value of $t$ such that the vector\n\\[t \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}\\]lies on the line passing through $\\mathbf{a}$ and $\\mathbf{b}... |
A sequence of real numbers (a_n) satisfies the recursion
\[a_{n+1}=\sqrt{a_n^2+1}\]
for all non-negative integers n. If a_0=0, what is a_{10}? | A sequence of real numbers (a_n) satisfies the recursion
\[a_{n+1}=\sqrt{a_n^2+1}\]
for all non-negative integers n. If a_0=0, what is a_{10}? | To solve the recursion defined by
\[
a_{n+1} = \sqrt{a_n^2 + 1}
\]
starting with \(a_0 = 0\), we will compute the first few terms in the sequence to observe a pattern.
1. For \(n=0\):
\[
a_0 = 0
\]
2. For \(n=1\):
\[
a_1 = \sqrt{a_0^2 + 1} = \sqrt{0^2 + 1} = \sqrt{1} = 1
\]
3. For \(n=2\):
\[
a_2 = \sqrt{a_1^2... | 10,349 | [
{
"from": "human",
"value": "A sequence of real numbers (a_n) satisfies the recursion\n\\[a_{n+1}=\\sqrt{a_n^2+1}\\]\nfor all non-negative integers n. If a_0=0, what is a_{10}?"
},
{
"from": "gpt",
"value": "To solve the recursion defined by \n\n\\[\na_{n+1} = \\sqrt{a_n^2 + 1}\n\\]\n\nstarting ... |
Mike earns $\$1800$for working 24 hours per week for 12 weeks during the summer. If he needs to make the same amount of money during the school year by working for 20 weeks, how many hours per week must he work? | Mike earns $\$1800$for working 24 hours per week for 12 weeks during the summer. If he needs to make the same amount of money during the school year by working for 20 weeks, how many hours per week must he work? | To determine how many hours per week Mike needs to work during the school year, we first calculate his hourly wage based on his summer earnings.
1. **Calculate total hours worked during the summer**:
\[
\text{Total hours} = 24 \text{ hours/week} \times 12 \text{ weeks} = 288 \text{ hours}
\]
2. **Determine M... | 10,350 | [
{
"from": "human",
"value": "Mike earns $\\$1800$for working 24 hours per week for 12 weeks during the summer. If he needs to make the same amount of money during the school year by working for 20 weeks, how many hours per week must he work?"
},
{
"from": "gpt",
"value": "To determine how many h... |
My club has 25 members. In how many ways can I choose members to form a 4-person executive committee? | My club has 25 members. In how many ways can I choose members to form a 4-person executive committee? | To form a 4-person executive committee from a club with 25 members, we need to choose 4 members from the 25 without regard to the order of selection. This is a combination problem, which can be solved using the binomial coefficient.
The number of ways to choose \( k \) members from \( n \) members is given by the form... | 10,351 | [
{
"from": "human",
"value": "My club has 25 members. In how many ways can I choose members to form a 4-person executive committee?"
},
{
"from": "gpt",
"value": "To form a 4-person executive committee from a club with 25 members, we need to choose 4 members from the 25 without regard to the ord... |
In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $BC$. What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction. | In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $BC$. What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction. | Let's denote the side length of square \(ABCD\) as \(s\). The vertices of the square can be defined as follows:
\[
A = (0, 0), \quad B = (s, 0), \quad C = (s, s), \quad D = (0, s)
\]
Point \(M\), being the midpoint of side \(AB\), has coordinates:
\[
M = \left(\frac{s+0}{2}, \frac{0+0}{2}\right) = \left(\frac{s}{2},... | 10,352 | [
{
"from": "human",
"value": "In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $BC$. What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "Let's denote th... |
Let $P$ be the point on the parabola $y = x^2$ such that the distance from $P$ to the vertex of the parabola is equal to the distance from $P$ to the point $(1, 0)$. Enter the coordinates of $P$. | Let $P$ be the point on the parabola $y = x^2$ such that the distance from $P$ to the vertex of the parabola is equal to the distance from $P$ to the point $(1, 0)$. Enter the coordinates of $P$. | Let the point \( P \) on the parabola \( y = x^2 \) be represented as \( P(x, y) = (x, x^2) \).
The distance from \( P \) to the vertex of the parabola, which is at the origin \( (0, 0) \), is given by the formula:
\[
d_1 = \sqrt{(x - 0)^2 + (x^2 - 0)^2} = \sqrt{x^2 + (x^2)^2} = \sqrt{x^2 + x^4} = \sqrt{x^2(1 + x^2)}... | 10,353 | [
{
"from": "human",
"value": "Let $P$ be the point on the parabola $y = x^2$ such that the distance from $P$ to the vertex of the parabola is equal to the distance from $P$ to the point $(1, 0)$. Enter the coordinates of $P$."
},
{
"from": "gpt",
"value": "Let the point \\( P \\) on the parabola... |
Find the maximum value of $x^2 + 2x + \frac{1}{x^2 + 2x + 1}$ over all real numbers $x.$ | Find the maximum value of $x^2 + 2x + \frac{1}{x^2 + 2x + 1}$ over all real numbers $x.$ | To maximize the expression
\[
y = x^2 + 2x + \frac{1}{x^2 + 2x + 1}
\]
we first simplify the expression. Notice that
\[
x^2 + 2x + 1 = (x+1)^2.
\]
Thus, we rewrite \(y\) as
\[
y = (x+1)^2 + \frac{1}{(x+1)^2}.
\]
Let us define a new variable:
\[
z = (x+1)^2.
\]
This means that \(y\) can be expressed in terms o... | 10,354 | [
{
"from": "human",
"value": "Find the maximum value of $x^2 + 2x + \\frac{1}{x^2 + 2x + 1}$ over all real numbers $x.$"
},
{
"from": "gpt",
"value": "To maximize the expression \n\n\\[\ny = x^2 + 2x + \\frac{1}{x^2 + 2x + 1}\n\\]\n\nwe first simplify the expression. Notice that \n\n\\[\nx^2 + 2x... |
In a standard 6-sided die, each side has a different color: red, blue, green, yellow, orange, and purple. If the die is rolled four times, what is the probability that each of the first three colors (red, blue, and green) appears at least once, and the fourth color (yellow, orange, or purple) appears exactly once? | In a standard 6-sided die, each side has a different color: red, blue, green, yellow, orange, and purple. If the die is rolled four times, what is the probability that each of the first three colors (red, blue, and green) appears at least once, and the fourth color (yellow, orange, or purple) appears exactly once? | To find the probability that when rolling a 6-sided die 4 times, each of the first three colors (red, blue, and green) appears at least once, and the fourth color (yellow, orange, or purple) appears exactly once, we can approach the problem step by step.
