problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k |
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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? | 80\% | 1. **Identify the total number of hits**: Judy had a total of 35 hits during the season.
2. **Determine the number of hits that were not singles**:
- Home runs: 1
- Triples: 1
- Doubles: 5
- Total non-single hits = 1 (home run) + 1 (triple) + 5 (doubles) = 7
3. **Calculate the number of singles**:
\[
... |
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 $ABCD$? | \frac{25}{32} | 1. **Assigning Side Lengths:**
Let the side length of square $ABCD$ be $8s$. This choice is made to simplify calculations, as $AE = 7 \cdot EB$ implies a division of side $AB$ into 8 equal parts.
2. **Locating Point E:**
Since $AE = 7 \cdot EB$, and if we let $EB = x$, then $AE = 7x$. Given $AB = 8s$, we have $A... |
If $991+993+995+997+999=5000-N$, then $N=$ | 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 ... |
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... | 17 | 1. **Establishing the Coordinate System and Points**:
Let's assume the side length of the hexagon is $6$ meters for simplicity. We place the hexagon in a 3D coordinate system with $A$ at the origin, i.e., $A = (0, 0, 0)$. The coordinates of $B$ and $C$ can be calculated based on the geometry of a regular hexagon:
... |
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? | 26 | 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... |
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 $S$? | 8 | 1. **Identify Symmetry Requirements**: The problem states that the set $S$ is symmetric about the origin, both coordinate axes, and the line $y=x$. This implies:
- Symmetry about the origin: If $(a, b) \in S$, then $(-a, -b) \in S$.
- Symmetry about the $x$-axis: If $(a, b) \in S$, then $(a, -b) \in S$.
- Symm... |
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... | 37 | 1. **Understanding the Sequence Property**: The sequence is such that the tens and units digits of each term become the hundreds and tens digits of the next term. This implies that each digit in a position (hundreds, tens, units) of one term will appear in the next position in the next term and eventually cycle back to... |
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 $a+b =$ | 7 | 1. **Identify the multiplication setup**: The multiplication setup is given as:
\[
\begin{array}{ccc}
& 2 & a\\
\times & b & 3\\
\hline
& 6 & 9\\
9 & 2\\
\hline
9 & 8 & 9
\end{array}
\]
This represents the multiplication of a two-digit number $2a$ by another two-digit number $b... |
In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?
$\begin{array}{cccc} & A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}$ | 1 | 1. **Identify the value of $A$:**
Given the multiplication problem:
\[
\begin{array}{cccc}
& A & B & A\\
\times & & C & D\\
\hline
C & D & C & D\\
\end{array}
\]
We observe that the product of $A$ and $D$ results in a number ending in $D$. This implies that $A \times D$ must be a num... |
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 $A+B$? | 17 | 1. **Objective**: We need to maximize the fraction $\frac{A+B}{C+D}$ where $A, B, C, D$ are distinct digits from $0$ to $9$. This involves maximizing $A+B$ and minimizing $C+D$.
2. **Maximizing $A+B$**:
- The maximum value for any digit is $9$. Thus, to maximize $A+B$, we should choose the largest available distin... |
What is the value of the expression $\sqrt{16\sqrt{8\sqrt{4}}}$? | 8 | To solve the expression $\sqrt{16\sqrt{8\sqrt{4}}}$, we will simplify the expression step-by-step:
1. **Simplify the innermost square root:**
\[
\sqrt{4} = 2
\]
2. **Substitute and simplify the next square root:**
\[
\sqrt{8\sqrt{4}} = \sqrt{8 \cdot 2} = \sqrt{16}
\]
3. **Simplify the result obtain... |
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? | 12 | 1. **Define the problem in terms of algebra**: Let the two-digit number Mary thought of be $x$. According to the problem, she performs the following operations on $x$:
- Multiplies by $3$: $3x$
- Adds $11$: $3x + 11$
- Switches the digits of the result.
2. **Analyze the range of the final number**: The switch... |
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... | 192 | 1. **Understanding the Problem**: We are given four circles with centers at $A$, $B$, $C$, and $D$. 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 similarly for circles $C$ and $D$. The distances $AB$ and $CD$ are both 39, and the length of segm... |
The number of digits in $4^{16}5^{25}$ (when written in the usual base $10$ form) is | 28 | 1. **Rewrite the expression using properties of exponents:**
The given expression is $4^{16}5^{25}$. We can express $4$ as $2^2$, so:
\[
4^{16} = (2^2)^{16} = 2^{32}
\]
Therefore, the expression becomes:
\[
4^{16}5^{25} = 2^{32}5^{25}
\]
2. **Combine powers of 2 and 5 to form powers of 10:** ... |
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? | 7 | 1. **Identify the sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula for the sum of an arithmetic series:
\[
S = \frac{n(n+1)}{2}
\]
2. **Expression for the average after erasing one number**: If one number $x$ is erased, the sum of the remaining numbe... |
A palindrome between $1000$ and $10000$ is chosen at random. What is the probability that it is divisible by $7$? | \frac{1}{5} | 1. **Identify the form of the palindrome**: A four-digit palindrome can be expressed in the form $\overline{abba}$, where $a$ and $b$ are digits, and $a \neq 0$ to ensure it is a four-digit number.
