problem stringlengths 11 5.15k | answer stringlengths 1 42 | solution stringlengths 0 8.75k |
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At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur within the group? | 245 | 1. **Identify the groups and their interactions**: We have two distinct groups at the gathering:
- Group A: 20 people who all know each other.
- Group B: 10 people who know no one in the gathering.
2. **Calculate handshakes between Group A and Group B**:
- Each person in Group B will shake hands with each pe... |
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? | 14 | Let $x$ be the total number of apples Bridget bought. According to the problem, Bridget first gave half of the apples to Ann. This means that Bridget was left with $\frac{x}{2}$ apples.
Next, Bridget gave Cassie 3 apples. After giving these apples to Cassie, the number of apples Bridget had left is:
\[
\frac{x}{2} - 3... |
Let $n$ be the least positive integer greater than $1000$ for which
\[\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.\]What is the sum of the digits of $n$? | 18 | 1. **Understanding the Problem:**
We need to find the smallest integer $n > 1000$ such that:
- $\gcd(63, n+120) = 21$
- $\gcd(n+63, 120) = 60$
2. **Using the Euclidean Algorithm:**
- For $\gcd(63, n+120) = 21$, we have:
\[
\gcd(63, n+120) = \gcd(63, n+120 - 63k_1) = 21
\]
This implies $... |
Consider the statements:
$\textbf{(1)}\ p\wedge \sim q\wedge r \qquad\textbf{(2)}\ \sim p\wedge \sim q\wedge r \qquad\textbf{(3)}\ p\wedge \sim q\wedge \sim r \qquad\textbf{(4)}\ \sim p\wedge q\wedge r$
where $p,q$, and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$? | 4 | To solve this problem, we need to evaluate each statement to see if it implies the truth of $(p \rightarrow q) \rightarrow r$. We start by understanding the implication $p \rightarrow q$, which is logically equivalent to $\sim p \vee q$. Then, we evaluate $(p \rightarrow q) \rightarrow r$ for each statement.
#### Stat... |
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? | 18 | 1. **Identify the factors of 128**: We know that the product of the ages of Kiana and her twin brothers is 128. Since 128 can be expressed as $128 = 2^7$, the ages must be factors of 128.
2. **List possible ages for the twins**: The twins are older than Kiana and have the same age. Possible factors of $128$ that could... |
In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar? | 9 | Let's denote the number of red marbles as $r$, green marbles as $g$, and blue marbles as $b$. According to the problem, we have the following equations based on the information given:
1. All but 6 marbles are red, so the number of non-red marbles (green and blue) is 6:
\[ g + b = 6 \]
2. All but 8 marbles are gree... |
The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn? | 15 | 1. **Identify the distance traveled by each biker after 4 hours**: According to the problem, Bjorn biked 45 miles and Alberto biked 60 miles in the same time period.
2. **Calculate the difference in miles traveled**: To find out how many more miles Alberto biked than Bjorn, subtract the distance biked by Bjorn from th... |
Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find... | 64 | 1. **Identify Key Properties of the Triangles**:
- Since $\triangle ABD$ and $\triangle BCE$ are equilateral, we have $AB = BD = 16$ and $BC = CE = 4$.
- The angles $\angle ABE$ and $\angle DBC$ are both $120^\circ$ because they are supplementary to the $60^\circ$ angles in the equilateral triangles.
2. **Congru... |
A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\frac{3}{4}$ of its former rate and arrives $3\tfrac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3$ hours late. The length of the trip in mile... | 600 | 1. **Define Variables:**
Let the speed of the train be $x$ miles per hour, and let $D$ miles be the total distance of the trip.
2. **Analyze the First Scenario:**
- The train travels for 1 hour before the accident, covering $x$ miles.
- After the accident, it is detained for 0.5 hours and then travels at $\fr... |
The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the ... | 2 | To solve this problem, we need to determine how many arrangements of the numbers $1, 2, 3, 4, 5$ in a circle are "bad," meaning that it is not possible to find a subset of numbers that appear consecutively on the circle that sum to every integer from $1$ to $15$.
