instruction_seed stringlengths 21 5.45k | response_seed stringlengths 21 5.45k ⌀ | reasoning stringlengths 484 91.2k | deepseek_solution stringlengths 18 13.3k | __original_row_idx int64 51 40k ⌀ | source stringclasses 2
values | final_reasoning_trace stringlengths 879 93k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
** Give an example of a polynomial $f(x)$ with integer coefficients, such that for $m$ different positive integers $x$, it yields $m$ different prime numbers. | null | Okay, so I need to come up with a polynomial f(x) with integer coefficients. The condition is that for m different positive integers x, the polynomial f(x) should produce m different prime numbers. Hmm, let me think about how to approach this.
First, I remember that there's a theorem related to primes and polynomials.... | To find a polynomial \( f(x) \) with integer coefficients such that for \( m \) different positive integers \( x \), it yields \( m \) different prime numbers, we can consider the simplest polynomial: the identity function.
1. **Polynomial Choice**: The polynomial \( f(x) = x \) has integer coefficients.
2. **Prime Ge... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to come up with a polynomial f(x) with integer coefficients. The condition is that for m different positive integers x, the polynomial f(x) should produce m different prime numbers. Hmm, let me think about how to approach this.
First, I remember that there's a theorem related to primes and poly... | [
{
"from": "human",
"value": "** Give an example of a polynomial $f(x)$ with integer coefficients, such that for $m$ different positive integers $x$, it yields $m$ different prime numbers."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to come up with a polynomial f(x) with integer coeff... |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 24.$ Find the largest possible value of $b.$ | Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 24.$ Find the largest possible value of $b.$ | Okay, so I need to find the largest possible value of \( b \) in a three-term geometric series where all terms are positive, and their sum is 24. Let me start by recalling what a geometric series is. In a geometric series, each term is multiplied by a common ratio to get the next term. So if the three terms are \( a \)... | Given a three-term geometric series \(a, b, c\) with all terms positive and their sum equal to 24, we need to find the largest possible value of \(b\).
Since \(a, b, c\) form a geometric series, we can write \(b = ar\) and \(c = ar^2\) where \(r\) is the common ratio. The sum of the terms is:
\[
a + ar + ar^2 = 24
\]
... | 19,277 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the largest possible value of \( b \) in a three-term geometric series where all terms are positive, and their sum is 24. Let me start by recalling what a geometric series is. In a geometric series, each term is multiplied by a common ratio to get the next term. So if the three terms are... | [
{
"from": "human",
"value": "Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 24.$ Find the largest possible value of $b.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest possible value of \\( b \\) in a three-te... |
q ' = 3 q - 3 , what is the value of ( 3 ' ) ' ? | null | Okay, let me try to figure out this problem. The question is: given that q' = 3q - 3, what is the value of (3')'? Hmm, primes can sometimes be confusing because they might mean different things in different contexts. But here, since it's defined as q' = 3q - 3, I think the prime symbol (') is being used to denote a fun... | Given the operation defined by \( q' = 3q - 3 \), we need to find the value of \( (3')' \).
First, we compute the inner prime operation on 3:
\[
3' = 3 \cdot 3 - 3 = 9 - 3 = 6
\]
Next, we apply the prime operation again to the result:
\[
6' = 3 \cdot 6 - 3 = 18 - 3 = 15
\]
Thus, the value of \( (3')' \) is \(\boxed{... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem. The question is: given that q' = 3q - 3, what is the value of (3')'? Hmm, primes can sometimes be confusing because they might mean different things in different contexts. But here, since it's defined as q' = 3q - 3, I think the prime symbol (') is being used to deno... | [
{
"from": "human",
"value": "q ' = 3 q - 3 , what is the value of ( 3 ' ) ' ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. The question is: given that q' = 3q - 3, what is the value of (3')'? Hmm, primes can sometimes be confusing because they might mean d... |
Simplify
\[\frac{\sin{10^\circ}+\sin{20^\circ}}{\cos{10^\circ}+\cos{20^\circ}}.\]Enter your answer is a trigonometric function evaluated at an integer, such as "sin 7". (The angle should be positive and as small as possible.) | Simplify
\[\frac{\sin{10^\circ}+\sin{20^\circ}}{\cos{10^\circ}+\cos{20^\circ}}.\]Enter your answer is a trigonometric function evaluated at an integer, such as "sin 7". (The angle should be positive and as small as possible.) | Okay, let's see. I need to simplify this trigonometric expression: (sin10° + sin20°)/(cos10° + cos20°). Hmm, the problem says to express it as a trigonometric function evaluated at an integer angle, like "sin 7", and the angle should be as small and positive as possible. Alright, so maybe I can use some trigonometric i... | To simplify the expression \(\frac{\sin{10^\circ}+\sin{20^\circ}}{\cos{10^\circ}+\cos{20^\circ}}\), we use the sum-to-product identities for sine and cosine.
For the numerator \(\sin{10^\circ} + \sin{20^\circ}\):
\[
\sin{10^\circ} + \sin{20^\circ} = 2 \sin{\left(\frac{10^\circ + 20^\circ}{2}\right)} \cos{\left(\frac{1... | 33,695 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to simplify this trigonometric expression: (sin10° + sin20°)/(cos10° + cos20°). Hmm, the problem says to express it as a trigonometric function evaluated at an integer angle, like "sin 7", and the angle should be as small and positive as possible. Alright, so maybe I can use some trigono... | [
{
"from": "human",
"value": "Simplify\n\\[\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\cos{20^\\circ}}.\\]Enter your answer is a trigonometric function evaluated at an integer, such as \"sin 7\". (The angle should be positive and as small as possible.)"
},
{
"from": "gpt",
"valu... |
Shyne can grow some eggplants in every seed packet and 10 sunflowers in every seed packet. She bought 4 seed packets of eggplants and 6 seed packets of sunflower. Shyne can grow 116 plants in her backyard. How many eggplants can she grow in every seed packet? | null | Okay, let's see. The problem says Shyne can grow some eggplants in every seed packet and 10 sunflowers in every seed packet. She bought 4 seed packets of eggplants and 6 seed packets of sunflower. She can grow 116 plants in her backyard. The question is, how many eggplants can she grow in every seed packet?
Hmm. Let m... | Let \( x \) be the number of eggplants in each seed packet. Shyne bought 4 seed packets of eggplants and 6 seed packets of sunflowers. Each sunflower packet contains 10 seeds. The total number of plants she can grow is 116.
The total number of eggplants from 4 packets is \( 4x \).
The total number of sunflowers from 6... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. The problem says Shyne can grow some eggplants in every seed packet and 10 sunflowers in every seed packet. She bought 4 seed packets of eggplants and 6 seed packets of sunflower. She can grow 116 plants in her backyard. The question is, how many eggplants can she grow in every seed packet?
Hm... | [
{
"from": "human",
"value": "Shyne can grow some eggplants in every seed packet and 10 sunflowers in every seed packet. She bought 4 seed packets of eggplants and 6 seed packets of sunflower. Shyne can grow 116 plants in her backyard. How many eggplants can she grow in every seed packet?"
},
{
"from... |
A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshelf is currently empty, what is the least number of shelves that need to be used to store 24 books of different sizes, assuming that books of the same size must be placed on the same shelf? | A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshelf is currently empty, what is the least number of shelves that need to be used to store 24 books of different sizes, assuming that books of the same size must be placed on the same shelf? | Alright, so I need to figure out the least number of shelves required to store 24 books of different sizes on a bookshelf that has 5 shelves. Each shelf can hold at most 8 books of the same size. The catch here is that books of the same size must be on the same shelf. Since all 24 books are different sizes, that means ... | The problem requires determining the least number of shelves needed to store 24 books of different sizes on a bookshelf with 5 shelves, where each shelf can hold at most 8 books of the same size.
Key points:
1. Each shelf can hold at most 8 books of the same size.
2. All 24 books are of different sizes, meaning each ... | 3,003 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to figure out the least number of shelves required to store 24 books of different sizes on a bookshelf that has 5 shelves. Each shelf can hold at most 8 books of the same size. The catch here is that books of the same size must be on the same shelf. Since all 24 books are different sizes, tha... | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshelf is currently empty, what is the least number of shelves that need to be used to store 24 books of different sizes, assuming that books of the same size must be placed on the sam... |
A sector with an obtuse central angle $\phi$ is cut from a circle of radius 8. Find the radius of the circle circumscribed about the sector. | null | Okay, so I have this problem here: A sector with an obtuse central angle φ is cut from a circle of radius 8. Find the radius of the circle circumscribed about the sector. Hmm, let me try to visualize this. A sector is like a "slice" of a circle, right? So if the original circle has radius 8, the sector has two radii of... | To find the radius of the circle circumscribed about a sector with an obtuse central angle \(\phi\) cut from a circle of radius 8, we need to consider the circumradius of the triangle formed by the center of the circle (O) and the endpoints of the arc (A and B).
1. **Triangle OAB**: The triangle formed by the points O... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here: A sector with an obtuse central angle φ is cut from a circle of radius 8. Find the radius of the circle circumscribed about the sector. Hmm, let me try to visualize this. A sector is like a "slice" of a circle, right? So if the original circle has radius 8, the sector has two ... | [
{
"from": "human",
"value": "A sector with an obtuse central angle $\\phi$ is cut from a circle of radius 8. Find the radius of the circle circumscribed about the sector."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: A sector with an obtuse central angle φ is cut fro... |
Solve the following system of equations:
\begin{align*}
x+y+z&=6,\\
x+2y+3z&=12,\\
x+3y+4z&=17.
\end{align*}
Express your answer as an ordered triple $(x,y,z)$. | Solve the following system of equations:
\begin{align*}
x+y+z&=6,\\
x+2y+3z&=12,\\
x+3y+4z&=17.
\end{align*}
Express your answer as an ordered triple $(x,y,z)$. | Okay, so I need to solve this system of equations. Let me write them down again to make sure I have them right:
1) x + y + z = 6
2) x + 2y + 3z = 12
3) x + 3y + 4z = 17
Hmm, three equations with three variables. That means I can probably solve for each variable step by step. Let me think about how to approach this.... | To solve the system of equations:
\[
\begin{align*}
x + y + z &= 6, \\
x + 2y + 3z &= 12, \\
x + 3y + 4z &= 17,
\end{align*}
\]
we start by eliminating \(x\) from the second and third equations. Subtract the first equation from the second equation:
\[
(x + 2y + 3z) - (x + y + z) = 12 - 6 \implies y + 2z = 6. \quad \... | 6,462 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this system of equations. Let me write them down again to make sure I have them right:
1) x + y + z = 6
2) x + 2y + 3z = 12
3) x + 3y + 4z = 17
Hmm, three equations with three variables. That means I can probably solve for each variable step by step. Let me think about how to approa... | [
{
"from": "human",
"value": "Solve the following system of equations:\n\n\\begin{align*}\nx+y+z&=6,\\\\\nx+2y+3z&=12,\\\\\nx+3y+4z&=17.\n\\end{align*}\n\nExpress your answer as an ordered triple $(x,y,z)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this system of equations.... |
Let's prove that
$$
183^{n}-126^{n}
$$
is divisible by 1957 if $n$ is an even number. | null | Alright, so I need to prove that \(183^{n} - 126^{n}\) is divisible by 1957 when \(n\) is even. Hmm, okay. Let me start by breaking down the problem.
First, 1957 seems like a composite number. Maybe I should factor it to see if I can apply the Chinese Remainder Theorem or check divisibility by its prime factors. Let ... | To prove that \(183^{n} - 126^{n}\) is divisible by 1957 if \(n\) is an even number, we start by factoring 1957.
1957 factors into the primes 19 and 103. We need to show that \(183^{n} - 126^{n}\) is divisible by both 19 and 103 when \(n\) is even.
**Modulo 19:**
- Calculate \(183 \mod 19\) and \(126 \mod 19\):
- ... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I need to prove that \(183^{n} - 126^{n}\) is divisible by 1957 when \(n\) is even. Hmm, okay. Let me start by breaking down the problem.
First, 1957 seems like a composite number. Maybe I should factor it to see if I can apply the Chinese Remainder Theorem or check divisibility by its prime facto... | [
{
"from": "human",
"value": "Let's prove that\n\n$$\n183^{n}-126^{n}\n$$\n\nis divisible by 1957 if $n$ is an even number."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to prove that \\(183^{n} - 126^{n}\\) is divisible by 1957 when \\(n\\) is even. Hmm, okay. Let me start by breaki... |
1.17. Let $x_{1}$ and $x_{2}$ be the roots of the quadratic equation $x^{2} + a x + b = 0$, and let $s_{n} = x_{1}^{n} + x_{2}^{n}$. Prove that
$$
\frac{s_{n}}{n} = \sum_{m} (-1)^{n+m} \frac{(n-m-1)!}{m!(n-2m)!} a^{n-2m} b^{m}
$$
where the summation is over all integers $m$ such that $0 \leqslant m \leqslant n / 2$ (... | null | Okay, so I have this problem here about quadratic equations and something called Waring's formula. Let me try to parse what it's asking. The problem states that for a quadratic equation \(x^2 + a x + b = 0\) with roots \(x_1\) and \(x_2\), we define \(s_n = x_1^n + x_2^n\). Then, we need to prove that \(\frac{s_n}{n}\)... | To prove Waring's formula for the sum \( \frac{s_n}{n} \) where \( s_n = x_1^n + x_2^n \) and \( x_1, x_2 \) are the roots of the quadratic equation \( x^2 + ax + b = 0 \), we use generating functions and series expansions.
1. **Recurrence Relation**: Using Vieta's formulas, we know \( x_1 + x_2 = -a \) and \( x_1 x_2... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here about quadratic equations and something called Waring's formula. Let me try to parse what it's asking. The problem states that for a quadratic equation \(x^2 + a x + b = 0\) with roots \(x_1\) and \(x_2\), we define \(s_n = x_1^n + x_2^n\). Then, we need to prove that \(\frac{s... | [
{
"from": "human",
"value": "1.17. Let $x_{1}$ and $x_{2}$ be the roots of the quadratic equation $x^{2} + a x + b = 0$, and let $s_{n} = x_{1}^{n} + x_{2}^{n}$. Prove that\n\n$$\n\\frac{s_{n}}{n} = \\sum_{m} (-1)^{n+m} \\frac{(n-m-1)!}{m!(n-2m)!} a^{n-2m} b^{m}\n$$\n\nwhere the summation is over all intege... |
[ Permutations and substitutions (other). ] [ Regular polygons $]
In the vertices of a regular 1983-gon, the numbers $1,2, \ldots, 1983$ are arranged. Any axis of symmetry divides the numbers not lying on it into two sets. We call an arrangement "good" relative to a given axis of symmetry if each number in one set is... | null | Okay, let's tackle this problem. So, we have a regular 1983-gon with the numbers 1 through 1983 arranged at its vertices. The question is about whether there exists an arrangement such that for any axis of symmetry of the polygon, the numbers not on the axis are split into two sets by the axis, and each number in one s... | To determine if there exists an arrangement of the numbers \(1, 2, \ldots, 1983\) on the vertices of a regular 1983-gon that is "good" relative to any axis of symmetry, we need to consider the constraints imposed by each axis. A "good" arrangement requires that for any axis of symmetry, each number in one set is greate... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's tackle this problem. So, we have a regular 1983-gon with the numbers 1 through 1983 arranged at its vertices. The question is about whether there exists an arrangement such that for any axis of symmetry of the polygon, the numbers not on the axis are split into two sets by the axis, and each number ... | [
{
"from": "human",
"value": "[ Permutations and substitutions (other). ] [ Regular polygons $]\n\nIn the vertices of a regular 1983-gon, the numbers $1,2, \\ldots, 1983$ are arranged. Any axis of symmetry divides the numbers not lying on it into two sets. We call an arrangement \"good\" relative to a given... |
Triangle $ABC$ has vertices $A(0, 10)$, $B(4, 0)$, $C(10, 0)$. A horizontal line with equation $y=t$ intersects line segment $\overline{AB}$ at $T$ and line segment $\overline{AC}$ at $U$, forming $\triangle ATU$ with area 15. Compute $t$. | null | Okay, so I have this problem here about triangle ABC with vertices at A(0,10), B(4,0), and C(10,0). A horizontal line y = t intersects segments AB and AC at points T and U, forming triangle ATU with area 15. I need to find the value of t. Hmm, let me think step by step how to approach this.
First, I should probably vi... | To solve for \( t \) where the horizontal line \( y = t \) intersects line segments \( \overline{AB} \) and \( \overline{AC} \) of triangle \( ABC \) with vertices \( A(0, 10) \), \( B(4, 0) \), and \( C(10, 0) \), forming triangle \( ATU \) with area 15, we follow these steps:
1. **Find the equations of lines \( AB \... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here about triangle ABC with vertices at A(0,10), B(4,0), and C(10,0). A horizontal line y = t intersects segments AB and AC at points T and U, forming triangle ATU with area 15. I need to find the value of t. Hmm, let me think step by step how to approach this.
First, I should pro... | [
{
"from": "human",
"value": "Triangle $ABC$ has vertices $A(0, 10)$, $B(4, 0)$, $C(10, 0)$. A horizontal line with equation $y=t$ intersects line segment $\\overline{AB}$ at $T$ and line segment $\\overline{AC}$ at $U$, forming $\\triangle ATU$ with area 15. Compute $t$."
},
{
"from": "gpt",
"va... |
Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x - 19,$ the remainder is 99, and when $P(x)$ is divided by $x - 99,$ the remainder is 19. What is the remainder when $P(x)$ is divided by $(x - 19)(x - 99)$? | Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x - 19,$ the remainder is 99, and when $P(x)$ is divided by $x - 99,$ the remainder is 19. What is the remainder when $P(x)$ is divided by $(x - 19)(x - 99)$? | Okay, so I have this polynomial P(x), and I need to find the remainder when it's divided by (x - 19)(x - 99). They told me that when P(x) is divided by (x - 19), the remainder is 99, and when divided by (x - 99), the remainder is 19. Hmm, okay, so this seems related to the Remainder Theorem. Let me recall... The Remain... | Given the polynomial \( P(x) \), we know that when \( P(x) \) is divided by \( x - 19 \), the remainder is 99, and when divided by \( x - 99 \), the remainder is 19. By the Remainder Theorem, this means \( P(19) = 99 \) and \( P(99) = 19 \).