2. **Total number of palindromes**: Since $a$ can be any digit from 1 to 9 (9 choices) and $b$ can be any digit from 0 to... |
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 $n$. | 1 | To solve for $n$, we start by analyzing the polynomial $P(x)$ given its values at specific points and its degree. We use Lagrange Interpolation Formula to express $P(x)$, and then evaluate it at $x = 3n+1$ to find $n$.
1. **Constructing the Polynomial Using Lagrange Interpolation:**
The polynomial $P(x)$ is defined... |
What is the hundreds digit of $(20! - 15!)?$ | 0 | To find the hundreds digit of $(20! - 15!)$, we need to analyze the factorials and their properties modulo $1000$.
1. **Factorial Properties**:
- $n!$ (where $n \geq 5$) contains at least one factor of $5$ and at least one factor of $2$, making it divisible by $10$.
- $n!$ (where $n \geq 10$) contains at least t... |
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 $m\sqrt{n}$... | 194 | 1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\overline{AB}\parallel\overline{CD}$ and $BC=CD=43$. Also, $\overline{AD}\perp\overline{BD}$, which implies that $\triangle ABD$ is a right triangle.
2. **Examine the diagonals and intersection**: The diagonals $\overline{AC}$ and ... |
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... | 23 | 1. **Initialization**: The program starts with $X = 3$ and $S = 0$.
2. **Loop Execution**:
- **Instruction 2**: Increase the value of $X$ by $2$.
- **Instruction 3**: Increase the value of $S$ by the current value of $X$.
- **Instruction 4**: Check if $S \geq 10000$. If true, go to instruction 5; otherwise, ... |
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... | 2 | 1. **Define the areas**: Let $A_{\text{outer}}$ be the area of the entire floor and $A_{\text{inner}}$ be the area of the painted rectangle. Given that the unpainted border is 1 foot wide, the dimensions of the painted rectangle are $(a-2)$ and $(b-2)$.
Therefore, we have:
\[
A_{\text{outer}} = ab
\]
... |
What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$? | 22 | To find the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$, we first need to determine the maximum number of digits that a base-seven number can have without exceeding $2019$ in decimal.
1. **Convert $2019$ to base-seven:**
- The largest power of $7$ less... |
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{... | 17 | 1. **Define Variables:**
Let the number of faces on the two dice be $a$ and $b$, respectively, with $a \geq b$. Assume each die has distinct integers from $1$ to the number of faces on that die.
2. **Analyze the Probability of Rolling a Sum of $7$:**
Since each die has at least $6$ faces, there are always $6$ wa... |
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? | 90 | 1. **Understanding the Cube Configuration**:
- A 3x3x3 cube consists of 27 smaller cubes.
- The smaller cubes on the corners have three faces visible.
- The smaller cubes on the edges have two faces visible.
- The smaller cubes in the center of each face have one face visible.
2. **Counting Visible Faces**... |
In counting $n$ colored balls, some red and some black, it was found that $49$ of the first $50$ counted were red.
Thereafter, $7$ out of every $8$ counted were red. If, in all, $90$ % or more of the balls counted were red, the maximum value of $n$ is: | 210 | 1. **Understanding the Problem:**
We are given that $49$ of the first $50$ balls are red, and thereafter, $7$ out of every $8$ balls counted are red. We need to find the maximum number of balls, $n$, such that at least $90\%$ of them are red.
2. **Setting Up the Equation:**
Let $x$ be the number of batches of $8... |
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? | 0.4 | 1. **Identify the ratio of volumes between the actual water tower and the miniature model**:
The actual water tower holds 100,000 liters, and Logan's miniature holds 0.1 liters. The ratio of the volumes is:
\[
\frac{100000 \text{ liters}}{0.1 \text{ liters}} = 1000000
\]
2. **Relate the volume ratio to th... |
Before the district play, the Unicorns had won $45$% of their basketball games. During district play, they won six more games and lost two, to finish the season having won half their games. How many games did the Unicorns play in all? | 48 | 1. Let $y$ be the total number of games the Unicorns played before the district play, and $x$ be the number of games they won. According to the problem, they won 45% of these games, so we have the equation:
\[
\frac{x}{y} = 0.45
\]
Multiplying both sides by $y$ gives:
\[
x = 0.45y
\]
2. During the... |
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? | 143 | 1. **Understanding the problem**: Claire calculated the sum of the interior angles of a convex polygon as $2017^\circ$. However, she missed one angle. We need to find the measure of this forgotten angle.