### Step 1: Total Outcomes
Since we are rolling a 6-sided die 4 ... | 10,355 | [
{
"from": "human",
"value": "In a standard 6-sided die, each side has a different color: red, blue, green, yellow, orange, and purple. If the die is rolled four times, what is the probability that each of the first three colors (red, blue, and green) appears at least once, and the fourth color (yellow, oran... |
Problem:
Evaluate $\int_{0}^{\frac{\pi}{2}} \sin(x) \cos(x) dx.$ Express your answer in a simplified form.
(Note: This problem requires integration, whereas the original problem required knowledge of inverse trigonometric functions. Both problems involve trigonometric functions, but in different contexts.) | Problem:
Evaluate $\int_{0}^{\frac{\pi}{2}} \sin(x) \cos(x) dx.$ Express your answer in a simplified form.
(Note: This problem requires integration, whereas the original problem required knowledge of inverse trigonometric functions. Both problems involve trigonometric functions, but in different contexts.) | We start with the integral
\[
\int_{0}^{\frac{\pi}{2}} \sin(x) \cos(x) \, dx.
\]
We can simplify the integration using a known trigonometric identity. Notice that
\[
\sin(x) \cos(x) = \frac{1}{2} \sin(2x).
\]
Thus, we can rewrite the integral as
\[
\int_{0}^{\frac{\pi}{2}} \sin(x) \cos(x) \, dx = \int_{0}^{\fra... | 10,356 | [
{
"from": "human",
"value": "Problem:\nEvaluate $\\int_{0}^{\\frac{\\pi}{2}} \\sin(x) \\cos(x) dx.$ Express your answer in a simplified form.\n\n(Note: This problem requires integration, whereas the original problem required knowledge of inverse trigonometric functions. Both problems involve trigonometric ... |
Let's define $f(a) = (a^2+2^a)^2 - 4 \cdot 2^a \cdot a^2$ and $g(a) = (a^2+2^a)^2 + 4 \cdot 2^a \cdot a^2$. If $2 \le a \le 500$, how many integral values of $a$ are there such that $f(a) \cdot g(a)$ is a perfect square? | Let's define $f(a) = (a^2+2^a)^2 - 4 \cdot 2^a \cdot a^2$ and $g(a) = (a^2+2^a)^2 + 4 \cdot 2^a \cdot a^2$. If $2 \le a \le 500$, how many integral values of $a$ are there such that $f(a) \cdot g(a)$ is a perfect square? | We start with the functions \( f(a) \) and \( g(a) \) defined as follows:
\[
f(a) = (a^2 + 2^a)^2 - 4 \cdot 2^a \cdot a^2
\]
\[
g(a) = (a^2 + 2^a)^2 + 4 \cdot 2^a \cdot a^2
\]
Next, we can factor the product \( f(a) \cdot g(a) \) by noticing that they resemble the difference of squares and the sum of squares respect... | 10,357 | [
{
"from": "human",
"value": "Let's define $f(a) = (a^2+2^a)^2 - 4 \\cdot 2^a \\cdot a^2$ and $g(a) = (a^2+2^a)^2 + 4 \\cdot 2^a \\cdot a^2$. If $2 \\le a \\le 500$, how many integral values of $a$ are there such that $f(a) \\cdot g(a)$ is a perfect square?"
},
{
"from": "gpt",
"value": "We start... |
Count the number of distinct terms in the expansion of $(x+y+z)^{5}$. | Count the number of distinct terms in the expansion of $(x+y+z)^{5}$. | To determine the number of distinct terms in the expansion of \((x+y+z)^5\), we can apply the multinomial theorem. According to the multinomial theorem, the expansion of \((x_1 + x_2 + \cdots + x_m)^n\) can be expressed in terms of the coefficients given by the multinomial coefficients. Specifically, the number of term... | 10,358 | [
{
"from": "human",
"value": "Count the number of distinct terms in the expansion of $(x+y+z)^{5}$."
},
{
"from": "gpt",
"value": "To determine the number of distinct terms in the expansion of \\((x+y+z)^5\\), we can apply the multinomial theorem. According to the multinomial theorem, the expansi... |
Determine the number of integers $n$ that satisfy the inequality $\sqrt[3]{n} < \log_{2}(n) < \sqrt[5]{n}$. | Determine the number of integers $n$ that satisfy the inequality $\sqrt[3]{n} < \log_{2}(n) < \sqrt[5]{n}$. | To solve the inequality \( \sqrt[3]{n} < \log_{2}(n) < \sqrt[5]{n} \), we will break it down into two parts:
1. \( \sqrt[3]{n} < \log_{2}(n) \)
2. \( \log_{2}(n) < \sqrt[5]{n} \)
### Step 1: Solve \( \sqrt[3]{n} < \log_{2}(n) \)
Rewriting the inequality, we have:
\[
\sqrt[3]{n} = n^{1/3}
\]
Thus, the inequality ca... | 10,359 | [
{
"from": "human",
"value": "Determine the number of integers $n$ that satisfy the inequality $\\sqrt[3]{n} < \\log_{2}(n) < \\sqrt[5]{n}$."
},
{
"from": "gpt",
"value": "To solve the inequality \\( \\sqrt[3]{n} < \\log_{2}(n) < \\sqrt[5]{n} \\), we will break it down into two parts:\n\n1. \\( \... |
Consider a torus (a doughnut-shaped surface) formed by rotating a circle of radius $r$ around a horizontal axis that lies outside the circle. If the center of the rotated circle is located $h$ units above the rotation axis and the height of the torus is $b$, what is the volume of the torus in terms of $r$, $h$, and $b$... | Consider a torus (a doughnut-shaped surface) formed by rotating a circle of radius $r$ around a horizontal axis that lies outside the circle. If the center of the rotated circle is located $h$ units above the rotation axis and the height of the torus is $b$, what is the volume of the torus in terms of $r$, $h$, and $b$... | To calculate the volume of the torus generated by rotating a circle of radius \( r \) around a horizontal axis that lies outside the circle, we need to establish the geometric relationships involved.
1. **Understanding the Shape**: The problem states that the circle of radius \( r \) is located at a height \( h \) abo... | 10,360 | [
{
"from": "human",
"value": "Consider a torus (a doughnut-shaped surface) formed by rotating a circle of radius $r$ around a horizontal axis that lies outside the circle. If the center of the rotated circle is located $h$ units above the rotation axis and the height of the torus is $b$, what is the volume o... |
Find the sum of the divisors of 72. | Find the sum of the divisors of 72. | To find the sum of the divisors of a number, we start with its prime factorization. The prime factorization of \( 72 \) is:
\[
72 = 2^3 \times 3^2
\]
To use the formula for the sum of the divisors based on its prime factorization, we use the following formula:
\[
\sigma(n) = (p_1^{k_1+1} - 1)/(p_1 - 1) \times (p_2^{... | 10,361 | [
{
"from": "human",
"value": "Find the sum of the divisors of 72."