#### Step 1: Count the total number of distinct arrange... |
Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he discovers that all his money is gone after cros... | 35 | Let's denote the amount of money Fox has at the beginning as $x$ coins. We will analyze the changes in the amount of money Fox has after each crossing and payment.
1. **First Crossing:**
- Before crossing: Fox has $x$ coins.
- After doubling: $2x$ coins.
- After paying toll: $2x - 40$ coins.
2. **Second Cros... |
Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$? | 200 | 1. **Understanding the problem**: We are given that $a$ is $150\%$ of $b$. This can be expressed as:
\[
a = 150\% \times b = 1.5b
\]
2. **Finding $3b$ in terms of $a$**: We need to find what percent $3b$ is of $a$. First, express $3b$ using the relationship between $a$ and $b$:
\[
3b = 3 \times b
\]
... |
The number of distinct pairs of integers $(x, y)$ such that $0<x<y$ and $\sqrt{1984}=\sqrt{x}+\sqrt{y}$ is | 3 | 1. **Simplify the given equation**: We start by simplifying $\sqrt{1984}$. We factorize 1984 as $1984 = 64 \times 31 = 8^2 \times 31$. Therefore, $\sqrt{1984} = \sqrt{8^2 \times 31} = 8\sqrt{31}$.
2. **Rewrite the equation**: The equation $\sqrt{1984} = \sqrt{x} + \sqrt{y}$ can now be rewritten using the simplified fo... |
The number of diagonals that can be drawn in a polygon of 100 sides is: | 4850 | To find the number of diagonals in a polygon, we use the formula:
\[
\frac{n(n-3)}{2}
\]
where $n$ is the number of sides of the polygon. This formula arises because each vertex of the polygon can connect to $n-3$ other vertices to form a diagonal (it cannot connect to itself or its two adjacent vertices, hence $n-3$).... |
The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first term is an integer, and $n>1$, then the number of possible values for $n$ is: | 5 | 1. **Identify the formula for the sum of an arithmetic progression (AP)**:
The sum $S_n$ of the first $n$ terms of an AP where the first term is $a$ and the common difference is $d$ is given by:
\[
S_n = \frac{n}{2} \left(2a + (n-1)d\right)
\]
Given $d = 2$ and $S_n = 153$, substituting these values give... |
The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log{b^n}$. What is $n$? | 112 |
1. **Define the terms in terms of logarithms**:
Let $A = \log(a)$ and $B = \log(b)$. The given terms of the sequence can be rewritten using logarithmic properties:
\[
\begin{align*}
\log(a^3b^7) &= 3A + 7B, \\
\log(a^5b^{12}) &= 5A + 12B, \\
\log(a^8b^{15}) &= 8A + 15B.
\end{align*}
\]
2. **Us... |
The entries in a $3 \times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? | 42 | To solve this problem, we need to consider the constraints given by the increasing order in rows and columns. We will analyze the possible placements of the numbers, focusing on the central number and leveraging symmetry.
#### Step 1: Analyze the constraints
Each number from 1 to 9 must be used exactly once in the $3 ... |
A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1... | 4 | To solve this problem, we need to determine the smallest number of cubes that can be snapped together such that all the protruding snaps are hidden and only the receptacle holes are visible.
1. **Understanding the Cube Configuration**: Each cube has one protruding snap and five receptacle holes. The protruding snap pr... |
A man travels $m$ feet due north at $2$ minutes per mile. He returns due south to his starting point at $2$ miles per minute. The average rate in miles per hour for the entire trip is: | 48 | To find the average rate in miles per hour for the entire trip, we need to calculate the total distance traveled and the total time taken, and then use the formula for average speed:
\[
\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}
\]
1. **Convert the distance from feet to miles:**
Since $... |
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? | 729 | 1. **Identify the constraints and setup the problem:**
- The chef has to prepare desserts for 7 days starting from Sunday.
- The desserts options are cake, pie, ice cream, or pudding.
- The same dessert cannot be served on consecutive days.
- Cake must be served on Friday due to a birthday.
2. **Determine ... |
In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of quadrilateral $ABCD$? | 36 | 1. **Identify the triangles and their properties:**
- Connect point $B$ to point $D$.