2. **Using the formula for the sum of interior angles**: The sum of the interior angles of an $n$-sided polygon is ... |
If $f(x)=\frac{x^4+x^2}{x+1}$, then $f(i)$, where $i=\sqrt{-1}$, is equal to | 0 | To find $f(i)$ for the function $f(x) = \frac{x^4 + x^2}{x + 1}$, we substitute $i$ for $x$:
1. **Calculate the numerator**:
\[
i^4 + i^2 = 1 - 1 = 0
\]
Here, $i^4 = (i^2)^2 = (-1)^2 = 1$ and $i^2 = -1$.
2. **Check the denominator**:
\[
i + 1 \neq 0
\]
The denominator $i + 1$ is not zero beca... |
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? | 24 | 1. **Define the recursive function**: Let $f(n)$ represent the number of ways Jo can climb to the $n$-th step. We start by defining the base cases:
- $f(0) = 1$: There is only one way to be on the ground (by starting there).
- $f(1) = 1$: There is only one way to reach the first step, which is by taking a single ... |
In a magic triangle, each of the six whole numbers $10-15$ is placed in one of the circles so that the sum, $S$, of the three numbers on each side of the triangle is the same. The largest possible value for $S$ is
[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(... | 39 | 1. **Assign Variables to Circles**: Let the numbers in the circles be $a$, $b$, $c$, $d$, $e$, and $f$ starting from the top circle and moving clockwise.
2. **Set Up Equations for Each Side of the Triangle**:
- The sum of the numbers on the first side is $S = a + b + c$.
- The sum of the numbers on the second si... |
Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels, dimes, and quarters, whose total value is $8.35. It does not necessarily contain coins of all three types. What is the difference between the largest and smallest number of dimes that could be in the bank? | 64 | Let $a$, $b$, and $c$ represent the number of nickels, dimes, and quarters, respectively. We are given two conditions:
1. The total number of coins is 100.
2. The total value of the coins is $8.35, which is equivalent to 835 cents.
From these conditions, we can set up the following equations:
\[
\begin{align*}
a + b +... |
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? | 996 |
1. **Understanding the Problem:**
We need to find the largest three-digit integer $n$ such that the sum of the first $n$ positive integers is not a divisor of the product of the first $n$ positive integers.
2. **Sum and Product Formulas:**
- The sum of the first $n$ positive integers is given by the formula:
... |
What is the value of $\frac{(2112-2021)^2}{169}$? | 49 | 1. **Calculate the difference in the numerator**:
\[
2112 - 2021 = 91
\]
This is the exact value, not an approximation.
2. **Square the difference**:
\[
(2112 - 2021)^2 = 91^2 = 8281
\]
3. **Divide by the denominator**:
\[
\frac{8281}{169}
\]
To simplify this, we can either perform t... |
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? | 12 | 1. **Identify the total number of unrestricted arrangements:**
Let's denote the marbles as $A$ (Aggie), $B$ (Bumblebee), $S$ (Steelie), and $T$ (Tiger). Without any restrictions, the total number of ways to arrange these four marbles is given by the factorial of the number of marbles, which is $4!$.
\[
4! = 4... |
What is the greatest number of consecutive integers whose sum is $45?$ | 90 | To find the greatest number of consecutive integers whose sum is $45$, we need to consider sequences of integers, both positive and negative.
1. **Understanding the sum of consecutive integers**:
The sum of $N$ consecutive integers starting from $a$ can be expressed as:
\[
a + (a+1) + (a+2) + \cdots + (a+N-1)... |
There are $120$ seats in a row. What is the fewest number of seats that must be occupied so the next person to be seated must sit next to someone? | 40 | To solve this problem, we need to find the minimum number of people that must be seated such that any additional person must sit next to someone already seated. We aim to maximize the number of empty seats between seated people under this constraint.
1. **Understanding the Pattern**:
- If we place a person in a se... |
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? | 331 | To find the greatest number of unit cubes that can be seen from a single point, we consider viewing the cube from one of its corners. From this vantage point, three faces of the cube are visible: the top face, the front face, and the side face.
1. **Counting the visible unit cubes on each face:**
- Each face of the... |
If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is | 18 | 1. **Identify the Midpoints**: Points E and G are the midpoints of sides AD and CD, respectively, in rectangle ABCD.
2. **Properties of Midpoints in a Rectangle**: Since E and G are midpoints, segment EG is parallel to sides AB and CD, and its length is half the length of AB (or CD). Similarly, since F is the midpoint... |
The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner? | 14 | To solve this problem, we need to understand how a cube is formed and how the numbers on the faces are arranged. In a standard die, opposite faces sum up to 7. This means:
- If one face shows 1, the opposite face shows 6.
- If one face shows 2, the opposite face shows 5.
- If one face shows 3, the opposite face shows 4... |
Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$. The ratio of the largest element of the set to the sum of the other ten elements of the set is closest to which integer? | 9 | To solve the problem, we need to find the ratio of the largest element in the set $\{1, 10, 10^2, 10^3, \ldots, 10^{10}\}$ to the sum of all other elements in the set. The largest element in this set is $10^{10}$.