When dividing \( P(x) \) by the quadratic polynomial \( (x - 19)(x - 99) \), ... | 23,143 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this polynomial P(x), and I need to find the remainder when it's divided by (x - 19)(x - 99). They told me that when P(x) is divided by (x - 19), the remainder is 99, and when divided by (x - 99), the remainder is 19. Hmm, okay, so this seems related to the Remainder Theorem. Let me recall... Th... | [
{
"from": "human",
"value": "Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x - 19,$ the remainder is 99, and when $P(x)$ is divided by $x - 99,$ the remainder is 19. What is the remainder when $P(x)$ is divided by $(x - 19)(x - 99)$?"
},
{
"from": "gpt",
"value": "<think>\nOka... |
Consider an ellipse given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$. Let $F$ be its right focus, $A$ be the intersection point of the line $x = \frac{a^2}{c}$ and the $x$-axis. If there exists a point $P$ on the ellipse such that the perpendicular bisector of segment $AP$ passes through point ... | null | Okay, so I need to solve this problem about an ellipse. Let me first recall some basic properties of an ellipse. The standard equation is x²/a² + y²/b² = 1, where a > b > 0. The eccentricity e is given by e = c/a, where c is the distance from the center to each focus, and c² = a² - b². So, the foci are located at (±c, ... | Given an ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) with \(a > b > 0\), we need to find the range of the eccentricity \(e\) such that there exists a point \(P\) on the ellipse where the perpendicular bisector of segment \(AP\) passes through the right focus \(F\).
1. **Key Points and Distances**:
- The righ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this problem about an ellipse. Let me first recall some basic properties of an ellipse. The standard equation is x²/a² + y²/b² = 1, where a > b > 0. The eccentricity e is given by e = c/a, where c is the distance from the center to each focus, and c² = a² - b². So, the foci are located ... | [
{
"from": "human",
"value": "Consider an ellipse given by the equation $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$. Let $F$ be its right focus, $A$ be the intersection point of the line $x = \\frac{a^2}{c}$ and the $x$-axis. If there exists a point $P$ on the ellipse such that the perpendicular bi... |
Given an unequal-sided $\triangle ABC$ satisfying $\angle BCA = 90^{\circ}$, $D$ is the foot of the perpendicular from $C$ to side $AB$. Take a point $X$ on segment $CD$, and let $K$ be a point on segment $AX$ such that $BK = BC$. Similarly, let $L$ be a point on segment $BX$ such that $AL = AC$. Point $T$ on segment $... | null | Okay, so I need to prove that angle ACT is equal to angle BCT. That means point T has to be such that CT is the angle bisector of angle ACB. But wait, angle ACB is 90 degrees, so if CT bisects it, then each of angles ACT and BCT would be 45 degrees. Hmm, interesting. But the problem doesn't state that T is the midpoint... | Given an unequal-sided right-angled triangle \( \triangle ABC \) with \( \angle BCA = 90^\circ \), we need to prove that \( \angle ACT = \angle BCT \) where \( T \) is a point on segment \( AB \) such that points \( K, L, T, D \) are concyclic. Here, \( D \) is the foot of the perpendicular from \( C \) to \( AB \), \(... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that angle ACT is equal to angle BCT. That means point T has to be such that CT is the angle bisector of angle ACB. But wait, angle ACB is 90 degrees, so if CT bisects it, then each of angles ACT and BCT would be 45 degrees. Hmm, interesting. But the problem doesn't state that T is the ... | [
{
"from": "human",
"value": "Given an unequal-sided $\\triangle ABC$ satisfying $\\angle BCA = 90^{\\circ}$, $D$ is the foot of the perpendicular from $C$ to side $AB$. Take a point $X$ on segment $CD$, and let $K$ be a point on segment $AX$ such that $BK = BC$. Similarly, let $L$ be a point on segment $BX$... |
Simplify \( \sqrt{360000} \). | null | Okay, so I need to simplify the square root of 360,000. Hmm, let's see. I remember that simplifying a square root involves breaking it down into factors that are perfect squares. That way, you can take the square root of those perfect squares and multiply them outside the radical. But I'm a bit rusty, so let me take it... | To simplify \( \sqrt{360000} \), we can break down the number into factors that are perfect squares.
First, note that:
\[
360000 = 3600 \times 100
\]
Both 3600 and 100 are perfect squares. The square roots of these factors are:
\[
\sqrt{3600} = 60 \quad \text{and} \quad \sqrt{100} = 10
\]
Multiplying these results to... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to simplify the square root of 360,000. Hmm, let's see. I remember that simplifying a square root involves breaking it down into factors that are perfect squares. That way, you can take the square root of those perfect squares and multiply them outside the radical. But I'm a bit rusty, so let me... | [
{
"from": "human",
"value": "Simplify \\( \\sqrt{360000} \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to simplify the square root of 360,000. Hmm, let's see. I remember that simplifying a square root involves breaking it down into factors that are perfect squares. That way, you c... |
In a triangular array of numbers, each number is the sum of the two numbers directly above it. The first row consists of the numbers 1, 2, and 3. The second row consists of the numbers 3, 5, and 7. The third row consists of the numbers 6, 10, and 14. And so on. Which row has the greatest number of entries that are divi... | In a triangular array of numbers, each number is the sum of the two numbers directly above it. The first row consists of the numbers 1, 2, and 3. The second row consists of the numbers 3, 5, and 7. The third row consists of the numbers 6, 10, and 14. And so on. Which row has the greatest number of entries that are divi... | Okay, let's see. I need to figure out which row in this triangular array has the most entries divisible by 5. The first row is 1, 2, 3. The second row is 3, 5, 7. The third row is 6, 10, 14. And it continues on like that. Each number is the sum of the two numbers above it. So, similar to Pascal's triangle, but starting... | The problem involves a triangular array where each number is the sum of the two numbers directly above it. The first few rows are given, and we need to determine which row has the greatest number of entries divisible by 5.
1. **Understanding the Construction**:
- The first row is \(1, 2, 3\).
- The second row is... | 7,791 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to figure out which row in this triangular array has the most entries divisible by 5. The first row is 1, 2, 3. The second row is 3, 5, 7. The third row is 6, 10, 14. And it continues on like that. Each number is the sum of the two numbers above it. So, similar to Pascal's triangle, but ... | [
{
"from": "human",
"value": "In a triangular array of numbers, each number is the sum of the two numbers directly above it. The first row consists of the numbers 1, 2, and 3. The second row consists of the numbers 3, 5, and 7. The third row consists of the numbers 6, 10, and 14. And so on. Which row has the... |
A fair six-sided die is rolled twice. What is the expected value of the product of the numbers on the two rolls? | A fair six-sided die is rolled twice. What is the expected value of the product of the numbers on the two rolls? | Okay, so I need to find the expected value of the product of two rolls of a fair six-sided die. Hmm, let me start by recalling what expected value means. The expected value is like the average outcome we would expect if we did an experiment many times. So in this case, if I roll a die twice, multiply the two numbers, a... | To find the expected value of the product of the numbers on two rolls of a fair six-sided die, we start by noting that each roll is an independent random variable. Let \(X\) and \(Y\) represent the outcomes of the first and second rolls, respectively. Both \(X\) and \(Y\) can take values from 1 to 6 with equal probabil... | 7,087 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the expected value of the product of two rolls of a fair six-sided die. Hmm, let me start by recalling what expected value means. The expected value is like the average outcome we would expect if we did an experiment many times. So in this case, if I roll a die twice, multiply the two nu... | [
{
"from": "human",
"value": "A fair six-sided die is rolled twice. What is the expected value of the product of the numbers on the two rolls?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the expected value of the product of two rolls of a fair six-sided die. Hmm, let me start ... |
A set of 8 coins consists of 2 pennies, 3 nickels, and 3 dimes. Two of the coins are selected at random and put into a box. If the identity of each coin (penny, nickel, or dime) and the total value of the coins are known, how many distinct possible combinations of coins could be in the box? | A set of 8 coins consists of 2 pennies, 3 nickels, and 3 dimes. Two of the coins are selected at random and put into a box. If the identity of each coin (penny, nickel, or dime) and the total value of the coins are known, how many distinct possible combinations of coins could be in the box? | Okay, let's see. So, we have a set of 8 coins: 2 pennies, 3 nickels, and 3 dimes. We need to pick two coins at random and put them into a box. Then, if we know the identity (type) of each coin and the total value, how many distinct possible combinations could be in the box? Hmm, the question is asking for the number of... | We have a set of 8 coins consisting of 2 pennies (P), 3 nickels (N), and 3 dimes (D). We need to determine the number of distinct possible combinations of coins that could be in the box when two coins are selected at random, considering both the identity of each coin and the total value.
First, we list all possible co... | 35,716 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. So, we have a set of 8 coins: 2 pennies, 3 nickels, and 3 dimes. We need to pick two coins at random and put them into a box. Then, if we know the identity (type) of each coin and the total value, how many distinct possible combinations could be in the box? Hmm, the question is asking for the n... | [
{
"from": "human",
"value": "A set of 8 coins consists of 2 pennies, 3 nickels, and 3 dimes. Two of the coins are selected at random and put into a box. If the identity of each coin (penny, nickel, or dime) and the total value of the coins are known, how many distinct possible combinations of coins could be... |
Find the value of \( k \) if \((x^2 - k)(x + 3k) = x^3 + k(x^2 - 2x - 8)\), given that \( k \neq 0 \). | null | Okay, let's see. I need to find the value of \( k \) such that when I expand the left side of the equation \( (x^2 - k)(x + 3k) \), it equals the right side \( x^3 + k(x^2 - 2x - 8) \). And \( k \) isn't zero, so that's a hint that maybe there's a specific solution. Alright, let's start by expanding the left side and t... | To find the value of \( k \) such that \((x^2 - k)(x + 3k) = x^3 + k(x^2 - 2x - 8)\) given that \( k \neq 0 \), we start by expanding both sides of the equation and comparing coefficients.
First, expand the left-hand side (LHS):
\[
(x^2 - k)(x + 3k) = x^3 + 3kx^2 - kx - 3k^2
\]
Next, expand the right-hand side (RHS):... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find the value of \( k \) such that when I expand the left side of the equation \( (x^2 - k)(x + 3k) \), it equals the right side \( x^3 + k(x^2 - 2x - 8) \). And \( k \) isn't zero, so that's a hint that maybe there's a specific solution. Alright, let's start by expanding the left si... | [
{
"from": "human",
"value": "Find the value of \\( k \\) if \\((x^2 - k)(x + 3k) = x^3 + k(x^2 - 2x - 8)\\), given that \\( k \\neq 0 \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the value of \\( k \\) such that when I expand the left side of the equation \\( (x^2... |
Let's consider a function $g(x) = \sqrt{x^2 - 2x + 1} + c$, where $c$ is a constant. If the graph of $g(x)$ intersects the line $y = x$ at exactly one point, what is the value of $c$? | Let's consider a function $g(x) = \sqrt{x^2 - 2x + 1} + c$, where $c$ is a constant. If the graph of $g(x)$ intersects the line $y = x$ at exactly one point, what is the value of $c$? | Okay, so I need to find the value of c such that the graph of g(x) = sqrt(x² - 2x + 1) + c intersects the line y = x at exactly one point. Hmm, let's start by understanding the problem step by step.
First, let me rewrite the function g(x) to see if I can simplify it. The expression under the square root is x² - 2x + 1... | To find the value of \( c \) such that the graph of \( g(x) = \sqrt{x^2 - 2x + 1} + c \) intersects the line \( y = x \) at exactly one point, we start by simplifying the function \( g(x) \).
The expression inside the square root, \( x^2 - 2x + 1 \), is a perfect square:
\[
\sqrt{x^2 - 2x + 1} = |x - 1|
\]
Thus, the f... | 22,156 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of c such that the graph of g(x) = sqrt(x² - 2x + 1) + c intersects the line y = x at exactly one point. Hmm, let's start by understanding the problem step by step.
First, let me rewrite the function g(x) to see if I can simplify it. The expression under the square root is x² ... | [
{
"from": "human",
"value": "Let's consider a function $g(x) = \\sqrt{x^2 - 2x + 1} + c$, where $c$ is a constant. If the graph of $g(x)$ intersects the line $y = x$ at exactly one point, what is the value of $c$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of c suc... |
in an election between two candidates , one got 55 % of the total valid votes . 20 % of the votes were invalid . if the total number of votes was 7500 , what was the number of valid votes that the other candidate got ? | null | Okay, let's see. There's an election between two candidates. One got 55% of the total valid votes. 20% of the votes were invalid. The total number of votes was 7500. We need to find the number of valid votes that the other candidate got. Hmm, let me break this down step by step.
First, the total number of votes is 750... | The total number of votes is 7500. Since 20% of the votes were invalid, we need to calculate the number of valid votes.
First, we find 80% of 7500 (since 100% - 20% = 80%):
\[
7500 \times 0.8 = 6000
\]
So, there are 6000 valid votes.
One candidate received 55% of these valid votes. Therefore, the other candidate rec... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. There's an election between two candidates. One got 55% of the total valid votes. 20% of the votes were invalid. The total number of votes was 7500. We need to find the number of valid votes that the other candidate got. Hmm, let me break this down step by step.
First, the total number of vote... | [
{
"from": "human",
"value": "in an election between two candidates , one got 55 % of the total valid votes . 20 % of the votes were invalid . if the total number of votes was 7500 , what was the number of valid votes that the other candidate got ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, le... |
Pentagon $A B C D E$ is inscribed in a circle. The distances from point $A$ to the lines $B C, C D$, and $D E$ are $a, b$, and $c$, respectively.
Find the distance from vertex $A$ to the line... | null | Okay, so I have this problem where there's a pentagon ABCDE inscribed in a circle. That means all five vertices lie on a single circle, right? So it's a cyclic pentagon. The question says that the distances from point A to the lines BC, CD, and DE are a, b, and c respectively. And I need to find the distance from verte... | Given a cyclic pentagon \(ABCDE\) inscribed in a circle, we need to find the distance from vertex \(A\) to the line \(BE\). The distances from point \(A\) to the lines \(BC\), \(CD\), and \(DE\) are given as \(a\), \(b\), and \(c\) respectively.
To solve this, we use the properties of cyclic polygons and the relations... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem where there's a pentagon ABCDE inscribed in a circle. That means all five vertices lie on a single circle, right? So it's a cyclic pentagon. The question says that the distances from point A to the lines BC, CD, and DE are a, b, and c respectively. And I need to find the distance fr... | [
{
"from": "human",
"value": "\n\nPentagon $A B C D E$ is inscribed in a circle. The distances from point $A$ to the lines $B C, C D$, and $D E$ are $a, b$, and $c$, respectively.\n\n... |
In a class of 50 students, 18 take chorus, 26 take band, and 2 take both chorus and band. How many students in the class are not enrolled in either chorus or band? | null | Okay, so there's this problem here: In a class of 50 students, 18 take chorus, 26 take band, and 2 take both chorus and band. The question is asking how many students aren't enrolled in either chorus or band. Hmm, alright, let's see. I remember something about sets and Venn diagrams from math class. Maybe I need to use... | To determine how many students in the class are not enrolled in either chorus or band, we use the principle of inclusion-exclusion.
First, we calculate the number of students enrolled in either chorus or band or both:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
where \( |A| \) is the number of students in chorus, \( |B... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so there's this problem here: In a class of 50 students, 18 take chorus, 26 take band, and 2 take both chorus and band. The question is asking how many students aren't enrolled in either chorus or band. Hmm, alright, let's see. I remember something about sets and Venn diagrams from math class. Maybe I nee... | [
{
"from": "human",
"value": "In a class of 50 students, 18 take chorus, 26 take band, and 2 take both chorus and band. How many students in the class are not enrolled in either chorus or band?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so there's this problem here: In a class of 50 students, ... |
$$\sin 30^o + \sin 45^o + \sin 60^o + \sin 90^o + \cos 120^o + \cos 135^o + \cos 150^o + \cos 180^o = ?$$
| null | Okay, so I have this trigonometric expression to evaluate: sin30° + sin45° + sin60° + sin90° + cos120° + cos135° + cos150° + cos180°. Hmm, let's see. I need to calculate each term individually and then add them all up. Let me start by recalling the values of these sine and cosine functions.
Starting with the sine term... | To evaluate the expression \(\sin 30^\circ + \sin 45^\circ + \sin 60^\circ + \sin 90^\circ + \cos 120^\circ + \cos 135^\circ + \cos 150^\circ + \cos 180^\circ\), we start by converting each trigonometric function to its exact value:
1. Sine terms:
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\sin 45^\circ = \frac{\sqrt... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this trigonometric expression to evaluate: sin30° + sin45° + sin60° + sin90° + cos120° + cos135° + cos150° + cos180°. Hmm, let's see. I need to calculate each term individually and then add them all up. Let me start by recalling the values of these sine and cosine functions.
Starting with the s... | [
{
"from": "human",
"value": "$$\\sin 30^o + \\sin 45^o + \\sin 60^o + \\sin 90^o + \\cos 120^o + \\cos 135^o + \\cos 150^o + \\cos 180^o = ?$$\n\n"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this trigonometric expression to evaluate: sin30° + sin45° + sin60° + sin90° + cos120° + cos1... |
For how many primes $p$, $|p^4-86|$ is also prime?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$ | null | Okay, so I have this problem here: For how many primes \( p \), is \( |p^4 - 86| \) also prime? The answer choices are from A to E, with E being 4. Hmm, interesting. Let me try to work through this step by step.