- Observe that $\triangle BCD$ is a right triangle because $\angle BCD = 90^\circ$. Given $BC = 4$ and $CD = 3$, we can use the Pythagorean theorem to find $BD$:
\[
BD = \sqrt{BC^2 + CD^2} = \sqrt{4^2 + 3^2} = \sqrt{16... |
It is given that one root of $2x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is $3+2i (i = \sqrt{-1})$. The value of $s$ is: | 26 | 1. **Identify the nature of the roots:** Given that one root of the quadratic equation $2x^2 + rx + s = 0$ is $3+2i$, and knowing that the coefficients $r$ and $s$ are real numbers, the other root must be the complex conjugate of $3+2i$. Therefore, the other root is $3-2i$.
2. **Use Vieta's formulas:** Vieta's formula... |
Star lists the whole numbers $1$ through $30$ once. Emilio copies Star's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Star adds her numbers and Emilio adds his numbers. How much larger is Star's sum than Emilio's? | 103 | 1. **Identify the numbers affected by the digit change**: Star lists numbers from $1$ to $30$. Emilio replaces each occurrence of the digit $2$ with the digit $1$. We need to identify where the digit $2$ appears in these numbers.
2. **Count occurrences of the digit $2$**:
- **As a tens digit**: The numbers $20$ thr... |
Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? | 56 | To solve this problem, we need to calculate the expected number of coins that show heads after up to three tosses. We will use the concept of probability and expected value to find the solution.
1. **Probability of a coin showing heads after each toss:**
- The probability of a coin landing heads on the first toss i... |
The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point? | 8 | 1. **Identify the center and radius of the circle:**
Given points $(0,0)$ and $(8,6)$ are endpoints of a diameter. The center of the circle, $(h,k)$, is the midpoint of the diameter. Using the midpoint formula:
\[
h = \frac{0+8}{2} = 4, \quad k = \frac{0+6}{2} = 3
\]
Therefore, the center of the circle i... |
The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? | 10 | To solve this problem, we need to find the common sum of the numbers in each row, column, and diagonal of a $5 \times 5$ square matrix using the integers from $-10$ to $14$ inclusive.
1. **Calculate the total sum of all integers from $-10$ to $14$:**
The sum of an arithmetic series can be calculated using the formu... |
For real numbers $x$ and $y$, define $x\spadesuit y = (x + y)(x - y)$. What is $3\spadesuit(4\spadesuit 5)$? | -72 | To solve the problem, we first need to evaluate $4 \spadesuit 5$ using the operation defined as $x \spadesuit y = (x + y)(x - y)$.
1. **Calculate $4 \spadesuit 5$:**
\[
4 \spadesuit 5 = (4 + 5)(4 - 5) = 9 \times (-1) = -9
\]
2. **Next, calculate $3 \spadesuit (-9)$:**
\[
3 \spadesuit (-9) = (3 + (-9))(... |
If $x^4 + 4x^3 + 6px^2 + 4qx + r$ is exactly divisible by $x^3 + 3x^2 + 9x + 3$, the value of $(p + q)r$ is: | 15 | 1. **Define the polynomials and their roots:**
Let $f(x) = x^3 + 3x^2 + 9x + 3$ and $g(x) = x^4 + 4x^3 + 6px^2 + 4qx + r$. Assume $f(x)$ has roots $r_1, r_2, r_3$. Since $f(x)$ divides $g(x)$, $g(x)$ must have roots $r_1, r_2, r_3$, and an additional root $r_4$.
2. **Apply Vieta's formulas to $f(x)$:**
- The sum... |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex? | 12 | 1. **Identify the coordinates of the vertices**: Let the $x$-coordinates of the vertices of the quadrilateral be $n, n+1, n+2, n+3$. Then, the coordinates of the vertices are $(n,\ln(n))$, $(n+1,\ln(n+1))$, $(n+2,\ln(n+2))$, $(n+3,\ln(n+3))$.
