problem string | response string | correct int64 | pass@1 int64 | mean@1 float64 | pass@2 int64 | mean@2 float64 | pass@4 int64 | mean@4 float64 | pass@8 int64 | mean@8 float64 | pass@16 int64 | mean@16 float64 | pass@32 int64 | mean@32 float64 | pass@64 int64 | mean@64 float64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
The operation $\otimes$ is defined for all nonzero numbers by $a \otimes b = \frac{a^{2}}{b}$. Determine $[(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]$. | To solve the problem, we need to evaluate the expression \([(1 \otimes 2) \otimes 3] - [1 \otimes (2 \otimes 3)]\) using the given operation \(a \otimes b = \frac{a^2}{b}\).
First, we need to evaluate \(1 \otimes 2\):
\[
1 \otimes 2 = \frac{1^2}{2} = \frac{1}{2}
\]
Next, we use the result to evaluate \((1 \otimes 2) \... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Doug constructs a square window using $8$ equal-size panes of glass. 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? | Let's analyze the problem step by step. We need to construct a square window using 8 equal-size panes of glass, and we are given that the ratio of the height to width of each pane is 5:2. Additionally, there is a 2-inch border around and between the panes.
First, let's determine the arrangement of the panes. Since the... | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.25 | 1 | 0.25 | 1 | 0.21875 | 1 | 0.25 |
Let $P(x)$ be a polynomial of degree $3n$ such that
\begin{align*} P(0) = P(3) = \dots = P(3n) &= 2, \\ P(1) = P(4) = \dots = P(3n+1-2) &= 1, \\ P(2) = P(5) = \dots = P(3n+2-2) &= 0. \end{align*}
Also, $P(3n+1) = 730$. Determine $n$. | To determine the value of \( n \), let's denote \( P(x) \) as a polynomial of degree \( 3n \) satisfying the given conditions.
Given:
\[
P(0) = P(3) = \cdots = P(3n) = 2,
\]
\[
P(1) = P(4) = \cdots = P(3n+1-2) = 1,
\]
\[
P(2) = P(5) = \cdots = P(3n+2-2) = 0,
\]
and
\[
P(3n+1) = 730.
\]
Firstly, to ensure a better un... | 0 | 0 | 0 | 0 | 0 | 1 | 0.25 | 1 | 0.125 | 1 | 0.1875 | 1 | 0.125 | 1 | 0.078125 |
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 $a=$ | To find the value of \(a\) such that \(f(f(\sqrt{2})) = -\sqrt{2}\) for the function \(f(x) = ax^2 - \sqrt{2}\), we will proceed step by step.
First, we need to evaluate \(f(\sqrt{2})\):
\[
f(\sqrt{2}) = a(\sqrt{2})^2 - \sqrt{2} = 2a - \sqrt{2}
\]
Next, we need to evaluate \(f(f(\sqrt{2}))\). We already know that \(f... | 1 | 1 | 1 | 1 | 0.5 | 1 | 0.75 | 1 | 0.75 | 1 | 0.75 | 1 | 0.6875 | 1 | 0.6875 |
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... | To determine the total number of mathematics students at Euclid Middle School taking the AMC 8 contest, we need to add the number of students from each teacher's class who are taking the contest. Here are the steps:
1. Identify the number of students in Mrs. Germain's class taking the contest: \(11\).
2. Identify the ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
If $991+993+995+997+999=5000-N$, then $N=$ | To solve the equation \(991 + 993 + 995 + 997 + 999 = 5000 - N\), we first need to find the sum of the numbers on the left-hand side.
The numbers \(991, 993, 995, 997, 999\) form an arithmetic sequence with the first term \(a = 991\) and the common difference \(d = 2\). The sum \(S_n\) of the first \(n\) terms of an a... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.96875 |
The total in-store price for an appliance is $99.99$. A television commercial advertises the same product for three easy payments of $29.98$ and a one-time shipping and handling charge of $9.98$. How many cents are saved by buying the appliance from the television advertiser? | To determine how much is saved by buying the appliance from the televisionến, we need to compare the total cost of the appliance from both the store and the нагрузкpeople.
First, let's calculate the total cost of the appliance from the store:
\[ \text{Total cost from store} = \$99.99 \]
Next, let's calculate the tota... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.9375 | 1 | 0.9375 | 1 | 0.890625 |
Points $A,B,C,D,E$ and $F$ lie, in that order, on $\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\overline{GD}$, and point $J$ lies on $\overline{GF}$. The line segments $\overline{HC}, \overline{JE},$ and $\overline{AG}$ are parallel. Find $HC/JE$. | To solve the problem, we start by understanding the geometric configuration and using the properties of similar triangles. The key observation is that since \(\overline{HC}\), \(\overline{opez}\), and \(\overline{AG}\) are parallel, the triangles formed by these lines are similar.
Given that points \(A, B, C, D, E, F\... | 0 | 0 | 0 | 1 | 0.5 | 1 | 0.25 | 1 | 0.125 | 1 | 0.0625 | 1 | 0.0625 | 1 | 0.0625 |
During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single? | To determine what percent of Judy's hits were singles, we first need to find out how many of her hits were singles. We know that Judy had a total of 35 hits, and among these hits were 1 home run, 1 triple, and 5 doubles. The rest of her hits were singles.