1. **Calculate the sum of the other elements:**
The sum of the other elements is $1 + 10 + 10^2 + 10^3... |
The digits 1, 2, 3, 4 and 9 are each used once to form the smallest possible even five-digit number. The digit in the tens place is | 9 | 1. **Identify the requirement for the number to be even**: The number must end in an even digit. The available even digits are 2 and 4.
2. **Determine the smallest possible even digit for the units place**: To form the smallest number, we prefer the smallest digits in the higher place values. Since the number must be ... |
The set $\{3,6,9,10\}$ is augmented by a fifth element $n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $n$? | 26 | We are given the set $\{3,6,9,10\}$ and an additional element $n$, which is distinct from the other elements. We need to find the sum of all possible values of $n$ such that the median and the mean of the augmented set are equal.
#### Case 1: Median is $6$
For $6$ to be the median, $n$ must be less than or equal to $6... |
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... | 50 | Let the listed price be $P$, where $P > 100$. We can express $P$ as $P = 100 + p$ where $p > 0$.
1. **Calculate the savings from each coupon:**
- **Coupon A:** This coupon gives $15\%$ off the listed price. Therefore, the savings from Coupon A are:
\[
0.15P = 0.15(100 + p) = 15 + 0.15p
\]
- **Coup... |
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 $p + q$. | 2016 |
To solve this problem, we need to find the arithmetic mean of the least elements of all 1000-element subsets of the set $\{1, 2, 3, \ldots, 2015\}$. We will use combinatorial arguments to derive the solution.
#### Step 1: Counting subsets with a fixed least element
Let $i$ be the least element of a 1000-element subse... |
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... | 10 | Let's denote the amount of money each person has as follows:
- Ashley: $A$
- Betty: $B$
- Carlos: $C$
- Dick: $D$
- Elgin: $E$
From the problem, we have the following equations based on the absolute differences:
1. $|A - B| = 19$
2. $|B - C| = 7$
3. $|C - D| = 5$
4. $|D - E| = 4$
5. $|E - A| = 11$
Additionally, we kn... |
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 $x_0 = x_5$? | 31 | 1. **Understanding the Sequence**: The sequence defined by $x_n$ is a binary sequence where each term is generated by doubling the previous term and subtracting 1 if the result is at least 1. This can be interpreted as a shift and truncate operation in binary representation.
2. **Binary Representation**: Let's represe... |
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and the... | 18 | 1. **Identify the segments of Samantha's route:**
- From her house to the southwest corner of City Park.
- Through City Park from the southwest corner to the northeast corner.
- From the northeast corner of City Park to her school.
2. **Calculate the number of ways from her house to the southwest corner of Ci... |
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? | 117 | 1. **Identify Similar Triangles**: Notice that $\triangle CRB \sim \triangle BAP$ by AA similarity (both have a right angle and share angle $BRP$).
2. **Set Up Ratio of Sides**: From the similarity, we have the ratio of corresponding sides:
\[
\frac{CB}{CR} = \frac{PB}{AB}
\]
Since $CB = AB = s$ (side leng... |
Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth? | $1.15 | 1. Let $n$ represent the number of nickels Patty has, and $d$ represent the number of dimes. Since Patty has a total of 20 coins, we can express the number of dimes in terms of nickels:
\[
d = 20 - n
\]
2. Calculate the total value of the coins when nickels and dimes are in their original form. The value of a... |
Points $A$ and $B$ are $5$ units apart. How many lines in a given plane containing $A$ and $B$ are $2$ units from $A$ and $3$ units from $B$? | 3 | To solve this problem, we need to consider the geometric configuration of two circles centered at points $A$ and $B$ with radii $2$ and $3$ units, respectively. We are looking for lines that are tangent to both circles.
1. **Identify the circles**:
- Circle centered at $A$ (denoted as $C_A$) has radius $2$ units.
... |
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 $y$? | 90 | 1. **Understanding the Circle and Arcs**: The circle is divided into 12 equal arcs, and each arc corresponds to a central angle at the center $O$. Since the circle's total degrees is $360^\circ$, each central angle measures:
\[
\frac{360^\circ}{12} = 30^\circ
\]
2. **Central Angles for $x$ and $y$**:
- If... |
The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used? | 24000 | 1. **Total Miles Driven by All Tires**: The car has five tires, but only four tires are used at any given time. Therefore, over the course of $30,000$ miles, the total number of tire-miles (the sum of the miles driven by each individual tire) is calculated by multiplying the total miles driven by the number of tires us... |
How many positive integer factors of $2020$ have more than $3$ factors? | 7 | To solve this problem, we first need to find the prime factorization of $2020$. We have:
\[ 2020 = 2^2 \times 5 \times 101. \]
Next, we use the formula for the number of divisors of a number given its prime factorization. If $n = p^a \times q^b \times r^c \times \ldots$, then the number of divisors of $n$, denoted as ... |
In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids? | 5 | 1. **Identify the total area of the large triangle**: Given in the problem, the area of the outer equilateral triangle is $16$ square units.
