instruction_seed stringlengths 17 5.02k | response_seed stringlengths 25 4.3k ⌀ | reasoning stringlengths 698 82.8k | deepseek_solution stringlengths 42 2.97k | __original_row_idx int64 1 40k ⌀ | source stringclasses 2
values | final_reasoning_trace stringlengths 1.01k 84.8k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | Okay, let's see. I need to find f(3) given that the function f(x) satisfies the equation f(x - y) = f(x)f(y) for all real numbers x and y, and also that f(x) is never zero. Hmm, functional equations can be tricky, but maybe I can figure this out by plugging in some specific values for x and y. Let me start by understan... | Given the function \( f(x) \) satisfies the equation \( f(x - y) = f(x) f(y) \) for all real numbers \( x \) and \( y \), and \( f(x) \neq 0 \) for all real numbers \( x \), we need to find \( f(3) \).
First, by setting \( x = y \), we get:
\[
f(0) = f(x - x) = f(x) f(x) = [f(x)]^2
\]
This implies \( [f(x)]^2 = f(0) \... | 11,083 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find f(3) given that the function f(x) satisfies the equation f(x - y) = f(x)f(y) for all real numbers x and y, and also that f(x) is never zero. Hmm, functional equations can be tricky, but maybe I can figure this out by plugging in some specific values for x and y. Let me start by u... | [
{
"from": "human",
"value": "The function $f(x)$ satisfies\n\\[f(x - y) = f(x) f(y)\\]for all real numbers $x$ and $y,$ and $f(x) \\neq 0$ for all real numbers $x.$ Find $f(3).$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find f(3) given that the function f(x) satisfies t... |
1. Let the complex number $z$ satisfy $|z|=1$. Then the maximum value of $\mid z^{3}+3 z+2 \mathrm{i}$ is ( ).
(A) $4 \sqrt{2}$
(B) $3 \sqrt{3}$
(C) $2 \sqrt{5}$
(D) $4 \sqrt{3}$ | null | Okay, so I have this complex number problem here. Let me read it again: Let the complex number z satisfy |z| = 1. Then the maximum value of |z³ + 3z + 2i| is one of the options given. The options are A) 4√2, B) 3√3, C) 2√5, D) 4√3. Hmm, I need to figure out which one is the maximum value.
First, since |z| = 1, z is a ... | Given the complex number \( z \) satisfying \( |z| = 1 \), we need to find the maximum value of \( |z^3 + 3z + 2i| \).
1. **Parametrization**: Since \( |z| = 1 \), we can write \( z = e^{i\theta} \) where \( \theta \) is the argument of \( z \). This gives \( z^3 = e^{i3\theta} \) and \( 3z = 3e^{i\theta} \).
2. **Ex... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this complex number problem here. Let me read it again: Let the complex number z satisfy |z| = 1. Then the maximum value of |z³ + 3z + 2i| is one of the options given. The options are A) 4√2, B) 3√3, C) 2√5, D) 4√3. Hmm, I need to figure out which one is the maximum value.
First, since |z| = 1,... | [
{
"from": "human",
"value": "1. Let the complex number $z$ satisfy $|z|=1$. Then the maximum value of $\\mid z^{3}+3 z+2 \\mathrm{i}$ is ( ).\n(A) $4 \\sqrt{2}$\n(B) $3 \\sqrt{3}$\n(C) $2 \\sqrt{5}$\n(D) $4 \\sqrt{3}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this complex number pr... |
Let $m$ and $n$ be positive integers. A $2 m \times 2 n$ grid of squares is coloured in the usual chessboard fashion. Find the number of ways of placing $m n$ counters on the white squares, at most one counter per square, so that no two counters are on white squares that are diagonally adjacent. An example of a way to ... | null | Okay, so I need to find the number of ways to place mn counters on the white squares of a 2m x 2n chessboard-colored grid, such that no two counters are diagonally adjacent. The chessboard coloring means that adjacent squares are alternately colored black and white. Each white square can have at most one counter, and t... | To solve the problem of finding the number of ways to place \( mn \) counters on the white squares of a \( 2m \times 2n \) chessboard such that no two counters are diagonally adjacent, we start by analyzing the structure of the chessboard and the placement constraints.
1. **Chessboard Coloring**: The chessboard is col... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the number of ways to place mn counters on the white squares of a 2m x 2n chessboard-colored grid, such that no two counters are diagonally adjacent. The chessboard coloring means that adjacent squares are alternately colored black and white. Each white square can have at most one counte... | [
{
"from": "human",
"value": "Let $m$ and $n$ be positive integers. A $2 m \\times 2 n$ grid of squares is coloured in the usual chessboard fashion. Find the number of ways of placing $m n$ counters on the white squares, at most one counter per square, so that no two counters are on white squares that are di... |
Given that $\sin \theta + \cos \theta = \frac{4}{3}$, with $\theta \in \left(0, \frac{\pi}{4}\right)$, find the value of $\sin \theta - \cos \theta$.
Options:
A) $\frac{\sqrt{2}}{3}$
B) $-\frac{\sqrt{2}}{3}$
C) $\frac{1}{3}$
D) $-\frac{1}{3}$ | null | Okay, let's see. I have this problem where sinθ + cosθ = 4/3, and θ is between 0 and π/4. I need to find sinθ - cosθ. The options are given, so maybe I can use some trigonometric identities to relate these expressions. Hmm.
First, I remember that squaring both sides of an equation can sometimes help in these situation... | Given that \(\sin \theta + \cos \theta = \frac{4}{3}\) with \(\theta \in \left(0, \frac{\pi}{4}\right)\), we need to find the value of \(\sin \theta - \cos \theta\).
First, we square both sides of the given equation:
\[
(\sin \theta + \cos \theta)^2 = \left(\frac{4}{3}\right)^2
\]
Expanding the left side, we get:
\[
\... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I have this problem where sinθ + cosθ = 4/3, and θ is between 0 and π/4. I need to find sinθ - cosθ. The options are given, so maybe I can use some trigonometric identities to relate these expressions. Hmm.
First, I remember that squaring both sides of an equation can sometimes help in these s... | [
{
"from": "human",
"value": "Given that $\\sin \\theta + \\cos \\theta = \\frac{4}{3}$, with $\\theta \\in \\left(0, \\frac{\\pi}{4}\\right)$, find the value of $\\sin \\theta - \\cos \\theta$. \nOptions: \nA) $\\frac{\\sqrt{2}}{3}$ \nB) $-\\frac{\\sqrt{2}}{3}$ \nC) $\\frac{1}{3}$ \nD) $-\\frac{1}{3}$"... |
3. As shown in Figure 1, in $\triangle A B C$, $D$ and $E$ are points on $A C$ and $A B$ respectively. Connect $D E$, $B D$, and $C E$, and let $O$ be the intersection of $B D$ and $C E$. If we set $m$
$$
=\frac{S_{\triangle A D E}}{S_{\triangle A B C}}, n=\frac{S_{\triangle O D E}}{S_{\triangle O B C}} \text {, }
$$
... | null | Alright, so I've got this geometry problem here. Let me try to visualize it first. There's triangle ABC, with points D on AC and E on AB. Then DE is connected, as well as BD and CE, which intersect at point O. The problem defines two ratios: m is the ratio of the area of triangle ADE to the area of triangle ABC, and n ... | Given triangle \( \triangle ABC \) with points \( D \) on \( AC \) and \( E \) on \( AB \). Lines \( DE \), \( BD \), and \( CE \) intersect at point \( O \). We need to determine the relationship between \( m = \frac{S_{\triangle ADE}}{S_{\triangle ABC}} \) and \( n = \frac{S_{\triangle ODE}}{S_{\triangle OBC}} \).
1... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I've got this geometry problem here. Let me try to visualize it first. There's triangle ABC, with points D on AC and E on AB. Then DE is connected, as well as BD and CE, which intersect at point O. The problem defines two ratios: m is the ratio of the area of triangle ADE to the area of triangle ABC... | [
{
"from": "human",
"value": "3. As shown in Figure 1, in $\\triangle A B C$, $D$ and $E$ are points on $A C$ and $A B$ respectively. Connect $D E$, $B D$, and $C E$, and let $O$ be the intersection of $B D$ and $C E$. If we set $m$\n$$\n=\\frac{S_{\\triangle A D E}}{S_{\\triangle A B C}}, n=\\frac{S_{\\tria... |
One endpoint of a line segment is (7, 2) and its midpoint is (5, 6). What is the difference between the coordinates of the other endpoint? | One endpoint of a line segment is (7, 2) and its midpoint is (5, 6). What is the difference between the coordinates of the other endpoint? | Okay, so I need to find the other endpoint of a line segment when one endpoint is (7, 2) and the midpoint is (5, 6). Then, I have to find the difference between the coordinates of that other endpoint. Let's see, how do I approach this?
First, I remember that the midpoint of a line segment is the average of the coordin... | Given one endpoint of a line segment as \((7, 2)\) and its midpoint as \((5, 6)\), we need to find the other endpoint and then determine the difference between its coordinates.
Using the midpoint formula, we know that the coordinates of the midpoint \((M_x, M_y)\) are the averages of the coordinates of the endpoints \... | 27,515 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the other endpoint of a line segment when one endpoint is (7, 2) and the midpoint is (5, 6). Then, I have to find the difference between the coordinates of that other endpoint. Let's see, how do I approach this?
First, I remember that the midpoint of a line segment is the average of the... | [
{
"from": "human",
"value": "One endpoint of a line segment is (7, 2) and its midpoint is (5, 6). What is the difference between the coordinates of the other endpoint?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the other endpoint of a line segment when one endpoint is (7, 2)... |
Given the hyperbola $E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0).$ The four vertices of rectangle $ABCD$ are on $E$, and the midpoints of $AB$ and $CD$ are the two foci of $E$. Additionally, $2|AB| = 3|BC|.$ Find the eccentricity of $E$ ($\,\,\,$ ).
A: $2$
B: $\frac{3+\sqrt{13}}{2}$
C: $\frac{3}{2}$
D: $\f... | null | Okay, let's try to tackle this hyperbola problem step by step. The question is about finding the eccentricity of hyperbola E given certain conditions about a rectangle inscribed in it. The answer choices are given, so maybe that can help check our work along the way.
First, let's recall some basic facts about hyperbol... | Given the hyperbola \( E: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) with \( a > 0 \) and \( b > 0 \), we need to find its eccentricity given that the four vertices of rectangle \( ABCD \) are on \( E \), the midpoints of \( AB \) and \( CD \) are the foci of \( E \), and \( 2|AB| = 3|BC| \).
1. **Identify the coordinat... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to tackle this hyperbola problem step by step. The question is about finding the eccentricity of hyperbola E given certain conditions about a rectangle inscribed in it. The answer choices are given, so maybe that can help check our work along the way.
First, let's recall some basic facts about ... | [
{
"from": "human",
"value": "Given the hyperbola $E: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0).$ The four vertices of rectangle $ABCD$ are on $E$, and the midpoints of $AB$ and $CD$ are the two foci of $E$. Additionally, $2|AB| = 3|BC|.$ Find the eccentricity of $E$ ($\\,\\,\\,$ ).\n\nA: $2$\nB... |
Among the following judgments:
① $x^2 \neq y^2 \Leftrightarrow x \neq y$ or $x \neq -y$;
② If $x^2 + y^2 = 0$, then both $x$ and $y$ are zero;
③ The proposition "$\emptyset \subseteq \{1, 2\}$ or $-1 \in \mathbb{N}$" is true;
④ "$am^2 < bm^2$" is a necessary and sufficient condition for "$a < b$";
⑤ If $b \le... | null | Okay, so I need to check which of these five statements are correct. Let's take them one by one.
Starting with ①: "x² ≠ y² ⇔ x ≠ y or x ≠ -y". Hmm, let me parse this. The biconditional here is saying that x squared not equal to y squared if and only if either x is not equal to y or x is not equal to -y. Let me think ... | Let's analyze each statement one by one:
1. **Statement ①**: \( x^2 \neq y^2 \Leftrightarrow x \neq y \) or \( x \neq -y \)
- \( x^2 = y^2 \) implies \( x = y \) or \( x = -y \). Therefore, \( x^2 \neq y^2 \) implies \( x \neq y \) and \( x \neq -y \). The given statement uses "or" instead of "and", which is incorr... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to check which of these five statements are correct. Let's take them one by one.
Starting with ①: "x² ≠ y² ⇔ x ≠ y or x ≠ -y". Hmm, let me parse this. The biconditional here is saying that x squared not equal to y squared if and only if either x is not equal to y or x is not equal to -y. Let m... | [
{
"from": "human",
"value": "Among the following judgments: \n① $x^2 \\neq y^2 \\Leftrightarrow x \\neq y$ or $x \\neq -y$; \n② If $x^2 + y^2 = 0$, then both $x$ and $y$ are zero; \n③ The proposition \"$\\emptyset \\subseteq \\{1, 2\\}$ or $-1 \\in \\mathbb{N}$\" is true; \n④ \"$am^2 < bm^2$\" is a nece... |
What is the base-5 equivalent of the decimal number 219? | What is the base-5 equivalent of the decimal number 219? | Okay, so I need to figure out what the base-5 equivalent of the decimal number 219 is. Hmm, base conversion. Let me remember how that works. I think it involves dividing by the base and keeping track of the remainders. But let me make sure I do this right.
First, let me recall that in base-5, each digit represents a p... | To convert the decimal number 219 to base-5, we use the division method by keeping track of the remainders:
1. Divide 219 by 5:
- \( 219 \div 5 = 43 \) with a remainder of 4.
- Remainder: 4
2. Divide the quotient 43 by 5:
- \( 43 \div 5 = 8 \) with a remainder of 3.
- Remainder: 3
3. Divide the quotient ... | 6,281 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out what the base-5 equivalent of the decimal number 219 is. Hmm, base conversion. Let me remember how that works. I think it involves dividing by the base and keeping track of the remainders. But let me make sure I do this right.
First, let me recall that in base-5, each digit repres... | [
{
"from": "human",
"value": "What is the base-5 equivalent of the decimal number 219?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out what the base-5 equivalent of the decimal number 219 is. Hmm, base conversion. Let me remember how that works. I think it involves dividing ... |
What is the largest power of 5 that divides the product of all positive integers from 1 to 25, inclusive? | What is the largest power of 5 that divides the product of all positive integers from 1 to 25, inclusive? | Okay, so I need to find the largest power of 5 that divides the product of all positive integers from 1 to 25. Hmm, that product is 25 factorial, written as 25!. The question is essentially asking for the highest exponent \( k \) such that \( 5^k \) divides 25!.
I remember that when finding the highest power of a pri... | To find the largest power of 5 that divides the product of all positive integers from 1 to 25 (i.e., 25!), we use Legendre's formula. This formula states that the exponent of a prime \( p \) in the prime factorization of \( n! \) is given by:
\[
\sum_{i=1}^{\infty} \left\lfloor \frac{n}{p^i} \right\rfloor
\]
For \( n... | 17,073 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the largest power of 5 that divides the product of all positive integers from 1 to 25. Hmm, that product is 25 factorial, written as 25!. The question is essentially asking for the highest exponent \( k \) such that \( 5^k \) divides 25!.
I remember that when finding the highest power ... | [
{
"from": "human",
"value": "What is the largest power of 5 that divides the product of all positive integers from 1 to 25, inclusive?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest power of 5 that divides the product of all positive integers from 1 to 25. Hmm, that ... |
What is the period of $y = \cos 2x$? | What is the period of $y = \cos 2x$? | Okay, so I need to find the period of the function y = cos(2x). Hmm, let's start by recalling what the period of a cosine function is. The basic cosine function, y = cos(x), has a period of 2π. That means it takes 2π radians for the cosine wave to complete one full cycle, right? So, if I graph y = cos(x), it starts at ... | To find the period of the function \( y = \cos 2x \), we start by recalling that the period of the basic cosine function \( y = \cos x \) is \( 2\pi \). When the cosine function is modified to \( \cos(Bx) \), the period is given by \( \frac{2\pi}{|B|} \).
For the function \( y = \cos 2x \), the coefficient \( B \) is ... | 3,771 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the period of the function y = cos(2x). Hmm, let's start by recalling what the period of a cosine function is. The basic cosine function, y = cos(x), has a period of 2π. That means it takes 2π radians for the cosine wave to complete one full cycle, right? So, if I graph y = cos(x), it st... | [
{
"from": "human",
"value": "What is the period of $y = \\cos 2x$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the period of the function y = cos(2x). Hmm, let's start by recalling what the period of a cosine function is. The basic cosine function, y = cos(x), has a period of... |
What is the domain of the function $$j(x) = \frac{1}{x+8} + \frac{1}{x^2+8} + \frac{1}{x^3+8}~?$$Express your answer as a union of intervals. | What is the domain of the function $$j(x) = \frac{1}{x+8} + \frac{1}{x^2+8} + \frac{1}{x^3+8}~?$$Express your answer as a union of intervals. | Okay, so I need to find the domain of the function j(x) which is given by three fractions added together: 1/(x+8), 1/(x² +8), and 1/(x³ +8). Hmm, the domain of a function includes all the real numbers x for which the function is defined. Since this function has three terms, each of which is a rational expression, the f... | To find the domain of the function \( j(x) = \frac{1}{x+8} + \frac{1}{x^2+8} + \frac{1}{x^3+8} \), we need to determine the values of \( x \) for which the function is defined. This involves checking the denominators of each term to ensure they are not zero.
1. **First term denominator \( x + 8 \)**:
- Solving \( x... | 18,314 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the domain of the function j(x) which is given by three fractions added together: 1/(x+8), 1/(x² +8), and 1/(x³ +8). Hmm, the domain of a function includes all the real numbers x for which the function is defined. Since this function has three terms, each of which is a rational expressio... | [
{
"from": "human",
"value": "What is the domain of the function $$j(x) = \\frac{1}{x+8} + \\frac{1}{x^2+8} + \\frac{1}{x^3+8}~?$$Express your answer as a union of intervals."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the domain of the function j(x) which is given by three fr... |
$(1)$ Calculate: $[(x^{2}+y^{2})-(x-y)^{2}+2y(x-y)]\div 4y$.