First, we calculate the total number of hits that were not sing... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.875 | 1 | 0.875 | 1 | 0.8125 | 1 | 0.859375 |
The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$? | To find the equation of the graph \( G' \) obtained by rotating the graph \( G \) of \( y = \log_{10} x \) by \( 90^\circ \) counter-clockwise about the origin, we need to follow these steps:
1. **Understand the rotation transformation**: Rotating a point \((x, y)\) by \( 90^\circ \) counter-clockwise about the origin... | 0 | 0 | 0 | 0 | 0 | 1 | 0.25 | 1 | 0.625 | 1 | 0.5 | 1 | 0.5625 | 1 | 0.46875 |
A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased? | Let's denote the set of consecutive positive integers starting from $1$ as $\{1, 2, 3, \ldots, n\}$. The sum of these integers can be represented by the formula for the sum of the first $n$ natural numbers: $$S = \frac{n(n+1)}{2}$$
Given that after erasing one number, the average of the remaining numbers is $35\frac{7... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.0625 | 1 | 0.03125 | 1 | 0.015625 |
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
[asy]
unitsize(2mm); defaultpen(... | To determine the fraction of the yard occupied by the flower beds, we first need to find the total area of the yard and the area of the two flower beds combined.
Let the dimensions of each isosceles right triangle be represented by \(x\). The area of one isosceles right triangle is given by:
\[
\text{Area of one trian... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.875 | 1 | 0.90625 | 1 | 0.71875 |
What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\cos 40^\circ,\sin 40^\circ)$, $(\cos 60^\circ,\sin 60^\circ)$, and $(\cos t^\circ,\sin t^\circ)$ is isosceles? | To determine the sum of all possible values of \( t \) between \( 0^\circ \) and \( 360^\circ \) such that the triangle with vertices \((\cos 40^\circ, \sin 40^\circ)\), \((\cos 60^\circ, \sin 60^\circ)\), and \((\cos t^\circ, \sin t^\circ)\) is isosceles, we need to analyze the conditions under which the triangle can ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has len... | First, note that since \( AB \), \( BC \), and \( CD \) are diameters of circles \( O \), \( N \), and \( P \) respectively, each of these diameters is equal to twice the radius, which is \( 15 \times 2 = 30 \). Consequently, the lengths \( AB \), \( BC \), and \( CD \) are all \( 30 \).
Since \( B \) and \( C \) lie ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: | To determine the value of \(x\) in the arithmetic progression where the first three terms are \(x - 1\), \(x + 1\), and \(2x + 3\), we need to use the property of arithmetic progressions that the difference between consecutive terms is constant.
Let's denote the common difference of the arithmetic progression by \(d\)... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Alicia had two containers. The first was $\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the sec... | Let the volume of the first container be \( V_1 \) and the volume of the second container be \( V_2 \). According to the problem, the first container is \(\frac{5}{6}\) full of water, so the amount of water in the first container is \(\frac{5}{6} V_1\).
When Alicia pours all the water from the first container into the... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.9375 | 1 | 0.90625 |
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, $B$, and $C$ are $12$, $9$, and $10$ meter... | To solve for the height of the pillar at vertex \(E\) of the regular hexagon \(ABCDEF\), we need to understand the symmetry and relationships among the heights of the pillars. Since the hexagon is regular, it can be divided into equilateral triangles, and the vertices can be expressed in terms of their coordinates with... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are): | To determine the value(s) of \( k \) such that the points \((2, -3)\), \((4, 3)\), and \((5, k/2)\) lie on the same straight line, we need to ensure that the slope between any two pairs of these points is the same.
First, let's calculate the slope between the points \((2, -3)\) and \((4, 3)\). The formula for the slop... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$ | Let's denote the price of gasoline at the beginning of January as \( P \).
**Step 1: Calculate the price at the end of January.**
The price rose by \( 20\% \) during January.
\[
\text{Price at the end of January} = P + 0.20P = 1.20P
\]
**Step 2: Calculate the price at the end of February.**
The price fell by \( 20\% ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.96875 | 1 | 0.96875 |
Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | To find the area of quadrilateral \(ABCD\), we start by noting that \(\angle ABC = 90^\circ\) and \(\angle ACD = 90^\circ\). This means that triangles \(ABC\) and \(ACD\) are right triangles. Given \(AC = 20\) and \(CD = 30\), we need to find the coordinates of the points and the length of the other sides to calculate ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y=3x$ has equation $y=kx.$ What is $k?$ | To find the equation of the angle bisector of the acute angle formed at the origin by the lines \(y = x\) and \(y = 3x\), we can use the angle bisector theorem in the context of lines.
The general formula for the angle bisector of two lines \(y = m_1 x\) and \(y = m_2 x\) is given by:
\[
y = \frac{m_1 + m_2 \pm \sqrt{... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$? | To determine the smallest number of points in the set \(S\) that is symmetric about the origin, both coordinate axes, and the line \(y=x\), we need to consider the implications of each symmetry condition.
1. **Symmetry about the origin**: If a point \((a, b)\) is in \(S\), then the point \((-a, -b)\) is also in \(S\).... | 1 | 0 | 0 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.4375 | 1 | 0.46875 |
Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$? | To determine the average of the five consecutive integers starting with \(a\), we first express these integers as \(a, a+1, a+2, a+3, a+4\). The average \(b\) of these five integers is given by the formula for the average of an arithmetic sequence:
\[
b = \frac{a + (a+1) + (a+2) + (a+3) + (a+4)}{5} = \frac{5a + 10}{5}... | 0 | 0 | 0 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.625 | 1 | 0.75 | 1 | 0.828125 |
At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values... | To solve the problem, let's define the variables and set up the equations based on the given information.
Let:
- \( M \) be the temperature in Minneapolis at noon.
- \( S \) be the temperature in St. Louis at noon.
