problem stringlengths 73 2.75k | solution stringlengths 335 3.59k | r1_distill_70b_response stringlengths 590 92.9k | __original_row_idx int64 0 7.5k |
|---|---|---|---|
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the co... | 1. **Identify the number of students in each class:**
- Mrs. Germain's class: 11 students
- Mr. Newton's class: 8 students
- Mrs. Young's class: 9 students
2. **Assumption of no overlap in students:**
Since the problem does not mention any students being in more than one class, we assume that all students ... | <think>
To determine the total number of mathematics students at Euclid Middle School taking the AMC 8 contest, I will add the number of students from each teacher's class.
First, there are 11 students in Mrs. Germain's class.
Next, there are 8 students in Mr. Newton's class.
Finally, there are 9 students in Mrs. Yo... | 12 |
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the co... | 1. **Identify the number of students in each class:**
- Mrs. Germain's class: 11 students
- Mr. Newton's class: 8 students
- Mrs. Young's class: 9 students
2. **Assumption of no overlap in students:**
Since the problem does not mention any students being in more than one class, we assume that all students ... | <think>
First, note the number of students in each mathematics teacher's AMC 8 class:
- Mrs. Germain has 11 students.
- Mr. Newton has 8 students.
- Mrs. Young has 9 students.
Next, add these numbers together to find the total number of mathematics students taking the contest:
11 + 8 + 9 = 28.
Therefore, there are 28... | 13 |
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students at Euclid Middle School are taking the co... | 1. **Identify the number of students in each class:**
- Mrs. Germain's class: 11 students
- Mr. Newton's class: 8 students
- Mrs. Young's class: 9 students
2. **Assumption of no overlap in students:**
Since the problem does not mention any students being in more than one class, we assume that all students ... | <think>
To determine the total number of mathematics students at Euclid Middle School taking the AMC 8 contest, I'll add the number of students from each teacher's class.
There are 11 students in Mrs. Germain's class, 8 in Mr. Newton's class, and 9 in Mrs. Young's class.
Adding these together: 11 + 8 + 9 equals 28.
... | 14 |
The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3}$ | 1. **Apply the operation $\otimes$ to the innermost parentheses first:**
\[
(1 \otimes 2) \otimes 3 = \left(\frac{1^2}{2}\right) \otimes 3 = \frac{1}{2} \otimes 3
\]
\[
1 \otimes (2 \otimes 3) = 1 \otimes \left(\frac{2^2}{3}\right) = 1 \otimes \frac{4}{3}
\]
2. **Calculate each part using the definit... | <think>
Alright, so I need to solve this problem where they define an operation called "otimes," which is given by \( a \otimes b = \frac{a^2}{b} \). The question asks me to compute \( [(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)] \) and the options are given from A to E.
Okay, let's break this down step by st... | 1 |
The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3}$ | 1. **Apply the operation $\otimes$ to the innermost parentheses first:**
\[
(1 \otimes 2) \otimes 3 = \left(\frac{1^2}{2}\right) \otimes 3 = \frac{1}{2} \otimes 3
\]
\[
1 \otimes (2 \otimes 3) = 1 \otimes \left(\frac{2^2}{3}\right) = 1 \otimes \frac{4}{3}
\]
2. **Calculate each part using the definit... | <think>
Alright, so I've got this problem here with a special operation defined as \( a \otimes b = \frac{a^2}{b} \). It asks me to find \( [(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)] \). Hmm, operation inside another operation, seems like I need to compute each part step by step.
First, I need to understand... | 0 |
The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b =\frac{a^{2}}{b}$. Determine $[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\frac{2}{3}\qquad\text{(B)}\ -\frac{1}{4}\qquad\text{(C)}\ 0\qquad\text{(D)}\ \frac{1}{4}\qquad\text{(E)}\ \frac{2}{3}$ | 1. **Apply the operation $\otimes$ to the innermost parentheses first:**
\[
(1 \otimes 2) \otimes 3 = \left(\frac{1^2}{2}\right) \otimes 3 = \frac{1}{2} \otimes 3
\]
\[
1 \otimes (2 \otimes 3) = 1 \otimes \left(\frac{2^2}{3}\right) = 1 \otimes \frac{4}{3}
\]
2. **Calculate each part using the definit... | <think>
Okay, so I have this problem here about an operation called "otimes," which is defined for all nonzero numbers as \( a \otimes b = \frac{a^2}{b} \). The question is asking me to find the value of \( [(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)] \). The answer choices are A through E: -2/3, -1/4, 0, 1/4,... | 2 |
If $991+993+995+997+999=5000-N$, then $N=$
$\text{(A)}\ 5 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 25$ | 1. **Identify the problem and express each term in a form that reveals a pattern:**
\[
991+993+995+997+999=5000-N
\]
We can rewrite each term as $1000$ minus a small number:
\[
(1000-9) + (1000-7) + (1000-5) + (1000-3) + (1000-1)
\]
2. **Simplify the expression by factoring out $1000$ and summing ... | <think>
Okay, so I have this problem here: 991 + 993 + 995 + 997 + 999 equals 5000 minus N, and I need to find N. The options are A through E, with numbers 5, 10, 15, 20, and 25. Alright, let me think about how to approach this.