$(2)$ Solve the equation: $\frac{x}{x-2}-\frac{3}{x}=1$. | null | Okay, so I have two problems to solve here. Let me start with the first one:
**Problem 1:** Calculate \([(x^{2}+y^{2}) - (x - y)^{2} + 2y(x - y)] \div 4y\).
Hmm, this looks like an algebraic expression that needs to be simplified. The goal is probably to simplify the numerator first and then divide by 4y. Let me tak... | (1) To simplify the expression \([(x^{2} + y^{2}) - (x - y)^{2} + 2y(x - y)] \div 4y\):
First, expand and simplify the numerator:
\[
(x^{2} + y^{2}) - (x - y)^{2} + 2y(x - y)
\]
Expanding \((x - y)^2\):
\[
(x - y)^2 = x^2 - 2xy + y^2
\]
Substituting back into the expression:
\[
(x^2 + y^2) - (x^2 - 2xy + y^2) + 2y(x -... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have two problems to solve here. Let me start with the first one:
**Problem 1:** Calculate \([(x^{2}+y^{2}) - (x - y)^{2} + 2y(x - y)] \div 4y\).
Hmm, this looks like an algebraic expression that needs to be simplified. The goal is probably to simplify the numerator first and then divide by 4y. Le... | [
{
"from": "human",
"value": "$(1)$ Calculate: $[(x^{2}+y^{2})-(x-y)^{2}+2y(x-y)]\\div 4y$. \n$(2)$ Solve the equation: $\\frac{x}{x-2}-\\frac{3}{x}=1$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have two problems to solve here. Let me start with the first one: \n\n**Problem 1:** Calcula... |
Given that $x$ is real and $x^4-\frac{1}{x^4}=47$, find $x^2+\frac{1}{x^2}$. | Given that $x$ is real and $x^4-\frac{1}{x^4}=47$, find $x^2+\frac{1}{x^2}$. | Okay, so I need to find the value of \( x^2 + \frac{1}{x^2} \) given that \( x \) is a real number and \( x^4 - \frac{1}{x^4} = 47 \). Hmm, let's see. I remember that expressions involving \( x^n \) and \( \frac{1}{x^n} \) can sometimes be simplified by using algebraic identities or substitutions. Maybe I can let \( y ... | Given that \( x \) is a real number and \( x^4 - \frac{1}{x^4} = 47 \), we need to find \( x^2 + \frac{1}{x^2} \).
First, we recognize that \( x^4 - \frac{1}{x^4} \) can be factored as a difference of squares:
\[
x^4 - \frac{1}{x^4} = \left( x^2 - \frac{1}{x^2} \right) \left( x^2 + \frac{1}{x^2} \right)
\]
Let \( y =... | 22,424 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of \( x^2 + \frac{1}{x^2} \) given that \( x \) is a real number and \( x^4 - \frac{1}{x^4} = 47 \). Hmm, let's see. I remember that expressions involving \( x^n \) and \( \frac{1}{x^n} \) can sometimes be simplified by using algebraic identities or substitutions. Maybe I can l... | [
{
"from": "human",
"value": "Given that $x$ is real and $x^4-\\frac{1}{x^4}=47$, find $x^2+\\frac{1}{x^2}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of \\( x^2 + \\frac{1}{x^2} \\) given that \\( x \\) is a real number and \\( x^4 - \\frac{1}{x^4} = 47 \\). Hmm, ... |
Problem:
A circle of radius 1 is divided into four equal parts by drawing two perpendicular diameters. Five points are chosen at random inside or on the circle. What is the smallest possible number a such that it is always possible to select one pair of points from these five such that the distance between them is equa... | Problem:
A circle of radius 1 is divided into four equal parts by drawing two perpendicular diameters. Five points are chosen at random inside or on the circle. What is the smallest possible number a such that it is always possible to select one pair of points from these five such that the distance between them is equa... | Okay, let's try to figure out this problem step by step. The problem says that we have a circle of radius 1 divided into four equal parts by two perpendicular diameters. So, basically, it's a circle divided into four quadrants like a coordinate system. Then, five points are chosen at random inside or on the circle. We ... | To solve the problem, we need to determine the smallest possible number \( a \) such that in any configuration of five points inside or on a circle of radius 1 divided into four quadrants, there exists at least one pair of points with a distance between them less than or equal to \( a \) times the length of the shorter... | 21,748 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to figure out this problem step by step. The problem says that we have a circle of radius 1 divided into four equal parts by two perpendicular diameters. So, basically, it's a circle divided into four quadrants like a coordinate system. Then, five points are chosen at random inside or on the cir... | [
{
"from": "human",
"value": "Problem:\nA circle of radius 1 is divided into four equal parts by drawing two perpendicular diameters. Five points are chosen at random inside or on the circle. What is the smallest possible number a such that it is always possible to select one pair of points from these five s... |
A square is divided into nine smaller squares of equal area. The center square is then divided into nine smaller squares of equal area and the pattern continues indefinitely. What fractional part of the figure is shaded? [asy]
import olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
void drawSquares(int n){... | A square is divided into nine smaller squares of equal area. The center square is then divided into nine smaller squares of equal area and the pattern continues indefinitely. What fractional part of the figure is shaded? [asy]
import olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
void drawSquares(int n){... | Okay, so I need to figure out the fractional part of the figure that's shaded. The problem says a square is divided into nine smaller squares of equal area, then the center square is divided again into nine even smaller squares, and this pattern continues indefinitely. The Asymptote code seems to be drawing this recurs... | The problem involves a square divided into nine smaller squares of equal area, with the center square then divided again into nine smaller squares, continuing indefinitely. The goal is to determine the fractional part of the figure that is shaded.
1. **Initial Shading**: The first division of the square shades the fou... | 22,834 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the fractional part of the figure that's shaded. The problem says a square is divided into nine smaller squares of equal area, then the center square is divided again into nine even smaller squares, and this pattern continues indefinitely. The Asymptote code seems to be drawing thi... | [
{
"from": "human",
"value": "A square is divided into nine smaller squares of equal area. The center square is then divided into nine smaller squares of equal area and the pattern continues indefinitely. What fractional part of the figure is shaded? [asy]\nimport olympiad; size(150); defaultpen(linewidth(0.... |
The relationship between the origin O(0,0) and the set A = {(x, y) | x + 2y - 1 ≥ 0, y ≤ x + 2, 2x + y - 5 ≤ 0} is __________, and the relationship between point M(1,1) and set A is __________. | null | Alright, let me try to figure out the relationships between the origin O(0,0), the point M(1,1), and the set A defined by those inequalities. Hmm, okay. So set A is a region in the coordinate plane defined by three inequalities: x + 2y - 1 ≥ 0, y ≤ x + 2, and 2x + y - 5 ≤ 0. I need to check if the origin and the point ... | To determine the relationship between the origin O(0,0) and the set A, we check if O satisfies all the inequalities defining A:
1. \( x + 2y - 1 \geq 0 \): Substituting \( x = 0 \) and \( y = 0 \) gives \( -1 \geq 0 \), which is false.
2. \( y \leq x + 2 \): Substituting \( x = 0 \) and \( y = 0 \) gives \( 0 \leq 2 \... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let me try to figure out the relationships between the origin O(0,0), the point M(1,1), and the set A defined by those inequalities. Hmm, okay. So set A is a region in the coordinate plane defined by three inequalities: x + 2y - 1 ≥ 0, y ≤ x + 2, and 2x + y - 5 ≤ 0. I need to check if the origin and th... | [
{
"from": "human",
"value": "The relationship between the origin O(0,0) and the set A = {(x, y) | x + 2y - 1 ≥ 0, y ≤ x + 2, 2x + y - 5 ≤ 0} is __________, and the relationship between point M(1,1) and set A is __________."
},
{
"from": "gpt",
"value": "<think>\nAlright, let me try to figure out... |
A pizza parlor sold large and small pizzas on Friday. Each large pizza was cut into 16 slices, and each small pizza was cut into 8 slices. On Saturday, the pizza parlor sold only large pizzas, and each large pizza was cut into 12 slices. If the total number of slices sold on both days was 176, and the number of slices ... | A pizza parlor sold large and small pizzas on Friday. Each large pizza was cut into 16 slices, and each small pizza was cut into 8 slices. On Saturday, the pizza parlor sold only large pizzas, and each large pizza was cut into 12 slices. If the total number of slices sold on both days was 176, and the number of slices ... | Okay, let's try to figure out this pizza problem. Hmm, so we have a pizza parlor that sold large and small pizzas on Friday, and only large pizzas on Saturday. Each large pizza on Friday was cut into 16 slices, each small into 8. On Saturday, the large pizzas were cut into 12 slices each. The total number of slices sol... | Let \( L \) be the number of large pizzas sold on Friday, \( S \) be the number of small pizzas sold on Friday, and \( R \) be the number of large pizzas sold on Saturday.
We know the following:
1. Each large pizza on Friday was cut into 16 slices, so the total number of slices from large pizzas on Friday is \( 16L \... | 16,665 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to figure out this pizza problem. Hmm, so we have a pizza parlor that sold large and small pizzas on Friday, and only large pizzas on Saturday. Each large pizza on Friday was cut into 16 slices, each small into 8. On Saturday, the large pizzas were cut into 12 slices each. The total number of sl... | [
{
"from": "human",
"value": "A pizza parlor sold large and small pizzas on Friday. Each large pizza was cut into 16 slices, and each small pizza was cut into 8 slices. On Saturday, the pizza parlor sold only large pizzas, and each large pizza was cut into 12 slices. If the total number of slices sold on bot... |
Given a circle $C$ that passes through points $A(1,4)$ and $B(3,2)$, and its center lies on the line $x+y-3=0$.
$(I)$ Find the equation of circle $C$;
$(II)$ If point $P(x,y)$ is on circle $C$, find the maximum value of $x+y$. | null | Okay, so I have this problem about a circle C that goes through points A(1,4) and B(3,2), and its center is on the line x + y - 3 = 0. There are two parts: first, find the equation of the circle, and second, find the maximum value of x + y for any point P(x,y) on the circle.
Let me start with part I. To find the equat... | Given a circle \( C \) that passes through points \( A(1,4) \) and \( B(3,2) \), and its center lies on the line \( x + y - 3 = 0 \):
(I) To find the equation of the circle \( C \):
1. The center \((h, k)\) of the circle lies on the line \( x + y - 3 = 0 \), so \( h + k = 3 \). Thus, \( k = 3 - h \).
2. The circle pas... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem about a circle C that goes through points A(1,4) and B(3,2), and its center is on the line x + y - 3 = 0. There are two parts: first, find the equation of the circle, and second, find the maximum value of x + y for any point P(x,y) on the circle.
Let me start with part I. To find t... | [
{
"from": "human",
"value": "Given a circle $C$ that passes through points $A(1,4)$ and $B(3,2)$, and its center lies on the line $x+y-3=0$.\n$(I)$ Find the equation of circle $C$;\n$(II)$ If point $P(x,y)$ is on circle $C$, find the maximum value of $x+y$."
},
{
"from": "gpt",
"value": "<think>... |
In triangle $ABC$, the angle bisector of $\angle BAC$ intersects side $BC$ at point $D$. The area of triangle $ABD$ is $15 \text{cm}^2$ and the area of triangle $ADC$ is $20 \text{cm}^2$. If the length of side $AC$ is $12 \text{cm}$, what is the length of side $AB$? | In triangle $ABC$, the angle bisector of $\angle BAC$ intersects side $BC$ at point $D$. The area of triangle $ABD$ is $15 \text{cm}^2$ and the area of triangle $ADC$ is $20 \text{cm}^2$. If the length of side $AC$ is $12 \text{cm}$, what is the length of side $AB$? | Okay, so I need to find the length of side AB in triangle ABC. The problem says that the angle bisector of angle BAC intersects side BC at point D. The areas of triangles ABD and ADC are given as 15 cm² and 20 cm² respectively, and AC is 12 cm. Hmm, let me think about how to approach this.
First, I remember that when ... | To find the length of side \( AB \) in triangle \( ABC \), we use the given information about the angle bisector and the areas of the triangles \( ABD \) and \( ADC \).
1. **Angle Bisector Theorem**: The angle bisector of \( \angle BAC \) intersects side \( BC \) at point \( D \). By the Angle Bisector Theorem, the ra... | 37,234 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the length of side AB in triangle ABC. The problem says that the angle bisector of angle BAC intersects side BC at point D. The areas of triangles ABD and ADC are given as 15 cm² and 20 cm² respectively, and AC is 12 cm. Hmm, let me think about how to approach this.
First, I remember th... | [
{
"from": "human",
"value": "In triangle $ABC$, the angle bisector of $\\angle BAC$ intersects side $BC$ at point $D$. The area of triangle $ABD$ is $15 \\text{cm}^2$ and the area of triangle $ADC$ is $20 \\text{cm}^2$. If the length of side $AC$ is $12 \\text{cm}$, what is the length of side $AB$?"
},
... |
In a sphere with a radius of $5$, there are four distinct points $A$, $B$, $C$, $D$. If $AB=AC=AD=2 \sqrt {5}$, the surface area of the shape formed by the intersection of the sphere and the plane $BCD$ is \_\_\_\_\_\_. | null | Okay, so I need to solve this geometry problem. Let me read it again to make sure I understand.
In a sphere with radius 5, there are four distinct points A, B, C, D. If AB = AC = AD = 2√5, the surface area of the shape formed by the intersection of the sphere and the plane BCD is _______.
Alright, so we have a sphere... | Given a sphere with radius 5 and four distinct points \(A\), \(B\), \(C\), \(D\) on the sphere, where \(AB = AC = AD = 2\sqrt{5}\), we need to find the surface area of the circle formed by the intersection of the sphere and the plane \(BCD\).
1. **Key Relationships**:
- The points \(B\), \(C\), and \(D\) lie on the... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this geometry problem. Let me read it again to make sure I understand.
In a sphere with radius 5, there are four distinct points A, B, C, D. If AB = AC = AD = 2√5, the surface area of the shape formed by the intersection of the sphere and the plane BCD is _______.
Alright, so we have ... | [
{
"from": "human",
"value": "In a sphere with a radius of $5$, there are four distinct points $A$, $B$, $C$, $D$. If $AB=AC=AD=2 \\sqrt {5}$, the surface area of the shape formed by the intersection of the sphere and the plane $BCD$ is \\_\\_\\_\\_\\_\\_."
},
{
"from": "gpt",
"value": "<think>\n... |
A standard eight-sided die is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$? | A standard eight-sided die is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$? | Okay, let's see. The problem is about an eight-sided die, which is rolled, and then P is the product of the seven numbers that are visible. We need to find the largest number that is certain to divide P, no matter which face is on the bottom. Hmm, okay.
First, I need to recall what a standard eight-sided die looks lik... | A standard eight-sided die is rolled, and \( P \) is the product of the seven numbers that are visible. We need to find the largest number that is certain to divide \( P \) regardless of which face is on the bottom.
First, we note that the product of all eight numbers on the die is \( 8! = 40320 \). When one number is... | 38,997 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about an eight-sided die, which is rolled, and then P is the product of the seven numbers that are visible. We need to find the largest number that is certain to divide P, no matter which face is on the bottom. Hmm, okay.
First, I need to recall what a standard eight-sided die l... | [
{
"from": "human",
"value": "A standard eight-sided die is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem is about an eight-sided die, which is ro... |
Point $M$ divides the side $B C$ of triangle $A B C$ in the ratio $B M: M C=2: 5$, It is known that $\overrightarrow{A B}=\vec{a}, \overrightarrow{A C}=$ $\vec{b}$. Find the vector $\overrightarrow{A M}$. | null | Okay, so I need to find the vector AM in triangle ABC, where point M divides BC in the ratio BM:MC = 2:5. They give me vectors AB as vector a and AC as vector b. Hmm, let's see. I remember that when a point divides a line segment in a certain ratio, we can use section formulas to find the vector. Is it the internal div... | To find the vector $\overrightarrow{A M}$ where point $M$ divides side $B C$ of triangle $A B C$ in the ratio $B M: M C=2: 5$, we use the section formula for internal division. Given $\overrightarrow{A B} = \vec{a}$ and $\overrightarrow{A C} = \vec{b}$, the position vectors of $B$ and $C$ with respect to $A$ are $\vec{... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the vector AM in triangle ABC, where point M divides BC in the ratio BM:MC = 2:5. They give me vectors AB as vector a and AC as vector b. Hmm, let's see. I remember that when a point divides a line segment in a certain ratio, we can use section formulas to find the vector. Is it the inte... | [
{
"from": "human",
"value": "Point $M$ divides the side $B C$ of triangle $A B C$ in the ratio $B M: M C=2: 5$, It is known that $\\overrightarrow{A B}=\\vec{a}, \\overrightarrow{A C}=$ $\\vec{b}$. Find the vector $\\overrightarrow{A M}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need t... |
The coordinates of the vertex of the parabola $y=x^{2}-4x+1$ are ____. | null | Okay, so I need to find the coordinates of the vertex of the parabola given by the equation y = x² - 4x + 1. Hmm, let me think. I remember there are a couple of ways to find the vertex of a parabola. One way is to use the vertex formula, which is for a quadratic equation in the form y = ax² + bx + c. The x-coordinate o... | To find the coordinates of the vertex of the parabola given by the equation \( y = x^2 - 4x + 1 \):
1. **Identify the coefficients**: The equation is in the form \( y = ax^2 + bx + c \) where \( a = 1 \), \( b = -4 \), and \( c = 1 \).