According to the problem:
1. At noon,streams is \( N \) degrees warmer than St. Louis.
\[
M = S +... | 1 | 0 | 0 | 1 | 0.5 | 1 | 0.75 | 1 | 0.625 | 1 | 0.6875 | 1 | 0.78125 | 1 | 0.828125 |
Consider all 1000-element subsets of the set $\{1, 2, 3, \dots , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | To find the arithmetic mean of all the least elements of the 1000-element subsets of the set \(\{1, 2, 3, \dots, 2015\}\), we need to determine the frequency of each element as the least element in these subsets.
Consider an element \( k \) in the set \(\{1, 2, 3, \dots, 2015\}\). For \( k \) to be the least element o... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.015625 |
A man on his way to dinner shortly after $6:00$ p.m. observes that the hands of his watch form an angle of $110^{\circ}$. Returning before $7:00$ p.m. he notices that again the hands of his watch form an angle of $110^{\circ}$. The number of minutes that he has been away is: | To solve the problem, we need to determine the times when the angle between the hour and minute hands of a clock is \(110^\circ\) between 6:00 p.m. and 7:00 p.m. and then find the difference between these two times.
First, let's express the angle between the hour and minute hands at any given time \(h\) hours and \(m\... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.625 | 1 | 0.75 | 1 | 0.78125 | 1 | 0.75 |
A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube? | To determine the smallest possible sum of all the values visible on the faces of a 3x3x3 cube made of 27 normal dice, we need to consider the arrangement of the dice and the properties of a standard die. Each die has opposite faces that sum to 7, so the possible values on each die are 1, 2, 3, 4, 5, and 6.
First, let'... | 1 | 1 | 1 | 1 | 0.5 | 1 | 0.25 | 1 | 0.125 | 1 | 0.0625 | 1 | 0.09375 | 1 | 0.046875 |
In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
$\begin{array}{cccc} & A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$ | To solve the given multiplication problem where \( A \), \( B \), \( C \), and \( D \) are different digits, we need to analyze the multiplication step by step.
The problem is:
\[
\begin{array}{cccc}
& A & B & A \\
\times & & C & D \\
\hline
C & D & C & D \\
\end{array}
\]
First, let's interpret the multiplication:... | 1 | 1 | 1 | 1 | 1 | 1 | 0.75 | 1 | 0.625 | 1 | 0.625 | 1 | 0.59375 | 1 | 0.546875 |
Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from t... | To solve the problem, we need to determine the speeds of Andrea and Lauren, and then calculate the time it takes for them to meet after Andrea stops biking due to a flat tire.
First, let's denote the speed of Lauren as \( v \) kilometers per minute. Since Andrea travels three times as fast as Lauren, Andrea's speed is... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.96875 | 1 | 0.96875 |
The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? | Let the two natural numbers be \( x \) and \( y \). According to the problem, the sum of these two numbers is:
\[ x + y = 17402 \]
Also, one of the numbers is divisible by 10, so let's assume \( x \) is the number divisible by 10. This means that the units digit of \( x \) is 0. When the units digit of \( x \) is erase... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.984375 |
What is the value of $2^{0^{1^9}} + (2^0)^{1^9}$? | To solve the expression \(2^{0^{1^9}} + (2^0)^{1^9}\), we will evaluate each part step by step.
First, let's evaluate the exponentiation inside the first term \(2^{0^{1^9}}\):
\[1^9 = 1.\]
So, the expression simplifies to:
\[0^1 = 0.\]
Now, we substitute back into the first term:
\[2^{0^{1^9}} = 2^0 = 1.\]
Next, let'... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
If $a$ and $b$ are digits for which
$\begin{array}{ccc}& 2 & a\ \times & b & 3\ \hline & 6 & 9\ 9 & 2 & \ \hline 9 & 8 & 9\end{array}$
then $a+b =$ | To solve the problem, we need to determine the values of the digits \(a\) and \(b\) such that the multiplication of the two-digit number \(2a\) by the two-digit number \(b3\) results in the three-digit number \(989\).
Let's represent the numbers in their expanded forms:
\[2a = 20 + a\]
\[b3 = 10b + 3\]
The product of ... | 1 | 1 | 1 | 1 | 1 | 1 | 0.75 | 1 | 0.75 | 1 | 0.5625 | 1 | 0.6875 | 1 | 0.703125 |
If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is: | To determine the fourth term of the sequence \(x, 2x+2, 3x+3, \dots\) given that it is in geometric progression, we need to use the property of geometric sequences that the ratio between consecutive terms is constant. Let's denote the common ratio by \(r\).
The common ratio \(r\) can be expressed as:
\[ r = \frac{2x+2... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
At $2:15$ o'clock, the hour and minute hands of a clock form an angle of: | To determine the angle between the hour and minute hands of a clock at 2:15, we need to calculate the positions of both hands separately and then find the difference between these positions.
### Step 1: Calculate the position of the minute hand
The minute hand moves 360 degrees in 60 minutes. Therefore, the position o... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first term. For example, such a sequence might be... | To solve the problem, let's denote the terms of the sequence as \(a_1, a_2, \ldots, a_n\), where each \(a_i\) is a three-digit number. We can express each term \(a_i\) in the form \(a_i = 100x_i + 10y_i + z_i\), where \(x_i, y_i, z_i\) are the hundreds, tens, and units digits of \(a_i\), respectively. The sequence has ... | 1 | 0 | 0 | 0 | 0 | 1 | 0.25 | 1 | 0.5 | 1 | 0.4375 | 1 | 0.34375 | 1 | 0.390625 |
Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$? | To solve the problem, we need to determine the ratio of the area of square \(EFGH\) to the area of square \(ABCD\). Let's start by setting up the problem with coordinates and using the given information.