First, primes are numbers greater than 1 that have no positive divisors other than 1 and themselves. The pr... | To determine how many primes \( p \) make \( |p^4 - 86| \) also prime, we start by checking small primes:
- For \( p = 2 \):
\[
|2^4 - 86| = |16 - 86| = 70 \quad (\text{composite})
\]
- For \( p = 3 \):
\[
|3^4 - 86| = |81 - 86| = 5 \quad (\text{prime})
\]
- For \( p = 5 \):
\[
|5^4 - 86| = |625 ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here: For how many primes \( p \), is \( |p^4 - 86| \) also prime? The answer choices are from A to E, with E being 4. Hmm, interesting. Let me try to work through this step by step.
First, primes are numbers greater than 1 that have no positive divisors other than 1 and themselves... | [
{
"from": "human",
"value": "For how many primes $p$, $|p^4-86|$ is also prime?\n\n$\\textbf{(A)}\\ 0 \\qquad\\textbf{(B)}\\ 1 \\qquad\\textbf{(C)}\\ 2 \\qquad\\textbf{(D)}\\ 3 \\qquad\\textbf{(E)}\\ 4$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: For how many prime... |
The third term of a geometric sequence of positive numbers is 27, and the ninth term is 3. What is the sixth term of the sequence? | null | Okay, so I need to find the sixth term of a geometric sequence where the third term is 27 and the ninth term is 3. Hmm, let's start by recalling what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, usually denot... | Given a geometric sequence where the third term is 27 and the ninth term is 3, we need to find the sixth term.
1. The third term of a geometric sequence is given by \( ar^2 = 27 \).
2. The ninth term is given by \( ar^8 = 3 \).
To find the common ratio \( r \), we divide the ninth term by the third term:
\[
\frac{ar^... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the sixth term of a geometric sequence where the third term is 27 and the ninth term is 3. Hmm, let's start by recalling what a geometric sequence is. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio, usual... | [
{
"from": "human",
"value": "The third term of a geometric sequence of positive numbers is 27, and the ninth term is 3. What is the sixth term of the sequence?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sixth term of a geometric sequence where the third term is 27 and th... |
For how many positive integers $x$ is $x^2 + 6x + 9$ between 20 and 40?
New problem:
Let $p(x) = x^2 + 6x + 9$. For how many positive integers $k$ is the equation $p(x) = k$ solvable in the interval $[2, 5]$? | For how many positive integers $x$ is $x^2 + 6x + 9$ between 20 and 40?
New problem:
Let $p(x) = x^2 + 6x + 9$. For how many positive integers $k$ is the equation $p(x) = k$ solvable in the interval $[2, 5]$? | Okay, let's see. I need to figure out for how many positive integers \( x \) the quadratic expression \( x^2 + 6x + 9 \) is between 20 and 40. Hmm, first, maybe I should rewrite the quadratic to make it easier to analyze. Wait, \( x^2 + 6x + 9 \) looks familiar. Oh, right! That's a perfect square. It factors into \( (x... | To determine how many positive integers \( k \) make the equation \( p(x) = k \) solvable in the interval \([2, 5]\) for \( p(x) = x^2 + 6x + 9 \), we start by recognizing that \( p(x) \) is a perfect square: \( p(x) = (x + 3)^2 \).
We need to find the range of \( p(x) \) when \( x \) is in the interval \([2, 5]\).
... | 820 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to figure out for how many positive integers \( x \) the quadratic expression \( x^2 + 6x + 9 \) is between 20 and 40. Hmm, first, maybe I should rewrite the quadratic to make it easier to analyze. Wait, \( x^2 + 6x + 9 \) looks familiar. Oh, right! That's a perfect square. It factors in... | [
{
"from": "human",
"value": "For how many positive integers $x$ is $x^2 + 6x + 9$ between 20 and 40?\n\nNew problem:\n\nLet $p(x) = x^2 + 6x + 9$. For how many positive integers $k$ is the equation $p(x) = k$ solvable in the interval $[2, 5]$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's ... |
What is the smallest positive integer that has exactly three factors that are also factors of 60? | What is the smallest positive integer that has exactly three factors that are also factors of 60? | Okay, let's see. The problem is asking for the smallest positive integer that has exactly three factors which are also factors of 60. Hmm, that sounds a bit tricky, but let me break it down step by step.
First, I need to recall that a factor of a number is another number that divides it without leaving a remainder. So... | To determine the smallest positive integer that has exactly three factors which are also factors of 60, we start by listing the factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
We need a number \( N \) such that exactly three of its factors are among these 12 numbers.
A number with exactly three factors mus... | 9,558 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is asking for the smallest positive integer that has exactly three factors which are also factors of 60. Hmm, that sounds a bit tricky, but let me break it down step by step.
First, I need to recall that a factor of a number is another number that divides it without leaving a remai... | [
{
"from": "human",
"value": "What is the smallest positive integer that has exactly three factors that are also factors of 60?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem is asking for the smallest positive integer that has exactly three factors which are also factors of... |
Ship A observes Ship B at a point A, with Ship B located to the north-east 60° direction from it, and the two ships are a nautical miles apart at point B. Ship B is moving north. If Ship A's speed is $\sqrt{3}$ times the speed of Ship B, then to catch up with Ship B as quickly as possible, Ship A should head in a direc... | null | Okay, so I need to figure out the angle Ship A should head in the north-east direction to catch up with Ship B as quickly as possible. Let me start by breaking down the problem.
First, Ship A is at point A, and Ship B is initially located at a point B, which is northeast 60° direction from Ship A. The distance between... | Ship A observes Ship B at a point B, located 60° east of north (north-east 60° direction) and a nautical miles away. Ship B is moving north, and Ship A's speed is \(\sqrt{3}\) times that of Ship B. We need to determine the optimal direction for Ship A to catch Ship B as quickly as possible.
1. **Coordinates and Veloci... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out the angle Ship A should head in the north-east direction to catch up with Ship B as quickly as possible. Let me start by breaking down the problem.
First, Ship A is at point A, and Ship B is initially located at a point B, which is northeast 60° direction from Ship A. The distance... | [
{
"from": "human",
"value": "Ship A observes Ship B at a point A, with Ship B located to the north-east 60° direction from it, and the two ships are a nautical miles apart at point B. Ship B is moving north. If Ship A's speed is $\\sqrt{3}$ times the speed of Ship B, then to catch up with Ship B as quickly ... |
The deck of 52 cards is randomly arranged. What is the probability that the third card is an ace, given that the first two cards are kings? | The deck of 52 cards is randomly arranged. What is the probability that the third card is an ace, given that the first two cards are kings? | Okay, so I need to find the probability that the third card is an ace, given that the first two cards are kings in a standard deck of 52 cards. Let me think about how to approach this.
First, since the problem is about conditional probability, I remember that the formula for conditional probability is P(A|B) = P(A an... | Given a standard deck of 52 cards, we need to find the probability that the third card is an ace, given that the first two cards are kings.
First, we recognize that this is a conditional probability problem. We use the formula for conditional probability \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( A \) is the ev... | 38,245 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that the third card is an ace, given that the first two cards are kings in a standard deck of 52 cards. Let me think about how to approach this.
First, since the problem is about conditional probability, I remember that the formula for conditional probability is P(A|B) ... | [
{
"from": "human",
"value": "The deck of 52 cards is randomly arranged. What is the probability that the third card is an ace, given that the first two cards are kings?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the probability that the third card is an ace, given that the f... |
Find the smallest whole number greater than 100 that has an odd number of positive factors and is also a perfect square. | Find the smallest whole number greater than 100 that has an odd number of positive factors and is also a perfect square. | Okay, so I need to find the smallest whole number greater than 100 that has an odd number of positive factors and is also a perfect square. Hmm, let's break this down step by step.
First, I remember that numbers with an odd number of positive factors are perfect squares. Wait, is that right? Let me think. A normal nu... | To find the smallest whole number greater than 100 that has an odd number of positive factors and is also a perfect square, we need to recognize that numbers with an odd number of positive factors are perfect squares. This is because the factors of a number are typically in pairs, but perfect squares have a repeated fa... | 29,632 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest whole number greater than 100 that has an odd number of positive factors and is also a perfect square. Hmm, let's break this down step by step.
First, I remember that numbers with an odd number of positive factors are perfect squares. Wait, is that right? Let me think. A n... | [
{
"from": "human",
"value": "Find the smallest whole number greater than 100 that has an odd number of positive factors and is also a perfect square."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest whole number greater than 100 that has an odd number of positive fact... |
For a 10-day period, a weather forecast predicts that for each day, there is a $50\%$ chance of sun, a $30\%$ chance of receiving 3 inches of rain, and a $20\%$ chance of receiving 6 inches of rain. Calculate the expected value of the total number of inches of rain which will fall over the 10-day period. Provide your a... | null | Okay, so I need to find the expected value of the total number of inches of rain over a 10-day period. Each day has three possible weather outcomes: sunny, 3 inches of rain, or 6 inches of rain, with probabilities 50%, 30%, and 20% respectively. Hmm, right.
First, I remember that the expected value for a single day i... | To calculate the expected value of the total number of inches of rain over a 10-day period, we first determine the expected rainfall for a single day. Each day has a 50% chance of sun (0 inches of rain), a 30% chance of 3 inches of rain, and a 20% chance of 6 inches of rain.
The expected inches of rain for one day is... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the expected value of the total number of inches of rain over a 10-day period. Each day has three possible weather outcomes: sunny, 3 inches of rain, or 6 inches of rain, with probabilities 50%, 30%, and 20% respectively. Hmm, right.
First, I remember that the expected value for a sing... | [
{
"from": "human",
"value": "For a 10-day period, a weather forecast predicts that for each day, there is a $50\\%$ chance of sun, a $30\\%$ chance of receiving 3 inches of rain, and a $20\\%$ chance of receiving 6 inches of rain. Calculate the expected value of the total number of inches of rain which will... |
a certain number of men can do a work in 20 days . if there were 4 men less it could be finished in 5 days more . how many men are there ? | null | Okay, let's see. There's this problem about some men doing a work in 20 days. If there were 4 fewer men, it would take 5 more days. I need to find out how many men there are. Hmm, work problems usually involve some kind of inverse proportionality between the number of workers and the time taken. Let me try to recall.
... | Let \( m \) be the number of men required to complete the work in 20 days. The total work can be considered as \( 20m \) man-days. If there were 4 fewer men, the number of men becomes \( m - 4 \) and the time taken would be 25 days. The total work in this scenario is \( 25(m - 4) \) man-days. Since the total work remai... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. There's this problem about some men doing a work in 20 days. If there were 4 fewer men, it would take 5 more days. I need to find out how many men there are. Hmm, work problems usually involve some kind of inverse proportionality between the number of workers and the time taken. Let me try to r... | [
{
"from": "human",
"value": "a certain number of men can do a work in 20 days . if there were 4 men less it could be finished in 5 days more . how many men are there ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. There's this problem about some men doing a work in 20 days. If there w... |
Given the function $f(x) = |\ln|x-1||$, let $x\_1$, $x\_2$, $x\_3$, $x\_4$ be the four zero points of $f(x) - m$, and $k = \frac{1}{x\_1} + \frac{1}{x\_2} + \frac{1}{x\_3} + \frac{1}{x\_4}$. Find the value of $f(k) - e^k$. | null | Okay, so I need to solve this problem where we have the function f(x) = |ln|x - 1||. Then, they mention x1, x2, x3, x4 are the four zero points of f(x) - m. So, zero points mean solutions to the equation f(x) - m = 0, which is equivalent to f(x) = m. So, essentially, we're looking for the x-values where the function f(... | Given the function \( f(x) = |\ln|x-1|| \), we need to find the value of \( f(k) - e^k \) where \( k = \frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \frac{1}{x_4} \) and \( x_1, x_2, x_3, x_4 \) are the four zero points of \( f(x) - m \).
1. **Understanding the function \( f(x) \)**:
- The function \( f(x) = |\ln... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this problem where we have the function f(x) = |ln|x - 1||. Then, they mention x1, x2, x3, x4 are the four zero points of f(x) - m. So, zero points mean solutions to the equation f(x) - m = 0, which is equivalent to f(x) = m. So, essentially, we're looking for the x-values where the fun... | [
{
"from": "human",
"value": "Given the function $f(x) = |\\ln|x-1||$, let $x\\_1$, $x\\_2$, $x\\_3$, $x\\_4$ be the four zero points of $f(x) - m$, and $k = \\frac{1}{x\\_1} + \\frac{1}{x\\_2} + \\frac{1}{x\\_3} + \\frac{1}{x\\_4}$. Find the value of $f(k) - e^k$."
},
{
"from": "gpt",
"value": "... |
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The length of the chord formed by the intersection of the line $\rho\sin \theta=1$ and $\rho=4\sin \theta$ is \_\_\_\_\_\_. | null | Okay, so I need to find the length of the chord formed by the intersection of two polar equations: ρ sin θ = 1 and ρ = 4 sin θ. Hmm, let's start by recalling what these equations represent in polar coordinates.
First, ρ sin θ = 1. In polar coordinates, ρ sin θ is equal to the y-coordinate in Cartesian coordinates. So... | To find the length of the chord formed by the intersection of the line \(\rho \sin \theta = 1\) and the circle \(\rho = 4 \sin \theta\):
1. Convert the polar equations to Cartesian coordinates:
- The line \(\rho \sin \theta = 1\) converts to \(y = 1\).
- The circle \(\rho = 4 \sin \theta\) converts to \(x^2 + (y... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the length of the chord formed by the intersection of two polar equations: ρ sin θ = 1 and ρ = 4 sin θ. Hmm, let's start by recalling what these equations represent in polar coordinates.
First, ρ sin θ = 1. In polar coordinates, ρ sin θ is equal to the y-coordinate in Cartesian coordin... | [
{
"from": "human",
"value": "In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The length of the chord formed by the intersection of the line $\\rho\\sin \\theta=1$ and $\\rho=4\\sin \\thet... |
2. If the solution set of the inequality
$|x-a|<|x|+|x+1|$ is all real numbers, then the range of values for $a$ is $\qquad$ | null | Okay, so I have this inequality: |x - a| < |x| + |x + 1|, and I need to find the range of values for a such that the solution set is all real numbers. That means no matter what x I pick, the inequality should hold true. Hmm, let's break this down step by step.
First, I remember that the triangle inequality says |A + B... | To determine the range of values for \( a \) such that the inequality \( |x - a| < |x| + |x + 1| \) holds for all real numbers \( x \), we analyze the inequality in three cases based on the value of \( x \).
**Case 1: \( x \geq 0 \)**
- The inequality simplifies to \( |x - a| < 2x + 1 \).
- Squaring both sides and sim... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this inequality: |x - a| < |x| + |x + 1|, and I need to find the range of values for a such that the solution set is all real numbers. That means no matter what x I pick, the inequality should hold true. Hmm, let's break this down step by step.
First, I remember that the triangle inequality say... | [
{
"from": "human",
"value": "2. If the solution set of the inequality\n$|x-a|<|x|+|x+1|$ is all real numbers, then the range of values for $a$ is $\\qquad$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this inequality: |x - a| < |x| + |x + 1|, and I need to find the range of values for... |
Given that $\left(a-2\right)x^{|a|-1}=-2$ is a linear equation in $x$, the value of $a$ is ____. | null | Okay, let's see. I need to find the value of 'a' such that the equation (a - 2)x^{|a| - 1} = -2 is a linear equation in x. Hmm, right. So, first of all, what defines a linear equation? A linear equation in x should be of the form px + q = 0, where p and q are constants. That means the variable x should be raised to the... | To determine the value of \( a \) such that the equation \( (a - 2)x^{|a| - 1} = -2 \) is linear in \( x \), we need to ensure two conditions:
1. The exponent of \( x \) must be 1.
2. The coefficient of \( x \) must not be zero.
First, we set the exponent \( |a| - 1 \) equal to 1:
\[
|a| - 1 = 1
\]
Solving this, we ge... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find the value of 'a' such that the equation (a - 2)x^{|a| - 1} = -2 is a linear equation in x. Hmm, right. So, first of all, what defines a linear equation? A linear equation in x should be of the form px + q = 0, where p and q are constants. That means the variable x should be raise... | [
{
"from": "human",
"value": "Given that $\\left(a-2\\right)x^{|a|-1}=-2$ is a linear equation in $x$, the value of $a$ is ____."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the value of 'a' such that the equation (a - 2)x^{|a| - 1} = -2 is a linear equation in x. Hmm, ... |
Simplify the expression: $[(3+4+5+6)\div2] + [(3\cdot6+9)\div4]$ | Simplify the expression: $[(3+4+5+6)\div2] + [(3\cdot6+9)\div4]$ | Okay, let's see. I need to simplify this expression: [(3 + 4 + 5 + 6) ÷ 2] + [(3·6 + 9) ÷ 4]. Hmm, let me break this down step by step.
First, I'll handle the left part of the addition, which is [(3 + 4 + 5 + 6) ÷ 2]. Let me compute the sum inside the brackets first. So, 3 + 4 is 7, then 5 + 6 is 11. Adding those tog... | First, simplify the left part of the expression:
\[
(3 + 4 + 5 + 6) \div 2
\]
Adding the numbers inside the brackets:
\[
3 + 4 + 5 + 6 = 18
\]
Dividing by 2:
\[
18 \div 2 = 9
\]
Next, simplify the right part of the expression:
\[
(3 \cdot 6 + 9) \div 4
\]
Performing the multiplication first:
\[
3 \cdot 6 = 18
\]
Then... | 16,649 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to simplify this expression: [(3 + 4 + 5 + 6) ÷ 2] + [(3·6 + 9) ÷ 4]. Hmm, let me break this down step by step.