},
{
"from": "gpt",
"value": "To find the sum of the divisors of a number, we start with its prime factorization. The prime factorization of \\( 72 \\) is:\n\n\\[\n72 = 2^3 \\times 3^2\n\\]\n\nTo use the formula for the sum of the... |
A number is divisible by 9 if the sum of its digits is divisible by 9.
What digit can be appended to the end of the five-digit number $23456$ to make it a multiple of 11? | A number is divisible by 9 if the sum of its digits is divisible by 9.
What digit can be appended to the end of the five-digit number $23456$ to make it a multiple of 11? | To determine the digit that can be appended to the end of the number \(23456\) to make it a multiple of \(11\), we need to apply the divisibility rule for \(11\). According to this rule, a number is divisible by \(11\) if the difference between the sum of its digits in odd positions and the sum of its digits in even po... | 10,362 | [
{
"from": "human",
"value": "A number is divisible by 9 if the sum of its digits is divisible by 9.\n\nWhat digit can be appended to the end of the five-digit number $23456$ to make it a multiple of 11?"
},
{
"from": "gpt",
"value": "To determine the digit that can be appended to the end of the ... |
Let $f(x)$ and $g(x)$ be two functions defined as follows:
$f(x) = x^3 - 2x^2 + x + 1$
$g(x) = x^2 - 3x + 2$
If $h(x)$ is defined as $h(x) = f(x) \text{ Y } g(x)$, where the operation Y is defined as $f(x) \text{ Y } g(x) = f(x) \cdot g(x) - 2 \cdot f(x) \cdot g'(x) + g''(x)$, find the value of $h(1)$. | Let $f(x)$ and $g(x)$ be two functions defined as follows:
$f(x) = x^3 - 2x^2 + x + 1$
$g(x) = x^2 - 3x + 2$
If $h(x)$ is defined as $h(x) = f(x) \text{ Y } g(x)$, where the operation Y is defined as $f(x) \text{ Y } g(x) = f(x) \cdot g(x) - 2 \cdot f(x) \cdot g'(x) + g''(x)$, find the value of $h(1)$. | To find \( h(1) \) defined by the operation \( h(x) = f(x) \text{ Y } g(x) \), we first need to compute the necessary derivatives and evaluate the functions at \( x = 1 \).
1. **Define the functions \( f(x) \) and \( g(x) \):**
\[
f(x) = x^3 - 2x^2 + x + 1
\]
\[
g(x) = x^2 - 3x + 2
\]
2. **Calculat... | 10,363 | [
{
"from": "human",
"value": "Let $f(x)$ and $g(x)$ be two functions defined as follows:\n\n$f(x) = x^3 - 2x^2 + x + 1$\n\n$g(x) = x^2 - 3x + 2$\n\nIf $h(x)$ is defined as $h(x) = f(x) \\text{ Y } g(x)$, where the operation Y is defined as $f(x) \\text{ Y } g(x) = f(x) \\cdot g(x) - 2 \\cdot f(x) \\cdot g'(x... |
A triangle has a side of length 6 cm, a side of length 8 cm and a right angle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth. | A triangle has a side of length 6 cm, a side of length 8 cm and a right angle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth. | Given a triangle with one right angle, we can apply the Pythagorean theorem. The triangle has sides of lengths 6 cm and 8 cm. The Pythagorean theorem states:
\[
a^2 + b^2 = c^2
\]
where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.
We consider two scenarios ... | 10,364 | [
{
"from": "human",
"value": "A triangle has a side of length 6 cm, a side of length 8 cm and a right angle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth."
},
{
"from": "gpt",
"value": "Given a... |
For what value of $n$ is the four-digit number $712n$, with units digit $n$, divisible by 18? | For what value of $n$ is the four-digit number $712n$, with units digit $n$, divisible by 18? | To determine the value of \( n \) such that the four-digit number \( 712n \) is divisible by 18, we must check two conditions since a number is divisible by 18 if and only if it is divisible by both 2 and 9.
1. **Divisibility by 2**:
A number is divisible by 2 if its last digit is even. Since the last digit of our ... | 10,365 | [
{
"from": "human",
"value": "For what value of $n$ is the four-digit number $712n$, with units digit $n$, divisible by 18?"
},
{
"from": "gpt",
"value": "To determine the value of \\( n \\) such that the four-digit number \\( 712n \\) is divisible by 18, we must check two conditions since a numb... |
An author finds that the time it takes her to write a book varies directly with the number of research papers she needs to read. When writing a book that requires reading 20 research papers, she can finish it in 6 months. How many months will it take her to write a book that requires reading 15 research papers? Express... | An author finds that the time it takes her to write a book varies directly with the number of research papers she needs to read. When writing a book that requires reading 20 research papers, she can finish it in 6 months. How many months will it take her to write a book that requires reading 15 research papers? Express... | To determine how many months it will take the author to write a book that requires reading 15 research papers, we start from the information that writing time varies directly with the number of research papers.
First, let's express the relationship mathematically:
Let \( T \) represent the time (in months) to write t... | 10,366 | [
{
"from": "human",
"value": "An author finds that the time it takes her to write a book varies directly with the number of research papers she needs to read. When writing a book that requires reading 20 research papers, she can finish it in 6 months. How many months will it take her to write a book that req... |
Compute the value of $abc + c + 2$ when $a$, $b$, and $c$ are the roots of the equation $x^3+2x^2-x+7=0$ | Compute the value of $abc + c + 2$ when $a$, $b$, and $c$ are the roots of the equation $x^3+2x^2-x+7=0$ | To solve the problem, we will use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots. The polynomial given is
\[
x^3 + 2x^2 - x + 7 = 0.