2. **Calculate the x-coordinate of the vertex**: Using the formula \( x = -\frac{b}... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the coordinates of the vertex of the parabola given by the equation y = x² - 4x + 1. Hmm, let me think. I remember there are a couple of ways to find the vertex of a parabola. One way is to use the vertex formula, which is for a quadratic equation in the form y = ax² + bx + c. The x-coor... | [
{
"from": "human",
"value": "The coordinates of the vertex of the parabola $y=x^{2}-4x+1$ are ____."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the coordinates of the vertex of the parabola given by the equation y = x² - 4x + 1. Hmm, let me think. I remember there are a coupl... |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$ | Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$ | Okay, let's try to solve this problem. We have two functions, f and g, such that f(g(x)) = x² and g(f(x)) = x³ for all x ≥ 1. Also, we know that g(16) = 16, and we need to find [g(4)]³. Hmm, interesting.
First, let me note down what is given:
1. f(g(x)) = x² for all x ≥ 1.
2. g(f(x)) = x³ for all x ≥ 1.
3. g(16) = 16... | Given the functions \( f(x) \) and \( g(x) \) which satisfy \( f(g(x)) = x^2 \) and \( g(f(x)) = x^3 \) for all \( x \ge 1 \), and knowing that \( g(16) = 16 \), we need to compute \([g(4)]^3\).
First, we derive the functional equations for \( g \) and \( f \):
1. From \( f(g(x)) = x^2 \), applying \( g \) to both si... | 16,097 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to solve this problem. We have two functions, f and g, such that f(g(x)) = x² and g(f(x)) = x³ for all x ≥ 1. Also, we know that g(16) = 16, and we need to find [g(4)]³. Hmm, interesting.
First, let me note down what is given:
1. f(g(x)) = x² for all x ≥ 1.
2. g(f(x)) = x³ for all x ≥ 1.
3. g(... | [
{
"from": "human",
"value": "Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \\ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to solve this problem. We have two functions, f and g... |
Let $a,$ $b,$ $c$ be the roots of $x^3 + px + q = 0.$ Compute the determinant
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}.\] | Let $a,$ $b,$ $c$ be the roots of $x^3 + px + q = 0.$ Compute the determinant
\[\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}.\] | Okay, so I have this problem where I need to compute the determinant of a 3x3 matrix where the rows are cyclic permutations of the roots a, b, c of the cubic equation x³ + px + q = 0. The matrix looks like this:
| a b c |
| b c a |
| c a b |
Hmm, determinants can sometimes be tricky, but maybe there's a p... | Given the roots \(a, b, c\) of the cubic equation \(x^3 + px + q = 0\), we need to compute the determinant of the matrix:
\[
\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}
\]
First, we use the standard formula for the determinant of a 3x3 matrix:
\[
\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & ... | 21,898 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem where I need to compute the determinant of a 3x3 matrix where the rows are cyclic permutations of the roots a, b, c of the cubic equation x³ + px + q = 0. The matrix looks like this:
| a b c |
| b c a |
| c a b |
Hmm, determinants can sometimes be tricky, but maybe the... | [
{
"from": "human",
"value": "Let $a,$ $b,$ $c$ be the roots of $x^3 + px + q = 0.$ Compute the determinant\n\\[\\begin{vmatrix} a & b & c \\\\ b & c & a \\\\ c & a & b \\end{vmatrix}.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem where I need to compute the determinant... |
A frustum of a right circular cone is formed by cutting off a smaller cone from the top of a larger one. The height of the frustum is 12 cm, the diameter of the lower base is 20 cm, and the diameter of the upper base is 8 cm. The slant height of the frustum is 20 cm. What is the volume of the frustum? | A frustum of a right circular cone is formed by cutting off a smaller cone from the top of a larger one. The height of the frustum is 12 cm, the diameter of the lower base is 20 cm, and the diameter of the upper base is 8 cm. The slant height of the frustum is 20 cm. What is the volume of the frustum? | Okay, so I need to find the volume of this frustum. Let me recall what a frustum is. It's like a cone that's been cut parallel to the base, removing the top part, right? So, the frustum has two circular bases, one bigger (lower) and one smaller (upper). The formula for the volume of a frustum, if I remember correctly, ... | To find the volume of the frustum, we use the formula for the volume of a frustum of a right circular cone:
\[
V = \frac{1}{3} \pi h (R^2 + Rr + r^2)
\]
where \( h \) is the height of the frustum, \( R \) is the radius of the lower base, and \( r \) is the radius of the upper base.
Given:
- The height of the frustum \... | 12,291 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the volume of this frustum. Let me recall what a frustum is. It's like a cone that's been cut parallel to the base, removing the top part, right? So, the frustum has two circular bases, one bigger (lower) and one smaller (upper). The formula for the volume of a frustum, if I remember cor... | [
{
"from": "human",
"value": "A frustum of a right circular cone is formed by cutting off a smaller cone from the top of a larger one. The height of the frustum is 12 cm, the diameter of the lower base is 20 cm, and the diameter of the upper base is 8 cm. The slant height of the frustum is 20 cm. What is the... |
A game show contestant has a $\frac{1}{5}$ probability of winning $\$1000$, a $\frac{1}{3}$ probability of winning $\$500$, and a $\frac{7}{15}$ probability of losing $\$200$. What is the expected worth of playing the game? Express your answer as a decimal rounded to the nearest hundredth. | A game show contestant has a $\frac{1}{5}$ probability of winning $\$1000$, a $\frac{1}{3}$ probability of winning $\$500$, and a $\frac{7}{15}$ probability of losing $\$200$. What is the expected worth of playing the game? Express your answer as a decimal rounded to the nearest hundredth. | Okay, so I need to figure out the expected worth of playing this game. Let me recall what expected value means. From what I remember, expected value is like the average outcome you'd expect if you were to play the game many times. It's calculated by multiplying each possible outcome by its probability and then adding a... | To find the expected worth of playing the game, we need to calculate the expected value by multiplying each outcome by its probability and summing the results. The probabilities and outcomes are given as follows:
1. A \(\frac{1}{5}\) probability of winning \(\$1000\).
2. A \(\frac{1}{3}\) probability of winning \(\$50... | 9,060 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the expected worth of playing this game. Let me recall what expected value means. From what I remember, expected value is like the average outcome you'd expect if you were to play the game many times. It's calculated by multiplying each possible outcome by its probability and then ... | [
{
"from": "human",
"value": "A game show contestant has a $\\frac{1}{5}$ probability of winning $\\$1000$, a $\\frac{1}{3}$ probability of winning $\\$500$, and a $\\frac{7}{15}$ probability of losing $\\$200$. What is the expected worth of playing the game? Express your answer as a decimal rounded to the n... |
For some integer $n$, the polynomial $x^3 - 2009x + n$ has the three integer roots $p$, $q$, and $r$. Find $|p| + |q| + |r|$. | null | Okay, so I need to find the integer roots p, q, r of the polynomial x³ - 2009x + n, where n is some integer. Then, once I find those roots, I have to calculate the sum of their absolute values, |p| + |q| + |r|.
First, let me recall that for a cubic polynomial of the form x³ + ax² + bx + c, the sum of the roots is -a,... | Given the polynomial \(x^3 - 2009x + n\) with integer roots \(p\), \(q\), and \(r\), we need to find \(|p| + |q| + |r|\).
Using Vieta's formulas, we know:
1. \(p + q + r = 0\)
2. \(pq + pr + qr = -2009\)
3. \(pqr = -n\)
Substituting \(r = -p - q\) into the second equation, we get:
\[
pq + p(-p - q) + q(-p - q) = -200... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the integer roots p, q, r of the polynomial x³ - 2009x + n, where n is some integer. Then, once I find those roots, I have to calculate the sum of their absolute values, |p| + |q| + |r|.
First, let me recall that for a cubic polynomial of the form x³ + ax² + bx + c, the sum of the root... | [
{
"from": "human",
"value": "For some integer $n$, the polynomial $x^3 - 2009x + n$ has the three integer roots $p$, $q$, and $r$. Find $|p| + |q| + |r|$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the integer roots p, q, r of the polynomial x³ - 2009x + n, where n is some i... |
The Leifeng Pagoda on Xizhao Mountain in Hangzhou is one of the Ten Scenes of Hangzhou. To measure the height of the Leifeng Pagoda (BC), the distance from the foot of the mountain (A) is measured to be $AC = 62\sqrt{2}m$. The angle of elevation to the top of the tower (B) is $\alpha = 45^{\circ}$, and the slope angle ... | null | Okay, so I need to find the height of the Leifeng Pagoda, which is BC. The given information includes AC = 62√2 meters, the angle of elevation to the top of the tower is 45 degrees, and the slope angle of the mountain is 15 degrees. Let me try to visualize this first.
I imagine there's a point A at the foot of the mou... | To find the height of the Leifeng Pagoda (BC), we start by noting the given information: the distance from the foot of the mountain (A) to the base of the pagoda (C) is \( AC = 62\sqrt{2} \) meters, the angle of elevation to the top of the pagoda (B) is \( \alpha = 45^\circ \), and the slope angle of the mountain is \(... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the height of the Leifeng Pagoda, which is BC. The given information includes AC = 62√2 meters, the angle of elevation to the top of the tower is 45 degrees, and the slope angle of the mountain is 15 degrees. Let me try to visualize this first.
I imagine there's a point A at the foot of... | [
{
"from": "human",
"value": "The Leifeng Pagoda on Xizhao Mountain in Hangzhou is one of the Ten Scenes of Hangzhou. To measure the height of the Leifeng Pagoda (BC), the distance from the foot of the mountain (A) is measured to be $AC = 62\\sqrt{2}m$. The angle of elevation to the top of the tower (B) is $... |
Julie works for 48 hours per week for 12 weeks during the summer, making X$. If she works for 48 weeks during the school year at the same rate of pay and needs to make another $\$5000$, She must work 12 hours per week. What is the value of unknown variable X? | null | Okay, let me try to figure out this problem. So Julie works 48 hours a week for 12 weeks during the summer and makes X dollars. Then during the school year, she works 48 weeks but needs to make an additional $5000, and she has to work 12 hours per week. We need to find X, which is her total earnings during the summer.
... | Julie works 48 hours per week for 12 weeks during the summer, earning a total of \( X \) dollars. During the school year, she works 12 hours per week for 48 weeks to earn an additional $5000 at the same hourly rate. We need to find the value of \( X \).
First, we calculate her hourly rate \( R \) using her school year... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem. So Julie works 48 hours a week for 12 weeks during the summer and makes X dollars. Then during the school year, she works 48 weeks but needs to make an additional $5000, and she has to work 12 hours per week. We need to find X, which is her total earnings during the ... | [
{
"from": "human",
"value": "Julie works for 48 hours per week for 12 weeks during the summer, making X$. If she works for 48 weeks during the school year at the same rate of pay and needs to make another $\\$5000$, She must work 12 hours per week. What is the value of unknown variable X?"
},
{
"fro... |
19. Eight numbers are written on the board. Two players take turns. In one move, it is necessary to replace two different numbers with two equal numbers with the same sum. If at some point it becomes possible to divide all the numbers into two sets of four with equal sums, the first player wins. Can the second player p... | null | Okay, so there's this problem where eight numbers are written on the board. Two players take turns. In each move, a player replaces two different numbers with two equal numbers that have the same total sum. The goal for the first player is to eventually make it possible to divide all the numbers into two sets of four w... | The problem involves two players taking turns replacing two different numbers on the board with two equal numbers such that the sum remains the same. The first player wins if it becomes possible to divide all numbers into two sets of four with equal sums. The question is whether the second player can prevent this.
Key... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so there's this problem where eight numbers are written on the board. Two players take turns. In each move, a player replaces two different numbers with two equal numbers that have the same total sum. The goal for the first player is to eventually make it possible to divide all the numbers into two sets o... | [
{
"from": "human",
"value": "19. Eight numbers are written on the board. Two players take turns. In one move, it is necessary to replace two different numbers with two equal numbers with the same sum. If at some point it becomes possible to divide all the numbers into two sets of four with equal sums, the f... |
When guests come to the house and you need to make tea, it takes 1 minute to wash the kettle, 10 minutes to boil water, 2 minutes to wash the tea cups, 1 minute to get the tea leaves, and 1 minute to brew the tea. What is the most reasonable arrangement for the minimum time required?
A: 10 minutes
B: 11 minutes
C: 12 m... | null | Okay, let's see. So the problem is about making tea when guests come over, and we need to figure out the minimum time required by arranging the tasks in the most efficient way. The answer choices are 10, 11, 12, or 13 minutes. Hmm.
First, let me list out all the tasks and their durations:
1. Wash the kettle: 1 minut... | The key to minimizing the time is to perform tasks in parallel where possible. Here's the optimal sequence:
1. **Wash the kettle (1 min)**
This must be done first to start boiling water.
2. **Boil water (10 min)**
While the water is boiling, perform the following tasks:
- **Wash tea cups (2 min)**
... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. So the problem is about making tea when guests come over, and we need to figure out the minimum time required by arranging the tasks in the most efficient way. The answer choices are 10, 11, 12, or 13 minutes. Hmm.
First, let me list out all the tasks and their durations:
1. Wash the kettle:... | [
{
"from": "human",
"value": "When guests come to the house and you need to make tea, it takes 1 minute to wash the kettle, 10 minutes to boil water, 2 minutes to wash the tea cups, 1 minute to get the tea leaves, and 1 minute to brew the tea. What is the most reasonable arrangement for the minimum time requ... |
How many zeros are in the expansion of $999,\!999,\!999,\!998^2$? | How many zeros are in the expansion of $999,\!999,\!999,\!998^2$? | Okay, let's see. I need to figure out how many zeros are in the expansion of 999,999,999,998 squared. Hmm, that's a pretty big number. Maybe there's a pattern when numbers close to 1 trillion (which is 10^12, right?) are squared. Let me start by thinking about squaring numbers that are just a little less than a power o... | To determine the number of zeros in the expansion of \(999,\!999,\!999,\!998^2\), we start by recognizing that \(999,\!999,\!999,\!998\) is \(10^{12} - 2\). Using the algebraic identity \((a - b)^2 = a^2 - 2ab + b^2\), we let \(a = 10^{12}\) and \(b = 2\). This gives us:
\[
(10^{12} - 2)^2 = (10^{12})^2 - 2 \cdot 10^{... | 17,002 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to figure out how many zeros are in the expansion of 999,999,999,998 squared. Hmm, that's a pretty big number. Maybe there's a pattern when numbers close to 1 trillion (which is 10^12, right?) are squared. Let me start by thinking about squaring numbers that are just a little less than a... | [
{
"from": "human",
"value": "How many zeros are in the expansion of $999,\\!999,\\!999,\\!998^2$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to figure out how many zeros are in the expansion of 999,999,999,998 squared. Hmm, that's a pretty big number. Maybe there's a pattern... |
A student has been studying for 3 hours. If she takes a 10-minute break every hour, how many breaks will she take in total if she continues studying for another 5 hours? | A student has been studying for 3 hours. If she takes a 10-minute break every hour, how many breaks will she take in total if she continues studying for another 5 hours? | Okay, so the problem is: A student has been studying for 3 hours. If she takes a 10-minute break every hour, how many breaks will she take in total if she continues studying for another 5 hours?
Hmm, let me try to figure this out. Let's break it down step by step. First, the student has already studied for 3 hours. No... | The student has been studying for 3 hours and continues studying for another 5 hours. She takes a 10-minute break every hour. To determine the total number of breaks she will take:
1. **Initial Study Period**: The student studied for 3 hours. If she takes a break after each hour of studying, she would have taken 3 bre... | 17,571 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so the problem is: A student has been studying for 3 hours. If she takes a 10-minute break every hour, how many breaks will she take in total if she continues studying for another 5 hours?
Hmm, let me try to figure this out. Let's break it down step by step. First, the student has already studied for 3 h... | [
{
"from": "human",
"value": "A student has been studying for 3 hours. If she takes a 10-minute break every hour, how many breaks will she take in total if she continues studying for another 5 hours?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem is: A student has been studying for ... |
at what rate percent on simple interest will rs . 750 amount to rs . 900 in 5 years ? | null | Okay, so I need to figure out the rate percent per annum for a simple interest problem. The principal is Rs. 750, and it amounts to Rs. 900 over 5 years. Hmm, let's recall the formula for simple interest.
Simple Interest (SI) is calculated as SI = (Principal × Rate × Time)/100. Right? And the amount is the principal ... | Given that the principal (P) is Rs. 750, the amount (A) after 5 years is Rs. 900. We need to find the rate percent (R) per annum on simple interest.
First, we calculate the simple interest (SI):
\[
\text{SI} = A - P = 900 - 750 = 150
\]
Using the simple interest formula:
\[
\text{SI} = \frac{P \times R \times T}{100}... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out the rate percent per annum for a simple interest problem. The principal is Rs. 750, and it amounts to Rs. 900 over 5 years. Hmm, let's recall the formula for simple interest.
Simple Interest (SI) is calculated as SI = (Principal × Rate × Time)/100. Right? And the amount is the pr... | [
{
"from": "human",
"value": "at what rate percent on simple interest will rs . 750 amount to rs . 900 in 5 years ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the rate percent per annum for a simple interest problem. The principal is Rs. 750, and it amounts to Rs. 900 o... |
What is the base 3 representation of the base 10 number 120? | What is the base 3 representation of the base 10 number 120? | Okay, so I need to figure out how to convert the number 120 from base 10 (which is regular decimal) to base 3. Hmm, base 3, right? That means each digit represents powers of 3 instead of powers of 10. Let me recall how base conversions work.