First, I'll handle the left part of the addition, which is [(3 + 4 + 5 + 6) ÷ 2]. Let me compute the sum inside the brackets first. So, 3 + 4 is 7, then 5 + 6 is 11. Adding t... | [
{
"from": "human",
"value": "Simplify the expression: $[(3+4+5+6)\\div2] + [(3\\cdot6+9)\\div4]$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to simplify this expression: [(3 + 4 + 5 + 6) ÷ 2] + [(3·6 + 9) ÷ 4]. Hmm, let me break this down step by step. \n\nFirst, I'll handle ... |
Example 3. Solve the system of integral equations
$$
\left\{\begin{array}{l}
\varphi_{1}(x)=1-2 \int_{0}^{x} e^{2(x-t)} \varphi_{1}(t) d t+\int_{0}^{x} \varphi_{2}(t) d t \\
\varphi_{2}(x)=4 x-\int_{0}^{x} \varphi_{1}(t) d t+4 \int_{0}^{x}(x-t) \varphi_{2}(t) d t
\end{array}\right.
$$ | null | Okay, so I need to solve this system of integral equations. Let me write them down again to make sure I have them right:
φ₁(x) = 1 - 2 ∫₀ˣ e²⁽ˣ⁻ᵗ⁾ φ₁(t) dt + ∫₀ˣ φ₂(t) dt
φ₂(x) = 4x - ∫₀ˣ φ₁(t) dt + 4 ∫₀ˣ (x - t) φ₂(t) dt
Hmm, integral equations. These look like Volterra equations of the second kind, but there are ... | To solve the system of integral equations:
\[
\left\{\begin{array}{l}
\varphi_{1}(x)=1-2 \int_{0}^{x} e^{2(x-t)} \varphi_{1}(t) d t+\int_{0}^{x} \varphi_{2}(t) d t \\
\varphi_{2}(x)=4 x-\int_{0}^{x} \varphi_{1}(t) d t+4 \int_{0}^{x}(x-t) \varphi_{2}(t) d t
\end{array}\right.
\]
we first differentiate both equations t... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this system of integral equations. Let me write them down again to make sure I have them right:
φ₁(x) = 1 - 2 ∫₀ˣ e²⁽ˣ⁻ᵗ⁾ φ₁(t) dt + ∫₀ˣ φ₂(t) dt
φ₂(x) = 4x - ∫₀ˣ φ₁(t) dt + 4 ∫₀ˣ (x - t) φ₂(t) dt
Hmm, integral equations. These look like Volterra equations of the second kind, but th... | [
{
"from": "human",
"value": "Example 3. Solve the system of integral equations\n\n$$\n\\left\\{\\begin{array}{l}\n\\varphi_{1}(x)=1-2 \\int_{0}^{x} e^{2(x-t)} \\varphi_{1}(t) d t+\\int_{0}^{x} \\varphi_{2}(t) d t \\\\\n\\varphi_{2}(x)=4 x-\\int_{0}^{x} \\varphi_{1}(t) d t+4 \\int_{0}^{x}(x-t) \\varphi_{2}(t... |
Example 1 Let $m$ be a given positive integer, and each term of the sequence $\left\{a_{n}\right\}$ is a positive integer, and for any $n \in \mathbf{N}^{*}$, we have $0<a_{n+1}-a_{n} \leqslant m$. Prove: There exist infinitely many pairs of positive integers $(p, q)$, such that $p<q$, and $a_{p} \mid a_{q}$. | null | Okay, so I need to prove that there are infinitely many pairs (p, q) with p < q such that a_p divides a_q. The sequence {a_n} has positive integers, and the difference between consecutive terms is at least 1 and at most m, where m is a given positive integer.
First, let me understand the problem. Each term of the seq... | To prove that there exist infinitely many pairs of positive integers \((p, q)\) with \(p < q\) such that \(a_p \mid a_q\) for a sequence \(\{a_n\}\) satisfying \(0 < a_{n+1} - a_n \leqslant m\), we can use the pigeonhole principle and properties of residues modulo \(a_p\):
**Step-by-Step Explanation and Proof:**
1. *... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that there are infinitely many pairs (p, q) with p < q such that a_p divides a_q. The sequence {a_n} has positive integers, and the difference between consecutive terms is at least 1 and at most m, where m is a given positive integer.
First, let me understand the problem. Each term of... | [
{
"from": "human",
"value": "Example 1 Let $m$ be a given positive integer, and each term of the sequence $\\left\\{a_{n}\\right\\}$ is a positive integer, and for any $n \\in \\mathbf{N}^{*}$, we have $0<a_{n+1}-a_{n} \\leqslant m$. Prove: There exist infinitely many pairs of positive integers $(p, q)$, su... |
Find the domain of the function \(f(x) = \sqrt[3]{2x - 3} + \sqrt[3]{5 - 2x}\). | null | Okay, so I need to find the domain of the function f(x) = cube root of (2x - 3) plus cube root of (5 - 2x). Hmm, domain means all the real numbers x for which the function is defined, right? Let me think.
First, let's recall what the cube root function looks like. Unlike square roots, cube roots can take any real num... | To find the domain of the function \(f(x) = \sqrt[3]{2x - 3} + \sqrt[3]{5 - 2x}\), we need to determine the set of all real numbers \(x\) for which the function is defined.
1. **Understanding Cube Roots**: The cube root function, \(\sqrt[3]{x}\), is defined for all real numbers \(x\). This means that the expressions i... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the domain of the function f(x) = cube root of (2x - 3) plus cube root of (5 - 2x). Hmm, domain means all the real numbers x for which the function is defined, right? Let me think.
First, let's recall what the cube root function looks like. Unlike square roots, cube roots can take any ... | [
{
"from": "human",
"value": "Find the domain of the function \\(f(x) = \\sqrt[3]{2x - 3} + \\sqrt[3]{5 - 2x}\\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the domain of the function f(x) = cube root of (2x - 3) plus cube root of (5 - 2x). Hmm, domain means all the real numb... |
Find the product of all values of $x$ for which the expression $\frac{x-2}{x^2+3x-10}$ is undefined. | Find the product of all values of $x$ for which the expression $\frac{x-2}{x^2+3x-10}$ is undefined. | Okay, so I need to find the product of all values of x where the expression (x - 2)/(x² + 3x - 10) is undefined. Hmm, let me think. I remember that a rational expression is undefined when the denominator is zero because you can't divide by zero. So first, I need to find the values of x that make the denominator x² + 3x... | To determine the product of all values of \( x \) for which the expression \(\frac{x-2}{x^2+3x-10}\) is undefined, we need to find the values of \( x \) that make the denominator zero.
The denominator is \( x^2 + 3x - 10 \). Factoring this quadratic expression, we look for two numbers that multiply to \(-10\) and add... | 10,893 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the product of all values of x where the expression (x - 2)/(x² + 3x - 10) is undefined. Hmm, let me think. I remember that a rational expression is undefined when the denominator is zero because you can't divide by zero. So first, I need to find the values of x that make the denominator... | [
{
"from": "human",
"value": "Find the product of all values of $x$ for which the expression $\\frac{x-2}{x^2+3x-10}$ is undefined."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the product of all values of x where the expression (x - 2)/(x² + 3x - 10) is undefined. Hmm, let me ... |
Given that $x = 2y - 5$, evaluate the expression $(3y - x + 2)$ when $y = 4$. | Given that $x = 2y - 5$, evaluate the expression $(3y - x + 2)$ when $y = 4$. | Okay, let's see. I need to evaluate the expression (3y - x + 2) when y = 4. But wait, x is given in terms of y: x = 2y - 5. Hmm, so first, maybe I should find out what x is when y is 4, and then plug both y and x into the expression?
Let me start by substituting y = 4 into the equation for x. So, x = 2*(4) - 5. Let me... | Given the equation \( x = 2y - 5 \), we need to evaluate the expression \( 3y - x + 2 \) when \( y = 4 \).
First, substitute \( y = 4 \) into the equation for \( x \):
\[
x = 2(4) - 5 = 8 - 5 = 3
\]
Next, substitute \( y = 4 \) and \( x = 3 \) into the expression \( 3y - x + 2 \):
\[
3(4) - 3 + 2 = 12 - 3 + 2 = 9 + 2... | 7,751 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to evaluate the expression (3y - x + 2) when y = 4. But wait, x is given in terms of y: x = 2y - 5. Hmm, so first, maybe I should find out what x is when y is 4, and then plug both y and x into the expression?
Let me start by substituting y = 4 into the equation for x. So, x = 2*(4) - 5... | [
{
"from": "human",
"value": "Given that $x = 2y - 5$, evaluate the expression $(3y - x + 2)$ when $y = 4$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to evaluate the expression (3y - x + 2) when y = 4. But wait, x is given in terms of y: x = 2y - 5. Hmm, so first, maybe I sh... |
Let $t$ be TNYWR.
At the start of 2018, the Canadian Excellent Mathematics Corporation had $t$ employees in its Moose Jaw office, 40 employees in its Okotoks office, and no other employees. During 2018, the number of employees in the Moose Jaw office increased by $25 \%$ and the number of employees in the Okotoks offi... | null | Okay, let's try to figure out this problem. So, the question is about the Canadian Excellent Mathematics Corporation and their employees in two offices: Moose Jaw and Okotoks. At the start of 2018, Moose Jaw had t employees, Okotoks had 40, and there were no other employees. During 2018, Moose Jaw's employees increased... | At the start of 2018, the Canadian Excellent Mathematics Corporation had \( t \) employees in its Moose Jaw office and 40 employees in its Okotoks office. During 2018, the number of employees in the Moose Jaw office increased by 25%, and the number of employees in the Okotoks office decreased by 35%.
To find the addi... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to figure out this problem. So, the question is about the Canadian Excellent Mathematics Corporation and their employees in two offices: Moose Jaw and Okotoks. At the start of 2018, Moose Jaw had t employees, Okotoks had 40, and there were no other employees. During 2018, Moose Jaw's employees i... | [
{
"from": "human",
"value": "Let $t$ be TNYWR.\n\nAt the start of 2018, the Canadian Excellent Mathematics Corporation had $t$ employees in its Moose Jaw office, 40 employees in its Okotoks office, and no other employees. During 2018, the number of employees in the Moose Jaw office increased by $25 \\%$ and... |
6. (3 points) A deck of playing cards has 54 cards, with the jokers considered as 0 points, $A$ as 1 point, $J$ as 11 points, $Q$ as 12 points, and $K$ as 13 points. If you draw several cards at random, disregarding the suits, and if it is required that each draw must include 2 cards whose points add up to 14, then at ... | null | Okay, let's tackle this problem step by step. So, we have a deck of 54 playing cards. The points system is a bit different: jokers are 0 points, A is 1, numbers 2 through 10 are their face values, J is 11, Q is 12, and K is 13. The question is asking, if we draw several cards at random, how many do we need to draw to g... | To determine the minimum number of cards that must be drawn to guarantee that at least two cards sum to 14 points, we use the Pigeonhole Principle. We need to consider the pairs of cards that add up to 14 and the maximum number of cards that can be drawn without forming such a pair.
1. **Identify pairs that sum to 14*... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's tackle this problem step by step. So, we have a deck of 54 playing cards. The points system is a bit different: jokers are 0 points, A is 1, numbers 2 through 10 are their face values, J is 11, Q is 12, and K is 13. The question is asking, if we draw several cards at random, how many do we need to d... | [
{
"from": "human",
"value": "6. (3 points) A deck of playing cards has 54 cards, with the jokers considered as 0 points, $A$ as 1 point, $J$ as 11 points, $Q$ as 12 points, and $K$ as 13 points. If you draw several cards at random, disregarding the suits, and if it is required that each draw must include 2 ... |
Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \[(x-3)^2 + (y-3)^2 = 6.\]
$\textbf{(A) }3 + 2 \sqrt 2\qquad
\textbf{(B) } 2 + \sqrt 3\qquad
\textbf{(C ) }3 \sqrt 3\qquad
\textbf{(D) }6\qquad
\textbf{(E) }6 + 2 \sqrt 3$ | null | Okay, so I need to find the largest value of y/x where (x, y) are real numbers that satisfy the equation (x - 3)^2 + (y - 3)^2 = 6. Hmm, let me think about how to approach this.
First, the equation given is a circle centered at (3, 3) with a radius of sqrt(6). The problem is asking for the maximum value of y/x among a... | To find the largest value of \(\frac{y}{x}\) for pairs of real numbers \((x, y)\) that satisfy \((x - 3)^2 + (y - 3)^2 = 6\), we start by setting \(k = \frac{y}{x}\). Substituting \(y = kx\) into the circle equation, we get:
\[
(x - 3)^2 + (kx - 3)^2 = 6
\]
Expanding and combining like terms, we obtain the quadratic ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the largest value of y/x where (x, y) are real numbers that satisfy the equation (x - 3)^2 + (y - 3)^2 = 6. Hmm, let me think about how to approach this.
First, the equation given is a circle centered at (3, 3) with a radius of sqrt(6). The problem is asking for the maximum value of y/x... | [
{
"from": "human",
"value": "Find the largest value for $\\frac{y}{x}$ for pairs of real numbers $(x,y)$ which satisfy \\[(x-3)^2 + (y-3)^2 = 6.\\]\n\n$\\textbf{(A) }3 + 2 \\sqrt 2\\qquad\n\\textbf{(B) } 2 + \\sqrt 3\\qquad\n\\textbf{(C ) }3 \\sqrt 3\\qquad\n\\textbf{(D) }6\\qquad\n\\textbf{(E) }6 + 2 \\sqr... |
A fair die is constructed by labelling the faces of a wooden cube with the numbers $1,1,1,2,3$, and 3 . If this die is rolled once, the probability of rolling an odd number is
(A) $\frac{5}{6}$
(B) $\frac{4}{6}$
(C) $\frac{3}{6}$
(D) $\frac{2}{6}$
(E) $\frac{1}{6}$ | null | Okay, let's see. I need to find the probability of rolling an odd number on this special die. The die is a cube, so it has six faces. The numbers on the faces are 1, 1, 1, 2, 3, and 3. Hmm, first, I should remember that probability is calculated by the number of favorable outcomes divided by the total number of possibl... | To find the probability of rolling an odd number on the given die, we start by noting that the die has six faces labeled with the numbers 1, 1, 1, 2, 3, and 3.
First, we determine the total number of possible outcomes, which is 6 since the die is a cube with six faces.
Next, we count the number of favorable outcomes... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find the probability of rolling an odd number on this special die. The die is a cube, so it has six faces. The numbers on the faces are 1, 1, 1, 2, 3, and 3. Hmm, first, I should remember that probability is calculated by the number of favorable outcomes divided by the total number of... | [
{
"from": "human",
"value": "A fair die is constructed by labelling the faces of a wooden cube with the numbers $1,1,1,2,3$, and 3 . If this die is rolled once, the probability of rolling an odd number is\n(A) $\\frac{5}{6}$\n(B) $\\frac{4}{6}$\n(C) $\\frac{3}{6}$\n(D) $\\frac{2}{6}$\n(E) $\\frac{1}{6}$"
... |
The function
\[f(z) = \frac{(1 - i \sqrt{2}) z + (3 \sqrt{2} - 5i)}{4}\]
represents a scaling and rotation around some complex number \(c\). Find \(c\). | null | Alright, so I have this problem here about a complex function representing a scaling and rotation around some complex number c, and I need to find c. The function is given by:
\[f(z) = \frac{(1 - i \sqrt{2}) z + (3 \sqrt{2} - 5i)}{4}\]
Hmm, okay. Let me recall. If a function represents a scaling and rotation around a... | To find the complex number \( c \) around which the function \( f(z) = \frac{(1 - i \sqrt{2}) z + (3 \sqrt{2} - 5i)}{4} \) represents a scaling and rotation, we need to determine the fixed point \( c \) such that \( f(c) = c \).
Starting with the equation \( f(c) = c \):
\[
\frac{(1 - i \sqrt{2}) c + (3 \sqrt{2} - 5i... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I have this problem here about a complex function representing a scaling and rotation around some complex number c, and I need to find c. The function is given by:
\[f(z) = \frac{(1 - i \sqrt{2}) z + (3 \sqrt{2} - 5i)}{4}\]
Hmm, okay. Let me recall. If a function represents a scaling and rotation ... | [
{
"from": "human",
"value": "The function\n\\[f(z) = \\frac{(1 - i \\sqrt{2}) z + (3 \\sqrt{2} - 5i)}{4}\\]\nrepresents a scaling and rotation around some complex number \\(c\\). Find \\(c\\)."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this problem here about a complex function r... |
A library charges a monthly membership fee of $10, and an additional $0.25 per book borrowed.
A book club decides to pay for each member's borrowings,
and to cover the costs, each member is charged a flat annual fee.
However, members who borrow more than 20 books per year will be charged an additional $0.10 per book... | A library charges a monthly membership fee of $10, and an additional $0.25 per book borrowed.
A book club decides to pay for each member's borrowings,
and to cover the costs, each member is charged a flat annual fee.
However, members who borrow more than 20 books per year will be charged an additional $0.10 per book... | Okay, let's see. The library charges a monthly membership fee of $10, so that's $10 per month. Since there are 12 months in a year, the annual membership fee would be 12 times $10, which is $120. Then, there's an additional $0.25 per book borrowed.
The book club is paying for each member's borrowings, but to cover th... | The library charges a monthly membership fee of $10, which amounts to $120 annually. Additionally, there is a fee of $0.25 per book borrowed. The book club wants to cover these costs by charging a flat annual fee and an additional $0.10 per book for members who borrow more than 20 books per year.
To break even, the fl... | 4,160 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The library charges a monthly membership fee of $10, so that's $10 per month. Since there are 12 months in a year, the annual membership fee would be 12 times $10, which is $120. Then, there's an additional $0.25 per book borrowed.
The book club is paying for each member's borrowings, but to ... | [
{
"from": "human",
"value": "A library charges a monthly membership fee of $10, and an additional $0.25 per book borrowed. \nA book club decides to pay for each member's borrowings, \nand to cover the costs, each member is charged a flat annual fee. \nHowever, members who borrow more than 20 books per year ... |
Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
[/asy] | Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);
dot(origin);
dot(dir(0));
dot(dir(60));
dot(dir(120));
dot(dir(180));
dot(dir(240));
dot(dir(300));
[/asy] | Okay, so the problem is about counting how many equilateral triangles can be formed with all three vertices in a hexagonal lattice, where each point is one unit apart from its nearest neighbors. The Asymptote code provided draws a small hexagon with a central point and six surrounding points, each at 60-degree angles f... | To determine the number of equilateral triangles in the hexagonal lattice shown, we first consider the structure of the lattice. The lattice consists of a central point and six surrounding points, each one unit away from the center and each adjacent pair of surrounding points also one unit apart.