First, adding these numbers directly might be a bit tedious since they are all close to 10... | 15 |
If $991+993+995+997+999=5000-N$, then $N=$
$\text{(A)}\ 5 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 25$ | 1. **Identify the problem and express each term in a form that reveals a pattern:**
\[
991+993+995+997+999=5000-N
\]
We can rewrite each term as $1000$ minus a small number:
\[
(1000-9) + (1000-7) + (1000-5) + (1000-3) + (1000-1)
\]
2. **Simplify the expression by factoring out $1000$ and summing ... | <think>
Alright, let me try to solve this problem. It says that 991 + 993 + 995 + 997 + 999 equals 5000 minus N, and we need to find N from the given options. Hmm, okay. So, first, I need to calculate the sum of those five numbers and then figure out how much more it is than 5000 minus N.
Let me jot down the numbers: ... | 17 |
The total in-store price for an appliance is $\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\textdollar 29.98$ and a one-time shipping and handling charge of $\textdollar 9.98$. How many cents are saved by buying the appliance from the television advertiser?
$\mathr... | 1. **Calculate the total cost of buying the appliance from the television advertiser:**
The advertisement mentions three payments of $\textdollar 29.98$ each and an additional one-time shipping and handling charge of $\textdollar 9.98$. First, calculate the total of the three payments:
\[
3 \times 29.98 = 89.9... | <think>
Okay, so I need to figure out how much money is saved by buying the appliance through the television advertiser instead of in-store. The in-store price is $99.99, and the TV ad is offering three payments of $29.98 each plus a one-time shipping and handling fee of $9.98. Hmm, sounds like I need to calculate the ... | 18 |
Given: $x > 0, y > 0, x > y$ and $z\ne 0$. The inequality which is not always correct is:
$\textbf{(A)}\ x + z > y + z \qquad\textbf{(B)}\ x - z > y - z \qquad\textbf{(C)}\ xz > yz$
$\textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$ | We are given that $x > y$ and both $x$ and $y$ are positive. We need to determine which of the given inequalities is not always correct.
1. **Option (A): $x + z > y + z$**
- Subtracting $z$ from both sides of the inequality:
\[
x + z - z > y + z - z \implies x > y
\]
- Since $x > y$ is given, this... | <think>
Alright, so I need to figure out which inequality isn't always correct given the conditions: \( x > 0 \), \( y > 0 \), \( x > y \), and \( z \neq 0 \). The options are labeled from A to E, and each presents a different inequality involving \( x \), \( y \), \( z \), and some operations. Hmm, let's go through th... | 52 |
Given: $x > 0, y > 0, x > y$ and $z\ne 0$. The inequality which is not always correct is:
$\textbf{(A)}\ x + z > y + z \qquad\textbf{(B)}\ x - z > y - z \qquad\textbf{(C)}\ xz > yz$
$\textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$ | We are given that $x > y$ and both $x$ and $y$ are positive. We need to determine which of the given inequalities is not always correct.
1. **Option (A): $x + z > y + z$**
- Subtracting $z$ from both sides of the inequality:
\[
x + z - z > y + z - z \implies x > y
\]
- Since $x > y$ is given, this... | <think>
Okay, so I need to figure out which inequality isn't always correct given the conditions. Let's see, we have x > 0, y > 0, x > y, and z ≠ 0. The options are A to E, each presenting an inequality derived from x, y, and z.
Let me go through each option one by one.
Starting with option A: x + z > y + z. Hmm, sin... | 53 |
If $991+993+995+997+999=5000-N$, then $N=$
$\text{(A)}\ 5 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 25$ | 1. **Identify the problem and express each term in a form that reveals a pattern:**
\[
991+993+995+997+999=5000-N
\]
We can rewrite each term as $1000$ minus a small number:
\[
(1000-9) + (1000-7) + (1000-5) + (1000-3) + (1000-1)
\]
2. **Simplify the expression by factoring out $1000$ and summing ... | <think>
Alright, so I came across this math problem, and at first glance, it seems straightforward, but I want to make sure I approach it methodically. Here's the problem:
If \(991 + 993 + 995 + 997 + 999 = 5000 - N\), then what is \(N\)? The options are (A) 5, (B) 10, (C) 15, (D) 20, and (E) 25.