From what I remember, to convert a decimal number to another base, you typic... | To convert the base 10 number 120 to base 3, we repeatedly divide by 3 and keep track of the remainders:
1. \( 120 \div 3 = 40 \) with remainder \( 0 \)
2. \( 40 \div 3 = 13 \) with remainder \( 1 \)
3. \( 13 \div 3 = 4 \) with remainder \( 1 \)
4. \( 4 \div 3 = 1 \) with remainder \( 1 \)
5. \( 1 \div 3 = 0 \) with r... | 36,185 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how to convert the number 120 from base 10 (which is regular decimal) to base 3. Hmm, base 3, right? That means each digit represents powers of 3 instead of powers of 10. Let me recall how base conversions work.
From what I remember, to convert a decimal number to another base, y... | [
{
"from": "human",
"value": "What is the base 3 representation of the base 10 number 120?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how to convert the number 120 from base 10 (which is regular decimal) to base 3. Hmm, base 3, right? That means each digit represents po... |
A cistern is 10 m long and 6 m wide. It contains water up to a certain breadth. The total area of the wet surface is 103.2 m². What is the breadth of the water in the cistern? | null | Okay, let's see. I need to find the breadth of the water in the cistern. The cistern is 10 meters long and 6 meters wide. The total area of the wet surface is given as 103.2 square meters. Hmm, wet surface area... So when the cistern has water in it, the surfaces that are in contact with the water are the bottom and th... | The cistern is 10 meters long and 6 meters wide. The total area of the wet surface is given as 103.2 m². We need to find the breadth of the water in the cistern.
1. **Interpretation of the Problem**:
- The term "breadth" here is likely referring to the depth of the water, given the context and the calculation requi... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find the breadth of the water in the cistern. The cistern is 10 meters long and 6 meters wide. The total area of the wet surface is given as 103.2 square meters. Hmm, wet surface area... So when the cistern has water in it, the surfaces that are in contact with the water are the botto... | [
{
"from": "human",
"value": "A cistern is 10 m long and 6 m wide. It contains water up to a certain breadth. The total area of the wet surface is 103.2 m². What is the breadth of the water in the cistern?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the breadth of the ... |
Z1) Prove that for every natural number $n \geq 3$ there are natural numbers $a_{1}<a_{2}<\ldots<a_{n}$ such that
$$
a_{k} \mid\left(a_{1}+a_{2}+\ldots+a_{n}\right)
$$
applies for every $k=1,2, \ldots, n$. | null | Okay, so I need to prove that for every natural number n ≥ 3, there exists a set of natural numbers a₁ < a₂ < ... < aₙ such that each aₖ divides the sum of all the numbers in the set. Hmm, interesting. Let me try to wrap my head around this.
First, let's parse the problem. We need to find a strictly increasing sequenc... | To prove that for every natural number \( n \geq 3 \) there exist natural numbers \( a_1 < a_2 < \ldots < a_n \) such that each \( a_k \) divides the sum \( a_1 + a_2 + \ldots + a_n \), we can use the following construction:
1. **Base Case (n = 3)**:
- The sequence \( 1, 2, 3 \) works because the sum is \( 6 \), an... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that for every natural number n ≥ 3, there exists a set of natural numbers a₁ < a₂ < ... < aₙ such that each aₖ divides the sum of all the numbers in the set. Hmm, interesting. Let me try to wrap my head around this.
First, let's parse the problem. We need to find a strictly increasing... | [
{
"from": "human",
"value": "Z1) Prove that for every natural number $n \\geq 3$ there are natural numbers $a_{1}<a_{2}<\\ldots<a_{n}$ such that\n\n$$\na_{k} \\mid\\left(a_{1}+a_{2}+\\ldots+a_{n}\\right)\n$$\n\napplies for every $k=1,2, \\ldots, n$."
},
{
"from": "gpt",
"value": "<think>\nOkay, ... |
What is the probability that Fatima gets fewer heads than tails if she flips 10 coins? | What is the probability that Fatima gets fewer heads than tails if she flips 10 coins? | Okay, so I need to find the probability that Fatima gets fewer heads than tails when she flips 10 coins. Hmm, let's start by understanding the problem. When flipping coins, each flip has two possible outcomes: heads (H) or tails (T). Since she's flipping 10 coins, the total number of possible outcomes is 2^10, which is... | To find the probability that Fatima gets fewer heads than tails when flipping 10 coins, we need to consider the cases where the number of heads (k) is 0, 1, 2, 3, or 4. The total number of possible outcomes is \(2^{10} = 1024\).
The probability of getting exactly k heads in 10 flips is given by the binomial coefficien... | 20,193 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that Fatima gets fewer heads than tails when she flips 10 coins. Hmm, let's start by understanding the problem. When flipping coins, each flip has two possible outcomes: heads (H) or tails (T). Since she's flipping 10 coins, the total number of possible outcomes is 2^10, ... | [
{
"from": "human",
"value": "What is the probability that Fatima gets fewer heads than tails if she flips 10 coins?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the probability that Fatima gets fewer heads than tails when she flips 10 coins. Hmm, let's start by understanding t... |
A box contains 9 red balls and 8 blue balls. If 8 balls are randomly selected from the box, what is the probability that exactly 7 of them are red? | A box contains 9 red balls and 8 blue balls. If 8 balls are randomly selected from the box, what is the probability that exactly 7 of them are red? | Okay, so I need to find the probability that exactly 7 out of 8 randomly selected balls are red. The box has 9 red balls and 8 blue balls, making a total of 17 balls. Hmm, probability problems can sometimes be tricky, but let's break it down step by step.
First, I remember that when dealing with probabilities involvin... | To find the probability that exactly 7 out of 8 randomly selected balls are red from a box containing 9 red balls and 8 blue balls, we use combinations to determine the number of favorable outcomes divided by the total number of possible outcomes.
1. **Number of ways to choose 7 red balls from 9 red balls:**
\[
... | 6,565 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that exactly 7 out of 8 randomly selected balls are red. The box has 9 red balls and 8 blue balls, making a total of 17 balls. Hmm, probability problems can sometimes be tricky, but let's break it down step by step.
First, I remember that when dealing with probabilities ... | [
{
"from": "human",
"value": "A box contains 9 red balls and 8 blue balls. If 8 balls are randomly selected from the box, what is the probability that exactly 7 of them are red?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the probability that exactly 7 out of 8 randomly select... |
What is the least common multiple of 12, 15, and 20?
Note: This problem is inspired by the original problem, but requires a different approach to solve. The numbers are not simply changed, but the problem is modified to require a different method of solution. In this case, the numbers are larger and more complex, requ... | What is the least common multiple of 12, 15, and 20?
Note: This problem is inspired by the original problem, but requires a different approach to solve. The numbers are not simply changed, but the problem is modified to require a different method of solution. In this case, the numbers are larger and more complex, requ... | Okay, so I need to find the least common multiple (LCM) of 12, 15, and 20. Hmm, LCM... right, that's the smallest number that all three of these numbers can divide into without leaving a remainder. I remember there are a couple of ways to find the LCM. One method is listing the multiples until you find a common one, bu... | To find the least common multiple (LCM) of 12, 15, and 20 using prime factorization, we start by breaking down each number into its prime factors:
- \(12 = 2^2 \times 3^1\)
- \(15 = 3^1 \times 5^1\)
- \(20 = 2^2 \times 5^1\)
Next, we identify the highest power of each prime number present in the factorizations:
- The... | 3,217 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the least common multiple (LCM) of 12, 15, and 20. Hmm, LCM... right, that's the smallest number that all three of these numbers can divide into without leaving a remainder. I remember there are a couple of ways to find the LCM. One method is listing the multiples until you find a common... | [
{
"from": "human",
"value": "What is the least common multiple of 12, 15, and 20?\n\nNote: This problem is inspired by the original problem, but requires a different approach to solve. The numbers are not simply changed, but the problem is modified to require a different method of solution. In this case, th... |
Evaluate $5^4-4^3-3^2-2^1-1^0$. | null | Okay, so I need to evaluate the expression 5⁴ - 4³ - 3² - 2¹ - 1⁰. Hmm, let's take it step by step. First, I remember that exponents mean multiplying the base by itself a certain number of times. Let me start with each term individually.
Starting with 5⁴. That's 5 multiplied by itself four times. Let me calculate that... | To evaluate \(5^4 - 4^3 - 3^2 - 2^1 - 1^0\), we start by calculating each term individually:
\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\]
\[
4^3 = 4 \times 4 \times 4 = 64
\]
\[
3^2 = 3 \times 3 = 9
\]
\[
2^1 = 2
\]
\[
1^0 = 1
\]
Substituting these values into the expression, we get:
\[
625 - 64 - 9 - 2 - 1
\]
Ne... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to evaluate the expression 5⁴ - 4³ - 3² - 2¹ - 1⁰. Hmm, let's take it step by step. First, I remember that exponents mean multiplying the base by itself a certain number of times. Let me start with each term individually.
Starting with 5⁴. That's 5 multiplied by itself four times. Let me calcul... | [
{
"from": "human",
"value": "Evaluate $5^4-4^3-3^2-2^1-1^0$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to evaluate the expression 5⁴ - 4³ - 3² - 2¹ - 1⁰. Hmm, let's take it step by step. First, I remember that exponents mean multiplying the base by itself a certain number of times.... |
6. Divide the lollipops between Peppa and George, so that both receive the same number of candies. Each person gets $\qquad$ lollipops. | null | Okay, so I need to divide the lollipops between Peppa and George so that both get the same number of candies. Hmm, let me think. First, I need to know how many lollipops there are in total. Wait, the problem doesn't specify the total number of lollipops. That's strange. How can I divide them equally if I don't know the... | To divide the lollipops between Peppa and George so that both receive the same number of candies, we need to know the total number of lollipops. However, the problem does not specify the total number.
Given that the problem is likely intended for young students and common totals in such problems are often even number... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to divide the lollipops between Peppa and George so that both get the same number of candies. Hmm, let me think. First, I need to know how many lollipops there are in total. Wait, the problem doesn't specify the total number of lollipops. That's strange. How can I divide them equally if I don't ... | [
{
"from": "human",
"value": "6. Divide the lollipops between Peppa and George, so that both receive the same number of candies. Each person gets $\\qquad$ lollipops."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to divide the lollipops between Peppa and George so that both get the same... |
3rd CaMO 1971 Problem 4 Find all real a such that x 2 + ax + 1 = x 2 + x + a = 0 for some real x. | null | Okay, so I need to find all real numbers a such that both quadratic equations x² + a x + 1 = 0 and x² + x + a = 0 have a common real solution x. Hmm, let's see.
First, if there's a real number x that satisfies both equations simultaneously, then that x must make both equations true. So, if I subtract one equation fro... | To find all real numbers \( a \) such that both quadratic equations \( x^2 + ax + 1 = 0 \) and \( x^2 + x + a = 0 \) have a common real solution \( x \), we start by subtracting the second equation from the first:
\[
(x^2 + ax + 1) - (x^2 + x + a) = 0
\]
Simplifying this, we get:
\[
(a - 1)x + (1 - a) = 0
\]
Factor... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find all real numbers a such that both quadratic equations x² + a x + 1 = 0 and x² + x + a = 0 have a common real solution x. Hmm, let's see.
First, if there's a real number x that satisfies both equations simultaneously, then that x must make both equations true. So, if I subtract one equa... | [
{
"from": "human",
"value": "3rd CaMO 1971 Problem 4 Find all real a such that x 2 + ax + 1 = x 2 + x + a = 0 for some real x."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find all real numbers a such that both quadratic equations x² + a x + 1 = 0 and x² + x + a = 0 have a common r... |
Three cylindrical vessels with capacities in the ratio of 2:4:7 are filled with water. The largest vessel is filled to 3/4 of its capacity, the middle vessel is filled to 2/3 of its capacity, and the smallest vessel is filled to 1/2 of its capacity. If 12 liters of water are poured from the largest vessel into the smal... | Three cylindrical vessels with capacities in the ratio of 2:4:7 are filled with water. The largest vessel is filled to 3/4 of its capacity, the middle vessel is filled to 2/3 of its capacity, and the smallest vessel is filled to 1/2 of its capacity. If 12 liters of water are poured from the largest vessel into the smal... | Okay, so I have this problem here about three cylindrical vessels with capacities in the ratio 2:4:7. They’re filled with water to different fractions of their capacities, and then some water is poured from the largest to the smallest. I need to find the percentage of the total capacity that’s now filled with water. Hm... | Given three cylindrical vessels with capacities in the ratio 2:4:7, we denote their capacities as \(2x\), \(4x\), and \(7x\) liters respectively. The vessels are filled as follows:
- The largest vessel (capacity \(7x\)) is filled to \(\frac{3}{4}\) of its capacity: \(\frac{3}{4} \times 7x = \frac{21x}{4}\) liters.
- Th... | 17,740 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I have this problem here about three cylindrical vessels with capacities in the ratio 2:4:7. They’re filled with water to different fractions of their capacities, and then some water is poured from the largest to the smallest. I need to find the percentage of the total capacity that’s now filled with w... | [
{
"from": "human",
"value": "Three cylindrical vessels with capacities in the ratio of 2:4:7 are filled with water. The largest vessel is filled to 3/4 of its capacity, the middle vessel is filled to 2/3 of its capacity, and the smallest vessel is filled to 1/2 of its capacity. If 12 liters of water are pou... |
In the Cartesian coordinate plane $(xOy)$, find the standard equation of the circle with the largest radius among all circles that are tangent to the line $2mx - y - 4m + 1 = 0 (m \in \mathbb{R})$ and have their center at point $(1, 0)$. | null | Okay, so I need to find the standard equation of the circle with the largest radius that's tangent to the line given by 2mx - y - 4m + 1 = 0, where m is a real number, and the center of the circle is at (1, 0). Hmm, let me think about how to approach this.
First, I remember that the standard equation of a circle with ... | To find the standard equation of the circle with the largest radius that is tangent to the line \(2mx - y - 4m + 1 = 0\) and has its center at \((1, 0)\), we start by noting that the given line can be rewritten in slope-intercept form as \(y = 2m(x - 2) + 1\). This shows that the line passes through the fixed point \((... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the standard equation of the circle with the largest radius that's tangent to the line given by 2mx - y - 4m + 1 = 0, where m is a real number, and the center of the circle is at (1, 0). Hmm, let me think about how to approach this.
First, I remember that the standard equation of a circ... | [
{
"from": "human",
"value": "In the Cartesian coordinate plane $(xOy)$, find the standard equation of the circle with the largest radius among all circles that are tangent to the line $2mx - y - 4m + 1 = 0 (m \\in \\mathbb{R})$ and have their center at point $(1, 0)$."
},
{
"from": "gpt",
"value... |
Let $a, b, c, d$ be positive integers such that $a d \neq b c$ and $\operatorname{gcd}(a, b, c, d)=1$. Prove that, as $n$ runs through the positive integers, the values $\operatorname{gcd}(a n+b, c n+d)$ may achieve form the set of all positive divisors of some integer.
Raul Alcantara, Peru | null | Okay, let's see. I need to prove that the set of all possible gcd(an + b, cn + d) values, as n ranges through positive integers, forms the set of all positive divisors of some integer. The given conditions are that a, b, c, d are positive integers with ad ≠ bc and gcd(a, b, c, d) = 1. Hmm.
First, let's recall that the... | Given positive integers \(a, b, c, d\) such that \(ad \neq bc\) and \(\operatorname{gcd}(a, b, c, d) = 1\), we need to prove that the set of values \(\operatorname{gcd}(an + b, cn + d)\) as \(n\) runs through positive integers forms the set of all positive divisors of some integer.
1. **Divisibility by \(ad - bc\)**:
... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to prove that the set of all possible gcd(an + b, cn + d) values, as n ranges through positive integers, forms the set of all positive divisors of some integer. The given conditions are that a, b, c, d are positive integers with ad ≠ bc and gcd(a, b, c, d) = 1. Hmm.
First, let's recall ... | [
{
"from": "human",
"value": "Let $a, b, c, d$ be positive integers such that $a d \\neq b c$ and $\\operatorname{gcd}(a, b, c, d)=1$. Prove that, as $n$ runs through the positive integers, the values $\\operatorname{gcd}(a n+b, c n+d)$ may achieve form the set of all positive divisors of some integer.\n\nRa... |
Can a triangle be constructed if the radius of the incircle corresponding to two given sides and the height to the third side are known? | null | Okay, so the question is: Can a triangle be constructed if we know the radius of the incircle corresponding to two given sides and the height to the third side? Hmm, let's try to break this down step by step.
First, let me recall some formulas related to triangles, incircle, and heights. The inradius (r) of a triangle... | Given two sides \(a\) and \(b\), the inradius \(r\), and the height \(h_c\) to the third side \(c\), we need to determine if a triangle can be constructed.
1. **Express the third side \(c\)**:
Using the relationship between the inradius \(r\), semi-perimeter \(s\), and area \(A\), we have:
\[
r \cdot s = \fra... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so the question is: Can a triangle be constructed if we know the radius of the incircle corresponding to two given sides and the height to the third side? Hmm, let's try to break this down step by step.
First, let me recall some formulas related to triangles, incircle, and heights. The inradius (r) of a ... | [
{
"from": "human",
"value": "Can a triangle be constructed if the radius of the incircle corresponding to two given sides and the height to the third side are known?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the question is: Can a triangle be constructed if we know the radius of the incir... |
Evaluate the sum
$$
\frac{1}{1+\tan 1^{\circ}}+\frac{1}{1+\tan 2^{\circ}}+\frac{1}{1+\tan 3^{\circ}}+\cdots+\frac{1}{1+\tan 89^{\circ}}
$$
(The tangent $(\tan )$ of an angle $\alpha$ is the ratio $B C / A C$ in a right triangle $A B C$ with $\angle C=90^{\circ}$ and $\angle A=\alpha$, and its value does not depend on... | null | Okay, so I need to evaluate this sum: 1 over (1 + tan 1°) plus 1 over (1 + tan 2°) plus ... all the way up to 1 over (1 + tan 89°). Hmm, that's 89 terms. That seems like a lot, but maybe there's a pattern or a symmetry here. Let me think.