1. **Small Equilatera... | 4,403 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so the problem is about counting how many equilateral triangles can be formed with all three vertices in a hexagonal lattice, where each point is one unit apart from its nearest neighbors. The Asymptote code provided draws a small hexagon with a central point and six surrounding points, each at 60-degree ... | [
{
"from": "human",
"value": "Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);\ndot(origin);\ndot(dir(0));\ndot(dir(60));\ndot(dir(120));\ndot(dir(180));\ndot(dir(240));\ndot(dir(300));\n[/as... |
A department store has a promotion where customers can get a 10% discount on their total purchase if they pay in cash. However, if they use a credit card, they will be charged a 2% processing fee. A customer purchases a TV and a sound system, with a total price of $\$800$. If they pay in cash, how much will they pay? I... | A department store has a promotion where customers can get a 10% discount on their total purchase if they pay in cash. However, if they use a credit card, they will be charged a 2% processing fee. A customer purchases a TV and a sound system, with a total price of $\$800$. If they pay in cash, how much will they pay? I... | Okay, let's tackle this problem step by step. So, the customer is buying a TV and a sound system that total $800. The store has a promotion where if you pay in cash, you get a 10% discount. But if you use a credit card, they charge a 2% processing fee instead. We need to find out how much the customer pays in each case... | If the customer pays in cash, they receive a 10% discount on the total purchase of \$800. The discount is calculated as:
\[ 0.10 \times 800 = 80 \]
Subtracting this discount from the original total:
\[ 800 - 80 = 720 \]
**Cash payment: \$720**
If the customer uses a credit card, a 2% processing fee is added to... | 129 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's tackle this problem step by step. So, the customer is buying a TV and a sound system that total $800. The store has a promotion where if you pay in cash, you get a 10% discount. But if you use a credit card, they charge a 2% processing fee instead. We need to find out how much the customer pays in e... | [
{
"from": "human",
"value": "A department store has a promotion where customers can get a 10% discount on their total purchase if they pay in cash. However, if they use a credit card, they will be charged a 2% processing fee. A customer purchases a TV and a sound system, with a total price of $\\$800$. If t... |
What is the largest positive integer $n$ for which $n^3+144$ is divisible by $n+12$? | null | Alright, so I have this problem here: "What is the largest positive integer \( n \) for which \( n^3 + 144 \) is divisible by \( n + 12 \)." Hmm, okay. Let me think about how to approach this.
First, divisibility. If \( n + 12 \) divides \( n^3 + 144 \), that means when we divide \( n^3 + 144 \) by \( n + 12 \), there... | To determine the largest positive integer \( n \) for which \( n^3 + 144 \) is divisible by \( n + 12 \), we use the Remainder Theorem and polynomial division.
First, by the Remainder Theorem, the remainder when \( n^3 + 144 \) is divided by \( n + 12 \) is given by substituting \( n = -12 \) into the polynomial:
\[
(... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I have this problem here: "What is the largest positive integer \( n \) for which \( n^3 + 144 \) is divisible by \( n + 12 \)." Hmm, okay. Let me think about how to approach this.
First, divisibility. If \( n + 12 \) divides \( n^3 + 144 \), that means when we divide \( n^3 + 144 \) by \( n + 12 \... | [
{
"from": "human",
"value": "What is the largest positive integer $n$ for which $n^3+144$ is divisible by $n+12$?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this problem here: \"What is the largest positive integer \\( n \\) for which \\( n^3 + 144 \\) is divisible by \\( n + 12 ... |
A circle is inscribed in a triangle whose sides have lengths 8, 15 and 17 units. Find the radius of the inscribed circle. | A circle is inscribed in a triangle whose sides have lengths 8, 15 and 17 units. Find the radius of the inscribed circle. | Okay, so I need to find the radius of the inscribed circle in a triangle with sides 8, 15, and 17 units. Hmm, inscribed circle... right, the radius of the incircle. I remember there's a formula for the radius of the incircle, something involving the area of the triangle and the semi-perimeter. Let me recall.
First, m... | To find the radius of the inscribed circle in a triangle with sides 8, 15, and 17 units, we first verify if the triangle is a right triangle. Checking the Pythagorean theorem:
\[
8^2 + 15^2 = 64 + 225 = 289 = 17^2
\]
Thus, it is a right triangle with legs 8 and 15, and hypotenuse 17. The area \(A\) of the triangle is... | 10,763 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the radius of the inscribed circle in a triangle with sides 8, 15, and 17 units. Hmm, inscribed circle... right, the radius of the incircle. I remember there's a formula for the radius of the incircle, something involving the area of the triangle and the semi-perimeter. Let me recall.
... | [
{
"from": "human",
"value": "A circle is inscribed in a triangle whose sides have lengths 8, 15 and 17 units. Find the radius of the inscribed circle."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the radius of the inscribed circle in a triangle with sides 8, 15, and 17 units. ... |
Consider two sets of positive integers: $A = \{20, 30, 40\}$ and $B = \{35, 42, 50\}$. Determine the number of elements in the symmetric difference of the sets of divisors of $A$ and $B$. In other words, find the number of positive integers that divide either one or more elements of $A$ or one or more elements of $B$, ... | Consider two sets of positive integers: $A = \{20, 30, 40\}$ and $B = \{35, 42, 50\}$. Determine the number of elements in the symmetric difference of the sets of divisors of $A$ and $B$. In other words, find the number of positive integers that divide either one or more elements of $A$ or one or more elements of $B$, ... | Okay, so I need to find the symmetric difference between the sets of divisors of A and B. A is {20, 30, 40} and B is {35, 42, 50}. The symmetric difference means numbers that divide elements of A or B but not both. Hmm, let me break this down step by step.
First, I think I need to find all the divisors of each element... | To determine the number of elements in the symmetric difference of the sets of divisors of \(A = \{20, 30, 40\}\) and \(B = \{35, 42, 50\}\), we follow these steps:
1. **Find the divisors of each element in \(A\):**
- Divisors of 20: \(1, 2, 4, 5, 10, 20\)
- Divisors of 30: \(1, 2, 3, 5, 6, 10, 15, 30\)
- Div... | 32,818 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the symmetric difference between the sets of divisors of A and B. A is {20, 30, 40} and B is {35, 42, 50}. The symmetric difference means numbers that divide elements of A or B but not both. Hmm, let me break this down step by step.
First, I think I need to find all the divisors of each... | [
{
"from": "human",
"value": "Consider two sets of positive integers: $A = \\{20, 30, 40\\}$ and $B = \\{35, 42, 50\\}$. Determine the number of elements in the symmetric difference of the sets of divisors of $A$ and $B$. In other words, find the number of positive integers that divide either one or more ele... |
What is the value of the following expression: $\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}$? Express your answer as a common fraction. | What is the value of the following expression: $\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}$? Express your answer as a common fraction. | Okay, let me try to figure out the value of this expression: 1/3 - 1/9 + 1/27 - 1/81 + 1/243. Hmm, it looks like a series where each term is a fraction with denominators that are powers of 3, right? Let me check the pattern here.
Starting with the first term, 1/3. The next term is -1/9, which is -1/(3^2). Then +1/27,... | The given expression is a finite alternating geometric series: \(\frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} + \frac{1}{243}\).
1. **Identify the parameters of the geometric series:**
- The first term \(a = \frac{1}{3}\)
- The common ratio \(r = -\frac{1}{3}\)
- The number of terms \(n = 5\)
2. *... | 18,958 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out the value of this expression: 1/3 - 1/9 + 1/27 - 1/81 + 1/243. Hmm, it looks like a series where each term is a fraction with denominators that are powers of 3, right? Let me check the pattern here.
Starting with the first term, 1/3. The next term is -1/9, which is -1/(3^2). The... | [
{
"from": "human",
"value": "What is the value of the following expression: $\\frac{1}{3}-\\frac{1}{9}+\\frac{1}{27}-\\frac{1}{81}+\\frac{1}{243}$? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out the value of this expression: 1/3 -... |
A convex polyhedron has a total of 36 edges and 18 vertices. If it has 8 triangular faces, how many quadrilateral faces does it have? | A convex polyhedron has a total of 36 edges and 18 vertices. If it has 8 triangular faces, how many quadrilateral faces does it have? | Okay, so I need to find out how many quadrilateral faces a convex polyhedron has, given that it has 36 edges, 18 vertices, 8 triangular faces, and the rest are quadrilateral faces. Hmm. Let me remember Euler's formula because that might help here. Euler's formula relates the number of vertices (V), edges (E), and faces... | Given a convex polyhedron with 36 edges, 18 vertices, and 8 triangular faces, we need to determine the number of quadrilateral faces.
First, we use Euler's formula for convex polyhedrons:
\[
V - E + F = 2
\]
Substituting the given values \(V = 18\) and \(E = 36\):
\[
18 - 36 + F = 2
\]
Solving for \(F\):
\[
-18 + F = ... | 32,147 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find out how many quadrilateral faces a convex polyhedron has, given that it has 36 edges, 18 vertices, 8 triangular faces, and the rest are quadrilateral faces. Hmm. Let me remember Euler's formula because that might help here. Euler's formula relates the number of vertices (V), edges (E), a... | [
{
"from": "human",
"value": "A convex polyhedron has a total of 36 edges and 18 vertices. If it has 8 triangular faces, how many quadrilateral faces does it have?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find out how many quadrilateral faces a convex polyhedron has, given that ... |
Apples are sold at the farmer's market at a rate of $\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples? | Apples are sold at the farmer's market at a rate of $\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples? | Okay, let's see here. The problem is about figuring out how much 15 pounds of apples cost if they're sold at $4 per five pounds. Hmm, so first, I need to understand the rate given. It says $4 for five pounds. That means for every five pounds of apples, it costs $4. So, I need to find out how much 15 pounds would cost. ... | Apples are sold at a rate of $4 per five pounds. To find the cost of 15 pounds of apples, we can use the following methods:
1. **Cost per pound method**:
- The cost per pound is calculated as:
\[
\text{Cost per pound} = \frac{\$4}{5 \text{ pounds}} = \$0.80 \text{ per pound}
\]
- The total cost fo... | 6,642 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see here. The problem is about figuring out how much 15 pounds of apples cost if they're sold at $4 per five pounds. Hmm, so first, I need to understand the rate given. It says $4 for five pounds. That means for every five pounds of apples, it costs $4. So, I need to find out how much 15 pounds woul... | [
{
"from": "human",
"value": "Apples are sold at the farmer's market at a rate of $\\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see here. The problem is about figuring out how much 15 p... |
Problem:
A circular fountain is surrounded by a circular path that is 2 meters wide. If the radius of the fountain is 3 meters, what is the area of the path? Express your answer in terms of $\pi$. | Problem:
A circular fountain is surrounded by a circular path that is 2 meters wide. If the radius of the fountain is 3 meters, what is the area of the path? Express your answer in terms of $\pi$. | Okay, so I need to find the area of the circular path around the fountain. Let me see. The fountain has a radius of 3 meters, and the path is 2 meters wide. Hmm. I think the path forms a ring around the fountain, right? So maybe I need to calculate the area of the bigger circle that includes both the fountain and the p... | To find the area of the circular path around the fountain, we need to calculate the area of the larger circle (which includes both the fountain and the path) and subtract the area of the fountain itself.
1. The radius of the fountain is 3 meters. The path is 2 meters wide, so the radius of the larger circle (fountain ... | 35,011 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the area of the circular path around the fountain. Let me see. The fountain has a radius of 3 meters, and the path is 2 meters wide. Hmm. I think the path forms a ring around the fountain, right? So maybe I need to calculate the area of the bigger circle that includes both the fountain a... | [
{
"from": "human",
"value": "Problem:\nA circular fountain is surrounded by a circular path that is 2 meters wide. If the radius of the fountain is 3 meters, what is the area of the path? Express your answer in terms of $\\pi$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the ... |
Jordan wants to divide his 24 chocolate bars into 8 boxes of equal weight. Each box can hold only whole bars, so some boxes may have more bars than others. If he gives one of these boxes to his friend Shaina, what is the maximum number of bars Shaina can get? | Jordan wants to divide his 24 chocolate bars into 8 boxes of equal weight. Each box can hold only whole bars, so some boxes may have more bars than others. If he gives one of these boxes to his friend Shaina, what is the maximum number of bars Shaina can get? | Okay, so Jordan has 24 chocolate bars, and he wants to divide them into 8 boxes with equal weight. Each box can only hold whole bars, so some boxes might have more bars than others. The question is, if he gives one of these boxes to his friend Shaina, what's the maximum number of bars Shaina can get?
Hmm, let's break ... | Jordan wants to divide his 24 chocolate bars into 8 boxes of equal weight, where each box can only hold whole bars. The goal is to determine the maximum number of bars Shaina can get if she receives one of these boxes.
Key considerations:
1. The chocolate bars can have different weights.
2. To maximize the number of b... | 30,781 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Jordan has 24 chocolate bars, and he wants to divide them into 8 boxes with equal weight. Each box can only hold whole bars, so some boxes might have more bars than others. The question is, if he gives one of these boxes to his friend Shaina, what's the maximum number of bars Shaina can get?
Hmm, let'... | [
{
"from": "human",
"value": "Jordan wants to divide his 24 chocolate bars into 8 boxes of equal weight. Each box can hold only whole bars, so some boxes may have more bars than others. If he gives one of these boxes to his friend Shaina, what is the maximum number of bars Shaina can get?"
},
{
"from... |
12 players are in a soccer team. During a practice session, every player passes the ball to every other player exactly once. How many passes are made in total? | 12 players are in a soccer team. During a practice session, every player passes the ball to every other player exactly once. How many passes are made in total? | Okay, so there's this problem about a soccer team with 12 players. During practice, every player passes the ball to every other player exactly once. We need to figure out how many passes are made in total. Hmm, let's think about how to approach this.
First, let's make sure I understand the problem correctly. There are... | To determine the total number of passes made by 12 players where each player passes the ball to every other player exactly once, we can consider the following reasoning:
1. Each player passes the ball to 11 other players.
2. With 12 players, each making 11 passes, the initial calculation is \(12 \times 11 = 132\).
Ho... | 17,652 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so there's this problem about a soccer team with 12 players. During practice, every player passes the ball to every other player exactly once. We need to figure out how many passes are made in total. Hmm, let's think about how to approach this.
First, let's make sure I understand the problem correctly. T... | [
{
"from": "human",
"value": "12 players are in a soccer team. During a practice session, every player passes the ball to every other player exactly once. How many passes are made in total?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so there's this problem about a soccer team with 12 players. ... |
A line is parameterized by $(x,y) = (\cos t + 3, \sin t - 2)$, where $t$ is a real number. Find the equation of the line. Enter the equation in the form "$y = mx + b$". | A line is parameterized by $(x,y) = (\cos t + 3, \sin t - 2)$, where $t$ is a real number. Find the equation of the line. Enter the equation in the form "$y = mx + b$". | Okay, let me try to figure out the equation of this line. The problem says it's parameterized by (x, y) = (cos t + 3, sin t - 2), where t is a real number. Hmm, parameterized lines usually involve a direction vector and a point, but here they're using cosine and sine. That makes me think of a circle, actually, because ... | The given parameterization is \((x, y) = (\cos t + 3, \sin t - 2)\). To determine the equation of the line, we start by expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\):
\[
\cos t = x - 3
\]
\[
\sin t = y + 2
\]
Using the Pythagorean identity \(\cos^2 t + \sin^2 t = 1\), we substitute the expressions ... | 22,243 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out the equation of this line. The problem says it's parameterized by (x, y) = (cos t + 3, sin t - 2), where t is a real number. Hmm, parameterized lines usually involve a direction vector and a point, but here they're using cosine and sine. That makes me think of a circle, actually, ... | [
{
"from": "human",
"value": "A line is parameterized by $(x,y) = (\\cos t + 3, \\sin t - 2)$, where $t$ is a real number. Find the equation of the line. Enter the equation in the form \"$y = mx + b$\"."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out the equation of this ... |
Evaluate $(-27)^{5/3}$. | Evaluate $(-27)^{5/3}$. | Okay, so I need to evaluate (-27)^{5/3}. Hmm, fractional exponents can sometimes be tricky, especially when dealing with negative bases. Let me recall the rules for exponents and radicals.
First, I remember that a fractional exponent like 5/3 can be broken down into a root and a power. Specifically, the denominator o... | To evaluate \((-27)^{5/3}\), we can break down the fractional exponent into a root and a power. The exponent \(\frac{5}{3}\) can be interpreted as taking the cube root first and then raising the result to the 5th power, or raising to the 5th power first and then taking the cube root. Both approaches yield the same resu... | 25,367 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to evaluate (-27)^{5/3}. Hmm, fractional exponents can sometimes be tricky, especially when dealing with negative bases. Let me recall the rules for exponents and radicals.
First, I remember that a fractional exponent like 5/3 can be broken down into a root and a power. Specifically, the denom... | [
{
"from": "human",
"value": "Evaluate $(-27)^{5/3}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to evaluate (-27)^{5/3}. Hmm, fractional exponents can sometimes be tricky, especially when dealing with negative bases. Let me recall the rules for exponents and radicals. \n\nFirst, I r... |
Which of the following operations is correct?
A: $a^{4}+a^{5}=a^{9}$
B: $a^{3}\cdot a^{3}\cdot a^{3}=3a^{3}$
C: $a^{4}\cdot a^{6}=a^{10}$
D: $(-a^{3})^{6}=a^{9}$ | null | Okay, so I need to figure out which of these operations is correct. Let's go through each option one by one. Maybe I should start by recalling some exponent rules. Let me remember... when you multiply like bases, you add the exponents, right? And when you raise a power to another power, you multiply the exponents. Also... | To determine which operation is correct, we need to apply exponent rules to each option:
- **Option A**: \(a^4 + a^5 = a^9\)
- Adding terms with different exponents is not possible. This is incorrect.