2. **Use the Shoelace Theorem**: The area $A$ of the quadrilateral can be ca... |
The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term? | 1 | 1. **Identify the pattern in the terms**: The given terms of the geometric progression are $\sqrt{3}$, $\sqrt[3]{3}$, and $\sqrt[6]{3}$. We can rewrite these terms using exponents:
- $\sqrt{3} = 3^{1/2} = 3^{3/6}$
- $\sqrt[3]{3} = 3^{1/3} = 3^{2/6}$
- $\sqrt[6]{3} = 3^{1/6} = 3^{1/6}$
2. **Recognize the seque... |
In the expression $\frac{x + 1}{x - 1}$ each $x$ is replaced by $\frac{x + 1}{x - 1}$. The resulting expression, evaluated for $x = \frac{1}{2}$, equals: | 3 | 1. **Substitute $x$ with $\frac{x+1}{x-1}$ in the expression:**
\[
\frac{\left(\frac{x+1}{x-1}\right) + 1}{\left(\frac{x+1}{x-1}\right) - 1}
\]
2. **Simplify the expression:**
- For the numerator:
\[
\frac{x+1}{x-1} + 1 = \frac{x+1}{x-1} + \frac{x-1}{x-1} = \frac{x+1 + x-1}{x-1} = \frac{2x}{x-1}
... |
The Blue Bird High School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible? | 12 | 1. **Positioning the Boys**: The problem specifies that a boy must sit at each end of the row. Since there are two boys, we can arrange them in $2!$ ways. This is because the first position can be filled by either of the two boys, and the second position is automatically filled by the remaining boy. Thus, we have:
\... |
How many whole numbers from $1$ through $46$ are divisible by either $3$ or $5$ or both? | 21 | 1. **Count numbers divisible by 3**:
To find how many numbers between 1 and 46 are divisible by 3, we calculate:
\[
\left\lfloor \frac{46}{3} \right\rfloor = 15
\]
This means there are 15 numbers divisible by 3.
2. **Count numbers divisible by 5**:
Similarly, to find how many numbers between 1 and ... |
Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is | 24 | 1. **Define Variables:**
Let $a$ and $b$ be the current ages of Ann and Barbara, respectively. We know that the sum of their ages is $a + b = 44$.
2. **Express Age Difference:**
Define $y$ as the difference in their ages, so $y = a - b$. Note that $y$ remains constant over time.
3. **Interpret the Problem State... |
The digit-sum of $998$ is $9+9+8=26$. How many 3-digit whole numbers, whose digit-sum is $26$, are even? | 1 | To find how many 3-digit whole numbers have a digit-sum of 26 and are even, we start by considering the constraints:
1. **Digit-Sum Requirement**: The sum of the digits of the number must be 26.
2. **Even Number Requirement**: The number must be even, which means its last digit must be an even number (0, 2, 4, 6, or 8... |
In the sequence
\[..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,...\]
each term is the sum of the two terms to its left. Find $a$. | -3 | To find the value of $a$, we will use the given property of the sequence that each term is the sum of the two terms to its left. We will calculate the values of $d$, $c$, $b$, and finally $a$ by working backwards from the known terms.
1. **Finding $d$:**
Given that the term after $d$ is $0$ and the term following $... |
The number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers? | 162 | To solve the problem, we first need to find the prime factorization of the number $6545$.
1. **Prime Factorization of $6545$:**
\[
6545 = 5 \times 1309
\]
We need to check if $1309$ is prime or can be further factored. Testing divisibility by small primes:
\[
1309 \div 7 = 187 \quad (\text{since } 13... |
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? | 137 | 1. **Calculate the total weight of the lemonade**:
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water. The total weight of the lemonade is:
\[
100 \text{ grams (lemon juice)} + 100 \text{ grams (sugar)} + 400 \text{ grams (water)} = 600 \text{ grams}
\]
2. **Calculate the ... |
The largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$, is: | 6 | To find the largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$, we can start by factoring the expression:
1. **Factorize the expression:**
\[
n^3 - n = n(n^2 - 1) = n(n-1)(n+1)
\]
Here, $n(n-1)(n+1)$ represents the product of three consecutive integers.
2... |
The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit | 0 | 1. **Identify the pattern in the set**: The set given is $\{9, 99, 999, 9999, \ldots, 999999999\}$. Each number in the set is composed of digits that are all 9s, and the number of 9s increases by one in each subsequent number.