First, I remember that tangent has some co-function identities. Like, tan(90° - ... | To evaluate the sum
$$
\frac{1}{1+\tan 1^{\circ}}+\frac{1}{1+\tan 2^{\circ}}+\frac{1}{1+\tan 3^{\circ}}+\cdots+\frac{1}{1+\tan 89^{\circ}},
$$
we use the co-function identity for tangent, which states that $\tan(90^\circ - x) = \cot x$. This allows us to pair terms from the beginning and the end of the sum.
Conside... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to evaluate this sum: 1 over (1 + tan 1°) plus 1 over (1 + tan 2°) plus ... all the way up to 1 over (1 + tan 89°). Hmm, that's 89 terms. That seems like a lot, but maybe there's a pattern or a symmetry here. Let me think.
First, I remember that tangent has some co-function identities. Like, ta... | [
{
"from": "human",
"value": "Evaluate the sum\n\n$$\n\\frac{1}{1+\\tan 1^{\\circ}}+\\frac{1}{1+\\tan 2^{\\circ}}+\\frac{1}{1+\\tan 3^{\\circ}}+\\cdots+\\frac{1}{1+\\tan 89^{\\circ}}\n$$\n\n(The tangent $(\\tan )$ of an angle $\\alpha$ is the ratio $B C / A C$ in a right triangle $A B C$ with $\\angle C=90^{... |
Let $a$ and $b$ be the roots of the equation $x^2-6x+2=0$. Evaluate $(a^2+b^2)(a^3-b^3)$. | Let $a$ and $b$ be the roots of the equation $x^2-6x+2=0$. Evaluate $(a^2+b^2)(a^3-b^3)$. | Okay, so I need to evaluate the expression (a² + b²)(a³ - b³) where a and b are the roots of the quadratic equation x² - 6x + 2 = 0. Hmm, let's start by recalling that for a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. In this case, the equation is x² - 6x + 2 = 0, so the su... | Given the quadratic equation \(x^2 - 6x + 2 = 0\) with roots \(a\) and \(b\), we need to evaluate \((a^2 + b^2)(a^3 - b^3)\).
First, we use Vieta's formulas to find the sum and product of the roots:
\[
a + b = 6 \quad \text{and} \quad ab = 2
\]
Next, we calculate \(a^2 + b^2\) using the identity:
\[
a^2 + b^2 = (a + ... | 28,149 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to evaluate the expression (a² + b²)(a³ - b³) where a and b are the roots of the quadratic equation x² - 6x + 2 = 0. Hmm, let's start by recalling that for a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. In this case, the equation is x² - 6x + 2 = 0, s... | [
{
"from": "human",
"value": "Let $a$ and $b$ be the roots of the equation $x^2-6x+2=0$. Evaluate $(a^2+b^2)(a^3-b^3)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to evaluate the expression (a² + b²)(a³ - b³) where a and b are the roots of the quadratic equation x² - 6x + 2 = 0. Hmm,... |
A circle and a regular hexagon both have circumferences of 50 inches. What is the ratio of the length of the radius of the circle to the length of the side of the hexagon? Express your answer as a common fraction. | A circle and a regular hexagon both have circumferences of 50 inches. What is the ratio of the length of the radius of the circle to the length of the side of the hexagon? Express your answer as a common fraction. | Okay, let me try to figure out this problem. So, we have a circle and a regular hexagon, both with a circumference of 50 inches. Wait, hold on, a regular hexagon has a circumference? Hmm, I thought circumference is a term usually used for circles. Maybe they mean the perimeter of the hexagon? That makes more sense beca... | To find the ratio of the radius of the circle to the side length of the regular hexagon, both having a circumference (or perimeter) of 50 inches, we start by using the formulas for circumference and perimeter.
For the circle, the circumference \( C \) is given by:
\[
C = 2\pi r
\]
Given \( C = 50 \), we solve for the ... | 37,440 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out this problem. So, we have a circle and a regular hexagon, both with a circumference of 50 inches. Wait, hold on, a regular hexagon has a circumference? Hmm, I thought circumference is a term usually used for circles. Maybe they mean the perimeter of the hexagon? That makes more se... | [
{
"from": "human",
"value": "A circle and a regular hexagon both have circumferences of 50 inches. What is the ratio of the length of the radius of the circle to the length of the side of the hexagon? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me ... |
I have eleven books, of which I want to bring two to read on vacation. How many different pairs can I choose? | I have eleven books, of which I want to bring two to read on vacation. How many different pairs can I choose? | Okay, let's see. The problem is: I have eleven books, and I want to bring two to read on vacation. How many different pairs can I choose? Hmm, alright. So, I need to figure out how many different ways I can pick two books out of eleven.
First, I remember that when the order doesn't matter in combinations, we use comb... | To determine the number of different pairs of books that can be chosen from 11 books, we use the combination formula. Since the order of selecting the books does not matter, we need to calculate the number of combinations of 11 books taken 2 at a time.
The formula for combinations is given by:
\[
\binom{n}{k} = \frac{... | 13,241 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is: I have eleven books, and I want to bring two to read on vacation. How many different pairs can I choose? Hmm, alright. So, I need to figure out how many different ways I can pick two books out of eleven.
First, I remember that when the order doesn't matter in combinations, we ... | [
{
"from": "human",
"value": "I have eleven books, of which I want to bring two to read on vacation. How many different pairs can I choose?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem is: I have eleven books, and I want to bring two to read on vacation. How many different... |
6 ) a marketing firm determined that , of 300 households surveyed , 80 used neither brand a nor brand b soap . 60 used only brand a soap and for every household that used both brands of soap , 3 used only brand b soap . how many of the 200 household surveyed used both brands of soap ? | null | Okay, so I need to figure out how many households used both brands of soap. Let's see. The problem says that out of 300 households surveyed, 80 used neither brand A nor brand B. So, that means 300 minus 80 households used at least one of the brands. Let me write that down.
Total households = 300
Neither A nor B = 80
T... | The marketing firm surveyed 300 households, out of which 80 used neither brand A nor brand B soap. Therefore, 300 - 80 = 220 households used at least one of the brands.
Given:
- 60 households used only brand A soap.
- For every household that used both brands of soap, 3 used only brand B soap.
Let \( x \) be the num... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out how many households used both brands of soap. Let's see. The problem says that out of 300 households surveyed, 80 used neither brand A nor brand B. So, that means 300 minus 80 households used at least one of the brands. Let me write that down.
Total households = 300
Neither A nor ... | [
{
"from": "human",
"value": "6 ) a marketing firm determined that , of 300 households surveyed , 80 used neither brand a nor brand b soap . 60 used only brand a soap and for every household that used both brands of soap , 3 used only brand b soap . how many of the 200 household surveyed used both brands of ... |
Task 1. The frog has a jump of $50 \mathrm{~cm}$, the rabbit $75 \mathrm{~cm}$, and the kangaroo $90 \mathrm{~cm}$. What is the smallest distance at which all will make an integer number of jumps. How many jumps will each of the animals then make? | null | Okay, let me try to figure out this problem. So, we have three animals: a frog, a rabbit, and a kangaroo. Each of them can jump a certain distance—50 cm, 75 cm, and 90 cm respectively. The question is asking for the smallest distance where all three animals will land exactly after an integer number of jumps. In other w... | To find the smallest distance at which all three animals (frog, rabbit, and kangaroo) will make an integer number of jumps, we need to determine the least common multiple (LCM) of their jump lengths: 50 cm, 75 cm, and 90 cm.
First, we break down each jump length into its prime factors:
- \(50 = 2^1 \times 5^2\)
- \(75... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem. So, we have three animals: a frog, a rabbit, and a kangaroo. Each of them can jump a certain distance—50 cm, 75 cm, and 90 cm respectively. The question is asking for the smallest distance where all three animals will land exactly after an integer number of jumps. In... | [
{
"from": "human",
"value": "Task 1. The frog has a jump of $50 \\mathrm{~cm}$, the rabbit $75 \\mathrm{~cm}$, and the kangaroo $90 \\mathrm{~cm}$. What is the smallest distance at which all will make an integer number of jumps. How many jumps will each of the animals then make?"
},
{
"from": "gpt",... |
Given a function $f(x)= \frac {-2^{x}-b}{2^{x+1}+2}$ defined on the domain of real numbers $\mathbb{R}$, which is an odd function.
(Ⅰ) Find the value of the real number $b$;
(Ⅱ) Determine and prove the monotonicity of the function $f(x)$;
(Ⅲ) If the equation $f(x)=m$ has a solution for $x$ in the interval $[0,1]$... | null | Alright, let's tackle this problem step by step. It's about an odd function, and there are three parts to solve. Let me start with part (I) where I need to find the value of the real number \( b \).
First, I remember that an odd function satisfies the property \( f(-x) = -f(x) \) for all \( x \) in its domain. Since t... | Given the function \( f(x) = \frac{-2^x - b}{2^{x+1} + 2} \) defined on the domain of real numbers \(\mathbb{R}\), which is an odd function, we need to solve the following parts:
Ⅰ. Find the value of the real number \( b \):
An odd function satisfies \( f(-x) = -f(x) \). We compute \( f(-x) \) and set it equal to \(-... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let's tackle this problem step by step. It's about an odd function, and there are three parts to solve. Let me start with part (I) where I need to find the value of the real number \( b \).
First, I remember that an odd function satisfies the property \( f(-x) = -f(x) \) for all \( x \) in its domain.... | [
{
"from": "human",
"value": "Given a function $f(x)= \\frac {-2^{x}-b}{2^{x+1}+2}$ defined on the domain of real numbers $\\mathbb{R}$, which is an odd function. \n(Ⅰ) Find the value of the real number $b$; \n(Ⅱ) Determine and prove the monotonicity of the function $f(x)$; \n(Ⅲ) If the equation $f(x)=m$ ... |
Calculate: $\frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \dots + \frac{11}{7} + \frac{12}{7}$. | Calculate: $\frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \dots + \frac{11}{7} + \frac{12}{7}$. | Okay, let me see. I need to calculate the sum of fractions starting from 1/7 and going up by 1/7 each time until 12/7. So, the series is 1/7 + 2/7 + 3/7 + ... + 12/7. Hmm, how do I approach this?
First, I remember that when adding fractions with the same denominator, you can just add the numerators and keep the denomi... | To calculate the sum \(\frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \dots + \frac{11}{7} + \frac{12}{7}\), we first note that all terms have the same denominator, 7. Therefore, we can factor out the denominator and sum the numerators:
\[
\frac{1}{7} + \frac{2}{7} + \frac{3}{7} + \dots + \frac{11}{7} + \frac{12}{7} = \fra... | 28,046 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me see. I need to calculate the sum of fractions starting from 1/7 and going up by 1/7 each time until 12/7. So, the series is 1/7 + 2/7 + 3/7 + ... + 12/7. Hmm, how do I approach this?
First, I remember that when adding fractions with the same denominator, you can just add the numerators and keep th... | [
{
"from": "human",
"value": "Calculate: $\\frac{1}{7} + \\frac{2}{7} + \\frac{3}{7} + \\dots + \\frac{11}{7} + \\frac{12}{7}$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me see. I need to calculate the sum of fractions starting from 1/7 and going up by 1/7 each time until 12/7. So, the se... |
Given the geometric progression 2, 6, 18,..., expand the product $(x+6)(x+18)$ using the properties of geometric progressions. | Given the geometric progression 2, 6, 18,..., expand the product $(x+6)(x+18)$ using the properties of geometric progressions. | Okay, so I need to expand the product (x + 6)(x + 18) using the properties of geometric progressions. Hmm, let's see. First, let me recall what a geometric progression is. It's a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. In the given sequen... | Given the geometric progression \(2, 6, 18, \ldots\), we need to expand the product \((x + 6)(x + 18)\) using the properties of geometric progressions.
First, we recognize that \(6\) and \(18\) are the second and third terms of the geometric progression. The first term \(a_1\) is \(2\) and the common ratio \(r\) is \(... | 27,848 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to expand the product (x + 6)(x + 18) using the properties of geometric progressions. Hmm, let's see. First, let me recall what a geometric progression is. It's a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. In the give... | [
{
"from": "human",
"value": "Given the geometric progression 2, 6, 18,..., expand the product $(x+6)(x+18)$ using the properties of geometric progressions."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to expand the product (x + 6)(x + 18) using the properties of geometric progressions... |
The extreme points of the function $f(x)= \frac {x^{4}}{4}- \frac {x^{3}}{3}$ are ( ).
A: $0$
B: $-1$
C: $0$ or $1$
D: $1$ | null | Okay, so I need to find the extreme points of the function f(x) = (x^4)/4 - (x^3)/3. The options given are A: 0, B: -1, C: 0 or 1, D: 1. Let me think about how to approach this.
First, I remember that extreme points of a function occur where the first derivative is zero or undefined, provided they are within the domai... | To find the extreme points of the function \( f(x) = \frac{x^4}{4} - \frac{x^3}{3} \), we start by computing the first derivative:
\[
f'(x) = x^3 - x^2
\]
Setting the first derivative equal to zero to find critical points:
\[
x^3 - x^2 = 0 \implies x^2(x - 1) = 0 \implies x = 0 \text{ or } x = 1
\]
Next, we use the... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the extreme points of the function f(x) = (x^4)/4 - (x^3)/3. The options given are A: 0, B: -1, C: 0 or 1, D: 1. Let me think about how to approach this.
First, I remember that extreme points of a function occur where the first derivative is zero or undefined, provided they are within t... | [
{
"from": "human",
"value": "The extreme points of the function $f(x)= \\frac {x^{4}}{4}- \\frac {x^{3}}{3}$ are ( ).\n\nA: $0$\n\nB: $-1$\n\nC: $0$ or $1$\n\nD: $1$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the extreme points of the function f(x) = (x^4)/4 - (x^3)/3. The o... |
Find $x$ such that $\log_{2x-1} 3 = \frac{1}{4}$ | Find $x$ such that $\log_{2x-1} 3 = \frac{1}{4}$ | Okay, so I need to solve this logarithmic equation: log base (2x - 1) of 3 equals 1/4. Hmm, let me recall how logarithms work. The general form is log_b a = c, which means that b^c = a. So applying that here, if log_{2x-1} 3 = 1/4, then (2x - 1)^{1/4} = 3. Wait, no, actually, the base raised to the result equals the nu... | To solve the equation \(\log_{2x-1} 3 = \frac{1}{4}\), we start by converting the logarithmic form to its exponential form. This gives us:
\[
(2x - 1)^{1/4} = 3
\]
Next, we eliminate the exponent by raising both sides of the equation to the 4th power:
\[
\left( (2x - 1)^{1/4} \right)^4 = 3^4
\]
This simplifies to:
... | 8,361 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve this logarithmic equation: log base (2x - 1) of 3 equals 1/4. Hmm, let me recall how logarithms work. The general form is log_b a = c, which means that b^c = a. So applying that here, if log_{2x-1} 3 = 1/4, then (2x - 1)^{1/4} = 3. Wait, no, actually, the base raised to the result equal... | [
{
"from": "human",
"value": "Find $x$ such that $\\log_{2x-1} 3 = \\frac{1}{4}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this logarithmic equation: log base (2x - 1) of 3 equals 1/4. Hmm, let me recall how logarithms work. The general form is log_b a = c, which means that... |
Find the number of consecutive zeroes in the expansion of $(3x^2+2x+1)^{2024}$, where the coefficients are read from left to right. | Find the number of consecutive zeroes in the expansion of $(3x^2+2x+1)^{2024}$, where the coefficients are read from left to right. | Okay, so I need to find the number of consecutive zeroes in the expansion of \((3x^2 + 2x + 1)^{2024}\) when the coefficients are read from left to right. Hmm, let's break this down step by step.
First, I know that expanding a polynomial raised to a power involves using the multinomial theorem. Each term in the expans... | To find the number of consecutive zeroes in the expansion of \((3x^2 + 2x + 1)^{2024}\), we need to determine if there are any gaps in the sequence of coefficients from \(x^0\) to \(x^{4048}\).
1. **Understanding the Expansion**:
- The polynomial \((3x^2 + 2x + 1)^{2024}\) is expanded using the multinomial theorem.... | 31,126 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the number of consecutive zeroes in the expansion of \((3x^2 + 2x + 1)^{2024}\) when the coefficients are read from left to right. Hmm, let's break this down step by step.
First, I know that expanding a polynomial raised to a power involves using the multinomial theorem. Each term in th... | [
{
"from": "human",
"value": "Find the number of consecutive zeroes in the expansion of $(3x^2+2x+1)^{2024}$, where the coefficients are read from left to right."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the number of consecutive zeroes in the expansion of \\((3x^2 + 2x + 1)... |
Three, (50 points) In an $8 \times 8$ grid, each cell contains a knight or a knave. Knights always tell the truth, and knaves always lie. It is known that each resident in the grid says: “The number of knaves in my column is (strictly) greater than the number of knaves in my row.” Determine the number of arrangements o... | null | Okay, so I have this problem about an 8x8 grid where each cell is either a knight or a knave. Knights always tell the truth, and knaves always lie. Every resident says, “The number of knaves in my column is strictly greater than the number of knaves in my row.” I need to figure out how many possible arrangements of kni... | Given an \(8 \times 8\) grid where each cell contains either a knight or a knave, we need to determine the number of valid arrangements such that each resident's statement about the number of knaves in their column being strictly greater than the number of knaves in their row is satisfied. Knights always tell the truth... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem about an 8x8 grid where each cell is either a knight or a knave. Knights always tell the truth, and knaves always lie. Every resident says, “The number of knaves in my column is strictly greater than the number of knaves in my row.” I need to figure out how many possible arrangement... | [
{
"from": "human",
"value": "Three, (50 points) In an $8 \\times 8$ grid, each cell contains a knight or a knave. Knights always tell the truth, and knaves always lie. It is known that each resident in the grid says: “The number of knaves in my column is (strictly) greater than the number of knaves in my ro... |
A circle with equation $(x-a)^2+(y-b)^2=r^2$ is rotated $90^{\circ}$ clockwise about the origin. The circle and its rotation are translated vertically three units in opposite directions to become the graphs of $x=f(y)$ and $x=g(y)$, respectively. Which of the following describes the graph of $x=(f+g)(y)$?