2. **Identify the area of the inner triangle**: The area of the inner equilateral triangle is given as $1$ square unit.
3. **Calculate the area between the inner and outer trian... |
In the table shown, the formula relating \(x\) and \(y\) is:
\[\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5\\ \hline y & 3 & 7 & 13 & 21 & 31\\ \hline\end{array}\] | y = x^2 + x + 1 | To find the correct formula relating $x$ and $y$, we will substitute the given values of $x$ into each formula choice and check if the resulting $y$ matches the values in the table.
#### Checking Choice (A) $y = 4x - 1$
1. For $x = 1$, $y = 4(1) - 1 = 3$
2. For $x = 2$, $y = 4(2) - 1 = 7$
3. For $x = 3$, $y = 4(3) - 1... |
Odell and Kershaw run for $30$ minutes on a circular track. Odell runs clockwise at $250 m/min$ and uses the inner lane with a radius of $50$ meters. Kershaw runs counterclockwise at $300 m/min$ and uses the outer lane with a radius of $60$ meters, starting on the same radial line as Odell. How many times after the sta... | 47 | 1. **Determine the Circumference of Each Track**:
- Odell's track radius = $50$ meters, so the circumference is $C_O = 2\pi \times 50 = 100\pi$ meters.
- Kershaw's track radius = $60$ meters, so the circumference is $C_K = 2\pi \times 60 = 120\pi$ meters.
2. **Calculate the Speed in Terms of Radians per Minute**... |
Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."
(1) It is prime.
(2) It is even.
(3) It is divisible by 7.
(4) One of its dig... | 8 | 1. **Analyze the given statements**: Isabella's house number is a two-digit number, and exactly three out of the four statements about it are true:
- (1) It is prime.
- (2) It is even.
- (3) It is divisible by 7.
- (4) One of its digits is 9.
2. **Determine the false statement**:
- If (1) is true (the ... |
Let $\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\circ$, $60^\circ$, and $60.001^\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-... | 15 | 1. **Define the angles and setup the problem:** Let $\triangle A_0B_0C_0$ be a triangle with angles $\angle C_0A_0B_0 = x_0 = 59.999^\circ$, $\angle A_0B_0C_0 = y_0 = 60^\circ$, and $\angle B_0C_0A_0 = z_0 = 60.001^\circ$. For each positive integer $n$, define $A_n$, $B_n$, and $C_n$ as the feet of the altitudes from t... |
Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How ma... | 6 | To solve this problem, we need to label each vertex of a cube with integers from $1$ to $8$ such that the sum of the numbers on the vertices of each face is the same. Additionally, we consider two arrangements the same if one can be obtained from the other by rotating the cube.
#### Step 1: Calculate the total sum and... |
Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ wer... | 3 | 1. **Substitute the given values into the expression ignoring parentheses:**
Larry ignored the parentheses, so he calculated the expression as:
\[
a - b - c - d + e = 1 - 2 - 3 - 4 + e
\]
Simplifying this, we get:
\[
-8 + e
\]
2. **Substitute the given values into the expression with correct ... |
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions? | 28 | We need to consider different seating arrangements for Alice, as her position affects the seating of the others due to her restrictions with Bob and Carla.
1. **Alice sits in the center chair (3rd position):**
- The 2nd and 4th chairs must be occupied by Derek and Eric in either order because Alice cannot sit next ... |
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | 19 | To solve this problem, we need to consider the possible number of intersection points formed by four distinct lines in a plane. Each pair of lines can intersect at most once, and the maximum number of intersection points is determined by the number of ways to choose 2 lines from 4, which is given by the binomial coeffi... |
Joy has $30$ thin rods, one each of every integer length from $1 \text{ cm}$ through $30 \text{ cm}$. She places the rods with lengths $3 \text{ cm}$, $7 \text{ cm}$, and $15 \text{cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How ma... | 17 | 1. **Identify the range for the fourth rod**: To form a quadrilateral, the sum of the lengths of any three sides must be greater than the length of the fourth side. This is known as the triangle inequality theorem. We apply this to the three rods of lengths $3 \text{ cm}$, $7 \text{ cm}$, and $15 \text{ cm}$.
2. **Cal... |
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? | 6 |
To solve this problem, we need to determine how many of the nine positions for the additional square allow the resulting figure to be folded into a cube with one face missing. We start by understanding the structure of the given figure and the implications of adding a square at each position.