- **Option B**: \(a^3 \cdot a^3 \cdot a^3 = 3a^3\)
- Multiplying exponents with the same base involves adding the... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out which of these operations is correct. Let's go through each option one by one. Maybe I should start by recalling some exponent rules. Let me remember... when you multiply like bases, you add the exponents, right? And when you raise a power to another power, you multiply the exponen... | [
{
"from": "human",
"value": "Which of the following operations is correct?\n\nA: $a^{4}+a^{5}=a^{9}$\n\nB: $a^{3}\\cdot a^{3}\\cdot a^{3}=3a^{3}$\n\nC: $a^{4}\\cdot a^{6}=a^{10}$\n\nD: $(-a^{3})^{6}=a^{9}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out which of these opera... |
A deck of 52 playing cards is shuffled, and three cards are drawn at random without replacement. What is the probability that the third card drawn is a spade, given that the first two cards drawn are not spades? Express your answer as a common fraction. | A deck of 52 playing cards is shuffled, and three cards are drawn at random without replacement. What is the probability that the third card drawn is a spade, given that the first two cards drawn are not spades? Express your answer as a common fraction. | Okay, so I need to find the probability that the third card drawn is a spade, given that the first two cards drawn are not spades. Hmm, let's think about this step by step.
First, the problem involves conditional probability. The formula for conditional probability is P(A|B) = P(A and B) / P(B). But in this case, sinc... | Given a deck of 52 playing cards, we need to find the probability that the third card drawn is a spade, given that the first two cards drawn are not spades.
1. **Initial Consideration**:
- The deck originally has 52 cards: 13 spades and 39 non-spades.
- After drawing two non-spades, the deck has 50 cards remaini... | 35,705 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that the third card drawn is a spade, given that the first two cards drawn are not spades. Hmm, let's think about this step by step.
First, the problem involves conditional probability. The formula for conditional probability is P(A|B) = P(A and B) / P(B). But in this ca... | [
{
"from": "human",
"value": "A deck of 52 playing cards is shuffled, and three cards are drawn at random without replacement. What is the probability that the third card drawn is a spade, given that the first two cards drawn are not spades? Express your answer as a common fraction."
},
{
"from": "gp... |
A set of five coins consists of two fair coins and three biased coins. The probability of getting heads on a single toss of a biased coin is $\frac{1}{3}$. If a coin is randomly selected from the set and flipped three times, what is the probability that at least two of the flips result in heads? Express your answer as ... | A set of five coins consists of two fair coins and three biased coins. The probability of getting heads on a single toss of a biased coin is $\frac{1}{3}$. If a coin is randomly selected from the set and flipped three times, what is the probability that at least two of the flips result in heads? Express your answer as ... | Okay, let me try to figure out this probability problem. So, we have a set of five coins. Two of them are fair, meaning the probability of getting heads is 1/2, and three are biased, where the probability of getting heads is 1/3. We need to find the probability that when we randomly pick a coin and flip it three times,... | A set of five coins consists of two fair coins and three biased coins. The probability of getting heads on a single toss of a biased coin is \(\frac{1}{3}\). We need to find the probability that a randomly selected coin flipped three times results in at least two heads.
1. **Probability of selecting a fair coin (2/5):... | 19,315 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this probability problem. So, we have a set of five coins. Two of them are fair, meaning the probability of getting heads is 1/2, and three are biased, where the probability of getting heads is 1/3. We need to find the probability that when we randomly pick a coin and flip it thre... | [
{
"from": "human",
"value": "A set of five coins consists of two fair coins and three biased coins. The probability of getting heads on a single toss of a biased coin is $\\frac{1}{3}$. If a coin is randomly selected from the set and flipped three times, what is the probability that at least two of the flip... |
If $2x - 3y = 15$ and $x + 2y = 7$, what is the value of $x$? | If $2x - 3y = 15$ and $x + 2y = 7$, what is the value of $x$? | Okay, so I need to solve this system of equations: 2x minus 3y equals 15 and x plus 2y equals 7. Hmm, let me think. There are two variables here, x and y, and two equations, so I should be able to find values for both. The question specifically asks for the value of x, so maybe I can focus on that.
Alright, let me re... | To solve the system of equations \(2x - 3y = 15\) and \(x + 2y = 7\), we can use the substitution method.
First, solve the second equation for \(x\):
\[
x + 2y = 7 \implies x = 7 - 2y
\]
Substitute \(x = 7 - 2y\) into the first equation:
\[
2(7 - 2y) - 3y = 15
\]
Simplify and solve for \(y\):
\[
14 - 4y - 3y = 15 \i... | 23,545 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this system of equations: 2x minus 3y equals 15 and x plus 2y equals 7. Hmm, let me think. There are two variables here, x and y, and two equations, so I should be able to find values for both. The question specifically asks for the value of x, so maybe I can focus on that.
Alright, l... | [
{
"from": "human",
"value": "If $2x - 3y = 15$ and $x + 2y = 7$, what is the value of $x$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this system of equations: 2x minus 3y equals 15 and x plus 2y equals 7. Hmm, let me think. There are two variables here, x and y, and two eq... |
Three positive integers $a,$ $b,$ and $y$ form a Modified O'Hara triple $(a,b,y)$ if $\sqrt{a}\cdot\sqrt{b}=y.$ For example, $(9,16,12)$ is a Modified O'Hara triple because $\sqrt{9}\cdot\sqrt{16}=12.$
If $(64,25,y)$ is a Modified O'Hara triple, determine the value of $y$ that satisfies the condition. | Three positive integers $a,$ $b,$ and $y$ form a Modified O'Hara triple $(a,b,y)$ if $\sqrt{a}\cdot\sqrt{b}=y.$ For example, $(9,16,12)$ is a Modified O'Hara triple because $\sqrt{9}\cdot\sqrt{16}=12.$
If $(64,25,y)$ is a Modified O'Hara triple, determine the value of $y$ that satisfies the condition. | Alright, let me try to figure out this problem. So, we need to find the value of y such that (64, 25, y) forms a Modified O'Hara triple. The definition says that three positive integers a, b, and y form such a triple if the product of the square roots of a and b equals y. The example given is (9, 16, 12) because sqrt(9... | To determine the value of \( y \) such that \( (64, 25, y) \) forms a Modified O'Hara triple, we need to check the condition \( \sqrt{a} \cdot \sqrt{b} = y \).
1. Calculate the square roots of the given integers \( a = 64 \) and \( b = 25 \):
- \( \sqrt{64} = 8 \)
- \( \sqrt{25} = 5 \)
2. Multiply these square ... | 30,180 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let me try to figure out this problem. So, we need to find the value of y such that (64, 25, y) forms a Modified O'Hara triple. The definition says that three positive integers a, b, and y form such a triple if the product of the square roots of a and b equals y. The example given is (9, 16, 12) becaus... | [
{
"from": "human",
"value": "Three positive integers $a,$ $b,$ and $y$ form a Modified O'Hara triple $(a,b,y)$ if $\\sqrt{a}\\cdot\\sqrt{b}=y.$ For example, $(9,16,12)$ is a Modified O'Hara triple because $\\sqrt{9}\\cdot\\sqrt{16}=12.$\n\nIf $(64,25,y)$ is a Modified O'Hara triple, determine the value of $... |
Lard decides to share a pizza with his friend Jelly. He takes a whole, uncut pepperoni pizza and cuts it in half. He then cuts these halves in half and gives one of the slices to Jelly. There are 9 slices of pepperoni on the slice Lard gives Jelly after 1 of them falls off the slice when Lard picks it up. How many slic... | null | Okay, let's try to figure out this pizza problem. So, Lard shares a pizza with Jelly. The pizza is initially whole and uncut. First, Lard cuts it in half. Then he cuts those halves in half again. So, after the first cut, we have two halves. Then cutting each half into half again would result in four equal slices, right... | Lard starts with a whole, uncut pizza and cuts it in half. He then cuts each half into half again, resulting in four equal slices. He gives one of these slices to Jelly. When Lard picks up the slice to give to Jelly, one pepperoni falls off, leaving 9 pepperonis on the slice.
Since the slice originally had 9 + 1 = 10... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to figure out this pizza problem. So, Lard shares a pizza with Jelly. The pizza is initially whole and uncut. First, Lard cuts it in half. Then he cuts those halves in half again. So, after the first cut, we have two halves. Then cutting each half into half again would result in four equal slice... | [
{
"from": "human",
"value": "Lard decides to share a pizza with his friend Jelly. He takes a whole, uncut pepperoni pizza and cuts it in half. He then cuts these halves in half and gives one of the slices to Jelly. There are 9 slices of pepperoni on the slice Lard gives Jelly after 1 of them falls off the s... |
What is the sum of the reciprocals of the first five positive integers? | What is the sum of the reciprocals of the first five positive integers? | Okay, so I need to find the sum of the reciprocals of the first five positive integers. Let me think... The first five positive integers are 1, 2, 3, 4, and 5. Their reciprocals would be 1/1, 1/2, 1/3, 1/4, and 1/5. So the problem is asking me to add these fractions together.
Hmm, adding fractions. I remember that to... | The first five positive integers are 1, 2, 3, 4, and 5. Their reciprocals are \( \frac{1}{1} \), \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{4} \), and \( \frac{1}{5} \). To find the sum of these reciprocals, we need a common denominator.
The least common multiple (LCM) of the denominators 1, 2, 3, 4, and 5 is ... | 9,193 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the sum of the reciprocals of the first five positive integers. Let me think... The first five positive integers are 1, 2, 3, 4, and 5. Their reciprocals would be 1/1, 1/2, 1/3, 1/4, and 1/5. So the problem is asking me to add these fractions together.
Hmm, adding fractions. I remember... | [
{
"from": "human",
"value": "What is the sum of the reciprocals of the first five positive integers?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of the reciprocals of the first five positive integers. Let me think... The first five positive integers are 1, 2, 3, 4, an... |
Eco-Friendly Water Bottle Company charges a $\$75$ set-up fee plus $\$15$ for each bottle produced. Green Bottle Company has no set-up fee but charges $\$20$ per bottle. What is the minimum number of bottles for which a customer saves money by using Eco-Friendly Water Bottle Company? | Eco-Friendly Water Bottle Company charges a $\$75$ set-up fee plus $\$15$ for each bottle produced. Green Bottle Company has no set-up fee but charges $\$20$ per bottle. What is the minimum number of bottles for which a customer saves money by using Eco-Friendly Water Bottle Company? | Okay, so I need to figure out the minimum number of bottles where Eco-Friendly Water Bottle Company becomes cheaper than Green Bottle Company. Let me start by understanding the problem again.
Eco-Friendly has a setup fee of $75 and charges $15 per bottle. Green Bottle has no setup fee but charges $20 per bottle. I nee... | To determine the minimum number of bottles for which a customer saves money by using Eco-Friendly Water Bottle Company instead of Green Bottle Company, we need to compare the total costs from both companies.
The total cost for Eco-Friendly Water Bottle Company is given by:
\[ 75 + 15n \]
where \( n \) is the number of... | 15,408 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the minimum number of bottles where Eco-Friendly Water Bottle Company becomes cheaper than Green Bottle Company. Let me start by understanding the problem again.
Eco-Friendly has a setup fee of $75 and charges $15 per bottle. Green Bottle has no setup fee but charges $20 per bottl... | [
{
"from": "human",
"value": "Eco-Friendly Water Bottle Company charges a $\\$75$ set-up fee plus $\\$15$ for each bottle produced. Green Bottle Company has no set-up fee but charges $\\$20$ per bottle. What is the minimum number of bottles for which a customer saves money by using Eco-Friendly Water Bottle ... |
Two positive integers, $a$ and $b$, have a greatest common divisor of 12 and a least common multiple of $ab/12$. If the difference between $a$ and $b$ is 60, what is the smallest possible value of the larger integer? | Two positive integers, $a$ and $b$, have a greatest common divisor of 12 and a least common multiple of $ab/12$. If the difference between $a$ and $b$ is 60, what is the smallest possible value of the larger integer? | Okay, so I need to find two positive integers, a and b, such that their greatest common divisor (GCD) is 12, their least common multiple (LCM) is ab divided by 12, and the difference between them is 60. The question is asking for the smallest possible value of the larger integer. Hmm, let me break this down step by ste... | Given two positive integers \(a\) and \(b\) such that their greatest common divisor (GCD) is 12, their least common multiple (LCM) is \(\frac{ab}{12}\), and the difference between them is 60, we need to find the smallest possible value of the larger integer.
1. **GCD and LCM Relationship**: Since the GCD of \(a\) and ... | 23,592 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find two positive integers, a and b, such that their greatest common divisor (GCD) is 12, their least common multiple (LCM) is ab divided by 12, and the difference between them is 60. The question is asking for the smallest possible value of the larger integer. Hmm, let me break this down ste... | [
{
"from": "human",
"value": "Two positive integers, $a$ and $b$, have a greatest common divisor of 12 and a least common multiple of $ab/12$. If the difference between $a$ and $b$ is 60, what is the smallest possible value of the larger integer?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I... |
Problem 5. Given a rectangle with sides $a \mathrm{~cm}$ and $b \mathrm{~cm}$. If the side with length $a \mathrm{~cm}$ is extended by $b \mathrm{~cm}$ and the side with length $b \mathrm{~cm}$ is extended by $a \mathrm{~cm}$, a square with area $100 \mathrm{~cm}^{2}$ is obtained. If the lengths of the sides of the giv... | null | Okay, let's try to solve this problem. So, we have a rectangle with sides a and b, both in centimeters. The problem says that if we extend the side of length a by b centimeters, and the side of length b by a centimeters, we get a square with area 100 cm². We need to find the rectangle with natural number sides that has... | Given a rectangle with sides \(a \mathrm{~cm}\) and \(b \mathrm{~cm}\), if the side with length \(a \mathrm{~cm}\) is extended by \(b \mathrm{~cm}\) and the side with length \(b \mathrm{~cm}\) is extended by \(a \mathrm{~cm}\), a square with area \(100 \mathrm{~cm}^{2}\) is obtained. We need to determine the rectangle ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to solve this problem. So, we have a rectangle with sides a and b, both in centimeters. The problem says that if we extend the side of length a by b centimeters, and the side of length b by a centimeters, we get a square with area 100 cm². We need to find the rectangle with natural number sides ... | [
{
"from": "human",
"value": "Problem 5. Given a rectangle with sides $a \\mathrm{~cm}$ and $b \\mathrm{~cm}$. If the side with length $a \\mathrm{~cm}$ is extended by $b \\mathrm{~cm}$ and the side with length $b \\mathrm{~cm}$ is extended by $a \\mathrm{~cm}$, a square with area $100 \\mathrm{~cm}^{2}$ is ... |
Find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is measured in inches and the y-axis is measured in feet. | Find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is measured in inches and the y-axis is measured in feet. | Okay, so I need to find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is in inches and the y-axis is in feet. Hmm, this seems a bit tricky because the units are different for each axis. Normally, if both axes are in the same units, I can just use the distance formula, right? T... | To find the distance between the points (3, 4) and (-2, 1) in a coordinate system where the x-axis is measured in inches and the y-axis is measured in feet, we need to convert the units to be consistent.
1. Convert the y-coordinates from feet to inches:
- The point (3, 4) has a y-coordinate of 4 feet, which conver... | 11,197 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is in inches and the y-axis is in feet. Hmm, this seems a bit tricky because the units are different for each axis. Normally, if both axes are in the same units, I can just use the distance formula, ... | [
{
"from": "human",
"value": "Find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is measured in inches and the y-axis is measured in feet."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the distance between the points (3,4) and (-2,1) in... |
Example 1.24. Perpendiculars are dropped from the point $P(2 ; 3 ;-5)$ to the coordinate planes. Find the equation of the plane passing through their feet. | null | Okay, so I need to find the equation of the plane that passes through the feet of the perpendiculars dropped from the point P(2, 3, -5) to the coordinate planes. Hmm, let me start by recalling what coordinate planes are. The coordinate planes in 3D are the XY-plane, YZ-plane, and XZ-plane. Each of these planes is defin... | To find the equation of the plane passing through the feet of the perpendiculars dropped from the point \( P(2, 3, -5) \) to the coordinate planes, we first determine the coordinates of these feet.
- The foot on the XY-plane (where \( z = 0 \)) is \( A(2, 3, 0) \).
- The foot on the YZ-plane (where \( x = 0 \)) is \(... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the equation of the plane that passes through the feet of the perpendiculars dropped from the point P(2, 3, -5) to the coordinate planes. Hmm, let me start by recalling what coordinate planes are. The coordinate planes in 3D are the XY-plane, YZ-plane, and XZ-plane. Each of these planes ... | [
{
"from": "human",
"value": "Example 1.24. Perpendiculars are dropped from the point $P(2 ; 3 ;-5)$ to the coordinate planes. Find the equation of the plane passing through their feet."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the equation of the plane that passes through t... |
In $\triangle XYZ$, the median from $X$ is given perpendicular to the median from $Y$. If $YZ=8$ and $XZ=10$, find the length of $XY$. | null | Okay, so I need to find the length of side XY in triangle XYZ. The problem says that the median from X is perpendicular to the median from Y. Also, they gave me YZ = 8 and XZ = 10. Hmm, medians and perpendicularity. Let me think about how to approach this.
First, let me recall what a median is. A median in a triangle ... | Given $\triangle XYZ$ with medians from $X$ and $Y$ being perpendicular, and knowing $YZ = 8$ and $XZ = 10$, we need to find the length of $XY$.
1. **Coordinate Setup**:
- Place $Z$ at the origin $(0, 0)$.
- Place $Y$ at $(8, 0)$ since $YZ = 8$.
- Let $X$ be at $(c, d)$ such that $XZ = 10$, giving $c^2 + d^2 ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the length of side XY in triangle XYZ. The problem says that the median from X is perpendicular to the median from Y. Also, they gave me YZ = 8 and XZ = 10. Hmm, medians and perpendicularity. Let me think about how to approach this.