\]
Let \( a, b, c \) be the roots of this polynomial. According to Vieta's formulas, we have the following relationships:
1. \( a + ... | 10,367 | [
{
"from": "human",
"value": "Compute the value of $abc + c + 2$ when $a$, $b$, and $c$ are the roots of the equation $x^3+2x^2-x+7=0$"
},
{
"from": "gpt",
"value": "To solve the problem, we will use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots... |
Given is a square $ABCD$ of side length 3. Two lines $l_1$ and $l_2$ intersect at $A$ and are perpendicular to each other. Line $l_1$ intersects $BC$ at $E$ and line $l_2$ intersects $CD$ at $F$. The distance from $E$ to $F$ is $\sqrt{10}$. What is the area of triangle $AEF$? | Given is a square $ABCD$ of side length 3. Two lines $l_1$ and $l_2$ intersect at $A$ and are perpendicular to each other. Line $l_1$ intersects $BC$ at $E$ and line $l_2$ intersects $CD$ at $F$. The distance from $E$ to $F$ is $\sqrt{10}$. What is the area of triangle $AEF$? | Let the coordinates of the vertices of square \(ABCD\) be:
- \(A(0, 0)\)
- \(B(3, 0)\)
- \(C(3, 3)\)
- \(D(0, 3)\)
We are given that the two lines \(l_1\) and \(l_2\) intersect at \(A (0, 0)\) and are perpendicular to each other. Let the equation of line \(l_1\) be \(y = m_1 x\) and the equation of line \(l_2\) be \(y... | 10,368 | [
{
"from": "human",
"value": "Given is a square $ABCD$ of side length 3. Two lines $l_1$ and $l_2$ intersect at $A$ and are perpendicular to each other. Line $l_1$ intersects $BC$ at $E$ and line $l_2$ intersects $CD$ at $F$. The distance from $E$ to $F$ is $\\sqrt{10}$. What is the area of triangle $AEF$?"
... |
Find the greatest common divisor of 23456, 35789, and 12345. | Find the greatest common divisor of 23456, 35789, and 12345. | To find the greatest common divisor (GCD) of the numbers 23456, 35789, and 12345, we can use the prime factorization method or the Euclidean algorithm. For practical purposes, we'll apply the Euclidean algorithm.
Let's denote the numbers as:
- \( a = 23456 \)
- \( b = 35789 \)
- \( c = 12345 \)
We will first find \( ... | 10,369 | [
{
"from": "human",
"value": "Find the greatest common divisor of 23456, 35789, and 12345."
},
{
"from": "gpt",
"value": "To find the greatest common divisor (GCD) of the numbers 23456, 35789, and 12345, we can use the prime factorization method or the Euclidean algorithm. For practical purposes,... |
A cylindrical water tank with a height of 20 feet and a radius of 5 feet is replaced by a new tank with the same volume but in the shape of a cone. If the new tank has a height of 10 feet, how many square feet smaller than the old tank is the base area of the new tank? | A cylindrical water tank with a height of 20 feet and a radius of 5 feet is replaced by a new tank with the same volume but in the shape of a cone. If the new tank has a height of 10 feet, how many square feet smaller than the old tank is the base area of the new tank? | To find out how many square feet smaller the base area of the new conical tank is compared to the base area of the old cylindrical tank, we first need to calculate the volume of the cylindrical tank, and then use that volume to determine the radius of the base of the new conical tank.
1. **Calculate the volume of the ... | 10,370 | [
{
"from": "human",
"value": "A cylindrical water tank with a height of 20 feet and a radius of 5 feet is replaced by a new tank with the same volume but in the shape of a cone. If the new tank has a height of 10 feet, how many square feet smaller than the old tank is the base area of the new tank?"
},
{... |
Find the equation of the circle that passes through the points (2,3), (4,5), and (6,7).
Let the equation of the circle be (x-a)^2 + (y-b)^2 = r^2, where (a,b) is the center of the circle and r is the radius.
We know that the circle passes through the points (2,3), (4,5), and (6,7), so we can plug these points into th... | Find the equation of the circle that passes through the points (2,3), (4,5), and (6,7).
Let the equation of the circle be (x-a)^2 + (y-b)^2 = r^2, where (a,b) is the center of the circle and r is the radius.
We know that the circle passes through the points (2,3), (4,5), and (6,7), so we can plug these points into th... | To find the equation of a circle that passes through the points (2,3), (4,5), and (6,7), we can use the method of establishing a system of equations based on the general form of the circle.
The general equation of a circle can be expressed as:
\[
(x - a)^2 + (y - b)^2 = r^2
\]
where \((a, b)\) is the center and \(r\... | 10,371 | [
{
"from": "human",
"value": "Find the equation of the circle that passes through the points (2,3), (4,5), and (6,7).\n\nLet the equation of the circle be (x-a)^2 + (y-b)^2 = r^2, where (a,b) is the center of the circle and r is the radius.\n\nWe know that the circle passes through the points (2,3), (4,5), a... |
What is the value of $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$? | What is the value of $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$? | The series \(\sum_{n=1}^{\infty}\frac{1}{n^2}\) is a famous series known as the Basel problem. The value of this series was solved by the mathematician Leonhard Euler in the 18th century.
Euler showed that:
\[
\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}
\]
This means that the value of the series is:
\[
\box... | 10,372 | [
{
"from": "human",
"value": "What is the value of $\\sum\\limits_{n=1}^{\\infty}\\frac{1}{n^2}$?"
},
{
"from": "gpt",
"value": "The series \\(\\sum_{n=1}^{\\infty}\\frac{1}{n^2}\\) is a famous series known as the Basel problem. The value of this series was solved by the mathematician Leonhard Eu... |
Given the vectors $\mathbf{a} = \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -2 \\ 5 \\ 1 \end{pmatrix}$,
find the projection of vector $\mathbf{a}$ onto vector $\mathbf{b}$. | Given the vectors $\mathbf{a} = \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -2 \\ 5 \\ 1 \end{pmatrix}$,
find the projection of vector $\mathbf{a}$ onto vector $\mathbf{b}$. | To find the projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\), we use the formula for the projection of \(\mathbf{a}\) onto \(\mathbf{b}\):
\[
\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}
\]
First, we need to compute the dot product \(\m... | 10,373 | [
{
"from": "human",
"value": "Given the vectors $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ -2 \\\\ 4 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} -2 \\\\ 5 \\\\ 1 \\end{pmatrix}$,\nfind the projection of vector $\\mathbf{a}$ onto vector $\\mathbf{b}$."
},
{
"from": "gpt",
"value": "To find the... |
What is the smallest prime divisor of $5^{23} + 7^{17}$? | What is the smallest prime divisor of $5^{23} + 7^{17}$? | To find the smallest prime divisor of \( 5^{23} + 7^{17} \), we will test small prime numbers to see if they divide the expression.
1. **Check divisibility by \(2\):**
- The expression \( 5^{23} \) is odd (since \(5\) is odd), and \( 7^{17} \) is also odd (since \(7\) is odd). Therefore, the sum of two odd numbers,... | 10,374 | [
{
"from": "human",
"value": "What is the smallest prime divisor of $5^{23} + 7^{17}$?"