#### Step 1: Understand ... |
The number of points common to the graphs of
$(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$ is: | 4 | 1. **Identify the equations of the lines**:
From the given equations, using the Zero Product Property, we have:
- From $(x-y+2)(3x+y-4)=0$:
- $x-y+2=0 \Rightarrow y = x - 2$
- $3x+y-4=0 \Rightarrow y = -3x + 4$
- From $(x+y-2)(2x-5y+7)=0$:
- $x+y-2=0 \Rightarrow y = -x + 2$
- $2x-5y+7=0 \R... |
A game is played with tokens according to the following rule. In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens,... | 37 |
We will analyze the game by observing the token distribution and the rules of the game. The key observation is that in each round, the player with the most tokens gives one token to each of the other two players and one token to the discard pile, effectively losing three tokens, while each of the other two players gai... |
The sum of all integers between 50 and 350 which end in 1 is | 5880 | 1. **Identify the sequence**: The problem asks for the sum of all integers between 50 and 350 that end in 1. These integers are $51, 61, 71, \ldots, 341$.
2. **Determine the sequence type**: The sequence is an arithmetic sequence where each term increases by 10.
3. **Find the first term ($a$) and common difference ($... |
At the end of 1994, Walter was half as old as his grandmother. The sum of the years in which they were born was 3838. How old will Walter be at the end of 1999? | 55 | 1. **Assign variables to ages**: Let Walter's age in 1994 be $x$. Then, his grandmother's age in 1994 is $2x$ because Walter is half as old as his grandmother.
2. **Set up the equation for their birth years**: Walter was born in $1994 - x$ and his grandmother was born in $1994 - 2x$. The sum of their birth years is gi... |
In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument? | 154 | Let's denote the number of seniors as $s$ and the number of non-seniors as $n$. Since there are $500$ students in total, we have:
\[ s + n = 500 \]
From the problem, $40\%$ of the seniors play a musical instrument, which implies that $60\%$ of the seniors do not play a musical instrument. Similarly, $30\%$ of the non-... |
A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $3\%$ to $2\%$ is approximately: | 10000 | 1. **Understanding the problem**: The problem states that a railroad needs to rise 600 feet to cross a mountain. The grade of the railroad, which is the ratio of the rise to the horizontal length, can be adjusted by changing the length of the track.
2. **Calculating the horizontal length for each grade**:
- The gra... |
Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? | \frac{3}{8} | We start by considering the different cases based on the number of heads that appear when two fair coins are tossed. For each head, a fair die is rolled. We need to find the probability that the sum of the die rolls is odd.
#### Case Analysis:
1. **Case 1: 0 Heads (2 Tails)**
- Probability of getting 2 tails: $\fra... |
The product $8 \times .25 \times 2 \times .125 =$ | $\frac{1}{2}$ | 1. **Convert Decimals to Fractions**: The given decimals are converted to fractions for easier multiplication.
- $0.25$ as a fraction is $\frac{1}{4}$.
- $0.125$ as a fraction is $\frac{1}{8}$.
2. **Set Up the Product**: Substitute the decimals with their fractional equivalents in the product.
\[
8 \times ... |
What is $10 \cdot \left(\frac{1}{2} + \frac{1}{5} + \frac{1}{10}\right)^{-1}$? | \frac{25}{2} | 1. **Simplify the expression inside the parentheses**:
We start by simplifying the sum inside the parentheses:
\[
\frac{1}{2} + \frac{1}{5} + \frac{1}{10}
\]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 2, 5, and 10 is 10. Thus, we rewrite each fraction with a... |
For each positive integer $n$, let
$a_n = \frac{(n+9)!}{(n-1)!}$.
Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is | 9 | 1. **Expression Simplification**:
Given $a_n = \frac{(n+9)!}{(n-1)!}$, we can simplify this as:
\[
a_n = n(n+1)(n+2)\cdots(n+9)
\]
This is the product of 10 consecutive integers starting from $n$.
2. **Factorization**:
We can express $a_n$ in terms of its prime factors as $2^{x_n} 5^{y_n} r_n$, where... |
Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$? | 18 | 1. **Identify the Geometry of the Problem**: Given that $A$, $B$, and $C$ are on the graph $y = x^2$, and $AB$ is parallel to the $x$-axis, we know that $A$ and $B$ have the same $y$-coordinate. Since $\triangle ABC$ is a right triangle with area $2008$, we need to determine the position of $C$.
2. **Determine the Rig... |
On Halloween Casper ate $\frac{1}{3}$ of his candies and then gave $2$ candies to his brother. The next day he ate $\frac{1}{3}$ of his remaining candies and then gave $4$ candies to his sister. On the third day he ate his final $8$ candies. How many candies did Casper have at the beginning? | 57 | Let $x$ represent the total number of candies Casper had at the beginning.
1. **First Day:**
- Casper ate $\frac{1}{3}$ of his candies, so he had $\frac{2}{3}x$ candies left.
- After giving $2$ candies to his brother, he had $\frac{2}{3}x - 2$ candies remaining.
2. **Second Day:**
- Casper ate $\frac{1}{3}$ ... |
When three different numbers from the set $\{ -3, -2, -1, 4, 5 \}$ are multiplied, the largest possible product is | 30 | To find the largest possible product when three different numbers from the set $\{ -3, -2, -1, 4, 5 \}$ are multiplied, we need to consider the signs and magnitudes of the products.