Is it:
(A) ... | A circle with equation $(x-a)^2+(y-b)^2=r^2$ is rotated $90^{\circ}$ clockwise about the origin. The circle and its rotation are translated vertically three units in opposite directions to become the graphs of $x=f(y)$ and $x=g(y)$, respectively. Which of the following describes the graph of $x=(f+g)(y)$?
Is it:
(A) ... | Okay, let me try to work through this problem step by step. Hmm, so we have a circle with equation (x - a)^2 + (y - b)^2 = r^2. Then it's rotated 90 degrees clockwise about the origin. After that, both the original circle and its rotated version are translated vertically three units in opposite directions. Then, these ... | The original circle is given by the equation \((x - a)^2 + (y - b)^2 = r^2\). When rotated \(90^\circ\) clockwise about the origin, the center \((a, b)\) is transformed to \((b, -a)\), resulting in the equation \((x - b)^2 + (y + a)^2 = r^2\).
Both the original circle and the rotated circle are translated vertically t... | 23,410 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to work through this problem step by step. Hmm, so we have a circle with equation (x - a)^2 + (y - b)^2 = r^2. Then it's rotated 90 degrees clockwise about the origin. After that, both the original circle and its rotated version are translated vertically three units in opposite directions. Then... | [
{
"from": "human",
"value": "A circle with equation $(x-a)^2+(y-b)^2=r^2$ is rotated $90^{\\circ}$ clockwise about the origin. The circle and its rotation are translated vertically three units in opposite directions to become the graphs of $x=f(y)$ and $x=g(y)$, respectively. Which of the following describe... |
A cylindrical tank with a height of 10 meters is to be constructed using a fixed amount of metal sheeting. The tank's radius and height must be natural numbers. What is the maximum volume of water that can be stored in the tank if the total surface area of the tank (including the top and bottom) is 300 square meters? | A cylindrical tank with a height of 10 meters is to be constructed using a fixed amount of metal sheeting. The tank's radius and height must be natural numbers. What is the maximum volume of water that can be stored in the tank if the total surface area of the tank (including the top and bottom) is 300 square meters? | Alright, let's tackle this problem step by step. So, we have a cylindrical tank that needs to be constructed with a fixed amount of metal sheeting, which gives us a total surface area of 300 square meters. The tank's radius and height must be natural numbers, and we need to find the maximum volume of water it can store... | The problem involves constructing a cylindrical tank with a fixed surface area of 300 square meters, where the radius and height must be natural numbers. The key steps are:
1. **Surface Area Equation**: The total surface area of the cylinder is given by \(2\pi r^2 + 2\pi rh = 300\). Simplifying, we get \(r(r + h) = \f... | 20,842 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's tackle this problem step by step. So, we have a cylindrical tank that needs to be constructed with a fixed amount of metal sheeting, which gives us a total surface area of 300 square meters. The tank's radius and height must be natural numbers, and we need to find the maximum volume of water it c... | [
{
"from": "human",
"value": "A cylindrical tank with a height of 10 meters is to be constructed using a fixed amount of metal sheeting. The tank's radius and height must be natural numbers. What is the maximum volume of water that can be stored in the tank if the total surface area of the tank (including th... |
Find the smallest positive integer $a$ such that $a^2+1$ has exactly four positive divisors, and the largest positive integer $b$ less than $200$ such that $b^2-1$ has exactly six positive divisors. What is $a+b$? | Find the smallest positive integer $a$ such that $a^2+1$ has exactly four positive divisors, and the largest positive integer $b$ less than $200$ such that $b^2-1$ has exactly six positive divisors. What is $a+b$? | Okay, so I need to find two numbers: the smallest positive integer \( a \) such that \( a^2 + 1 \) has exactly four positive divisors, and the largest positive integer \( b \) less than 200 such that \( b^2 - 1 \) has exactly six positive divisors. Then, add them together for the answer. Let me start with the first par... | To find the smallest positive integer \( a \) such that \( a^2 + 1 \) has exactly four positive divisors, we need \( a^2 + 1 \) to be a product of two distinct primes (a semiprime). Testing small values of \( a \):
- \( a = 1 \): \( 1^2 + 1 = 2 \) (prime, two divisors)
- \( a = 2 \): \( 2^2 + 1 = 5 \) (prime, two divi... | 27,214 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find two numbers: the smallest positive integer \( a \) such that \( a^2 + 1 \) has exactly four positive divisors, and the largest positive integer \( b \) less than 200 such that \( b^2 - 1 \) has exactly six positive divisors. Then, add them together for the answer. Let me start with the f... | [
{
"from": "human",
"value": "Find the smallest positive integer $a$ such that $a^2+1$ has exactly four positive divisors, and the largest positive integer $b$ less than $200$ such that $b^2-1$ has exactly six positive divisors. What is $a+b$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I ne... |
Problem 9.5. Show that for any non-zero natural number $n$ there exists a sequence of $2 n+1$ consecutive natural numbers such that the sum of the squares of the first $n+1$ of them is equal to the sum of the squares of the next $n$ of them. Verify whether there exists such a sequence with the middle number equal to 20... | null | Okay, so I need to show that for any non-zero natural number \( n \), there's a sequence of \( 2n + 1 \) consecutive natural numbers. The key part is that the sum of the squares of the first \( n + 1 \) numbers should equal the sum of the squares of the next \( n \) numbers. And then check if such a sequence can have 2... | To show that for any non-zero natural number \( n \), there exists a sequence of \( 2n + 1 \) consecutive natural numbers such that the sum of the squares of the first \( n + 1 \) of them is equal to the sum of the squares of the next \( n \) of them, we start by denoting the middle number as \( m \). The sequence rang... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to show that for any non-zero natural number \( n \), there's a sequence of \( 2n + 1 \) consecutive natural numbers. The key part is that the sum of the squares of the first \( n + 1 \) numbers should equal the sum of the squares of the next \( n \) numbers. And then check if such a sequence ca... | [
{
"from": "human",
"value": "Problem 9.5. Show that for any non-zero natural number $n$ there exists a sequence of $2 n+1$ consecutive natural numbers such that the sum of the squares of the first $n+1$ of them is equal to the sum of the squares of the next $n$ of them. Verify whether there exists such a se... |
Find the integer $m$, $0 \le m \le 9$, such that \[m \equiv -1567 \pmod{10}.\] | Find the integer $m$, $0 \le m \le 9$, such that \[m \equiv -1567 \pmod{10}.\] | Okay, so I need to find an integer m between 0 and 9 inclusive, such that m is congruent to -1567 modulo 10. Hmm, let me think about what this means.
First, congruence modulo 10 usually refers to the remainder when a number is divided by 10. So, when the problem says m ≡ -1567 mod 10, it means that m is the remainder... | To find the integer \( m \) such that \( 0 \le m \le 9 \) and \( m \equiv -1567 \pmod{10} \), we need to determine the remainder when \(-1567\) is divided by 10.
1. **Understanding Modulo with Negative Numbers**: When dealing with negative numbers, the remainder must still be a non-negative integer between 0 and 9.
... | 22,096 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find an integer m between 0 and 9 inclusive, such that m is congruent to -1567 modulo 10. Hmm, let me think about what this means.
First, congruence modulo 10 usually refers to the remainder when a number is divided by 10. So, when the problem says m ≡ -1567 mod 10, it means that m is the r... | [
{
"from": "human",
"value": "Find the integer $m$, $0 \\le m \\le 9$, such that \\[m \\equiv -1567 \\pmod{10}.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find an integer m between 0 and 9 inclusive, such that m is congruent to -1567 modulo 10. Hmm, let me think about what this... |
1. Form all possible subsets for a set consisting of: a) two elements; b) three elements. | null | Okay, so I need to figure out how to form all possible subsets for sets with two and three elements. Let me start with part a), which is a set with two elements. Hmm, subsets... right, a subset is any collection of elements from the original set, including the empty set and the set itself.
First, let me recall that i... | To form all possible subsets for a set with two elements and a set with three elements, we start by noting that the number of subsets of a set with \( n \) elements is \( 2^n \).
### Part a) Set with two elements
For a set \(\{a, b\}\):
- The empty set: \(\{\}\)
- Subsets with one element: \(\{a\}\), \(\{b\}\)
- The s... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out how to form all possible subsets for sets with two and three elements. Let me start with part a), which is a set with two elements. Hmm, subsets... right, a subset is any collection of elements from the original set, including the empty set and the set itself.
First, let me recal... | [
{
"from": "human",
"value": "1. Form all possible subsets for a set consisting of: a) two elements; b) three elements."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how to form all possible subsets for sets with two and three elements. Let me start with part a), which is ... |
A square has two of its vertices at points (3,4) and (7,8). Find the coordinates of the center of the square. | A square has two of its vertices at points (3,4) and (7,8). Find the coordinates of the center of the square. | Okay, let me try to figure this out. So the problem is: a square has two of its vertices at points (3,4) and (7,8). Find the coordinates of the center of the square. Hmm, squares can be placed in different orientations, so I need to make sure I consider all possibilities. Wait, but maybe there's only one possible squar... | To find the coordinates of the center of the square with vertices at points (3,4) and (7,8), we first consider the distance between these points. The distance is calculated as follows:
\[
\text{Distance} = \sqrt{(7-3)^2 + (8-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]
This distance could be either the side leng... | 24,881 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure this out. So the problem is: a square has two of its vertices at points (3,4) and (7,8). Find the coordinates of the center of the square. Hmm, squares can be placed in different orientations, so I need to make sure I consider all possibilities. Wait, but maybe there's only one possib... | [
{
"from": "human",
"value": "A square has two of its vertices at points (3,4) and (7,8). Find the coordinates of the center of the square."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure this out. So the problem is: a square has two of its vertices at points (3,4) and (7,8). F... |
If $f(x) = \frac{1}{x} + x$ and $g(x) = x^2 - 2$, find $a$ such that $f(g(a)) = g(f(a))$. | If $f(x) = \frac{1}{x} + x$ and $g(x) = x^2 - 2$, find $a$ such that $f(g(a)) = g(f(a))$. | Okay, let's tackle this problem step by step. The question is asking to find a value 'a' such that f(g(a)) equals g(f(a)). The functions given are f(x) = 1/x + x and g(x) = x² - 2. Hmm, composition of functions, right? So, first, I need to understand what f(g(a)) and g(f(a)) mean.
Let me start by writing down the fun... | To find \( a \) such that \( f(g(a)) = g(f(a)) \) given \( f(x) = \frac{1}{x} + x \) and \( g(x) = x^2 - 2 \):
1. Compute \( f(g(a)) \):
\[
f(g(a)) = f(a^2 - 2) = \frac{1}{a^2 - 2} + (a^2 - 2)
\]
2. Compute \( g(f(a)) \):
\[
g(f(a)) = g\left(\frac{1}{a} + a\right) = \left(\frac{1}{a} + a\right)^2 - 2
... | 8,029 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's tackle this problem step by step. The question is asking to find a value 'a' such that f(g(a)) equals g(f(a)). The functions given are f(x) = 1/x + x and g(x) = x² - 2. Hmm, composition of functions, right? So, first, I need to understand what f(g(a)) and g(f(a)) mean.
Let me start by writing down... | [
{
"from": "human",
"value": "If $f(x) = \\frac{1}{x} + x$ and $g(x) = x^2 - 2$, find $a$ such that $f(g(a)) = g(f(a))$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's tackle this problem step by step. The question is asking to find a value 'a' such that f(g(a)) equals g(f(a)). The functions... |
Solve for $n$: $|n + 6| = 2 - n$. | Solve for $n$: $|n + 6| = 2 - n$. | Okay, so I need to solve the equation |n + 6| = 2 - n. Hmm, absolute value equations can sometimes be tricky because of the cases involved. Let me recall that for |A| = B, we have two possibilities: either A = B or A = -B, but only if B is non-negative because absolute value can't be negative. So, in this case, the rig... | To solve the equation \( |n + 6| = 2 - n \), we need to consider the cases for the absolute value and ensure the right-hand side is non-negative.
First, we check that \( 2 - n \geq 0 \), which implies \( n \leq 2 \).
**Case 1: \( n + 6 \geq 0 \)**
Solve \( n + 6 = 2 - \n \):
\[
\begin{align*}
n + 6 &= 2 - n \\
2n &=... | 11,045 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve the equation |n + 6| = 2 - n. Hmm, absolute value equations can sometimes be tricky because of the cases involved. Let me recall that for |A| = B, we have two possibilities: either A = B or A = -B, but only if B is non-negative because absolute value can't be negative. So, in this case,... | [
{
"from": "human",
"value": "Solve for $n$: $|n + 6| = 2 - n$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve the equation |n + 6| = 2 - n. Hmm, absolute value equations can sometimes be tricky because of the cases involved. Let me recall that for |A| = B, we have two possibili... |
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the gre... | null | Okay, let's try to tackle this problem step by step. It's a bit complex, but if I break it down, maybe I can figure it out. So, we have positive integers a, b, c (nonzero digits), and for each n, A_n is an n-digit number with all digits a. Similarly, B_n is n-digit with all b's, and C_n is a 2n-digit number with all c'... | Given a positive integer \( n \) and nonzero digits \( a \), \( b \), and \( c \), we need to find the greatest possible value of \( a + b + c \) such that there are at least two values of \( n \) for which \( C_n - B_n = A_n^2 \). Here, \( A_n \), \( B_n \), and \( C_n \) are \( n \)-digit integers with digits \( a \)... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to tackle this problem step by step. It's a bit complex, but if I break it down, maybe I can figure it out. So, we have positive integers a, b, c (nonzero digits), and for each n, A_n is an n-digit number with all digits a. Similarly, B_n is n-digit with all b's, and C_n is a 2n-digit number wit... | [
{
"from": "human",
"value": "For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whos... |
In writing the integers from 10 through 99 inclusive, how many times is the digit 6 written? | In writing the integers from 10 through 99 inclusive, how many times is the digit 6 written? | Okay, so I need to figure out how many times the digit 6 is written when listing all the integers from 10 through 99. Hmm, let's start by understanding the problem.
First, the numbers from 10 to 99 are all the two-digit numbers. Each number has a tens place and a ones place. For example, in 16, the tens digit is 1 an... | To determine how many times the digit 6 is written when listing all integers from 10 through 99, we need to consider both the tens place and the ones place separately.
1. **Counting 6s in the ones place:**
- The ones digit cycles from 0 to 9 for each tens group (10-19, 20-29, ..., 90-99).
- In each group, the di... | 13,141 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out how many times the digit 6 is written when listing all the integers from 10 through 99. Hmm, let's start by understanding the problem.
First, the numbers from 10 to 99 are all the two-digit numbers. Each number has a tens place and a ones place. For example, in 16, the tens digit... | [
{
"from": "human",
"value": "In writing the integers from 10 through 99 inclusive, how many times is the digit 6 written?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how many times the digit 6 is written when listing all the integers from 10 through 99. Hmm, let's start... |
Mark constructs a cube of pure gold. The cube is 6 cm on each side. The density of gold is 19 grams per cubic centimeter. He buys the gold for a certain price per gram and sells it for 1.5 times its gold value. His profit was $123,120. What was the price per gram he bought the gold for? | null | Okay, let's try to figure out this problem step by step. So, Mark built a cube made of pure gold. The cube has sides that are 6 cm each. The density of gold is given as 19 grams per cubic centimeter. He buys the gold at a certain price per gram and sells it for 1.5 times that price. His total profit was $123,120. We ne... | Mark constructs a cube of pure gold with each side measuring 6 cm. The volume of the cube is calculated as follows:
\[
6 \, \text{cm} \times 6 \, \text{cm} \times 6 \, \text{cm} = 216 \, \text{cm}^3
\]
Given the density of gold is 19 grams per cubic centimeter, the mass of the gold cube is:
\[
19 \, \text{g/cm}^3 \tim... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to figure out this problem step by step. So, Mark built a cube made of pure gold. The cube has sides that are 6 cm each. The density of gold is given as 19 grams per cubic centimeter. He buys the gold at a certain price per gram and sells it for 1.5 times that price. His total profit was $123,12... | [
{
"from": "human",
"value": "Mark constructs a cube of pure gold. The cube is 6 cm on each side. The density of gold is 19 grams per cubic centimeter. He buys the gold for a certain price per gram and sells it for 1.5 times its gold value. His profit was $123,120. What was the price per gram he bought the g... |
Simplify $3(7x+2y)-2(5x-3y)+(x+2y)$. | Simplify $3(7x+2y)-2(5x-3y)+(x+2y)$. | Okay, so I need to simplify this expression: 3(7x + 2y) - 2(5x - 3y) + (x + 2y). Hmm, let's take it step by step. First, I remember that simplifying expressions like this involves using the distributive property, which means multiplying the terms outside the parentheses by each term inside. Then, combining like terms. ... | To simplify the expression \(3(7x + 2y) - 2(5x - 3y) + (x + 2y)\), we start by applying the distributive property:
1. Distribute the 3 in \(3(7x + 2y)\):
\[
3 \cdot 7x + 3 \cdot 2y = 21x + 6y
\]
2. Distribute the -2 in \(-2(5x - 3y)\):
\[
-2 \cdot 5x + (-2) \cdot (-3y) = -10x + 6y
\]
3. The term \(... | 27,000 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to simplify this expression: 3(7x + 2y) - 2(5x - 3y) + (x + 2y). Hmm, let's take it step by step. First, I remember that simplifying expressions like this involves using the distributive property, which means multiplying the terms outside the parentheses by each term inside. Then, combining like... | [
{
"from": "human",
"value": "Simplify $3(7x+2y)-2(5x-3y)+(x+2y)$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to simplify this expression: 3(7x + 2y) - 2(5x - 3y) + (x + 2y). Hmm, let's take it step by step. First, I remember that simplifying expressions like this involves using the ... |
Jessica is making an apple pie. She knows that each serving requires 1.5 apples and she has 12 quests. She plans to make 3 pies, which each contain 8 servings. If her guests finish all the pie, on average , how many apples does each guest eat? | null | Okay, let's see. Jessica is making apple pies, and I need to figure out how many apples each guest eats on average. Let me start by breaking down the problem step by step.