1. **Define the coordinates of square \(ABCD\):**
Let the side length of square \(ABCD\) be \(s\). Place the squ... | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.125 | 1 | 0.25 | 1 | 0.28125 | 1 | 0.296875 |
Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns $16$ dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper rout... | To determine how many weeks Chandler needs to save in order to buy the mountain bike, we first need to calculate the total amount of money he already has from his birthday and then figure out how much more he needs to save.
1. Calculate the total amount of money Chandler has from his birthday:
\[
50 + 35 + 15 = ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4? | To find the harmonic mean of the numbers 1, 2, and 4, we need to follow these steps:
1. Calculate the reciprocals of the numbers.
2. Find the average of these reciprocals.
3. Take the reciprocal of this average to get the harmonic mean.
First, let's find the reciprocals of the numbers 1, 2, and 4:
\[ \frac{1}{1} = 1,... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, what number of words would be an appropriate length for her speech? | To determine the appropriate length for Julie's speech in terms of the number of words, we need to calculate the total number of words she would speak given the constraints on the duration and her ideal speech rate.
First, let's convert the duration of the speech from hours to minutes, since the rate is given in words... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the... | Let's start by setting the side length of the equilateral triangle \(ABC\) to be \(s\). The area of an equilateral triangle with side length \(s\) is given by:
\[
\text{Area of } \triangle ABC = \frac{\sqrt{3}}{4} s^2
\]
Next, we need to find the side length of the triangle \(A'B'C'\). Since \(BB' = 3 \cdot AB\), \(B... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.5 | 1 | 0.5 | 1 | 0.5 | 1 | 0.609375 |
If the following instructions are carried out by a computer, what value of \(X\) will be printed because of instruction \(5\)?
1. START \(X\) AT \(3\) AND \(S\) AT \(0\).
2. INCREASE THE VALUE OF \(X\) BY \(2\).
3. INCREASE THE VALUE OF \(S\) BY THE VALUE OF \(X\).
4. IF \(S\) IS AT LEAST \(10000\),
TH... | Let's go through the instructions step by step to determine the value of \(X\) printed because of instruction 5.
1. **start \(X\) at 3 and \(S\) at 0.**
\[
X = 3, \quad S = 0
\]
2. **increase the value of \(X\) by 2.**
\[
X = X + 2 = 3 + 2 = 5
\]
3. **increase the value of \(S\) by the value of \(X... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Letters $A, B, C,$ and $D$ represent four different digits selected from $0, 1, 2, \ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$? | To find the value of \(A + B\) such that \(\frac{A+B}{C+D}\) is an integer and as large as possible, we need to maximize \(A + B\) and minimize \(C + D\) while ensuring that \(A + B\) is divisible by \(C + D\).
First, we choose the largest possible values for \(A\) and \(B\). The largest digits available are 9 and 8. ... | 1 | 1 | 1 | 1 | 1 | 1 | 0.75 | 1 | 0.625 | 1 | 0.6875 | 1 | 0.8125 | 1 | 0.78125 |
A shopper plans to purchase an item that has a listed price greater than $\$100$ and can use any one of the three coupons. Coupon A gives $15\%$ off the listed price, Coupon B gives $\$30$ off the listed price, and Coupon C gives $25\%$ off the amount by which the listed price exceeds
$\$100$.
Let $x$ and $y$ be the s... | To determine the smallest and largest prices \( x \) and \( y \) for which Coupon A saves at least as many dollars as Coupon B or C, we need to compare the savings from each coupon.
Let's denote the listed price of the item by \( P \). The savings from each coupon can be calculated as follows:
- Coupon A: \( 0.15P \)
... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.75 | 1 | 0.875 | 1 | 0.9375 | 1 | 0.890625 |
In $\triangle ABC$, $\angle ABC=45^\circ$. Point $D$ is on $\overline{BC}$ so that $2 \cdot BD=CD$ and $\angle DAB=15^\circ$. Find $\angle ACB.$ | Given \(\triangle ABC\) with \(\angle ABC = 45^\circ\) and point \(D\) on \(\overline{BC}\) such that \(2 \cdot BD = CD\) and \(\angle DAB = 15^\circ\), we need to find \(\angle ACB\).
First, let's set \(\angle ACB = \theta\). Since \(\angle ABC = 45^\circ\), we can express \(\angle BAC\) as \(180^\circ - 45^\circ - \... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.875 | 1 | 0.6875 | 1 | 0.59375 | 1 | 0.6875 |
Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters? | To find the average book width, we need to follow these steps:
1. Calculate the sum of the widths of all the books.
2. Determine the number of books.
3. Divide the sum of the widths by the number of books.
First, let's list the widths of the books again: \(6\), \(\frac{1}{2}\), \(1\), \(2.5\), and \(10\).
Next, we c... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is | To solve the problem, we need to understand the definitions of the greatest integer less than or equal to \( x \) and the least integer greater than or equal to \( x \). Let's denote these as follows:
- The greatest integer less than or equal to \( x \) is denoted by \( \lfloor x \rfloor \).
- The least integer greater... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately: | To determine how far the powderman had run before hearing the blast, we need to consider both the time it took for the sound of the blast to reach him and the distance the powderman traveled during that time. Let's break this problem down step by step.
1. **Convert the speed of sound to yards per second:**
The spee... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person plannin... | To solve the problem, we need to determine the number of students on the student council and then use that information to find the number of ways to select a three-person planning committee.
First, let's determine the number of students on the student council. We know that there are exactly 10 ways to select a two-per... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? | Let's denote Pete's current age by \( P \) and Claire's current age by \( C \). We'll start by setting up the equations based on the given conditions.