First, let me recall what a median is. A median in a t... | [
{
"from": "human",
"value": "In $\\triangle XYZ$, the median from $X$ is given perpendicular to the median from $Y$. If $YZ=8$ and $XZ=10$, find the length of $XY$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the length of side XY in triangle XYZ. The problem says that the me... |
Find the total number of different integers the function
\[ f(x) = \left[x \right] + \left[2 \cdot x \right] + \left[\frac{5 \cdot x}{3} \right] + \left[3 \cdot x \right] + \left[4 \cdot x \right] \]
takes for $0 \leq x \leq 100.$ | null | Okay, so I have this problem where I need to find the total number of different integers that the function f(x) = [x] + [2x] + [5x/3] + [3x] + [4x] takes for x in the interval [0, 100]. The brackets here represent the floor function, right? So [y] is the greatest integer less than or equal to y.
First, I need to unde... | To find the total number of different integers the function \( f(x) = \left\lfloor x \right\rfloor + \left\lfloor 2x \right\rfloor + \left\lfloor \frac{5x}{3} \right\rfloor + \left\lfloor 3x \right\rfloor + \left\lfloor 4x \right\rfloor \) takes for \( 0 \leq x \leq 100 \), we need to consider the discontinuity points ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem where I need to find the total number of different integers that the function f(x) = [x] + [2x] + [5x/3] + [3x] + [4x] takes for x in the interval [0, 100]. The brackets here represent the floor function, right? So [y] is the greatest integer less than or equal to y.
First, I need... | [
{
"from": "human",
"value": "Find the total number of different integers the function\r\n\r\n\\[ f(x) = \\left[x \\right] + \\left[2 \\cdot x \\right] + \\left[\\frac{5 \\cdot x}{3} \\right] + \\left[3 \\cdot x \\right] + \\left[4 \\cdot x \\right] \\]\r\n\r\ntakes for $0 \\leq x \\leq 100.$"
},
{
"... |
A bookstore has three shelves, each containing a different type of book: fiction, non-fiction, and mystery. If 1/5 of the fiction books, 1/3 of the non-fiction books, and 3/5 of the mystery books are sold, and the remaining books fill 2/3 of the total shelf space, what fraction of the books were sold? | A bookstore has three shelves, each containing a different type of book: fiction, non-fiction, and mystery. If 1/5 of the fiction books, 1/3 of the non-fiction books, and 3/5 of the mystery books are sold, and the remaining books fill 2/3 of the total shelf space, what fraction of the books were sold? | Okay, let's see. The problem is about a bookstore with three shelves: fiction, non-fiction, and mystery. Each shelf has a different type of book. Then, fractions of each type are sold: 1/5 of fiction, 1/3 of non-fiction, and 3/5 of mystery. After those sales, the remaining books take up 2/3 of the total shelf space. Th... | Let \( F \) be the number of fiction books, \( N \) be the number of non-fiction books, and \( M \) be the number of mystery books.
When the fractions of each type of book are sold, the remaining books are:
- Fiction: \( \frac{4}{5}F \)
- Non-fiction: \( \frac{2}{3}N \)
- Mystery: \( \frac{2}{5}M \)
The total remain... | 11,682 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about a bookstore with three shelves: fiction, non-fiction, and mystery. Each shelf has a different type of book. Then, fractions of each type are sold: 1/5 of fiction, 1/3 of non-fiction, and 3/5 of mystery. After those sales, the remaining books take up 2/3 of the total shelf s... | [
{
"from": "human",
"value": "A bookstore has three shelves, each containing a different type of book: fiction, non-fiction, and mystery. If 1/5 of the fiction books, 1/3 of the non-fiction books, and 3/5 of the mystery books are sold, and the remaining books fill 2/3 of the total shelf space, what fraction ... |
To understand the performance of a certain type of tire, 8 were randomly selected for testing. Their maximum mileage (unit: 1000km) were: 96, 112, 97, 108, 99, 104, 86, 98. The median of these values is ( )
A: 100
B: 99
C: 98.5
D: 98 | null | Okay, let's see. I need to find the median of these tire mileage data. The numbers given are: 96, 112, 97, 108, 99, 104, 86, 98. Hmm, the options are A: 100, B: 99, C: 98.5, D: 98.
First, I remember that the median is the middle value when the numbers are arranged in order. If there's an even number of observations, ... | To find the median of the given tire mileage data: 96, 112, 97, 108, 99, 104, 86, 98, we need to follow these steps:
1. **Sort the data in ascending order**:
- Original data: 96, 112, 97, 108, 99, 104, 86, 98
- Sorted data: 86, 96, 97, 98, 99, 104, 108, 112
2. **Determine the positions of the middle values**:
... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find the median of these tire mileage data. The numbers given are: 96, 112, 97, 108, 99, 104, 86, 98. Hmm, the options are A: 100, B: 99, C: 98.5, D: 98.
First, I remember that the median is the middle value when the numbers are arranged in order. If there's an even number of observ... | [
{
"from": "human",
"value": "To understand the performance of a certain type of tire, 8 were randomly selected for testing. Their maximum mileage (unit: 1000km) were: 96, 112, 97, 108, 99, 104, 86, 98. The median of these values is ( )\nA: 100\nB: 99\nC: 98.5\nD: 98"
},
{
"from": "gpt",
"value":... |
For all real numbers $a$ and $b$, define the mathematical operation $\ast$ such that the following conditions apply: $a\ \ast\ 0 = a, a\ \ast\ b = b\ \ast\ a$, and $(a + 1)\ \ast\ b = (a\ \ast\ b) + a + b$. What is the value of $7\ \ast\ 3$? | For all real numbers $a$ and $b$, define the mathematical operation $\ast$ such that the following conditions apply: $a\ \ast\ 0 = a, a\ \ast\ b = b\ \ast\ a$, and $(a + 1)\ \ast\ b = (a\ \ast\ b) + a + b$. What is the value of $7\ \ast\ 3$? | Okay, so I need to figure out the value of 7 ∗ 3 using the given conditions. Let me start by writing down what's given:
1. For any real number a, a ∗ 0 = a. That's our identity property, maybe?
2. The operation is commutative: a ∗ b = b ∗ a.
3. And there's this recursive-like formula: (a + 1) ∗ b = (a ∗ b) + a + b.
H... | Given the operation \( \ast \) defined for all real numbers \( a \) and \( b \) with the conditions:
1. \( a \ast 0 = a \)
2. \( a \ast b = b \ast a \)
3. \( (a + 1) \ast b = (a \ast b) + a + b \)
We need to find the value of \( 7 \ast 3 \).
First, we use the third condition recursively to express \( 7 \ast 3 \) in t... | 5,176 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the value of 7 ∗ 3 using the given conditions. Let me start by writing down what's given:
1. For any real number a, a ∗ 0 = a. That's our identity property, maybe?
2. The operation is commutative: a ∗ b = b ∗ a.
3. And there's this recursive-like formula: (a + 1) ∗ b = (a ∗ b) + a... | [
{
"from": "human",
"value": "For all real numbers $a$ and $b$, define the mathematical operation $\\ast$ such that the following conditions apply: $a\\ \\ast\\ 0 = a, a\\ \\ast\\ b = b\\ \\ast\\ a$, and $(a + 1)\\ \\ast\\ b = (a\\ \\ast\\ b) + a + b$. What is the value of $7\\ \\ast\\ 3$?"
},
{
"fro... |
Example 4 As shown in Figure $4, O_{B}$ and $O_{C}$ are the centers of the excircles of $\triangle A B C$ opposite to vertices $B$ and $C$, respectively. The incircle $\odot I$ of $\triangle A B C$ touches the lines $A B$ and $A C$ at points $F$ and $E$, respectively. $O_{C} E$ and $O_{B} F$ intersect at point $P$. Pro... | null | Okay, let's try to tackle this geometry problem. It seems a bit involved, but I'll take it step by step. The problem states:
"As shown in Figure 4, \( O_{B} \) and \( O_{C} \) are the centers of the excircles of \( \triangle ABC \) opposite to vertices \( B \) and \( C \), respectively. The incircle \( \odot I \) of \... | To prove that \( PI \perp BC \) where \( P \) is the intersection of \( O_C E \) and \( O_B F \), we start by considering the properties of the incenter \( I \) and the excenters \( O_B \) and \( O_C \).
1. **Notations and Definitions:**
- \( I \) is the incenter of \( \triangle ABC \), and \( O_B \), \( O_C \) are... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to tackle this geometry problem. It seems a bit involved, but I'll take it step by step. The problem states:
"As shown in Figure 4, \( O_{B} \) and \( O_{C} \) are the centers of the excircles of \( \triangle ABC \) opposite to vertices \( B \) and \( C \), respectively. The incircle \( \odot I... | [
{
"from": "human",
"value": "Example 4 As shown in Figure $4, O_{B}$ and $O_{C}$ are the centers of the excircles of $\\triangle A B C$ opposite to vertices $B$ and $C$, respectively. The incircle $\\odot I$ of $\\triangle A B C$ touches the lines $A B$ and $A C$ at points $F$ and $E$, respectively. $O_{C} ... |
Let $d$ and $e$ denote the solutions of $3x^2+10x-25=0$. Find $(d-e)^2$. | Let $d$ and $e$ denote the solutions of $3x^2+10x-25=0$. Find $(d-e)^2$. | Alright, so I have this quadratic equation here: 3x² + 10x - 25 = 0. The problem says that d and e are the solutions, and I need to find (d - e)². Hmm, okay. Let me think about how to approach this.
First, I remember that for a quadratic equation of the form ax² + bx + c = 0, the solutions can be found using the quadr... | To find \((d - e)^2\) where \(d\) and \(e\) are the solutions of the quadratic equation \(3x^2 + 10x - 25 = 0\), we can use Vieta's formulas and properties of quadratic equations.
First, using Vieta's formulas, we know the sum and product of the roots \(d\) and \(e\):
\[
d + e = -\frac{b}{a} = -\frac{10}{3}
\]
\[
d \c... | 12,272 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I have this quadratic equation here: 3x² + 10x - 25 = 0. The problem says that d and e are the solutions, and I need to find (d - e)². Hmm, okay. Let me think about how to approach this.
First, I remember that for a quadratic equation of the form ax² + bx + c = 0, the solutions can be found using t... | [
{
"from": "human",
"value": "Let $d$ and $e$ denote the solutions of $3x^2+10x-25=0$. Find $(d-e)^2$."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I have this quadratic equation here: 3x² + 10x - 25 = 0. The problem says that d and e are the solutions, and I need to find (d - e)². Hmm, ok... |
What is the ratio of $x$ to $y$ if: $\frac{10x-3y}{13x-2y} = \frac{3}{5}$? Express your answer as a common fraction. | What is the ratio of $x$ to $y$ if: $\frac{10x-3y}{13x-2y} = \frac{3}{5}$? Express your answer as a common fraction. | Okay, let's see. I need to find the ratio of x to y given the equation (10x - 3y)/(13x - 2y) = 3/5. Hmm, ratios usually mean I have to find x/y or something like that. Let me think. So, first, maybe cross-multiply to get rid of the fraction. Yeah, that seems like a standard step.
So, if I cross-multiply, that would gi... | Given the equation \(\frac{10x - 3y}{13x - 2y} = \frac{3}{5}\), we need to find the ratio of \(x\) to \(y\).
First, we cross-multiply to eliminate the fraction:
\[
5(10x - 3y) = 3(13x - 2y)
\]
Expanding both sides, we get:
\[
50x - 15y = 39x - 6y
\]
Next, we move all \(x\) terms to one side and \(y\) terms to the ot... | 30,701 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find the ratio of x to y given the equation (10x - 3y)/(13x - 2y) = 3/5. Hmm, ratios usually mean I have to find x/y or something like that. Let me think. So, first, maybe cross-multiply to get rid of the fraction. Yeah, that seems like a standard step.
So, if I cross-multiply, that ... | [
{
"from": "human",
"value": "What is the ratio of $x$ to $y$ if: $\\frac{10x-3y}{13x-2y} = \\frac{3}{5}$? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the ratio of x to y given the equation (10x - 3y)/(13x - 2y) = 3/5. Hmm, rat... |
during a sale , the price of a pair of shoes is marked down 20 % from the regular price . after the sale ends , the price goes back to the original price . what is the percent of increase to the nearest percent from the sale price back to the regular price for the shoes ? | null | Okay, so I need to figure out the percentage increase from the sale price back to the regular price of the shoes. Let me start by breaking down the problem step by step.
First, the original price of the shoes is marked down by 20% during the sale. Then, after the sale ends, the price goes back up to the original price... | Let the regular price of the shoes be \( P \). During the sale, the price is marked down by 20%, so the sale price is 80% of the regular price.
1. **Calculating the sale price**:
\[
\text{Sale Price} = P - 0.20P = 0.80P
\]
2. **Calculating the percentage increase from the sale price back to the regular pric... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out the percentage increase from the sale price back to the regular price of the shoes. Let me start by breaking down the problem step by step.
First, the original price of the shoes is marked down by 20% during the sale. Then, after the sale ends, the price goes back up to the origin... | [
{
"from": "human",
"value": "during a sale , the price of a pair of shoes is marked down 20 % from the regular price . after the sale ends , the price goes back to the original price . what is the percent of increase to the nearest percent from the sale price back to the regular price for the shoes ?"
},
... |
Determine the range of the function $$H(x) = |x^2-4|-|x^2-9|.$$ Express your answer in interval notation. | Determine the range of the function $$H(x) = |x^2-4|-|x^2-9|.$$ Express your answer in interval notation. | Okay, so I need to find the range of the function H(x) = |x² - 4| - |x² - 9|. Hmm, let's see. The function involves absolute values of quadratic expressions. Since absolute value functions can sometimes be tricky, maybe I should break it down piece by piece. Let me start by recalling that the absolute value function |A... | To determine the range of the function \( H(x) = |x^2 - 4| - |x^2 - 9| \), we consider different cases based on the value of \( x^2 \):
1. **Case 1: \( x^2 < 4 \) (i.e., \( |x| < 2 \))**
- Both \( x^2 - 4 \) and \( x^2 - 9 \) are negative.
- \( H(x) = (-x^2 + 4) - (-x^2 + 9) = -5 \).
2. **Case 2: \( 4 \leq x^2 ... | 14,849 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the range of the function H(x) = |x² - 4| - |x² - 9|. Hmm, let's see. The function involves absolute values of quadratic expressions. Since absolute value functions can sometimes be tricky, maybe I should break it down piece by piece. Let me start by recalling that the absolute value fun... | [
{
"from": "human",
"value": "Determine the range of the function $$H(x) = |x^2-4|-|x^2-9|.$$ Express your answer in interval notation."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the range of the function H(x) = |x² - 4| - |x² - 9|. Hmm, let's see. The function involves absol... |
Lily purchased a new laptop and paid $15\%$ of the cost upfront, which was $\$200$. What was the price of the laptop? | Lily purchased a new laptop and paid $15\%$ of the cost upfront, which was $\$200$. What was the price of the laptop? | Okay, so Lily bought a new laptop and paid 15% of the cost upfront, and that amount was $200. The question is asking for the total price of the laptop. Hmm, let's see. I need to figure out the total price based on the fact that 15% of it is $200.
Alright, let me start by recalling some basic percentage concepts. If 1... | Lily paid 15% of the laptop's cost upfront, which was $200. To find the total price of the laptop, we set up the equation where 15% of the total price \( P \) equals $200:
\[
0.15 \times P = 200
\]
To isolate \( P \), we divide both sides by 0.15:
\[
P = \frac{200}{0.15}
\]
Converting the division into a fraction f... | 18,666 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Lily bought a new laptop and paid 15% of the cost upfront, and that amount was $200. The question is asking for the total price of the laptop. Hmm, let's see. I need to figure out the total price based on the fact that 15% of it is $200.
Alright, let me start by recalling some basic percentage concep... | [
{
"from": "human",
"value": "Lily purchased a new laptop and paid $15\\%$ of the cost upfront, which was $\\$200$. What was the price of the laptop?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so Lily bought a new laptop and paid 15% of the cost upfront, and that amount was $200. The question ... |
After the announcement of the Sichuan Province's college entrance examination scores in 2023, Shishi High School continued its glory. The scores of 12 students in a base class are (unit: points): 673, 673, 677, 679, 682, 682, 684, 685, 687, 691, 697, 705. The upper quartile of the scores of these 12 students is ______. | null | Okay, so I need to find the upper quartile of these 12 students' scores. Let me remember what upper quartile means. Quartiles divide the data into four equal parts. The upper quartile, also known as the third quartile (Q3), is the value below which 75% of the data falls. That means 25% of the data is above Q3 and 75% i... | To find the upper quartile (Q3) of the scores of the 12 students, we first note that the data is already sorted in ascending order. The upper quartile is the median of the upper half of the data.
Given the scores: 673, 673, 677, 679, 682, 682, 684, 685, 687, 691, 697, 705, we split the data into two halves. Since the... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the upper quartile of these 12 students' scores. Let me remember what upper quartile means. Quartiles divide the data into four equal parts. The upper quartile, also known as the third quartile (Q3), is the value below which 75% of the data falls. That means 25% of the data is above Q3 a... | [
{
"from": "human",
"value": "After the announcement of the Sichuan Province's college entrance examination scores in 2023, Shishi High School continued its glory. The scores of 12 students in a base class are (unit: points): 673, 673, 677, 679, 682, 682, 684, 685, 687, 691, 697, 705. The upper quartile of t... |
Harry and Sandy are on the diagonally opposite corners of a square field. Harry is standing at $(2,-3)$, and Sandy is standing at $(-4,5)$. They decide to meet at a point that is the midpoint of a line segment directly connecting both of them. What are the coordinates where they meet? | null | Okay, so Harry is at (2, -3) and Sandy is at (-4, 5). They want to meet at the midpoint of the line segment connecting their positions. Hmm, midpoints. Let me recall how to find the midpoint between two points in coordinate geometry.
I think the formula for the midpoint is average of the x-coordinates and the average... | Harry is standing at \((2, -3)\) and Sandy is standing at \((-4, 5)\). To find the midpoint of the line segment connecting their positions, we use the midpoint formula, which is the average of the x-coordinates and the average of the y-coordinates.