First, each serving requires 1.5 apples. She has 12 guests, and she's making 3 pies. Each pie contains 8 servings. So, the guests finish all the pi... | Jessica is making 3 pies, each containing 8 servings. Therefore, the total number of servings is calculated as follows:
\[
3 \text{ pies} \times 8 \text{ servings per pie} = 24 \text{ servings}
\]
Each serving requires 1.5 apples, so the total number of apples used is:
\[
24 \text{ servings} \times 1.5 \text{ apples p... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. Jessica is making apple pies, and I need to figure out how many apples each guest eats on average. Let me start by breaking down the problem step by step.
First, each serving requires 1.5 apples. She has 12 guests, and she's making 3 pies. Each pie contains 8 servings. So, the guests finish al... | [
{
"from": "human",
"value": "Jessica is making an apple pie. She knows that each serving requires 1.5 apples and she has 12 quests. She plans to make 3 pies, which each contain 8 servings. If her guests finish all the pie, on average , how many apples does each guest eat?"
},
{
"from": "gpt",
"v... |
A fair coin is flipped 5 times, and a player wins if the sequence of flips has exactly 2 heads. What is the probability that the player will win on the next flip after losing the first 4 attempts, given that the coin is flipped exactly once every 3 minutes and the player can only play for 15 minutes? | A fair coin is flipped 5 times, and a player wins if the sequence of flips has exactly 2 heads. What is the probability that the player will win on the next flip after losing the first 4 attempts, given that the coin is flipped exactly once every 3 minutes and the player can only play for 15 minutes? | Okay, so I need to figure out the probability that the player will win on the next flip after losing the first 4 attempts. Let me start by understanding the problem step by step.
First, the setup: A fair coin is flipped 5 times, and the player wins if the sequence of flips has exactly 2 heads. The coin is flipped onc... | The problem involves determining the probability that a player will win on the fifth coin flip after losing the first four attempts. The player wins if the sequence of five flips has exactly 2 heads. The coin is flipped once every 3 minutes, and the player can play for 15 minutes, resulting in exactly 5 flips.
Key ste... | 27,656 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the probability that the player will win on the next flip after losing the first 4 attempts. Let me start by understanding the problem step by step.
First, the setup: A fair coin is flipped 5 times, and the player wins if the sequence of flips has exactly 2 heads. The coin is fli... | [
{
"from": "human",
"value": "A fair coin is flipped 5 times, and a player wins if the sequence of flips has exactly 2 heads. What is the probability that the player will win on the next flip after losing the first 4 attempts, given that the coin is flipped exactly once every 3 minutes and the player can onl... |
Mark borrows $10$ dollars from Emily with a simple interest rate of $15\%$ everyday. What is the least integer number of days after which Mark will have to pay her back at least twice as much as he borrowed? | Mark borrows $10$ dollars from Emily with a simple interest rate of $15\%$ everyday. What is the least integer number of days after which Mark will have to pay her back at least twice as much as he borrowed? | Okay, so Mark borrowed $10 from Emily, and there's a simple interest rate of 15% every day. I need to figure out the least number of days after which Mark has to pay back at least twice the amount he borrowed. Twice the amount would be $20, right? Because $10 times 2 is $20.
First, let me recall what simple interest ... | Mark borrows $10 from Emily with a simple interest rate of 15% per day. We need to find the least integer number of days after which the total amount to be paid back is at least twice the borrowed amount, which is $20.
The formula for simple interest is \( I = P \cdot r \cdot t \), where:
- \( P = 10 \) dollars (princ... | 13,400 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Mark borrowed $10 from Emily, and there's a simple interest rate of 15% every day. I need to figure out the least number of days after which Mark has to pay back at least twice the amount he borrowed. Twice the amount would be $20, right? Because $10 times 2 is $20.
First, let me recall what simple i... | [
{
"from": "human",
"value": "Mark borrows $10$ dollars from Emily with a simple interest rate of $15\\%$ everyday. What is the least integer number of days after which Mark will have to pay her back at least twice as much as he borrowed?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so Mark borr... |
x can do a piece of work in 40 days. He works at it for 8 days and then y finished it in 36 days. How long will y take to complete the work on his own? | null | Okay, so I need to figure out how long Y will take to complete the work on his own. Let me start by understanding the problem step by step.
First, it says that X can do a piece of work in 40 days. That means X's work rate is 1 work per 40 days, right? So, in one day, X can complete 1/40 of the work.
Now, X works on ... | X can complete a piece of work in 40 days, so X's work rate is \(\frac{1}{40}\) per day.
X works for 8 days, completing:
\[
8 \times \frac{1}{40} = \frac{8}{40} = \frac{1}{5}
\]
of the work.
The remaining work is:
\[
1 - \frac{1}{5} = \frac{4}{5}
\]
Y finishes the remaining \(\frac{4}{5}\) of the work in 36 days. ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out how long Y will take to complete the work on his own. Let me start by understanding the problem step by step.
First, it says that X can do a piece of work in 40 days. That means X's work rate is 1 work per 40 days, right? So, in one day, X can complete 1/40 of the work.
Now, X w... | [
{
"from": "human",
"value": "x can do a piece of work in 40 days. He works at it for 8 days and then y finished it in 36 days. How long will y take to complete the work on his own?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how long Y will take to complete the work on ... |
Solve the following equation:
$\frac{4}{\sqrt{x}} = \frac{x}{2}$ | Solve the following equation:
$\frac{4}{\sqrt{x}} = \frac{x}{2}$ | Okay, so I need to solve the equation 4 over the square root of x equals x over 2. Let me write that down to make sure I have it right: 4/√x = x/2. Hmm, alright, let's see. I need to find the value of x that makes this equation true.
First, maybe I should get rid of the square root in the denominator because those ca... | To solve the equation \(\frac{4}{\sqrt{x}} = \frac{x}{2}\):
1. Multiply both sides by \(\sqrt{x}\) to eliminate the denominator:
\[
4 = \frac{x \cdot \sqrt{x}}{2}
\]
2. Simplify the right side by recognizing \(x \cdot \sqrt{x} = x^{3/2}\):
\[
4 = \frac{x^{3/2}}{2}
\]
3. Multiply both sides by 2:
... | 5,994 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to solve the equation 4 over the square root of x equals x over 2. Let me write that down to make sure I have it right: 4/√x = x/2. Hmm, alright, let's see. I need to find the value of x that makes this equation true.
First, maybe I should get rid of the square root in the denominator because ... | [
{
"from": "human",
"value": "Solve the following equation:\n\n$\\frac{4}{\\sqrt{x}} = \\frac{x}{2}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve the equation 4 over the square root of x equals x over 2. Let me write that down to make sure I have it right: 4/√x = x/2. Hmm, alr... |
The light glows for every some seconds. Between 1:57:58 and 3:20:47 am, the light glowed 276.05555555555554 times. What is the interval in seconds between each glow of the light? | null | Okay, so I need to figure out the interval in seconds between each glow of the light. Let me start by understanding the problem step by step.
First, the light glows repeatedly every certain number of seconds. The problem states that between 1:57:58 AM and 3:20:47 AM, the light glowed approximately 276.05555555555554 t... | First, we need to calculate the total time elapsed between 1:57:58 AM and 3:20:47 AM.
1. Convert the start time (1:57:58 AM) to seconds:
- 1 hour = 3600 seconds
- 57 minutes = 3420 seconds
- 58 seconds
- Total = 3600 + 3420 + 58 = 7078 seconds
2. Convert the end time (3:20:47 AM) to seconds:
- 3 hours... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out the interval in seconds between each glow of the light. Let me start by understanding the problem step by step.
First, the light glows repeatedly every certain number of seconds. The problem states that between 1:57:58 AM and 3:20:47 AM, the light glowed approximately 276.05555555... | [
{
"from": "human",
"value": "The light glows for every some seconds. Between 1:57:58 and 3:20:47 am, the light glowed 276.05555555555554 times. What is the interval in seconds between each glow of the light?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the interval in se... |
Find the constant term in the expansion of
\[(x^3 + 1)(x^6 + 2)(x^9 + 3) \dotsm (x^{30} + 10)(x^{33} + 11).\] | Find the constant term in the expansion of
\[(x^3 + 1)(x^6 + 2)(x^9 + 3) \dotsm (x^{30} + 10)(x^{33} + 11).\] | Okay, so I need to find the constant term in the expansion of this product: (x³ + 1)(x⁶ + 2)(x⁹ + 3)... all the way up to (x³³ + 11). Hmm, let's see.
First, let me make sure I understand the problem. The product is a sequence of terms starting with (x³ + 1), then (x⁶ + 2), then (x⁹ + 3), and each subsequent term incr... | To find the constant term in the expansion of \((x^3 + 1)(x^6 + 2)(x^9 + 3) \dotsm (x^{30} + 10)(x^{33} + 11)\), we need to consider the structure of each term in the product. Each term is of the form \((x^{3k} + k)\) where \(k\) ranges from 1 to 11.
The constant term in the expansion is obtained by selecting the con... | 32,939 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the constant term in the expansion of this product: (x³ + 1)(x⁶ + 2)(x⁹ + 3)... all the way up to (x³³ + 11). Hmm, let's see.
First, let me make sure I understand the problem. The product is a sequence of terms starting with (x³ + 1), then (x⁶ + 2), then (x⁹ + 3), and each subsequent t... | [
{
"from": "human",
"value": "Find the constant term in the expansion of\n\\[(x^3 + 1)(x^6 + 2)(x^9 + 3) \\dotsm (x^{30} + 10)(x^{33} + 11).\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the constant term in the expansion of this product: (x³ + 1)(x⁶ + 2)(x⁹ + 3)... all the w... |
If $\left( r + \frac{1}{r} \right)^2 = 3,$ then find $r^3 + \frac{1}{r^3}.$ | If $\left( r + \frac{1}{r} \right)^2 = 3,$ then find $r^3 + \frac{1}{r^3}.$ | Okay, so I need to find \( r^3 + \frac{1}{r^3} \) given that \( \left( r + \frac{1}{r} \right)^2 = 3 \). Hmm, let's start by expanding the left side of the equation they provided. Maybe that will help me find \( r + \frac{1}{r} \) first, and then I can use that to find the cube term.
Alright, expanding \( \left( r + \... | Given \(\left( r + \frac{1}{r} \right)^2 = 3\), we start by expanding the left-hand side:
\[
\left( r + \frac{1}{r} \right)^2 = r^2 + 2 + \frac{1}{r^2}
\]
This simplifies to:
\[
r^2 + 2 + \frac{1}{r^2} = 3
\]
Subtracting 2 from both sides, we get:
\[
r^2 + \frac{1}{r^2} = 1
\]
Next, we use the identity for the cu... | 12,852 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find \( r^3 + \frac{1}{r^3} \) given that \( \left( r + \frac{1}{r} \right)^2 = 3 \). Hmm, let's start by expanding the left side of the equation they provided. Maybe that will help me find \( r + \frac{1}{r} \) first, and then I can use that to find the cube term.
Alright, expanding \( \lef... | [
{
"from": "human",
"value": "If $\\left( r + \\frac{1}{r} \\right)^2 = 3,$ then find $r^3 + \\frac{1}{r^3}.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find \\( r^3 + \\frac{1}{r^3} \\) given that \\( \\left( r + \\frac{1}{r} \\right)^2 = 3 \\). Hmm, let's start by expanding the ... |
Find the smallest positive $a$ such that $a$ is a multiple of $6$ and $a$ is a multiple of $15.$ | null | Okay, so I need to find the smallest positive number a that is a multiple of both 6 and 15. Hmm, let me think. I remember something about least common multiples from math class. Maybe that's what this is about? Let me recall. The least common multiple (LCM) of two numbers is the smallest number that is a multiple of bo... | To find the smallest positive \( a \) that is a multiple of both 6 and 15, we need to determine the least common multiple (LCM) of 6 and 15.
First, we list the multiples of each number:
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
- Multiples of 15: 15, 30, 45, 60, 75, ...
The smallest common multiple... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the smallest positive number a that is a multiple of both 6 and 15. Hmm, let me think. I remember something about least common multiples from math class. Maybe that's what this is about? Let me recall. The least common multiple (LCM) of two numbers is the smallest number that is a multip... | [
{
"from": "human",
"value": "Find the smallest positive $a$ such that $a$ is a multiple of $6$ and $a$ is a multiple of $15.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the smallest positive number a that is a multiple of both 6 and 15. Hmm, let me think. I remember somethin... |
Given a sphere with center $(1,2,-1)$ and radius $\sqrt{7},$ find its equation in Cartesian coordinates. | Given a sphere with center $(1,2,-1)$ and radius $\sqrt{7},$ find its equation in Cartesian coordinates. | Okay, so I need to find the equation of a sphere in Cartesian coordinates. The problem gives me the center of the sphere as (1, 2, -1) and the radius as √7. Hmm, let's see. I remember that the general equation of a sphere in 3D space is something like (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) is the cent... | To find the equation of a sphere with center \((1, 2, -1)\) and radius \(\sqrt{7}\) in Cartesian coordinates, we use the standard form of the sphere's equation:
\[
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
\]
where \((h, k, l)\) is the center and \(r\) is the radius. Substituting the given center coordinates \(h = 1\),... | 11,246 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the equation of a sphere in Cartesian coordinates. The problem gives me the center of the sphere as (1, 2, -1) and the radius as √7. Hmm, let's see. I remember that the general equation of a sphere in 3D space is something like (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2, where (h, k, l) is ... | [
{
"from": "human",
"value": "Given a sphere with center $(1,2,-1)$ and radius $\\sqrt{7},$ find its equation in Cartesian coordinates."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the equation of a sphere in Cartesian coordinates. The problem gives me the center of the sphere ... |
A random number is generated from a set of positive integers that are less than or equal to 1000 and are divisible by 3. What is the expected number of digits in the number generated? Express your answer in a decimal. | A random number is generated from a set of positive integers that are less than or equal to 1000 and are divisible by 3. What is the expected number of digits in the number generated? Express your answer in a decimal. | Okay, let's see. The problem is asking for the expected number of digits in a randomly generated number from the set of positive integers less than or equal to 1000 that are divisible by 3. Hmm. So first, I need to figure out what numbers are in this set. They have to be divisible by 3 and less than or equal to 1000. T... | To find the expected number of digits in a randomly generated number from the set of positive integers less than or equal to 1000 that are divisible by 3, we first identify the numbers in this set. These numbers are 3, 6, 9, ..., 999, forming an arithmetic sequence with the first term 3, common difference 3, and the la... | 35,707 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is asking for the expected number of digits in a randomly generated number from the set of positive integers less than or equal to 1000 that are divisible by 3. Hmm. So first, I need to figure out what numbers are in this set. They have to be divisible by 3 and less than or equal to... | [
{
"from": "human",
"value": "A random number is generated from a set of positive integers that are less than or equal to 1000 and are divisible by 3. What is the expected number of digits in the number generated? Express your answer in a decimal."
},
{
"from": "gpt",
"value": "<think>\nOkay, let... |
Find the coefficient of $x^2$ when $3(x^2 - x^3) +2(x - 2x^2 + 3x^5) -(4x^3 - x^2)$ is simplified. | Find the coefficient of $x^2$ when $3(x^2 - x^3) +2(x - 2x^2 + 3x^5) -(4x^3 - x^2)$ is simplified. | Okay, so I need to find the coefficient of \( x^2 \) when the expression \( 3(x^2 - x^3) +2(x - 2x^2 + 3x^5) -(4x^3 - x^2) \) is simplified. Hmm, let's see. I remember that to find the coefficient of a specific term, I need to expand the entire expression first and then combine like terms. Then, look for the term with ... | To find the coefficient of \( x^2 \) when \( 3(x^2 - x^3) + 2(x - 2x^2 + 3x^5) - (4x^3 - x^2) \) is simplified, we need to expand and combine like terms step by step.
First, expand each part of the expression:
1. \( 3(x^2 - x^3) = 3x^2 - 3x^3 \)
2. \( 2(x - 2x^2 + 3x^5) = 2x - 4x^2 + 6x^5 \)
3. \( -(4x^3 - x^2) = -4x^... | 9,435 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the coefficient of \( x^2 \) when the expression \( 3(x^2 - x^3) +2(x - 2x^2 + 3x^5) -(4x^3 - x^2) \) is simplified. Hmm, let's see. I remember that to find the coefficient of a specific term, I need to expand the entire expression first and then combine like terms. Then, look for the te... | [
{
"from": "human",
"value": "Find the coefficient of $x^2$ when $3(x^2 - x^3) +2(x - 2x^2 + 3x^5) -(4x^3 - x^2)$ is simplified."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the coefficient of \\( x^2 \\) when the expression \\( 3(x^2 - x^3) +2(x - 2x^2 + 3x^5) -(4x^3 - x^2) \\... |
What value of $x$ will give the minimum value for $x^2- 10x + 24$? | What value of $x$ will give the minimum value for $x^2- 10x + 24$? | Okay, so I need to find the value of x that gives the minimum value for the quadratic expression x² - 10x + 24. Hmm, let's see. I remember that quadratic functions have either a minimum or maximum value depending on the coefficient of the x² term. Since the coefficient here is 1, which is positive, the parabola opens u... | To find the value of \( x \) that gives the minimum value for the quadratic expression \( x^2 - 10x + 24 \), we use the vertex formula for a quadratic function in standard form \( ax^2 + bx + c \). The x-coordinate of the vertex is given by \( -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -10 \):
\[
x = -\frac{-10}{2... | 11,294 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of x that gives the minimum value for the quadratic expression x² - 10x + 24. Hmm, let's see. I remember that quadratic functions have either a minimum or maximum value depending on the coefficient of the x² term. Since the coefficient here is 1, which is positive, the parabola... | [
{
"from": "human",
"value": "What value of $x$ will give the minimum value for $x^2- 10x + 24$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of x that gives the minimum value for the quadratic expression x² - 10x + 24. Hmm, let's see. I remember that quadratic functi... |
Every morning when Tim wakes up, he groggily searches around his sock drawer and picks two socks randomly. If he has 10 gray-bottomed socks and 8 white-bottomed socks in his drawer, what is the probability that he picks a matching pair? | Every morning when Tim wakes up, he groggily searches around his sock drawer and picks two socks randomly. If he has 10 gray-bottomed socks and 8 white-bottomed socks in his drawer, what is the probability that he picks a matching pair? | Okay, so Tim has a sock drawer with 10 gray-bottomed socks and 8 white-bottomed socks. Every morning he picks two socks at random, and we need to find the probability that he picks a matching pair. Hmm, let me think.