First, two years ago, Pete was three times as old as Claire. So, two years ago, Pete's age was \( P - 2 \) and Claire's age was \( C - 2 \). This gives us the equation:... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.9375 | 1 | 0.96875 | 1 | 0.90625 |
A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? | To solve the problem, we start by analyzing the initial square piece of paper and the steps of folding and cutting it.
1. **Initial Square Piece:**
- The side length of the square is 4 inches.
- The perimeter of the square is calculated as:
\[
P_{\text{square}} = 4 \times 4 = 16 \text{ inches}
\]
... | 0 | 1 | 1 | 1 | 1 | 1 | 0.75 | 1 | 0.375 | 1 | 0.25 | 1 | 0.1875 | 1 | 0.21875 |
Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection? | To find the $x$-coordinates of the points where the graphs of $y = ax + 5$ and $y = 3x + b$ intersect the $x$-axis, we need to determine the $x$-intercepts of both lines. The $x$-intercept occurs when $y = 0$.
For the line $y = ax + 5$, setting $y = 0$ gives:
\[0 = ax + 5 \implies ax = -5 \implies x = -\frac{5}{a}.\]
... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.75 | 1 | 0.875 | 1 | 0.90625 | 1 | 0.9375 |
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ... | Given four circles with centers at \(A\), \(B\), \(C\), and \(D\), and points \(P\) and \(Q\) lying on all four circles, we need to find \(AR + BR + CR + DR\) where \(R\) is the midpoint of \(\overline{PQ}\). The radii of the circles are such that the radius of circle \(A\) is \(\frac{5}{8}\) times the radius of circle... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.015625 |
Medians $AD$ and $CE$ of $\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$.
Let the area of $\triangle MNE$ be $k$ times the area of $\triangle ABC$. Then $k$ equals: | To solve the problem, we need to understand the properties of medians and the centroid of a triangle. The centroid \(M\) of a triangle divides each median into a ratio of 2:1, with the longer segment being closer to the vertex.
Given that \(AD\) and \(CE\) are medians of \(\triangle ABC\), the centroid \(M\) is the po... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.125 | 1 | 0.125 | 1 | 0.15625 |
Find the minimum value of $\sqrt{x^2+y^2}$ if $5x+12y=60$. | To find the minimum value of \(\sqrt{x^2 + y^2}\) subject to the constraint \(5x + 12y = 60\), we can use the method of Lagrange multipliers or geometric interpretation. Here, we will use the geometric interpretation.
The problem is to find the minimum distance from the origin \((0,0)\) to the line \(5x + 12y = 60\). ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.9375 | 1 | 0.90625 | 1 | 0.90625 |
On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell? | To determine the number of sedans sold based on the given ratio and the number of sports cars sold, we can follow these steps:
1. Identify the ratio of sports cars to sedans. The problem states that for every 4 sports cars sold, 7 sedans are sold. This can be written as:
\[
\text{Ratio of sports cars to sedans} ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{... | To solve the problem, let's denote the number of faces on the two dice as \(a\) and \(b\) respectively. Each die has distinct integers from 1 to \(a\) and 1 to \(b\). The total number of outcomes when rolling the two dice is \(ab\).
First, let's determine the probability of rolling a sum of 7. The pairs that sum to 7 ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to re... | Let's analyze the problem as the larger tube rolls over the circumference of the smaller tube without slipping. The smaller tube's circumference is given by:
\[
C_{\text{small}} = 2\pi \times 24 = 48\pi
\]
Since the larger tube rolls over the smaller tube, it will have to cover a distance equivalent to its own circumfe... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The number $2.5252525\ldots$ can be written as a fraction.
When reduced to lowest terms the sum of the numerator and denominator of this fraction is: | The number \(2.5252525\ldots\) can be written as a repeating decimal, where the digits "52" repeat indefinitely. Let's denote this repeating decimal by \(x\). So we have:
\[x = 2.5252525\ldots\]
To convert this repeating decimal into a fraction, we first separate the non-repeating part from the repeating part. Let's ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)$ equals | To solve the given expression \(\left(x - \frac{1}{x}\right)\left(y + \frac{1}{y}\right)\) for \(x\) and \(y\) such that \(x = \frac{1}{y}\), we will follow these steps:
1. **Substitute \(y\) in terms of \(x\):**
Given \(x = \frac{1}{y}\), we can express \(y\) as \(y = \frac{1}{x}\).
2. **Simplify the expression i... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: | To determine the values of \( k \) for which the quadratic equation \( 2x^2 - kx + x + 8 = 0 \) has real and equal roots, we need to use the discriminant of the quadratic equation. A quadratic equation \( ax^2 + bx + c = 0 \) has real and equal roots if and only if its discriminant is zero. The discriminant \(\Delta\) ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive? | To determine how many perfect cubes lie between \(2^8 + 1\) and \(2^{18} + 1\), inclusive, we will first find the range of the integers whose cubes fall within this interval.
1. Calculate the lower bound:
\[
2^8 + 1 = 256 + 1 = 257
\]
We need to find the smallest integer \(n\) such that \(n^3 \geq 257\). B... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.625 | 1 | 0.6875 | 1 | 0.71875 | 1 | 0.71875 |
A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \frac{\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle? | To solve the problem, we need to find the equations of the lines that form the equilateral triangle and then calculate the perimeter of the triangle.
1. **Identify the lines and their intersections:**
- The first line is \( x = 1 \).
- The second line is \( y = 1 + \frac{\sqrt{3}}{3} x \).
