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amc_aime | For a positive integer $n\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available; two from each point. ... | 1. **Labeling and Initial Setup**:
We label the points in clockwise order as $1, 2, 3, \ldots, n$, where point $A$ is the same as point $1$. We start and end at point $1$, and we must cross over it, either by visiting it again, or else by making the move from point $n$ to point $2$.
2. **Interpretation as Tiling**... | [
{
"content": "For a positive integer $n\\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available... |
amc_aime | You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points
$(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$. If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$,
which of the following is necessarily the slope of the line which best fits the data?
"Best fits" means that the su... | To find the slope of the line that best fits the data points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, we can use the formula for the slope ($m$) of the regression line for a simple linear regression model, which is given by:
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
where $\bar{x... | [
{
"content": "You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points \n$(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$. If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$, \nwhich of the following is necessarily the slope of the line which best fits the data? \n\"Be... |
amc_aime | A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
$\textbf{(A) }\frac{1}{6} \qquad \textbf{(B) }\frac{1}{5} \qquad \textbf{(C) }\frac{1}{4} \qquad \textbf{(D) }\frac{1}{3} \qquad \textbf{(E) }\frac{1}{2}$ |
To find the fraction of the grid covered by the triangle with vertices $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$, we first need to calculate the area of the triangle and then compare it to the area of the grid.
#### Step 1: Calculate the area of the triangle using the Shoelace Theorem
The Shoelace Theorem provides a formul... | [
{
"content": "A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\\times5$ grid. What fraction of the grid is covered by the triangle?\n$\\textbf{(A) }\\frac{1}{6} \\qquad \\textbf{(B) }\\frac{1}{5} \\qquad \\textbf{(C) }\\frac{1}{4} \\qquad \\textbf{(D) }\\frac{1}{3} \\qquad \\te... |
amc_aime | A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is
$\textbf{(A)}\ 36\pi-34 \qquad \textbf{(B)}\ 30\pi - 15 \qquad \textbf{(C)}\ 36\pi - 33 \qquad$
$\textbf{(D)}\ 35\pi ... | 1. **Identify the dimensions of the plot and path**: The circular grass plot has a diameter of 12 feet, so its radius \( r \) is \( \frac{12}{2} = 6 \) feet. The gravel path is 3 feet wide.
2. **Calculate the area of the entire circle**: The area \( A \) of a circle is given by the formula \( A = \pi r^2 \). Substitut... | [
{
"content": "A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is \n$\\textbf{(A)}\\ 36\\pi-34 \\qquad \\textbf{(B)}\\ 30\\pi - 15 \\qquad \\textbf{(C)}\\ 36\\pi ... |
amc_aime | The graphs of $y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,$ and $y=\log_x \dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ ... | To solve this problem, we need to analyze the intersections of the given logarithmic functions. We start by rewriting each function in terms of a common logarithmic base or expression for easier comparison.
1. **Rewriting the functions:**
- $y = \log_3 x$
- $y = \log_x 3$
- $y = \log_{\frac{1}{3}} x$
- $y ... | [
{
"content": "The graphs of $y=\\log_3 x, y=\\log_x 3, y=\\log_\\frac{1}{3} x,$ and $y=\\log_x \\dfrac{1}{3}$ are plotted on the same set of axes. How many points in the plane with positive $x$-coordinates lie on two or more of the graphs? \n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qq... |
amc_aime | Amanda Reckonwith draws five circles with radii $1, 2, 3, 4$ and $5$. Then for each circle she plots the point $(C,A)$,
where $C$ is its circumference and $A$ is its area. Which of the
following could be her graph?
$\textbf{(A)}$
$\textbf{(B)}$
$\textbf{(C)}$
$\textbf{(D)}$
$\textbf{(E)}$ | To solve this problem, we need to calculate the circumference and area for each circle with radii $1, 2, 3, 4,$ and $5$, and then analyze how these values relate to each other graphically.
1. **Calculate the Circumference and Area:**
- The formula for the circumference ($C$) of a circle is $C = 2\pi r$.
- The fo... | [
{
"content": "Amanda Reckonwith draws five circles with radii $1, 2, 3, 4$ and $5$. Then for each circle she plots the point $(C,A)$,\nwhere $C$ is its circumference and $A$ is its area. Which of the\nfollowing could be her graph?\n$\\textbf{(A)}$\n\n$\\textbf{(B)}$\n\n$\\textbf{(C)}$\n\n$\\textbf{(D)}$\n\n$\\t... |
amc_aime | The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?
$\textbf{(A... | 1. **Identify the larger triangle and its dimensions**: The problem describes a larger triangle with vertices at points A, B, and D. The base of this triangle (AD) is along Main Street and measures 6 miles. The height of this triangle (from point B to line AD) is along Brown Road and measures 3 miles.
2. **Calculate t... | [
{
"content": "The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of ... |
amc_aime | Jorge's teacher asks him to plot all the ordered pairs $(w. l)$ of positive integers for which $w$ is the width and $l$ is the length of a rectangle with area 12. What should his graph look like?
$\textbf{(A)}$
$\textbf{(B)}$
$\textbf{(C)}$
$\textbf{(D)}$
$\textbf{(E)}$ | To solve this problem, we need to find all the ordered pairs $(w, l)$ of positive integers where $w$ is the width and $l$ is the length of a rectangle, and the area of the rectangle is 12 square units. The relationship between the width and the length can be expressed by the equation:
\[ w \cdot l = 12 \]
1. **Identif... | [
{
"content": "Jorge's teacher asks him to plot all the ordered pairs $(w. l)$ of positive integers for which $w$ is the width and $l$ is the length of a rectangle with area 12. What should his graph look like?\n$\\textbf{(A)}$ \n$\\textbf{(B)}$ \n$\\textbf{(C)}$ \n$\\textbf{(D)}$ \n$\\textbf{(E)}$",
"role":... |
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