Alright, so let me d... | 16 |
Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$
$\text{(A) } \frac{2-\sqrt{2}}{2}\quad \text{(B) } \frac{1}{2}\quad \text{(C) } 2-\sqrt{2}\quad \text{(D) } \frac{\sqrt{2}}{2}\quad \text{(E) } \frac{2+\sqrt{2}}{2}$ | 1. **Define the function and apply it to $\sqrt{2}$:**
Given the function $f(x) = ax^2 - \sqrt{2}$, we first find $f(\sqrt{2})$:
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}.
\]
2. **Apply the function $f$ to the result of $f(\sqrt{2})$:**
Next, we need to find $f(f(\sqrt{2}))$:
\[
f(... | <think>
Okay, so I need to find the value of 'a' in the function f(x) = a x² - √2. The condition given is that when we apply the function f twice to √2, we get -√2. That is, f(f(√2)) = -√2. Hmm, let me step through this step by step.
First, let me write down the function: f(x) = a x² - √2. So, it's a quadratic functio... | 11 |
Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$
$\text{(A) } \frac{2-\sqrt{2}}{2}\quad \text{(B) } \frac{1}{2}\quad \text{(C) } 2-\sqrt{2}\quad \text{(D) } \frac{\sqrt{2}}{2}\quad \text{(E) } \frac{2+\sqrt{2}}{2}$ | 1. **Define the function and apply it to $\sqrt{2}$:**
Given the function $f(x) = ax^2 - \sqrt{2}$, we first find $f(\sqrt{2})$:
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}.
\]
2. **Apply the function $f$ to the result of $f(\sqrt{2})$:**
Next, we need to find $f(f(\sqrt{2}))$:
\[
f(... | <think>
Alright, so I have this problem here where I need to find the value of 'a' in the function f(x) = ax² - √2. The condition given is that when I apply the function f twice on √2, I get -√2. In other words, f(f(√2)) = -√2. Hmm, okay, let's break this down step by step. I hope I'm not too rusty on function composit... | 9 |
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?
[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill(... | 1. **Identify the dimensions of each pane**: Given that the ratio of the height to the width of each pane is $5:2$, let the height of each pane be $5x$ inches and the width be $2x$ inches.
2. **Calculate the total dimensions of the window**: The window is constructed with $8$ panes arranged in $2$ rows and $4$ columns... | <think>
Alright, let me try to figure out this window problem. So Doug made a square window with 8 equal-sized panes of glass. Each pane has a height to width ratio of 5:2. There are borders around and between the panes, each 2 inches wide. I need to find the side length of the square window in inches.
First, since th... | 4 |
Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$
$\text{(A) } \frac{2-\sqrt{2}}{2}\quad \text{(B) } \frac{1}{2}\quad \text{(C) } 2-\sqrt{2}\quad \text{(D) } \frac{\sqrt{2}}{2}\quad \text{(E) } \frac{2+\sqrt{2}}{2}$ | 1. **Define the function and apply it to $\sqrt{2}$:**
Given the function $f(x) = ax^2 - \sqrt{2}$, we first find $f(\sqrt{2})$:
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}.
\]
2. **Apply the function $f$ to the result of $f(\sqrt{2})$:**
Next, we need to find $f(f(\sqrt{2}))$:
\[
f(... | <think>
Okay, let's tackle this problem step by step. So, we have a function \( f(x) = ax^2 - \sqrt{2} \), where \( a \) is a positive constant. The problem states that \( f(f(\sqrt{2})) = -\sqrt{2} \), and we need to find the value of \( a \) from the given options.
First, I'll try to understand what the composition ... | 10 |
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 504 \qquad \textbf{(D)}\ 1008 \qquad \textbf{(E)}\ 2015$ | 1. **Identify the formula for the harmonic mean of two numbers, $a$ and $b$:**
\[
\text{Harmonic Mean} = \frac{2ab}{a+b}
\]
Here, $a = 1$ and $b = 2016$.
2. **Substitute the values into the formula:**
\[
\text{Harmonic Mean} = \frac{2 \times 1 \times 2016}{1 + 2016} = \frac{4032}{2017}
\]
3. **Si... | <think>
Okay, so I need to find the harmonic mean of 1 and 2016, and then figure out which integer it's closest to. The options given are A) 2, B) 45, C) 504, D) 1008, E) 2015. Hmm, harmonic mean... I remember it's different from the arithmetic mean. Let me think.
The problem says the harmonic mean of two numbers can ... | 58 |
The graph of the line $y=mx+b$ is shown. Which of the following is true?