- The third line pa... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Alicia earns 20 dollars per hour, of which $1.45\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? | To determine how many cents per hour Alicia's wages are used to pay local taxes, we need to follow these steps:
1. Calculate the amount of local taxes Alicia pays per hour.
2. Convert the amount of local taxes from dollars to cents.
First, we calculate the amount of local taxes Alicia pays per hour. She earns $20 per... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
How many positive factors of 36 are also multiples of 4? | To determine how many positive factors of 36 are also multiples of 4, we will follow these steps:
1. **Find the prime factorization of 36.**
\[
36 = 2^2 \times 3^2
\]
2. **Identify the general form of a factor of 36.**
Any factor of 36 can be written in the form \(2^a \times 3^b\) where \(0 \leq a \leq 2\... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:
1. The largest isn't first, but it is in one of the first three places.
2. The smallest isn't last, but it is in one of the last three places.
3. The median isn't first or last.
What is the average of the first and l... | To solve the problem, we need to rearrange the numbers \(-2, 4, 6, 9,\) and \(12\) according to the given rules. Let's break down the rules and find a valid sequence step by step.
1. **The largest isn't first, but it is in one of the first three places.**
- The largest number is \(12\). Therefore, \(12\) cannot be ... | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.125 | 1 | 0.25 | 1 | 0.3125 | 1 | 0.359375 |
The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$? | To solve the problem, we start by understanding the division of the circle and the meaning of the angles \(x\) and \(y\).
1. **Understanding the Circle Division**:
The circumference of the circle is divided into 12 equal arcs. Since a full circle is 360 degrees, each of these arcs measures:
\[
\frac{360^\circ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.0625 | 1 | 0.03125 | 1 | 0.078125 |
Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? | Let's denote the two-digit number Mary thought of as \( x \). According to the problem, she multiplied this number by 3 and added 11, so the expression becomes \( 3x + 11 \). After this, she switched the digits of the result and obtained a number between 71 and 75, inclusive. Let's denote the result after adding 11 as ... | 1 | 0 | 0 | 1 | 0.5 | 1 | 0.25 | 1 | 0.375 | 1 | 0.5625 | 1 | 0.5 | 1 | 0.53125 |
Trapezoid $ABCD$ has $\overline{AB} \parallel \overline{CD}, BC=CD=43$, and $\overline{AD} \perp \overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$... | To solve for the length of \(AD\) in the trapezoid \(ABCD\) with the given conditions, we will use coordinate geometry and properties of similar triangles. Let's place the trapezoid in the coordinate plane for方便.
1. **Assign Coordinates:**
- Place \(B\) at \((0, 0)\).
- Place \(A\) at \((a, 0)\).
- Place \(D\... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
If $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^8$ is divided by $x + \frac{1}{2}$, and if $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x + \frac{1}{2}$, then $r_2$ equals | To find the remainder \( r_2 \) when the quotient \( q_1(x) \) is divided by \( x + \frac{1}{2} \), we first need to determine \( q_1(x) \) and \( r_1 \) when \( x^8 \) is divided by \( x + \frac{1}{2} \).
Using the Remainder Theorem, the remainder \( r_1 \) when \( x^8 \) is divided by \( x + \frac{1}{2} \) is given ... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.125 | 1 | 0.0625 | 1 | 0.03125 | 1 | 0.03125 |
The expression $\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to: | To simplify the expression \(\frac{1^{4y-1}}{5^{-1}+3^{-1}}\), we will follow these steps:
1. Simplify the numerator \(1^{4y-1}\).
2. Simplify the denominator \(5^{-1} + 3^{-1}\).
3. Divide the simplified numerator by the simplified denominator.
**Step 1: Simplify the numerator \(1^{4y-1}\).**
The expression \(1^{4y... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect to harvest? | To determine how many strawberries Carrie can expect to harvest, we need to follow a step-by-step approach:
1. Calculate the area of the garden.
2. Determine the total number of strawberry plants.
3. Calculate the total number of strawberries harvested.
**Step 1: Calculate the area of the garden.**
The garden is a r... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$? | To determine the smallest possible value of \( s \) for a triangle with sides \( 6.5 \), \( 10 \), and \( s \), we need to use the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. Therefo... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of segments $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$ | To determine the ratio \( R \) of the area of the cross-section \( EJCI \) to the area of one of the faces of the cube, we start by placing the cube in a coordinate system. Let the cube have side length 1 and the coordinates of its vertices be as follows:
- \( A = (0,0,0) \)
- \( B = (1,0,0) \)
- \( C = (1,1,0) \)
- \(... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
A square and an equilateral triangle have the same perimeter. Let $A$ be the area of the circle circumscribed about the square and $B$ the area of the circle circumscribed around the triangle. Find $A/B$. | Given that a square and an equilateral triangle have the same perimeter, let's denote the side length of the square by \( s \) and the side length of the equilateral triangle by \( t \). The perimeter of the square is \( 4s \) and the perimeter of the equilateral triangle is \( 3t \). Since these perimeters are equal, ... | 1 | 1 | 1 | 1 | 0.5 | 1 | 0.75 | 1 | 0.625 | 1 | 0.75 | 1 | 0.875 | 1 | 0.921875 |
Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$? | To determine the measure of \(\angle BAD\) in the given quadrilateral \(ABCD\) where \(AB = BC = CD\), \(\angle ABC = 70^\circ\), and \(\angle BCD = 170^\circ\), we can proceed as follows:
First, note that since \(AB = BC\), \(\triangle ABC\) is isosceles with \(\angle BAC = \angle BCA\). Let \(\angle BAC = \angle BCA... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\] | To solve the problem, we first need to express the complex number \(2+i\) in polar form. The magnitude of \(2+i\) is given by:
\[
|2+i| = \sqrt{2^2 + 1^2} = \sqrt{5}.