},
{
"from": "gpt",
"value": "To find the smallest prime divisor of \\( 5^{23} + 7^{17} \\), we will test small prime numbers to see if they divide the expression.\n\n1. **Check divisibility by \\(2\\):**\n ... |
Given the polynomial expression:
3(x^2 + 5x - 1) - 2(x^3 - 2x^2 - x + 3)
After fully simplifying this expression, find the product of the coefficients of all the terms. | Given the polynomial expression:
3(x^2 + 5x - 1) - 2(x^3 - 2x^2 - x + 3)
After fully simplifying this expression, find the product of the coefficients of all the terms. | To simplify the expression \( 3(x^2 + 5x - 1) - 2(x^3 - 2x^2 - x + 3) \), we will start by distributing the constants into their respective polynomials.
First, distribute \( 3 \) in the first part:
\[
3(x^2 + 5x - 1) = 3x^2 + 15x - 3
\]
Next, distribute \( -2 \) in the second part:
\[
-2(x^3 - 2x^2 - x + 3) = -2x^3... | 10,375 | [
{
"from": "human",
"value": "Given the polynomial expression: \n3(x^2 + 5x - 1) - 2(x^3 - 2x^2 - x + 3)\nAfter fully simplifying this expression, find the product of the coefficients of all the terms."
},
{
"from": "gpt",
"value": "To simplify the expression \\( 3(x^2 + 5x - 1) - 2(x^3 - 2x^2 - ... |
A deck of 40 cards has 4 suits (hearts, diamonds, clubs, and spades), each containing 10 cards. The cards are numbered from 1 to 10 in each suit. If two cards are drawn randomly without replacement, what is the probability that the sum of the numbers on the two cards is equal to 11? | A deck of 40 cards has 4 suits (hearts, diamonds, clubs, and spades), each containing 10 cards. The cards are numbered from 1 to 10 in each suit. If two cards are drawn randomly without replacement, what is the probability that the sum of the numbers on the two cards is equal to 11? | To determine the probability that the sum of the numbers on two randomly drawn cards equals 11, we first need to identify all the possible pairs of numbered cards that can achieve this sum.
The pairs of numbers that sum to 11 from the cards numbered 1 through 10 are:
- (1, 10)
- (2, 9)
- (3, 8)
- (4, 7)
- (5, 6)
- (6,... | 10,376 | [
{
"from": "human",
"value": "A deck of 40 cards has 4 suits (hearts, diamonds, clubs, and spades), each containing 10 cards. The cards are numbered from 1 to 10 in each suit. If two cards are drawn randomly without replacement, what is the probability that the sum of the numbers on the two cards is equal to... |
The roots of the quadratic equation $x^2+bx+c=0$ are the reciprocals of the roots of the equation $ax^2+dx+e=0$. If $b=2d$ and $c=-\frac{1}{2}e$, find the ordered pair $(a,e)$. | The roots of the quadratic equation $x^2+bx+c=0$ are the reciprocals of the roots of the equation $ax^2+dx+e=0$. If $b=2d$ and $c=-\frac{1}{2}e$, find the ordered pair $(a,e)$. | Given the quadratic equations \(x^2 + bx + c = 0\) and \(ax^2 + dx + e = 0\), we know that the roots of the first equation are the reciprocals of the roots of the second equation. Let \(r_1\) and \(r_2\) be the roots of the second equation \(ax^2 + dx + e = 0\). The roots of the first equation can then be represented a... | 10,377 | [
{
"from": "human",
"value": "The roots of the quadratic equation $x^2+bx+c=0$ are the reciprocals of the roots of the equation $ax^2+dx+e=0$. If $b=2d$ and $c=-\\frac{1}{2}e$, find the ordered pair $(a,e)$."
},
{
"from": "gpt",
"value": "Given the quadratic equations \\(x^2 + bx + c = 0\\) and \... |
Let \[A=111111\]and \[B=142857\]Find a positive integer $N$ with six or fewer digits such that $N$ is the multiplicative inverse of $AB$ modulo 1,000,000. | Let \[A=111111\]and \[B=142857\]Find a positive integer $N$ with six or fewer digits such that $N$ is the multiplicative inverse of $AB$ modulo 1,000,000. | To find a positive integer \( N \) that is the multiplicative inverse of \( AB \) modulo \( 1,000,000 \), we start by calculating \( A \) and \( B \):
\[
A = 111111
\]
\[
B = 142857
\]
Next, we calculate \( AB \):
\[
AB = 111111 \times 142857
\]
To perform this multiplication, let's break it down. We can simplify \... | 10,378 | [
{
"from": "human",
"value": "Let \\[A=111111\\]and \\[B=142857\\]Find a positive integer $N$ with six or fewer digits such that $N$ is the multiplicative inverse of $AB$ modulo 1,000,000."
},
{
"from": "gpt",
"value": "To find a positive integer \\( N \\) that is the multiplicative inverse of ... |
Consider a positive integer to be "fortunate" if its digits are arranged in increasing order from left to right. What is the least positive multiple of 5 that is not a fortunate integer? | Consider a positive integer to be "fortunate" if its digits are arranged in increasing order from left to right. What is the least positive multiple of 5 that is not a fortunate integer? | To determine the least positive multiple of 5 that is not a fortunate integer, we first establish the definition of a fortunate integer: a positive integer is fortunate if its digits are arranged in increasing order from left to right.
Now, any positive integer that is a multiple of 5 must end in either 0 or 5. We wi... | 10,379 | [
{
"from": "human",
"value": "Consider a positive integer to be \"fortunate\" if its digits are arranged in increasing order from left to right. What is the least positive multiple of 5 that is not a fortunate integer?"
},
{
"from": "gpt",
"value": "To determine the least positive multiple of 5 t... |
The sum of the fractions $\frac{1}{a}$ and $\frac{1}{b}$ is equal to $\frac{2}{3}$. What is the value of $a$ if $b = 2a$? | The sum of the fractions $\frac{1}{a}$ and $\frac{1}{b}$ is equal to $\frac{2}{3}$. What is the value of $a$ if $b = 2a$? | We start with the equation given by the sum of the fractions:
\[
\frac{1}{a} + \frac{1}{b} = \frac{2}{3}
\]
Next, we use the condition that \( b = 2a \). Substituting \( b \) into our equation, we have:
\[
\frac{1}{a} + \frac{1}{2a} = \frac{2}{3}
\]
To combine the fractions on the left, we find a common denominator... | 10,380 | [
{
"from": "human",
"value": "The sum of the fractions $\\frac{1}{a}$ and $\\frac{1}{b}$ is equal to $\\frac{2}{3}$. What is the value of $a$ if $b = 2a$?"