1. **Identify the possible combinations**:
- Three positive numbers: Not possible as there are only two positive numbers in the set (4... |
Three times Dick's age plus Tom's age equals twice Harry's age.
Double the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age.
Their respective ages are relatively prime to each other. The sum of the squares of their ages is | 42 | We are given two equations involving the ages of Dick, Tom, and Harry, denoted as $d$, $t$, and $h$ respectively:
1. \(3d + t = 2h\)
2. \(2h^3 = 3d^3 + t^3\)
We start by expressing $t$ in terms of $d$ and $h$ from the first equation:
\[ t = 2h - 3d \]
Substitute this expression for $t$ into the second equation:
\[ 2h... |
How many unordered pairs of edges of a given cube determine a plane? | 42 | 1. **Understanding the Problem**: We need to find how many unordered pairs of edges in a cube determine a plane. Two edges determine a plane if they are either parallel or intersecting (not skew).
2. **Total Number of Edges in a Cube**: A cube has 12 edges.
3. **Choosing One Edge**: Choose one edge arbitrarily. There... |
An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes? | \frac{10a}{r} | 1. **Identify the rate of travel**: The automobile travels $\frac{a}{6}$ feet in $r$ seconds. Thus, the rate of travel is:
\[
\text{Rate} = \frac{\frac{a}{6} \text{ feet}}{r \text{ seconds}}
\]
2. **Convert the rate to yards per second**: Since there are 3 feet in a yard, we convert feet to yards:
\[
\t... |
If $1998$ is written as a product of two positive integers whose difference is as small as possible, then the difference is | 17 | 1. **Objective**: Find two positive integers whose product is $1998$ and whose difference is minimized.
2. **Calculate the approximate square root of $1998$**:
\[
\sqrt{1998} \approx \sqrt{2000} = \sqrt{4 \times 500} = 2 \times \sqrt{500} \approx 2 \times 22.36 \approx 44.72
\]
Thus, the integers should be... |
Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\frac ab,$ where $a$ and $b$ are relatively prime positi... | 85 | 1. **Understanding the Problem:**
The problem asks us to find the possible values of the slope $m$ of a line such that exactly $300$ lattice points from the set $S$ (where $S$ consists of points $(x,y)$ with $1 \leq x, y \leq 30$) lie on or below the line $y = mx$. The total number of lattice points in $S$ is $30 \t... |
The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is: | $n(n+2)$ | 1. **Identify the sequence**: The problem asks for the sum of numbers of the form $2k + 1$ where $k$ ranges from $1$ to $n$. This forms a sequence of odd numbers starting from $3$ (when $k=1$, $2k+1=3$) up to $2n+1$ (when $k=n$, $2k+1=2n+1$).
2. **Write out the sequence explicitly**: The sequence is $3, 5, 7, \ldots, ... |
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 | 2 | 1. **Given Information and Equations:**
- The equations of lines $L_1$ and $L_2$ are $y = mx$ and $y = nx$ respectively.
- $L_1$ makes twice as large of an angle with the horizontal as does $L_2$.
- $L_1$ has 4 times the slope of $L_2$.
- $L_1$ is not horizontal.
2. **Relating Slopes to Angles:**
- Let ... |
A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$ | 4 | 1. **Identify the conditions for a number to be flippy and divisible by 15**:
- A flippy number alternates between two distinct digits.
- A number is divisible by 15 if it is divisible by both 3 and 5.
2. **Condition for divisibility by 5**:
- The last digit must be either 0 or 5.
3. **Eliminate the possibil... |
What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?
[asy]
unitsize(12);
draw((0,0)--(10,0));
draw((-1.5,1.5)--(-1.5,2.5));
draw((-1,2)--(-2,2));
draw((1,1)--(3,1)--(3,3)--(1,3)--cycle);
draw((1,4)--(3,4)... | 1047 | 1. **Identify the problem**: We need to find the smallest sum of two 3-digit numbers formed by the digits 4, 5, 6, 7, 8, and 9, each used exactly once.
2. **Understand the sum of two numbers**: If the two numbers are $\overline{abc}$ and $\overline{def}$, their sum is given by:
\[
100(a+d) + 10(b+e) + (c+f)
\... |
A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, the median of the trapezoid is: | 9 \text{ inches} | 1. **Identify the formula for the area of the triangle and trapezoid:**
- The area of a triangle is given by the formula:
\[
\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}bh
\]
- The area of a trapezoid is given by the formula:
\[
\text{A... |
A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | \frac{1}{12} | 1. **Understanding the problem**: We need to find the probability that the area of a circle is less than its circumference, given that the diameter $d$ of the circle is determined by the sum of two 6-sided dice.