\]
The argument \(\theta\) is given by:
\[
\theta = \tan^{-1}\left(\frac{1}{2}\right).
\]
Thus, we can write \(2+i\) in polar form as:
\[
2+i = \sqrt{5}\... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An $11 \times 11 \times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? | To determine the greatest number of unit cubes that can be seen from a single point, we need to consider the visible faces of the $11 \times 11 \times 11$ cube when viewed from a corner.
First, let's analyze the structure of the cube:
- The cube has a side length of 11 units.
- Each face of the cube is a $11 \times 11... | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0.125 | 1 | 0.125 | 1 | 0.15625 | 1 | 0.140625 |
If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is: | To determine the possible number of points of intersection when two lines intersect a hyperbola, we need to consider the geometry of the situation and the properties of a hyperbola.
A hyperbola is defined as the set of all points in a plane such that the absolute value of the difference of the distances from two fixed... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds? | Let's define the variables for the problem:
- Let \( x \) be the number of two-legged birds.
- Let \( y \) be the number of four-legged mammals.
From the problem, we know two things:
1. The total number of heads is 200. Since each animal has one head, we can write the equation:
\[
x + y = 200
\]
2. The total ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
If $\frac{1}{x} - \frac{1}{y} = \frac{1}{z}$, then $z$ equals: | We start with the given equation:
\[
\frac{1}{x} - \frac{1}{y} = \frac{1}{z}
\]
First, we find a common denominator for the fractions on the left-hand side. The common denominator for \(x\) and \(y\) is \(xy\). Thus, we rewrite the equation as:
\[
\frac{y - x}{xy} = \frac{1}{z}
\]
Next, we take the reciprocal of bo... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies hal... | To solve the problem, we need to determine the dimensions of the floor and the painted rectangle such that the unpainted border occupies half of the area of the entire floor. Let's denote the dimensions of the painted rectangle as \( (a-2) \) by \( (b-2) \) because the border is 1 foot wide on all sides.
The area of t... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.9375 | 1 | 0.890625 |
Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty and Carlos had was $7$ dollars, between Car... | To solve the problem, let's define the amounts of money that Ashley, Betty, Carlos, Dick, and Elgin had as \(A\), \(B\), \(C\), \(D\), and \(E\) respectively. We are given the following information:
1. \(A + B + C + D + E = 56\)
2. \(|A - B| = 19\)
3. \(|B - C| = 7\)
4. \(|C - D| = 5\)
5. \(|D - E| = 4\)
6. \(|E - A| ... | 1 | 0 | 0 | 0 | 0 | 1 | 0.25 | 1 | 0.25 | 1 | 0.1875 | 1 | 0.15625 | 1 | 0.125 |
A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$? | To solve this problem, we start by identifying all the four-digit palindromic numbers between \(1000\) and \(10000\). A four-digit palindrome can be represented in the form \( \overline{abba} \), where \(a\) and \(b\) are digits.
First, we determine the total number of four-digit palindromes:
The digit \(a\) (first di... | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.875 | 1 | 0.90625 | 1 | 0.828125 |
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers? | The sum of the first \( n \) positive integers is given by the formula:
\[
1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}
\]
and the product of the first \( n \) positive integers is:
\[
n! = 1 \cdot 2 \cdot 3 \cdots n
\]
We need to find the greatest three-digit \( n \) such that \(\frac{n(n+1)}{2}\) is not a divisor of \( ... | 1 | 0 | 0 | 0 | 0 | 1 | 0.25 | 1 | 0.25 | 1 | 0.1875 | 1 | 0.25 | 1 | 0.1875 |
In $\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, then the area of $\triangle ABC$ is | Given a right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), the altitude \( CH \) and the median \( CM \) trisect the right angle \( \angle ACB \). This means that \( \angle HCM = 30^\circ \) and \( \angle➠ = \angle MCB = 30^\circ \).
Let's denote the lengths of the legs \( AC \) and \( BC \) as \( a \)... | 1 | 0 | 0 | 1 | 0.5 | 1 | 0.25 | 1 | 0.125 | 1 | 0.3125 | 1 | 0.40625 | 1 | 0.328125 |
Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle? | To determine the degree measure of the forgotten angle, we first need to understand the sum of the interior angles of a convex polygon. The formula for the sum of the interior angles of a polygon with \( n \) sides is given by:
\[
180(n-2)
\]
Claire's initial sum of the interior angles is 2017 degrees, but she forgot... | 1 | 1 | 1 | 1 | 1 | 1 | 0.75 | 1 | 0.625 | 1 | 0.6875 | 1 | 0.71875 | 1 | 0.703125 |
Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this? | To determine the number of ways Zara can arrange her marbles such that the Steelie and the Tiger are not next to each other, we can follow these steps:
1. Calculate the total number of arrangements of the 4 marbles without any restrictions.
2. Subtract the number of arrangements where the Steelie and the Tiger are nex... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.96875 | 1 | 0.984375 |
Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\triangle ABC$? | To find the area of triangle \(ABC\) with vertices \(A = (3,0)\), \(B = (0,3)\), and \(C\) on the line \(x + y = 7\), we can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - ... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Everyday at school, Jo climbs a flight of $6$ stairs. Jo can take the stairs $1$, $2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs? | To determine the number of ways Jo can climb a flight of 6 stairs, where Jo can take 1, 2, or 3 steps at a time, we can use a recursive approach. Let \(a_n\) represent the number of ways to climb \(n\) stairs. We can express \(a_n\) in terms of smaller subproblems:
\[ a_n = a_{n-1} + a_{n-2} + a_{n-3} \]
This is beca... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.875 | 1 | 0.9375 | 1 | 0.96875 | 1 | 0.96875 |
A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \%$ of its games for the season? | Let's denote the number of remaining games that the team must win as \( x \). The team has already won 40 out of its first 50 games, and there are 40 games left in the season, so the total number of games in the season is \( 50 + 40 = 90 \).