},
{
"from": "gpt",
"value": "We start with the equation given by the sum of the fractions:\n\n\\[\n\\frac{1}{a} + \\frac{1}{b} = \\frac{2}{3... |
Given that the quadratic equation $ax^2 + bx + c = 0$ has the solutions $x_1$ and $x_2$, determine $a$, $b$, and $c$ if $x_1 + x_2 = -1$, $x_1 - x_2 = 2$, and $x_1x_2 = -2$. | Given that the quadratic equation $ax^2 + bx + c = 0$ has the solutions $x_1$ and $x_2$, determine $a$, $b$, and $c$ if $x_1 + x_2 = -1$, $x_1 - x_2 = 2$, and $x_1x_2 = -2$. | To solve for \(a\), \(b\), and \(c\) in the quadratic equation \(ax^2 + bx + c = 0\) given the properties of the roots \(x_1\) and \(x_2\), we can use Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.
Given:
1. \(x_1 + x_2 = -1\)
2. \(x_1 - x_2 = 2\)
3. \(x_1 x_2 = -2\)... | 10,381 | [
{
"from": "human",
"value": "Given that the quadratic equation $ax^2 + bx + c = 0$ has the solutions $x_1$ and $x_2$, determine $a$, $b$, and $c$ if $x_1 + x_2 = -1$, $x_1 - x_2 = 2$, and $x_1x_2 = -2$."
},
{
"from": "gpt",
"value": "To solve for \\(a\\), \\(b\\), and \\(c\\) in the quadratic eq... |
Let $\log_{4}3=x$. Then $\log_{2}27=kx$. Find $k$. | Let $\log_{4}3=x$. Then $\log_{2}27=kx$. Find $k$. | To solve for \( k \) in the equation \( \log_2 27 = kx \) given \( \log_4 3 = x \), we start by expressing \( x \) in terms of base 2 logarithms.
First, we can use the change of base formula to convert \( \log_4 3 \) to base 2:
\[
\log_4 3 = \frac{\log_2 3}{\log_2 4}
\]
Since \( \log_2 4 = 2 \), this simplifies to:
\[... | 10,382 | [
{
"from": "human",
"value": "Let $\\log_{4}3=x$. Then $\\log_{2}27=kx$. Find $k$."
},
{
"from": "gpt",
"value": "To solve for \\( k \\) in the equation \\( \\log_2 27 = kx \\) given \\( \\log_4 3 = x \\), we start by expressing \\( x \\) in terms of base 2 logarithms.\n\nFirst, we can use the ch... |
A spiral staircase turns $270^\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth. | A spiral staircase turns $270^\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth. | To determine the length of the handrail along a spiral staircase that turns \(270^\circ\) while rising \(10\) feet, we can visualize the problem as a helix on a cylindrical structure.
1. **Convert the angle from degrees to radians**:
\[
270^\circ = \frac{270 \pi}{180} = \frac{3\pi}{2} \text{ radians}
\]
2. *... | 10,383 | [
{
"from": "human",
"value": "A spiral staircase turns $270^\\circ$ as it rises 10 feet. The radius of the staircase is 3 feet. What is the number of feet in the length of the handrail? Express your answer as a decimal to the nearest tenth."
},
{
"from": "gpt",
"value": "To determine the lengt... |
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 $$1+4+7+10+13+16+\cdots+91+94+97 \equiv n\pmod{21},$$where $0\le n<21$, 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 $n$?
If $$1+4+7+10+13+16+\cdots+91+94+97 \equiv n\pmod{21},$$where $0\le n<21$, what is the value of $n$? | To find the value of \( n \) in the first sum \( 1 + 6 + 11 + 16 + 21 + 26 + \cdots + 91 + 96 + 101 \mod 15 \), we start by identifying the structure of the sequence.
This sequence is an arithmetic series where the first term \( a = 1 \) and the common difference \( d = 5 \). We can represent the \( n \)-th term of th... | 10,384 | [
{
"from": "human",
"value": "If\n$$1+6+11+16+21+26+\\cdots+91+96+101 \\equiv n\\pmod{15},$$\nwhere $0\\le n<15$, what is the value of $n$?\n\n\n\nIf $$1+4+7+10+13+16+\\cdots+91+94+97 \\equiv n\\pmod{21},$$where $0\\le n<21$, what is the value of $n$?"
},
{
"from": "gpt",
"value": "To find the va... |
Determine the sum of all positive integer values of $a$ such that the quadratic equation $6x^2 - 7x + a = 0$ has two distinct real roots. | Determine the sum of all positive integer values of $a$ such that the quadratic equation $6x^2 - 7x + a = 0$ has two distinct real roots. | To determine the values of \( a \) such that the quadratic equation \( 6x^2 - 7x + a = 0 \) has two distinct real roots, we use the condition on the discriminant of the quadratic equation.
The general form of a quadratic equation is given by \( Ax^2 + Bx + C = 0 \), where the discriminant \( D \) is calculated as:
\[... | 10,385 | [
{
"from": "human",
"value": "Determine the sum of all positive integer values of $a$ such that the quadratic equation $6x^2 - 7x + a = 0$ has two distinct real roots."
},
{
"from": "gpt",
"value": "To determine the values of \\( a \\) such that the quadratic equation \\( 6x^2 - 7x + a = 0 \\) ha... |
What is the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice? | What is the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice? | To find the minimum number of times we must throw three fair six-sided dice to ensure that the same sum is rolled twice, we first need to determine the range of possible sums that can be obtained when throwing three dice.
When rolling three six-sided dice:
- The minimum sum occurs when all dice show a 1, which is \(1... | 10,386 | [
{
"from": "human",
"value": "What is the minimum number of times you must throw three fair six-sided dice to ensure that the same sum is rolled twice?"
},
{
"from": "gpt",
"value": "To find the minimum number of times we must throw three fair six-sided dice to ensure that the same sum is rolled ... |
A certain company has a hierarchical structure with three levels: interns, employees, and managers. Each year, the current interns are promoted to employees, each employee is assigned a new intern to supervise, and a certain number of new managers are hired from outside the company. Initially, there are 20 employees, 5... | A certain company has a hierarchical structure with three levels: interns, employees, and managers. Each year, the current interns are promoted to employees, each employee is assigned a new intern to supervise, and a certain number of new managers are hired from outside the company. Initially, there are 20 employees, 5... | To solve the problem, we need to keep track of the number of interns, employees, and managers each year, considering the promotions and new hires.
### Year 0 (initial values)
- **Interns**: 10
- **Employees**: 20
- **Managers**: 5
### Year 1
1. **Promotion**: All interns become employees.