2. **Relating area and circumference**: The formula for the circumference $C$ of a circle is $C = \pi d$, a... |
If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is | 1 | 1. **Formulate the given condition**: The problem states that four times the reciprocal of the circumference of a circle equals the diameter of the circle. Let the radius of the circle be $r$. The circumference of the circle is $2\pi r$, and the diameter is $2r$. Thus, the equation becomes:
\[
4 \cdot \frac{1}{2\... |
Quadrilateral $ABCD$ has $AB = BC = CD$, $m\angle ABC = 70^\circ$ and $m\angle BCD = 170^\circ$. What is the degree measure of $\angle BAD$? | 85 | 1. **Assign the unknown and draw necessary diagonals**: Let $\angle BAD = x$. Draw diagonals $BD$ and $AC$. Let $I$ be the intersection of diagonals $BD$ and $AC$.
2. **Analyze the isosceles triangles**: Since $AB = BC = CD$, triangles $\triangle ABC$ and $\triangle BCD$ are isosceles. Therefore, $\angle DBC = \angle ... |
A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is: | 0 | 1. **Expressing $N$ in different bases**: Let $N$ be represented as $\overline{abc}_7$ in base $7$ and as $\overline{cba}_9$ in base $9$. This means:
- In base $7$: $N = 49a + 7b + c$
- In base $9$: $N = 81c + 9b + a$
2. **Setting up the equation**: Since both expressions represent the same number $N$, we equate... |
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 $ABCD?$ | 360 | 1. **Assign Variables and Use Pythagorean Theorem in $\triangle ABC$:**
Let $AB = x$ and $BC = y$. Since $\angle ABC = 90^\circ$, by the Pythagorean theorem, we have:
\[
x^2 + y^2 = AC^2 = 20^2 = 400.
\]
2. **Calculate Area of $\triangle ACD$:**
Since $\angle ACD = 90^\circ$, the area of $\triangle ACD$... |
Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of $\triangle CEF$ is $200$ square inches. Then the number of inches in $BE$ is: | 12 | 1. **Identify the properties of the square**: Given that $ABCD$ is a square with an area of $256$ square inches, we can find the side length of the square:
\[
s^2 = 256 \implies s = 16 \text{ inches}
\]
Therefore, $AB = BC = CD = DA = 16$ inches.
2. **Analyze the triangle and use the area information**: Th... |
Three generous friends, each with some money, redistribute the money as followed:
Amy gives enough money to Jan and Toy to double each amount has.
Jan then gives enough to Amy and Toy to double their amounts.
Finally, Toy gives enough to Amy and Jan to double their amounts.
If Toy had 36 dollars at the beginning and 3... | 252 | 1. **Initial Setup**: Let's denote the initial amounts of money that Amy, Jan, and Toy have as $a$, $j$, and $t$ respectively. According to the problem, Toy starts with $t = 36$ dollars.
2. **After Amy's Redistribution**: Amy gives enough money to Jan and Toy to double their amounts. This means:
- Toy's new amount ... |
Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\%$ of the difference between $A$ and the cost of his movie ticket. To ... | 23\% | 1. **Define Variables:**
Let $m$ represent the cost of the movie ticket and $s$ represent the cost of the soda.
2. **Set Up Equations:**
According to the problem, we have:
\[ m = 0.20(A - s) \]
\[ s = 0.05(A - m) \]
3. **Convert to Fractional Form:**
These equations can be rewritten as:
\[ m = \frac... |
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 $HC/JE$. | \frac{5}{3} | 1. **Identify Key Points and Relationships**:
- Points $A, B, C, D, E,$ and $F$ are collinear on line $\overline{AF}$, and each segment between consecutive points is of length 1.
- Point $G$ is not on line $AF$, and points $H$ and $J$ lie on lines $\overline{GD}$ and $\overline{GF}$ respectively.
- Lines $\ov... |
Placing no more than one X in each small square, what is the greatest number of X's that can be put on the grid shown without getting three X's in a row vertically, horizontally, or diagonally?
[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b)); } [/asy] | 6 | 1. **Understanding the Grid**: The grid is a $4 \times 4$ square grid, which means there are 16 small squares in total.
2. **Objective**: We need to place the maximum number of $\text{X}$'s such that no three $\text{X}$'s are aligned vertically, horizontally, or diagonally.
3. **Using the Pigeonhole Principle**: If w... |
A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? | \frac{1}{3} |
We are given that the probability that a ball is tossed into bin $k$ is $2^{-k}$ for $k = 1, 2, 3, \ldots$. We need to find the probability that the red ball is tossed into a higher-numbered bin than the green ball.
#### Step-by-step Analysis:
1. **Probability of Landing in the Same Bin:**
Let's first calculate t... |
If $\log_{10}{m}= b-\log_{10}{n}$, then $m=$ | \frac{10^{b}}{n} | 1. Start by expressing $b$ in terms of logarithm base 10:
\[
b = \log_{10}{10^b}
\]
This follows from the property of logarithms that $\log_b{b^x} = x$.
2. Substitute this expression for $b$ into the given equation:
\[
\log_{10}{m} = \log_{10}{10^b} - \log_{10}{n}
\]
3. Apply the logarithmic prop... |
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