The team needs to win exactly 70% of its games for the season. Therefore, the... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\frac{1}{2}$% of itself, while the lengths of ... | To determine the percent decrease in the area of the square when the length of segment \(AD\) is decreased by \(12.5\%\) of itself, we start by finding the original side length of the square.
The area of the square is given as \(32\) square inches. Let the side length of the square be \(s\). The area of a square is gi... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \cdot a - a \cdot b$? | To find the smallest possible value for the expression \(2 \cdot a - a \cdot b\) where \(a\) and \(b\) are positive integers each less than 6, we start by rewriting the expression in a simpler form:
\[2a - ab = a(2 - b).\]
We need to minimize this expression. Since \(a\) and \(b\) are positive integers less than 6, th... | 1 | 0 | 0 | 1 | 0.5 | 1 | 0.75 | 1 | 0.75 | 1 | 0.6875 | 1 | 0.6875 | 1 | 0.71875 |
In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR = 6$ and $PR = 7$. What is the area of the square? | Let the side length of the square \(ABCD\) be \(s\). Place the square in the coordinate plane with \(A = (0, s)\), \(B = (s, s)\), \(C = (s, 0)\), and \(D = (0, 0)\). Let the coordinates of point \(P\) be \((0, y)\) and the coordinates of point \(Q\) be \((x, s)\).
The equations of the lines \(BP\) and \(CQ\) are:
\[B... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The equations of $L_1$ and $L_2$ are $y=mx$ and $y=nx$, respectively. Suppose $L_1$ makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does $L_2$, and that $L_1$ has 4 times the slope of $L_2$. If $L_1$ is not horizontal, then $mn$ is | To solve the problem, we need to use the relationship between the slopes of the lines and the angles they make with the horizontal axis.
First, recall that the slope of a line is the tangent of the angle it makes with the horizontal axis. If the angle that line \(L_2\) makes with the horizontal is \(\theta\), then the... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.75 | 1 | 0.875 | 1 | 0.9375 | 1 | 0.953125 |
An amusement park has a collection of scale models, with a ratio of $1: 20$, of buildings and other sights from around the country. The height of the United States Capitol is $289$ feet. What is the height in feet of its duplicate to the nearest whole number? | To determine the height of the scale model of the United States Capitol, we need to use the given ratio of the scale model to the actual building. The ratio is \(1:20\), which means that the height of the scale model is \(\frac{1}{20}\) of the height of the actual building.
The height of the actual United States Capit... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? | To determine the fraction of the area of the equilateral triangle \(ABC\) that lies outside the circle, we need to first understand the geometric configuration.
Given that \( \overline{AB} \) and \( \overline{AC} \) are tangent to the circle at points \( B \) and \( C \) respectively, it implies that the circle is ins... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Given $0 \le x_0 < 1$, let
\[x_n = \begin{cases} 2x_{n-1} & \text{ if } 2x_{n-1} < 1 \\ 2x_{n-1} - 1 & \text{ if } 2x_{n-1} \ge 1 \end{cases}\]for all integers $n > 0$. For how many $x_0$ is it true that $x_0 = x_5$? | To determine the number of \( x_0 \) such that \( x_0 = x_5 \), we need to analyze the behavior of the sequence \( \{x_n\} \) defined by:
\[ x_n = \begin{cases} 2x_{n-1} & \text{if } 2x_{n-1} < 1 \\ 2x_{n-1} - 1 & \text{if } 2x_{n-1} \ge 1 \end{cases} \]
This sequence essentially represents the fractional part of \( 2... | 1 | 0 | 0 | 1 | 0.5 | 1 | 0.5 | 1 | 0.375 | 1 | 0.375 | 1 | 0.4375 | 1 | 0.390625 |
Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? | To determine the height of Logan's scaled-down water tower, we need to understand the relationship between the volumes of the water towers and how this translates to the heights.
First, we note the volumes of the actual and scaled-down water towers:
- Actual water tower: 100,000 liters
- scaled-down water tower: 0.1 l... | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.9375 | 1 | 0.9375 | 1 | 0.890625 |
In $\triangle PQR$, $PR=15$, $QR=20$, and $PQ=25$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, and points $E$ and $F$ lie on $\overline{PR}$, with $PA=QB=QC=RD=RE=PF=5$. Find the area of hexagon $ABCDEF$. | To solve for the area of hexagon \(ABCDEF\) in \(\triangle PQR\), we start by identifying and calculating the area of \(\triangle PQR\) and then subtracting the areas of the three smaller triangles that are not part of the hexagon.
First, we note that \(\triangle PQR\) is a right triangle with \(PR = 15\), \(QR = 20\)... | 0 | 0 | 0 | 1 | 0.5 | 1 | 0.25 | 1 | 0.25 | 1 | 0.125 | 1 | 0.15625 | 1 | 0.203125 |
Adams plans a profit of $10$ % on the selling price of an article and his expenses are $15$ % of sales. The rate of markup on an article that sells for $ $5.00$ is: | To determine the rate of markup on an article that sells for $5.00, we need to follow these steps:
1. Identify the selling price and the coefficients for profit and expenses.
2. Calculate the cost price of the article.
3. Determine the markup amount and then the rate of markup.
Let's start with the given information:... | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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