- New employees = 10 (pre... | 10,387 | [
{
"from": "human",
"value": "A certain company has a hierarchical structure with three levels: interns, employees, and managers. Each year, the current interns are promoted to employees, each employee is assigned a new intern to supervise, and a certain number of new managers are hired from outside the comp... |
The quadratic function $f(x) = x^2 - 4x - 5$ has two real roots. Find the number of integers $n$ such that $f(n) > 0$ and $f(n+1) < 0$. | The quadratic function $f(x) = x^2 - 4x - 5$ has two real roots. Find the number of integers $n$ such that $f(n) > 0$ and $f(n+1) < 0$. | We begin with the quadratic function given by
\[
f(x) = x^2 - 4x - 5.
\]
Next, we will find the roots of this quadratic function using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1}.
\]
Calculating the discriminant:
\[
b^2 - 4ac = 16 ... | 10,388 | [
{
"from": "human",
"value": "The quadratic function $f(x) = x^2 - 4x - 5$ has two real roots. Find the number of integers $n$ such that $f(n) > 0$ and $f(n+1) < 0$."
},
{
"from": "gpt",
"value": "We begin with the quadratic function given by\n\n\\[\nf(x) = x^2 - 4x - 5.\n\\]\n\nNext, we will fin... |
The roots of the equation $x^4 + 4x^2 + 4 = 0$ can be expressed in the form $a + bi$, where $a$ and $b$ are real. Find the sum of the squares of the roots with $b > 0$. | The roots of the equation $x^4 + 4x^2 + 4 = 0$ can be expressed in the form $a + bi$, where $a$ and $b$ are real. Find the sum of the squares of the roots with $b > 0$. | To solve the equation \( x^4 + 4x^2 + 4 = 0 \), we can start by letting \( y = x^2 \). This changes the equation into:
\[
y^2 + 4y + 4 = 0
\]
Now we can factor this quadratic equation:
\[
(y + 2)^2 = 0
\]
Thus, we find a double root:
\[
y + 2 = 0 \implies y = -2
\]
Since \( y = x^2 \), we have:
\[
x^2 = -2
\]
T... | 10,389 | [
{
"from": "human",
"value": "The roots of the equation $x^4 + 4x^2 + 4 = 0$ can be expressed in the form $a + bi$, where $a$ and $b$ are real. Find the sum of the squares of the roots with $b > 0$."
},
{
"from": "gpt",
"value": "To solve the equation \\( x^4 + 4x^2 + 4 = 0 \\), we can start by l... |
A sequence $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, $\ldots$ of points in the coordinate plane satisfies
\[(x_{n + 1}, y_{n + 1}) = (x_n + 2y_n, -y_n)\]for $n = 1,2,3,\ldots$. Suppose that $(x_{123},y_{123}) = (14,7)$. What is $x_1 + y_1$? | A sequence $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, $\ldots$ of points in the coordinate plane satisfies
\[(x_{n + 1}, y_{n + 1}) = (x_n + 2y_n, -y_n)\]for $n = 1,2,3,\ldots$. Suppose that $(x_{123},y_{123}) = (14,7)$. What is $x_1 + y_1$? | We start by analyzing the recursive equations for the points in the sequence:
\[
(x_{n + 1}, y_{n + 1}) = (x_n + 2y_n, -y_n).
\]
The transformations can be separated into two components:
1. The \( x \)-coordinate update: \( x_{n+1} = x_n + 2y_n \)
2. The \( y \)-coordinate update: \( y_{n+1} = -y_n \)
From this rel... | 10,390 | [
{
"from": "human",
"value": "A sequence $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$, $\\ldots$ of points in the coordinate plane satisfies\n\\[(x_{n + 1}, y_{n + 1}) = (x_n + 2y_n, -y_n)\\]for $n = 1,2,3,\\ldots$. Suppose that $(x_{123},y_{123}) = (14,7)$. What is $x_1 + y_1$?"
},
{
"from": "gpt",
"v... |
Points $A$ and $B$ are on parabola $y=3x^2-2x-1$, and the origin is the midpoint of $\overline{AB}$. Find the square of the length of $\overline{AB}$. | Points $A$ and $B$ are on parabola $y=3x^2-2x-1$, and the origin is the midpoint of $\overline{AB}$. Find the square of the length of $\overline{AB}$. | Let the coordinates of points \( A \) and \( B \) be \( A(a, 3a^2 - 2a - 1) \) and \( B(b, 3b^2 - 2b - 1) \). The origin \((0,0)\) being the midpoint of \( \overline{AB} \) gives us the following equations:
\[
\frac{a + b}{2} = 0 \quad \Rightarrow \quad a + b = 0 \quad \Rightarrow \quad b = -a
\]
Next, we need to cal... | 10,391 | [
{
"from": "human",
"value": "Points $A$ and $B$ are on parabola $y=3x^2-2x-1$, and the origin is the midpoint of $\\overline{AB}$. Find the square of the length of $\\overline{AB}$."
},
{
"from": "gpt",
"value": "Let the coordinates of points \\( A \\) and \\( B \\) be \\( A(a, 3a^2 - 2a - 1) \\... |
A positive integer $n$ has exactly four positive divisors. What is the smallest possible value of the sum of these four divisors? | A positive integer $n$ has exactly four positive divisors. What is the smallest possible value of the sum of these four divisors? | To find a positive integer \( n \) that has exactly four positive divisors, we need to consider the possible forms of \( n \). A number has exactly four divisors if it is either of the form \( p^3 \) where \( p \) is a prime, or of the form \( pq \) where \( p \) and \( q \) are distinct primes.
1. **Case 1: \( n = p^... | 10,392 | [
{
"from": "human",
"value": "A positive integer $n$ has exactly four positive divisors. What is the smallest possible value of the sum of these four divisors?"
},
{
"from": "gpt",
"value": "To find a positive integer \\( n \\) that has exactly four positive divisors, we need to consider the poss... |
If the average of five numbers is 2.8, what is the sum of the five numbers? | If the average of five numbers is 2.8, what is the sum of the five numbers? | To find the sum of five numbers when their average is given, you can use the formula for the average:
\[
\text{Average} = \frac{\text{Sum of the numbers}}{\text{Number of numbers}}
\]
In this case, we have the average (2.8) and the number of numbers (5). We can rearrange the formula to find the sum:
\[
\text{Sum of ... | 10,393 | [
{
"from": "human",
"value": "If the average of five numbers is 2.8, what is the sum of the five numbers?"