[2004 AMC 12A Problem 5.png](https://artofproblemsolving.com/wiki/index.php/File:2004_AMC_12A_Problem_5.png)
$\mathrm {(A)} mb<-1 \qquad \mathrm {(B)} -1<mb<0 \qquad \mathrm {(C)} mb=0 \qquad \mathrm {(D)}$ $0<mb<1 \qquad \mathrm {(E)} mb>1$ | 1. **Identify the slope and y-intercept from the graph**:
From the graph, we observe that the slope $m$ of the line is negative and appears to be $-\frac{1}{2}$. The y-intercept $b$ is positive and appears to be $\frac{4}{5}$.
2. **Calculate the product $mb$**:
\[
m = -\frac{1}{2}, \quad b = \frac{4}{5}
\... | <think>
Okay, so I've come across this AMC problem, and it involves a graph of a line with equation y = mx + b. The question is asking which of the given options is true regarding the product mb. The options range from mb being less than -1, between -1 and 0, equal to 0, between 0 and 1, or greater than 1.
Hmm, since... | 55 |
What time was it $2011$ minutes after midnight on January 1, 2011?
$\textbf{(A)}\ \text{January 1 at 9:31PM}$
$\textbf{(B)}\ \text{January 1 at 11:51PM}$
$\textbf{(C)}\ \text{January 2 at 3:11AM}$
$\textbf{(D)}\ \text{January 2 at 9:31AM}$
$\textbf{(E)}\ \text{January 2 at 6:01PM}$ | 1. **Convert minutes to hours and minutes**:
We start by converting the total minutes (2011) into hours and minutes. Since there are 60 minutes in an hour, we perform the division:
\[
2011 \div 60 = 33 \text{ remainder } 31
\]
This tells us that 2011 minutes is equivalent to 33 hours and 31 minutes.
2.... | <think>
First, I'll convert the total minutes into hours and minutes. There are 60 minutes in an hour, so I'll divide 2011 by 60.
2011 divided by 60 is 33 with a remainder of 31. This means there are 33 hours and 31 minutes.
Next, I'll add these hours and minutes to the starting time of midnight on January 1, 2011.
... | 161 |
Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?
[asy] fill((0,0)--(2,0)--(2,26)--(0,26)--cycle,gray); fill(... | 1. **Identify the dimensions of each pane**: Given that the ratio of the height to the width of each pane is $5:2$, let the height of each pane be $5x$ inches and the width be $2x$ inches.
2. **Calculate the total dimensions of the window**: The window is constructed with $8$ panes arranged in $2$ rows and $4$ columns... | <think>
Okay, so Doug is constructing a square window with 8 equal-sized panes of glass. Each pane has a height to width ratio of 5:2, and the borders around and between the panes are 2 inches wide. I need to find the side length of the square window in inches.
Let me try to visualize the Asymptote figure they provide... | 3 |
Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?
$\textbf{(A)}\ \text{For all x}, x^2 < 0\qquad \textbf{(B)}\ \text{For all x}, x^2 \le 0\qquad \textbf{(C)}\ \text{For no x}, x^2>0\qquad \\ \textbf{(D)}\ \text{For some x}, x^2>0\qquad \textbf{(E)}\ \text{For some x}, x^2 \... | 1. **Identify the original statement and its structure**: The original statement is "For all $x$ of a certain set, $x^2 > 0$". This is a universal quantification statement, which asserts that a certain property (in this case, $x^2 > 0$) holds for every element $x$ in a specified set.
2. **Understand the negation of a ... | <think>
Okay, so I need to find the negation of the statement: "For all x of a certain set, x squared is greater than zero." Hmm, negation... Let me think.
First, I remember that the negation of a universal statement ("for all") is an existential statement ("there exists") with the negation of the inner statement. So... | 106 |
The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 504 \qquad \textbf{(D)}\ 1008 \qquad \textbf{(E)}\ 2015$ | 1. **Identify the formula for the harmonic mean of two numbers, $a$ and $b$:**
\[
\text{Harmonic Mean} = \frac{2ab}{a+b}
\]
Here, $a = 1$ and $b = 2016$.
2. **Substitute the values into the formula:**
\[
\text{Harmonic Mean} = \frac{2 \times 1 \times 2016}{1 + 2016} = \frac{4032}{2017}
\]
3. **Si... | <think>
Okay, so I need to find the harmonic mean of 1 and 2016 and figure out which integer it's closest to. The options are 2, 45, 504, 1008, and 2015. Hmm, harmonic mean—let me recall. I remember something about it involving reciprocals or something...
Ah, yes! I think the harmonic mean of two numbers is calculate... | 57 |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 14