1. Calculate the x-coordinate of the midpoint:
\[
\frac{2 + (-4)... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so Harry is at (2, -3) and Sandy is at (-4, 5). They want to meet at the midpoint of the line segment connecting their positions. Hmm, midpoints. Let me recall how to find the midpoint between two points in coordinate geometry.
I think the formula for the midpoint is average of the x-coordinates and the... | [
{
"from": "human",
"value": "Harry and Sandy are on the diagonally opposite corners of a square field. Harry is standing at $(2,-3)$, and Sandy is standing at $(-4,5)$. They decide to meet at a point that is the midpoint of a line segment directly connecting both of them. What are the coordinates where they... |
When 242 is divided by a certain divisor, the remainder obtained is 6. When 698 is divided by the same divisor, the remainder obtained is 13. However, when the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 5. What is the value of the divisor? | null | Okay, let's try to solve this problem step by step. So, we have three pieces of information here:
1. When 242 is divided by a certain divisor (let's call it 'd'), the remainder is 6.
2. When 698 is divided by the same divisor 'd', the remainder is 13.
3. When the sum of 242 and 698 (which is 940) is divided by 'd', th... | Given the problem, we need to find a divisor \( d \) such that:
1. When 242 is divided by \( d \), the remainder is 6.
2. When 698 is divided by \( d \), the remainder is 13.
3. When the sum of 242 and 698 (which is 940) is divided by \( d \), the remainder is 5.
We start by noting the following:
1. \( 242 - 6 = 236 \... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to solve this problem step by step. So, we have three pieces of information here:
1. When 242 is divided by a certain divisor (let's call it 'd'), the remainder is 6.
2. When 698 is divided by the same divisor 'd', the remainder is 13.
3. When the sum of 242 and 698 (which is 940) is divided by... | [
{
"from": "human",
"value": "When 242 is divided by a certain divisor, the remainder obtained is 6. When 698 is divided by the same divisor, the remainder obtained is 13. However, when the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 5. What is the value of the div... |
Given a point in polar coordinates (r, θ), convert it to rectangular coordinates if r = 10 and θ = 3π/4. | Given a point in polar coordinates (r, θ), convert it to rectangular coordinates if r = 10 and θ = 3π/4. | Okay, so I need to convert a point from polar coordinates to rectangular coordinates. The given point is (r, θ) where r is 10 and θ is 3π/4. Hmm, let me remember how to do this. I think the formulas involve cosine and sine, right?
First, I recall that in polar coordinates, a point is represented by its distance from ... | To convert the polar coordinates \((r, \theta) = (10, 3\pi/4)\) to rectangular coordinates \((x, y)\), we use the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\).
1. Calculate \(x\):
\[
x = 10 \cos \left( \frac{3\pi}{4} \right)
\]
The angle \(\frac{3\pi}{4}\) is in the second quadrant, where cosin... | 24,972 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to convert a point from polar coordinates to rectangular coordinates. The given point is (r, θ) where r is 10 and θ is 3π/4. Hmm, let me remember how to do this. I think the formulas involve cosine and sine, right?
First, I recall that in polar coordinates, a point is represented by its distan... | [
{
"from": "human",
"value": "Given a point in polar coordinates (r, θ), convert it to rectangular coordinates if r = 10 and θ = 3π/4."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to convert a point from polar coordinates to rectangular coordinates. The given point is (r, θ) where r is... |
Solve for $y$: $\frac{y+2y}{y-2y} = \frac{250}{300-(3y+4y)}$. | Solve for $y$: $\frac{y+2y}{y-2y} = \frac{250}{300-(3y+4y)}$. | Okay, let's see. I need to solve this equation: (y + 2y) divided by (y - 2y) equals 250 divided by (300 minus (3y + 4y)). Hmm, first, maybe I should simplify both sides of the equation step by step. Let me start with the left side.
The numerator is y + 2y. If I combine those like terms, y plus 2y is 3y. Then the denom... | First, simplify both sides of the equation:
Left side:
\[
\frac{y + 2y}{y - 2y} = \frac{3y}{-y} = -3
\]
Right side:
\[
\frac{250}{300 - (3y + 4y)} = \frac{250}{300 - 7y}
\]
This simplifies the equation to:
\[
-3 = \frac{250}{300 - 7y}
\]
Multiply both sides by \(300 - 7y\):
\[
-3(300 - 7y) = 250
\]
Expand and simp... | 16,047 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to solve this equation: (y + 2y) divided by (y - 2y) equals 250 divided by (300 minus (3y + 4y)). Hmm, first, maybe I should simplify both sides of the equation step by step. Let me start with the left side.
The numerator is y + 2y. If I combine those like terms, y plus 2y is 3y. Then t... | [
{
"from": "human",
"value": "Solve for $y$: $\\frac{y+2y}{y-2y} = \\frac{250}{300-(3y+4y)}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to solve this equation: (y + 2y) divided by (y - 2y) equals 250 divided by (300 minus (3y + 4y)). Hmm, first, maybe I should simplify both ... |
What is the smallest positive integer $n$ such that $n^3 \equiv 2 \pmod{17}$? | What is the smallest positive integer $n$ such that $n^3 \equiv 2 \pmod{17}$? | Okay, so I need to find the smallest positive integer n such that n cubed is congruent to 2 modulo 17. Hmm, let's see. I remember that when dealing with congruences like this, especially with exponents, maybe Fermat's Little Theorem could be useful? Fermat's Little Theorem says that if p is a prime number, then for any... | To find the smallest positive integer \( n \) such that \( n^3 \equiv 2 \pmod{17} \), we start by checking small integers:
\[
\begin{aligned}
1^3 &\equiv 1 \pmod{17}, \\
2^3 &\equiv 8 \pmod{17}, \\
3^3 &\equiv 10 \pmod{17}, \\
4^3 &\equiv 13 \pmod{17}, \\
5^3 &\equiv 6 \pmod{17}, \\
6^3 &\equiv 12 \pmod{17}, \\
7^3 &\... | 11,436 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the smallest positive integer n such that n cubed is congruent to 2 modulo 17. Hmm, let's see. I remember that when dealing with congruences like this, especially with exponents, maybe Fermat's Little Theorem could be useful? Fermat's Little Theorem says that if p is a prime number, then... | [
{
"from": "human",
"value": "What is the smallest positive integer $n$ such that $n^3 \\equiv 2 \\pmod{17}$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest positive integer n such that n cubed is congruent to 2 modulo 17. Hmm, let's see. I remember that when dealing... |
Jose has 85 peanuts. Kenya has more peanuts than Jose. Kenya has 133 peanuts. How many more peanuts does Kenya have than Jose? | null | Okay, let's see. Jose has 85 peanuts, and Kenya has 133. The question is asking how many more peanuts Kenya has than Jose. Hmm. So, this is a subtraction problem, right? Because when they say "how many more," that usually means you subtract the smaller number from the bigger one.
Wait, let me make sure. If Jose has 85... | Kenya has 133 peanuts and Jose has 85. To find how many more peanuts Kenya has, subtract Jose's peanuts from Kenya's:
\[
133 - 85 = 48
\]
**Answer:** Kenya has 48 more peanuts than Jose. | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. Jose has 85 peanuts, and Kenya has 133. The question is asking how many more peanuts Kenya has than Jose. Hmm. So, this is a subtraction problem, right? Because when they say "how many more," that usually means you subtract the smaller number from the bigger one.
Wait, let me make sure. If Jos... | [
{
"from": "human",
"value": "Jose has 85 peanuts. Kenya has more peanuts than Jose. Kenya has 133 peanuts. How many more peanuts does Kenya have than Jose?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. Jose has 85 peanuts, and Kenya has 133. The question is asking how many more peanut... |
6 A $4 \times 4$ chessboard composed of 16 small squares, 8 of which are to be colored black, such that each row and each column has exactly 2 black squares. There are $\qquad$ different ways to do this. | null | Okay, so I have this problem here: I need to figure out how many different ways there are to color 8 squares black on a 4x4 chessboard, such that each row and each column has exactly 2 black squares. Hmm, interesting. Let me try to work through this step by step.
First, let me visualize the chessboard. It's a 4x4 grid... | To determine the number of ways to color 8 squares black on a 4x4 chessboard such that each row and each column has exactly 2 black squares, we need to count the number of 4x4 (0,1)-matrices with exactly 2 ones in each row and column. This is a well-known combinatorial problem.
1. **Understanding the Problem**:
- W... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here: I need to figure out how many different ways there are to color 8 squares black on a 4x4 chessboard, such that each row and each column has exactly 2 black squares. Hmm, interesting. Let me try to work through this step by step.
First, let me visualize the chessboard. It's a ... | [
{
"from": "human",
"value": "6 A $4 \\times 4$ chessboard composed of 16 small squares, 8 of which are to be colored black, such that each row and each column has exactly 2 black squares. There are $\\qquad$ different ways to do this."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this ... |
There are constants $a$ and $b$ such that $\frac{x-a}{x+b} = \frac{x^2-120x+4554}{x^2+65x-2700}$. What is $a+b$? | There are constants $a$ and $b$ such that $\frac{x-a}{x+b} = \frac{x^2-120x+4554}{x^2+65x-2700}$. What is $a+b$? | Okay, so I need to find constants a and b such that (x - a)/(x + b) is equal to (x² - 120x + 4554)/(x² + 65x - 2700). Then, after finding a and b, I need to compute a + b. Hmm, let me think.
First, I remember that if two rational expressions are equal for all x (except where the denominator is zero), then their numer... | To find the constants \(a\) and \(b\) such that \(\frac{x-a}{x+b} = \frac{x^2 - 120x + 4554}{x^2 + 65x - 2700}\), we start by cross-multiplying and equating the coefficients of the resulting polynomials.
Cross-multiplying gives:
\[
(x - a)(x^2 + 65x - 2700) = (x + b)(x^2 - 120x + 4554)
\]
Expanding both sides, we get... | 26,981 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find constants a and b such that (x - a)/(x + b) is equal to (x² - 120x + 4554)/(x² + 65x - 2700). Then, after finding a and b, I need to compute a + b. Hmm, let me think.
First, I remember that if two rational expressions are equal for all x (except where the denominator is zero), then the... | [
{
"from": "human",
"value": "There are constants $a$ and $b$ such that $\\frac{x-a}{x+b} = \\frac{x^2-120x+4554}{x^2+65x-2700}$. What is $a+b$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find constants a and b such that (x - a)/(x + b) is equal to (x² - 120x + 4554)/(x² + 65x - 2... |
Given the parabola $C$: $y^{2}=2px(p > 0)$ with focus $F$, and a point $M(x_{0},2 \sqrt {2})$ on the parabola $C$, a circle $M$ is tangent to the $y$-axis and intersects the line segment $MF$ at point $A$. If $\dfrac {|MA|}{|AF|}=2$, then $p=$ ______. | null | Okay, let's tackle this problem step by step. First, I need to make sure I understand all the given information and what is being asked. The problem is about a parabola C defined by the equation \( y^2 = 2px \) where \( p > 0 \). The focus of this parabola is F. There's a point M with coordinates \( (x_0, 2\sqrt{2}) \)... | Given the parabola \( C \): \( y^2 = 2px \) with focus \( F \), and a point \( M(x_0, 2\sqrt{2}) \) on the parabola \( C \), a circle \( M \) is tangent to the \( y \)-axis and intersects the line segment \( MF \) at point \( A \). If \( \dfrac{|MA|}{|AF|} = 2 \), we need to find the value of \( p \).
1. **Identify th... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's tackle this problem step by step. First, I need to make sure I understand all the given information and what is being asked. The problem is about a parabola C defined by the equation \( y^2 = 2px \) where \( p > 0 \). The focus of this parabola is F. There's a point M with coordinates \( (x_0, 2\sqr... | [
{
"from": "human",
"value": "Given the parabola $C$: $y^{2}=2px(p > 0)$ with focus $F$, and a point $M(x_{0},2 \\sqrt {2})$ on the parabola $C$, a circle $M$ is tangent to the $y$-axis and intersects the line segment $MF$ at point $A$. If $\\dfrac {|MA|}{|AF|}=2$, then $p=$ ______."
},
{
"from": "gp... |
What is the product of the largest and smallest prime factors of 1872? | What is the product of the largest and smallest prime factors of 1872? | Okay, so I need to find the product of the largest and smallest prime factors of 1872. Hmm, let me start by recalling what prime factors are. Prime factors are the prime numbers that multiply together to give the original number. So, first, I need to factorize 1872 into its prime components and then identify the smalle... | To find the product of the largest and smallest prime factors of 1872, we start by performing prime factorization on 1872.
1. Divide 1872 by 2 repeatedly:
\[
1872 \div 2 = 936 \\
936 \div 2 = 468 \\
468 \div 2 = 234 \\
234 \div 2 = 117
\]
We have divided by 2 four times, resulting in 117.
2. Next... | 3,896 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the product of the largest and smallest prime factors of 1872. Hmm, let me start by recalling what prime factors are. Prime factors are the prime numbers that multiply together to give the original number. So, first, I need to factorize 1872 into its prime components and then identify th... | [
{
"from": "human",
"value": "What is the product of the largest and smallest prime factors of 1872?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the product of the largest and smallest prime factors of 1872. Hmm, let me start by recalling what prime factors are. Prime factors ... |
Simplify the following expression: $(3x+2)\cdot(6x^8-3x^7+2x^6+x^5)$.
Express the answer as a polynomial with the degrees of the terms in decreasing order. | Simplify the following expression: $(3x+2)\cdot(6x^8-3x^7+2x^6+x^5)$.
Express the answer as a polynomial with the degrees of the terms in decreasing order. | Okay, so I need to simplify the expression (3x + 2) multiplied by (6x^8 - 3x^7 + 2x^6 + x^5). Hmm, let me think. Alright, this is a polynomial multiplication problem. I remember that to multiply two polynomials, I have to use the distributive property, right? That means I need to multiply each term in the first polynom... | To simplify the expression \((3x + 2) \cdot (6x^8 - 3x^7 + 2x^6 + x^5)\), we use the distributive property (also known as the FOIL method for polynomials) and then combine like terms.
First, we distribute each term in the polynomial \(3x + 2\) across each term in the polynomial \(6x^8 - 3x^7 + 2x^6 + x^5\):
1. Multip... | 5,973 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to simplify the expression (3x + 2) multiplied by (6x^8 - 3x^7 + 2x^6 + x^5). Hmm, let me think. Alright, this is a polynomial multiplication problem. I remember that to multiply two polynomials, I have to use the distributive property, right? That means I need to multiply each term in the first... | [
{
"from": "human",
"value": "Simplify the following expression: $(3x+2)\\cdot(6x^8-3x^7+2x^6+x^5)$.\nExpress the answer as a polynomial with the degrees of the terms in decreasing order."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to simplify the expression (3x + 2) multiplied by (6x... |
Given a function $y=f(x), x∈(0,\frac{π}{2}), {f}'(x)$ is its derivative, and it is always true that ${f'}\left(x\right)\cos x \gt f\left(x\right)\sin x$, then which of the following statements is true?
A: $f(\frac{π}{3})＞\sqrt{2}f(\frac{π}{4})$
B: $2f(\frac{π}{4})＞\sqrt{6}f(\frac{π}{6})$
C: $\frac{1}{2}f(\frac{π}{6}... | null | Okay, let's tackle this problem step by step. So, we have a function y = f(x) defined on the interval (0, π/2). The derivative f'(x) exists, and we're given that the inequality f'(x)cosx > f(x)sinx holds for all x in that interval. We need to determine which of the statements A, B, C, or D is true based on this informa... | Given a function \( y = f(x) \) defined on the interval \( (0, \frac{\pi}{2}) \) with the condition that \( f'(x) \cos x > f(x) \sin x \) for all \( x \) in this interval, we need to determine which of the given statements are true.
First, we recognize that the given inequality can be rewritten using the product rule ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's tackle this problem step by step. So, we have a function y = f(x) defined on the interval (0, π/2). The derivative f'(x) exists, and we're given that the inequality f'(x)cosx > f(x)sinx holds for all x in that interval. We need to determine which of the statements A, B, C, or D is true based on this... | [
{
"from": "human",
"value": "Given a function $y=f(x), x∈(0,\\frac{π}{2}), {f}'(x)$ is its derivative, and it is always true that ${f'}\\left(x\\right)\\cos x \\gt f\\left(x\\right)\\sin x$, then which of the following statements is true?\n\nA: $f(\\frac{π}{3})＞\\sqrt{2}f(\\frac{π}{4})$\n\nB: $2f(\\frac{π}{... |
A line passing through vertex $A$ of regular triangle $ABC$ doesn’t intersect segment $BC$. Points $M$ and $N$ lie on this line, and $AM = AN = AB$ (point $B$ lies inside angle $MAC$). Prove that the quadrilateral formed by lines $AB, AC, BN, CM$ is cyclic. | null | Okay, so I have this geometry problem here. Let me read it again and make sure I understand all the parts.
We have a regular triangle ABC. So that's an equilateral triangle, right? All sides equal, all angles 60 degrees. A line passes through vertex A and doesn't intersect segment BC. So that line is going through A b... | Given a regular triangle \(ABC\) with a line passing through vertex \(A\) that does not intersect segment \(BC\). Points \(M\) and \(N\) lie on this line such that \(AM = AN = AB\) (with point \(B\) inside angle \(MAC\)). We need to prove that the quadrilateral formed by lines \(AB\), \(AC\), \(BN\), and \(CM\) is cycl... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this geometry problem here. Let me read it again and make sure I understand all the parts.
We have a regular triangle ABC. So that's an equilateral triangle, right? All sides equal, all angles 60 degrees. A line passes through vertex A and doesn't intersect segment BC. So that line is going thr... | [
{
"from": "human",
"value": "A line passing through vertex $A$ of regular triangle $ABC$ doesn’t intersect segment $BC$. Points $M$ and $N$ lie on this line, and $AM = AN = AB$ (point $B$ lies inside angle $MAC$). Prove that the quadrilateral formed by lines $AB, AC, BN, CM$ is cyclic."
},
{
"from":... |
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