},
{
"from": "gpt",
"value": "To find the sum of five numbers when their average is given, you can use the formula for the average:\n\n\\[\n\\text{Average} = \\frac{\\text{Sum of the numbers... |
Let $a_n = 1 + 2 + 2^2 + \dots + 2^{n - 1}$. Find the smallest $k$ such that $a_k$ is divisible by $11$ for $k \ge 3$. | Let $a_n = 1 + 2 + 2^2 + \dots + 2^{n - 1}$. Find the smallest $k$ such that $a_k$ is divisible by $11$ for $k \ge 3$. | The series \( a_n = 1 + 2 + 2^2 + \dots + 2^{n-1} \) is a geometric series that can be summed using the formula for the sum of a geometric series:
\[
S = a \frac{r^n - 1}{r - 1}
\]
In our case, the first term \( a = 1 \) and the common ratio \( r = 2 \). Therefore, we have:
\[
a_n = \frac{2^n - 1}{2 - 1} = 2^n - 1
\... | 10,394 | [
{
"from": "human",
"value": "Let $a_n = 1 + 2 + 2^2 + \\dots + 2^{n - 1}$. Find the smallest $k$ such that $a_k$ is divisible by $11$ for $k \\ge 3$."
},
{
"from": "gpt",
"value": "The series \\( a_n = 1 + 2 + 2^2 + \\dots + 2^{n-1} \\) is a geometric series that can be summed using the formula ... |
If $y=\frac{12x^4+4x^3+9x^2+5x+3}{3x^4+2x^3+8x^2+3x+1}$, at what value of $y$ will there be a horizontal asymptote? | If $y=\frac{12x^4+4x^3+9x^2+5x+3}{3x^4+2x^3+8x^2+3x+1}$, at what value of $y$ will there be a horizontal asymptote? | To determine the horizontal asymptote of the function
\[
y = \frac{12x^4 + 4x^3 + 9x^2 + 5x + 3}{3x^4 + 2x^3 + 8x^2 + 3x + 1},
\]
we need to analyze the degrees of the polynomial in the numerator and the denominator.
The numerator is \(12x^4 + 4x^3 + 9x^2 + 5x + 3\), which is a polynomial of degree 4. The leading t... | 10,395 | [
{
"from": "human",
"value": "If $y=\\frac{12x^4+4x^3+9x^2+5x+3}{3x^4+2x^3+8x^2+3x+1}$, at what value of $y$ will there be a horizontal asymptote?"
},
{
"from": "gpt",
"value": "To determine the horizontal asymptote of the function \n\n\\[\ny = \\frac{12x^4 + 4x^3 + 9x^2 + 5x + 3}{3x^4 + 2x^3 + 8... |
Given the polynomial $p(x, y) = ax^3 + bx^2y + cxy^2 + dy^3$, where $a, b, c,$ and $d$ are nonzero constants, find the point $(r, s)$ such that $p(r, s) = 0$ and the distance from $(r, s)$ to the line $x + y = 0$ is a minimum. | Given the polynomial $p(x, y) = ax^3 + bx^2y + cxy^2 + dy^3$, where $a, b, c,$ and $d$ are nonzero constants, find the point $(r, s)$ such that $p(r, s) = 0$ and the distance from $(r, s)$ to the line $x + y = 0$ is a minimum. | To find the point \((r, s)\) such that \(p(r, s) = 0\) and the distance from \((r, s)\) to the line \(x + y = 0\) is minimized, we start by analyzing the conditions we are given.
1. **Distance**: The distance \(D\) from a point \((r, s)\) to the line \(x + y = 0\) can be computed using the formula:
\[
D = \frac{|r + ... | 10,396 | [
{
"from": "human",
"value": "Given the polynomial $p(x, y) = ax^3 + bx^2y + cxy^2 + dy^3$, where $a, b, c,$ and $d$ are nonzero constants, find the point $(r, s)$ such that $p(r, s) = 0$ and the distance from $(r, s)$ to the line $x + y = 0$ is a minimum."
},
{
"from": "gpt",
"value": "To find t... |
A company has 36 employees and wants to divide them into teams of no more than 8 employees each. What is the minimum number of teams required to accomplish this? | A company has 36 employees and wants to divide them into teams of no more than 8 employees each. What is the minimum number of teams required to accomplish this? | To determine the minimum number of teams required to accommodate 36 employees, we need to divide the total number of employees by the maximum number of employees allowed per team.
Given:
- Total employees = 36
- Maximum team size = 8 employees
We can calculate the number of teams needed by dividing the total number o... | 10,397 | [
{
"from": "human",
"value": "A company has 36 employees and wants to divide them into teams of no more than 8 employees each. What is the minimum number of teams required to accomplish this?"
},
{
"from": "gpt",
"value": "To determine the minimum number of teams required to accommodate 36 employ... |
Expand $(2t^2 -3t+2)(-3t^2 + t-5)$. | Expand $(2t^2 -3t+2)(-3t^2 + t-5)$. | To expand the expression \((2t^2 - 3t + 2)(-3t^2 + t - 5)\), we will distribute each term from the first polynomial to each term in the second polynomial.
Let's start by distributing \(2t^2\):
\[
2t^2 \cdot (-3t^2) = -6t^4
\]
\[
2t^2 \cdot (t) = 2t^3
\]
\[
2t^2 \cdot (-5) = -10t^2
\]
Next, distribute \(-3t\):
\[
-3... | 10,398 | [
{
"from": "human",
"value": "Expand $(2t^2 -3t+2)(-3t^2 + t-5)$."
},
{
"from": "gpt",
"value": "To expand the expression \\((2t^2 - 3t + 2)(-3t^2 + t - 5)\\), we will distribute each term from the first polynomial to each term in the second polynomial.\n\nLet's start by distributing \\(2t^2\\):\... |
Let $f(x) = e^{x^2}$ and $g(x) = e^{1/x}$. Find the range of $f(x) + g(x)$ on the interval $(0, 1]$. | Let $f(x) = e^{x^2}$ and $g(x) = e^{1/x}$. Find the range of $f(x) + g(x)$ on the interval $(0, 1]$. | To find the range of the function \( h(x) = f(x) + g(x) = e^{x^2} + e^{1/x} \) on the interval \( (0, 1] \), we first need to analyze the behavior of each component in the interval and then find the minimum and maximum values of \( h(x) \).
1. **Behavior of \( f(x) = e^{x^2} \)**:
- As \( x \) approaches \( 0 \), \... | 10,399 | [
{
"from": "human",
"value": "Let $f(x) = e^{x^2}$ and $g(x) = e^{1/x}$. Find the range of $f(x) + g(x)$ on the interval $(0, 1]$."
},
{
"from": "gpt",
"value": "To find the range of the function \\( h(x) = f(x) + g(x) = e^{x^2} + e^{1/x} \\) on the interval \\( (0, 1] \\), we first need to analy... |
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