2. **Express each number as a sum of 9s**: Each number in the set can be expressed as a sum ... |
I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number? | 8 | 1. **Define the variables**: Let the two whole numbers be $x$ and $y$.
2. **Set up the equations based on the problem statement**:
- The product of the numbers is 24: \( xy = 24 \)
- The sum of the numbers is 11: \( x + y = 11 \)
3. **Express one variable in terms of the other**:
- From the sum equation, sol... |
The sum of $25$ consecutive even integers is $10,000$. What is the largest of these $25$ consecutive integers? | 424 | 1. **Define the sequence**: Let the smallest of the 25 consecutive even integers be $x$. Then the integers are $x, x+2, x+4, \ldots, x+48$.
2. **Formulate the sum**: The sum of these integers can be expressed as:
\[
x + (x+2) + (x+4) + \cdots + (x+48)
\]
This is an arithmetic sequence where the first term ... |
A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals: | 12 | 1. **Understanding the Problem**: We are given a regular polygon with $n$ sides inscribed in a circle of radius $R$. The area of the polygon is given as $3R^2$. We need to find the value of $n$.
2. **Area of a Regular Polygon**: The area $A$ of a regular polygon inscribed in a circle can be calculated using the formul... |
How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) | 24 | To solve this problem, we need to find the number of ways to schedule 3 mathematics courses (algebra, geometry, and number theory) in a 6-period day such that no two mathematics courses are taken in consecutive periods.
#### Step 1: Total ways to schedule without restrictions
First, we calculate the total number of wa... |
Kaashish has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? | 8 | 1. Let the first number be $x$ and the second number be $y$. According to the problem, Kaashish has written down one integer ($x$) two times and another integer ($y$) three times. The sum of these five numbers is given by:
\[
2x + 3y = 100
\]
2. We are also given that one of the numbers is $28$. We need to de... |
Each of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$. What is the area of $S_3$? | 4 | 1. **Understanding the problem**: We are given a square $S_1$ with area 16. Each side of $S_1$ is bisected, and a smaller square $S_2$ is constructed using these bisection points. This process is repeated to construct $S_3$ from $S_2$.
2. **Calculating the side length of $S_1$**:
- The area of $S_1$ is given as 16... |
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if neces... | 13 | To solve this problem, we need to find the number of steps in the staircase such that Dash takes 19 fewer jumps than Cozy. Let $n$ be the total number of steps in the staircase. Cozy jumps 2 steps at a time, and Dash jumps 5 steps at a time. We can express the number of jumps each takes as follows:
- Cozy's jumps: $\le... |
Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?
(a) the length of the segment $MN$
(b) the perimeter of $\triangle PAB$
(c) the area of $\triangle PAB$
(d) the area of trapezoid ... | 1 | To determine how many of the four quantities change as $P$ moves along a line parallel to side $AB$, we analyze each quantity individually:
1. **Length of segment $MN$:**
- Since $M$ and $N$ are midpoints of $PA$ and $PB$ respectively, segment $MN$ is parallel to $AB$ and half its length due to the midpoint theorem... |
A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$? | 24 |
#### Step 1: Understand the properties of $x$ and $x+32$
Given that $x$ is a three-digit palindrome and $x+32$ is a four-digit palindrome, we need to find the possible values of $x$ and $x+32$ that satisfy these conditions.
#### Step 2: Determine the range of $x$
Since $x$ is a three-digit number, $100 \leq x \leq 99... |
A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip? | 24 | 1. **Calculate the total distance of the round trip**:
The student drove $120$ miles to his home and $120$ miles back, so the total distance of the round trip is:
\[
120 \text{ miles} + 120 \text{ miles} = 240 \text{ miles}
\]
2. **Calculate the amount of gasoline used for each leg of the trip**:
- On ... |
Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$? | 5 | 1. **Define the variables and set up the equation:**
- Let $T$ be Tom's current age.
- Let $N$ be the number of years ago when Tom's age was twice the sum of his children's ages.
- The sum of the ages of Tom's three children is also $T$.