First, probability problems often involve figuring out the number of favorable outcomes divided by the... | Tim has 10 gray-bottomed socks and 8 white-bottomed socks in his drawer. We need to find the probability that he picks a matching pair when he randomly selects two socks.
1. **Total number of socks**: \(10 + 8 = 18\).
2. **Total number of ways to pick two socks**:
\[
\binom{18}{2} = \frac{18 \times 17}{2 \times... | 4,284 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Tim has a sock drawer with 10 gray-bottomed socks and 8 white-bottomed socks. Every morning he picks two socks at random, and we need to find the probability that he picks a matching pair. Hmm, let me think.
First, probability problems often involve figuring out the number of favorable outcomes divide... | [
{
"from": "human",
"value": "Every morning when Tim wakes up, he groggily searches around his sock drawer and picks two socks randomly. If he has 10 gray-bottomed socks and 8 white-bottomed socks in his drawer, what is the probability that he picks a matching pair?"
},
{
"from": "gpt",
"value": ... |
The combination lock on a safe consists of three dials, each containing six numbers (1 through 6), and two letters (A and B). The lock can be opened by setting the correct sequence of a number, followed by a letter, and then another number. If each number can only be used once in the sequence and the letters can be rep... | The combination lock on a safe consists of three dials, each containing six numbers (1 through 6), and two letters (A and B). The lock can be opened by setting the correct sequence of a number, followed by a letter, and then another number. If each number can only be used once in the sequence and the letters can be rep... | Alright, let's try to figure out this probability problem step by step. So, we have a combination lock with three dials. The first and third dials are numbers from 1 to 6, and the second dial is letters A and B. The combination is a sequence: number, letter, number. The rules are that each number can only be used once,... | To determine the probability that the combination lock starts with a prime number, followed by the letter A, and ends with an even number, we need to calculate both the total number of possible combinations and the number of successful outcomes.
**Total number of possible combinations:**
- The first number has 6 choic... | 4,956 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's try to figure out this probability problem step by step. So, we have a combination lock with three dials. The first and third dials are numbers from 1 to 6, and the second dial is letters A and B. The combination is a sequence: number, letter, number. The rules are that each number can only be us... | [
{
"from": "human",
"value": "The combination lock on a safe consists of three dials, each containing six numbers (1 through 6), and two letters (A and B). The lock can be opened by setting the correct sequence of a number, followed by a letter, and then another number. If each number can only be used once i... |
1. Determine the real numbers $a, b, c$ from the equality:
 | null | Okay, so I need to figure out the real numbers a, b, and c from the given equality. Hmm, let me start by recalling that when you have an equality of polynomials or rational functions, the coefficients of corresponding powers of x must be equal on both sides. That makes sense. But wait, the problem just mentions an equa... | To determine the real numbers \(a\), \(b\), and \(c\) from the given equality, we follow the method of partial fraction decomposition. Here are the key steps:
1. **Multiply both sides by the denominator**: This eliminates the fractions and results in a polynomial equation.
2. **Substitute specific values of \(x\)**: I... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out the real numbers a, b, and c from the given equality. Hmm, let me start by recalling that when you have an equality of polynomials or rational functions, the coefficients of corresponding powers of x must be equal on both sides. That makes sense. But wait, the problem just mentions... | [
{
"from": "human",
"value": "1. Determine the real numbers $a, b, c$ from the equality:\n\n"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the rea... |
103. The sum
$$
a^{2}+b^{2}+c^{2}+a_{1}^{2}+b_{1}^{2}+c_{1}^{2}
$$
of the squares of all edges of a tetrahedron \(ABCD\) is denoted by \(\delta\). Prove that for any tetrahedron
a) \(\mathrm{II} \leqslant \frac{\sqrt{6 \delta}}{2} ;\)
b) \(\Delta \leqslant \frac{\sqrt{3}}{6} \delta\).
In what case do these inequal... | null | Okay, so I need to prove these two inequalities for any tetrahedron. Let's start with part a) which states that II is less than or equal to sqrt(6δ)/2. Hmm, first, I need to recall what II and Δ denote in the context of a tetrahedron. Since the problem mentions the sum of the squares of all edges denoted by δ, and then... | To prove the inequalities for any tetrahedron:
a) \(\mathrm{II} \leqslant \frac{\sqrt{6 \delta}}{2}\)
b) \(\Delta \leqslant \frac{\sqrt{3}}{6} \delta\)
where \(\delta\) is the sum of the squares of all edges of the tetrahedron, and to determine the cases when these inequalities become equalities:
### Summary of Key... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove these two inequalities for any tetrahedron. Let's start with part a) which states that II is less than or equal to sqrt(6δ)/2. Hmm, first, I need to recall what II and Δ denote in the context of a tetrahedron. Since the problem mentions the sum of the squares of all edges denoted by δ, ... | [
{
"from": "human",
"value": "103. The sum\n\n$$\na^{2}+b^{2}+c^{2}+a_{1}^{2}+b_{1}^{2}+c_{1}^{2}\n$$\n\nof the squares of all edges of a tetrahedron \\(ABCD\\) is denoted by \\(\\delta\\). Prove that for any tetrahedron\n\na) \\(\\mathrm{II} \\leqslant \\frac{\\sqrt{6 \\delta}}{2} ;\\)\n\nb) \\(\\Delta \\le... |
Problem:
A deck of cards contains 5 blue cards, 3 red cards, and 2 green cards. You draw 3 cards at random without replacement. What is the probability that you will draw exactly 2 blue cards and 1 red card? Express your answer as a common fraction. | Problem:
A deck of cards contains 5 blue cards, 3 red cards, and 2 green cards. You draw 3 cards at random without replacement. What is the probability that you will draw exactly 2 blue cards and 1 red card? Express your answer as a common fraction. | Alright, let's tackle this probability problem. So, we have a deck with 5 blue cards, 3 red cards, and 2 green cards. That makes a total of 5 + 3 + 2 = 10 cards. We need to find the probability of drawing exactly 2 blue cards and 1 red card when we draw 3 cards without replacement. Hmm, okay. Let me think about how to ... | To find the probability of drawing exactly 2 blue cards and 1 red card from a deck of 10 cards (5 blue, 3 red, 2 green) when drawing 3 cards without replacement, we use combinations to calculate the number of favorable outcomes and the total number of possible outcomes.
1. **Total number of ways to draw 3 cards from 1... | 6,963 | ai2-adapt-dev/openmath-2-math | <think>
Alright, let's tackle this probability problem. So, we have a deck with 5 blue cards, 3 red cards, and 2 green cards. That makes a total of 5 + 3 + 2 = 10 cards. We need to find the probability of drawing exactly 2 blue cards and 1 red card when we draw 3 cards without replacement. Hmm, okay. Let me think about... | [
{
"from": "human",
"value": "Problem:\nA deck of cards contains 5 blue cards, 3 red cards, and 2 green cards. You draw 3 cards at random without replacement. What is the probability that you will draw exactly 2 blue cards and 1 red card? Express your answer as a common fraction."
},
{
"from": "gpt",... |
A department store held a sale to sell all of the 214 winter jackets that remained after the season ended. Until noon, each jacket in the store was priced at 31.95 dollars. At noon, the price of the jackets was further reduced to 18.95 dollars. After the last jacket was sold, the total receipts for the clearance sale w... | null | Okay, let's see. So there's this department store that had a sale to get rid of all their remaining 214 winter jackets. The problem says they first priced each jacket at $31.95 until noon. Then, after noon, they reduced the price to $18.95, and they sold 133 jackets after noon. We need to find out the total amount of m... | To determine the total amount of money earned from the clearance sale, we need to calculate the revenue from jackets sold before noon and after noon, then sum these amounts.
1. **Calculate the number of jackets sold before noon:**
\[
214 \text{ total jackets} - 133 \text{ jackets sold after noon} = 81 \text{ jac... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. So there's this department store that had a sale to get rid of all their remaining 214 winter jackets. The problem says they first priced each jacket at $31.95 until noon. Then, after noon, they reduced the price to $18.95, and they sold 133 jackets after noon. We need to find out the total amo... | [
{
"from": "human",
"value": "A department store held a sale to sell all of the 214 winter jackets that remained after the season ended. Until noon, each jacket in the store was priced at 31.95 dollars. At noon, the price of the jackets was further reduced to 18.95 dollars. After the last jacket was sold, th... |
It is said that there are 468 more white Go stones than black Go stones. There are 954 white stones. Find the numbers of all Go stones. | null | Okay, let's see. The problem says there are 468 more white Go stones than black ones, and there are 954 white stones. We need to find the total number of all Go stones. Hmm, let me break this down step by step.
First, let's make sure I understand the problem correctly. There are two colors of Go stones: black and whi... | Given that there are 468 more white Go stones than black Go stones, and there are 954 white stones, we need to find the total number of all Go stones.
1. Let \( B \) be the number of black stones and \( W \) be the number of white stones.
2. We know \( W = B + 468 \) and \( W = 954 \).
3. Substituting \( W = 954 \) in... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. The problem says there are 468 more white Go stones than black ones, and there are 954 white stones. We need to find the total number of all Go stones. Hmm, let me break this down step by step.
First, let's make sure I understand the problem correctly. There are two colors of Go stones: black... | [
{
"from": "human",
"value": "It is said that there are 468 more white Go stones than black Go stones. There are 954 white stones. Find the numbers of all Go stones."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. The problem says there are 468 more white Go stones than black ones, and t... |
Determine the value of $b$ so that the lines represented by the following equations are perpendicular:
\begin{align*}
y &= -3x+5 \\
4y-bx &= -2
\end{align*} | Determine the value of $b$ so that the lines represented by the following equations are perpendicular:
\begin{align*}
y &= -3x+5 \\
4y-bx &= -2
\end{align*} | Okay, so I need to find the value of \( b \) that makes the two lines perpendicular. Let me remember what makes two lines perpendicular. Hmm, right, the slopes of two perpendicular lines are negative reciprocals of each other. That means if one line has a slope \( m \), the other line should have a slope of \( -\frac{1... | To determine the value of \( b \) such that the lines represented by the equations \( y = -3x + 5 \) and \( 4y - bx = -2 \) are perpendicular, we need to ensure that their slopes are negative reciprocals of each other.
1. The first equation \( y = -3x + 5 \) is already in slope-intercept form, so its slope \( m_1 \) i... | 19,362 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the value of \( b \) that makes the two lines perpendicular. Let me remember what makes two lines perpendicular. Hmm, right, the slopes of two perpendicular lines are negative reciprocals of each other. That means if one line has a slope \( m \), the other line should have a slope of \( ... | [
{
"from": "human",
"value": "Determine the value of $b$ so that the lines represented by the following equations are perpendicular:\n\\begin{align*}\ny &= -3x+5 \\\\\n4y-bx &= -2\n\\end{align*}"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the value of \\( b \\) that makes the ... |
Find the number of integers between $1$ and $2^{20}$ that have exactly three $1$'s in their binary representation and are divisible by $5$. | Find the number of integers between $1$ and $2^{20}$ that have exactly three $1$'s in their binary representation and are divisible by $5$. | Okay, let's try to figure out how many integers between 1 and 2^20 have exactly three 1's in their binary representation and are also divisible by 5. Hmm, okay, so first, numbers with exactly three 1's in their binary form are combinations of choosing 3 bits out of 20, right? So the total number of such numbers is C(20... | To find the number of integers between 1 and \(2^{20}\) that have exactly three 1's in their binary representation and are divisible by 5, we start by noting that such numbers are combinations of three distinct powers of 2. We need these numbers to be congruent to 0 modulo 5.
The powers of 2 modulo 5 cycle every 4 exp... | 20,471 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's try to figure out how many integers between 1 and 2^20 have exactly three 1's in their binary representation and are also divisible by 5. Hmm, okay, so first, numbers with exactly three 1's in their binary form are combinations of choosing 3 bits out of 20, right? So the total number of such numbers... | [
{
"from": "human",
"value": "Find the number of integers between $1$ and $2^{20}$ that have exactly three $1$'s in their binary representation and are divisible by $5$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's try to figure out how many integers between 1 and 2^20 have exactly three 1... |
A florist had 5 roses. She sold some of them and then later picked 34 more. She now has 36 roses. How many roses did she sell? | null | Okay, let me try to figure out this florist problem. So, the florist starts with 5 roses. Then she sold some of them, and later she picked 34 more. Now she has 36 roses. The question is, how many roses did she sell?
Hmm, let's break it down step by step. First, she had 5 roses. Then she sold some. Let's call the numbe... | The florist started with 5 roses. Let \( x \) be the number of roses she sold. After selling \( x \) roses, she had \( 5 - x \) roses left. Later, she picked 34 more roses, which gave her \( (5 - x) + 34 \) roses. According to the problem, she now has 36 roses. We can set up the equation:
\[
(5 - x) + 34 = 36
\]
Simp... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this florist problem. So, the florist starts with 5 roses. Then she sold some of them, and later she picked 34 more. Now she has 36 roses. The question is, how many roses did she sell?
Hmm, let's break it down step by step. First, she had 5 roses. Then she sold some. Let's call t... | [
{
"from": "human",
"value": "A florist had 5 roses. She sold some of them and then later picked 34 more. She now has 36 roses. How many roses did she sell?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this florist problem. So, the florist starts with 5 roses. Then she s... |
5. An isosceles triangle $ABC$ has a base $\overline{BC}$ of length $12 \text{ cm}$ and a perimeter of $\frac{3}{5} \text{ m}$. At the midpoint $P$ of the leg $\overline{AB}$, a perpendicular to $\overline{AB}$ is drawn, intersecting the leg $\overline{AC}$ at point $D$. Calculate the perimeter of triangle $BCD$.
## E... | null | Okay, let me try to figure out this geometry problem step by step. So, we have an isosceles triangle ABC with base BC of length 12 cm. The perimeter of the triangle is 3/5 of a meter. First, I need to make sure all the units are consistent. The base is given in centimeters, and the perimeter is in meters. Let me conver... | Given an isosceles triangle \(ABC\) with base \(BC\) of length \(12 \text{ cm}\) and a perimeter of \(\frac{3}{5} \text{ m}\) (which converts to \(60 \text{ cm}\)), we determine the lengths of the legs \(AB\) and \(AC\). Since the triangle is isosceles, \(AB = AC\). The perimeter equation gives \(2x + 12 = 60\), solvin... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this geometry problem step by step. So, we have an isosceles triangle ABC with base BC of length 12 cm. The perimeter of the triangle is 3/5 of a meter. First, I need to make sure all the units are consistent. The base is given in centimeters, and the perimeter is in meters. Let m... | [
{
"from": "human",
"value": "5. An isosceles triangle $ABC$ has a base $\\overline{BC}$ of length $12 \\text{ cm}$ and a perimeter of $\\frac{3}{5} \\text{ m}$. At the midpoint $P$ of the leg $\\overline{AB}$, a perpendicular to $\\overline{AB}$ is drawn, intersecting the leg $\\overline{AC}$ at point $D$. ... |
6 Given non-zero vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$ satisfy $\left(\frac{\overrightarrow{A B}}{|\overrightarrow{A B}|}+\frac{\overrightarrow{A C}}{|\overrightarrow{A C}|}\right) \cdot \overrightarrow{B C}=0$ and $\frac{\overrightarrow{A B}}{|\overrightarrow{A B}|} \cdot \frac{\overrightarrow{A C}... | null | Okay, let me try to figure out this problem step by step. So, we have triangle ABC, and we're given two conditions involving vectors AB and AC. The first condition is that the sum of the unit vectors in the direction of AB and AC dotted with vector BC equals zero. The second condition is that the dot product of the uni... | Given non-zero vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) satisfy the conditions:
1. \(\left(\frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} + \frac{\overrightarrow{AC}}{|\overrightarrow{AC}|}\right) \cdot \overrightarrow{BC} = 0\)
2. \(\frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} \cdot \frac{\o... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem step by step. So, we have triangle ABC, and we're given two conditions involving vectors AB and AC. The first condition is that the sum of the unit vectors in the direction of AB and AC dotted with vector BC equals zero. The second condition is that the dot product of... | [
{
"from": "human",
"value": "6 Given non-zero vectors $\\overrightarrow{A B}$ and $\\overrightarrow{A C}$ satisfy $\\left(\\frac{\\overrightarrow{A B}}{|\\overrightarrow{A B}|}+\\frac{\\overrightarrow{A C}}{|\\overrightarrow{A C}|}\\right) \\cdot \\overrightarrow{B C}=0$ and $\\frac{\\overrightarrow{A B}}{|... |
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