2. **Calculate the ages $N$ years ago:**
- Tom's age $N$ years ago wa... |
When $x^{13}+1$ is divided by $x-1$, the remainder is: | 2 |
We are given the polynomial \(x^{13} + 1\) and we need to find the remainder when it is divided by \(x-1\).
**Solution 1: Using the Remainder Theorem**
The Remainder Theorem states that if a polynomial \(f(x)\) is divided by \(x - a\), the remainder is \(f(a)\). Here, \(f(x) = x^{13} + 1\) and \(a = 1\). Therefore, ... |
Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$? | 19 | To determine how many quadratic equations of the form $x^2 + bx + c = 0$ have real roots with $b$ and $c$ chosen from the set $\{1, 2, 3, 4, 5, 6\}$, we need to ensure that the discriminant of the quadratic equation is non-negative. The discriminant $\Delta$ for a quadratic equation $ax^2 + bx + c = 0$ is given by:
\[
... |
4(299) + 3(299) + 2(299) + 298 = | 2989 | 1. **Identify the common factor and apply the distributive property**: Notice that $299$ appears as a common factor in the first three terms. We can factor out $299$ from these terms:
\[
4(299) + 3(299) + 2(299) + 298 = (4+3+2)(299) + 298
\]
2. **Simplify the expression inside the parentheses**: Add the coeff... |
The function $f$ is given by the table
\[\begin{tabular}{|c||c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \ \hline f(x) & 4 & 1 & 3 & 5 & 2 \ \hline \end{tabular}\]
If $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \ge 0$, find $u_{2002}$ | 2 | 1. **Identify the sequence**: We are given a function $f$ defined by a table and a sequence defined recursively by $u_{n+1} = f(u_n)$ with $u_0 = 4$.
2. **Calculate the first few terms**:
- $u_1 = f(u_0) = f(4) = 5$
- $u_2 = f(u_1) = f(5) = 2$
- $u_3 = f(u_2) = f(2) = 1$
- $u_4 = f(u_3) = f(1) = 4$
3. **O... |
When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is | 420 | To find the least common denominator (LCD) for the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, we need to determine the least common multiple (LCM) of the denominators: $2, 3, 4, 5, 6, 7$.
1. **Prime Factorization**:
- $2$ is prime.
- $3$ is prime.
- $4 = 2^2$.
- $5$ is p... |
The difference in the areas of two similar triangles is $18$ square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triangle, in square feet, is an integer, and one of its sides is $3$ feet. The corresponding side of the larger triangle, in feet, is: | 6 | 1. **Identify the given information:**
- The difference in the areas of two similar triangles is $18$ square feet.
- The ratio of the larger area to the smaller is the square of an integer.
- The area of the smaller triangle is an integer.
- One side of the smaller triangle is $3$ feet.
2. **Set up the equ... |
What is the value of \[\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?\] | 21000 | 1. **Simplify the first sum:**
We start by simplifying each term in the first sum:
\[
\log_{5^k} 3^{k^2} = \frac{\log_5 3^{k^2}}{\log_5 5^k} = \frac{k^2 \log_5 3}{k \log_5 5} = k \log_5 3.
\]
This simplification uses the property of logarithms that $\log_b a^c = c \log_b a$ and the change of base formula... |
The area of a rectangle remains unchanged when it is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower, or when it is made $2 \frac{1}{2}$ inches shorter and $\frac{4}{3}$ inch wider. Its area, in square inches, is: | 20 | 1. **Set up the equations based on the problem statement:**
We are given that the area of the rectangle remains unchanged under two transformations:
- When the rectangle is made $2 \frac{1}{2}$ inches longer and $\frac{2}{3}$ inch narrower.
- When the rectangle is made $2 \frac{1}{2}$ inches shorter and $\frac... |
There are integers $a, b,$ and $c,$ each greater than $1,$ such that
\[\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}\]
for all $N \neq 1$. What is $b$? | 3 | 1. **Expression Simplification**:
Start by simplifying the left-hand side of the equation:
\[
\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = N^{\frac{1}{a} + \frac{1}{ab} + \frac{1}{abc}}
\]
This simplification comes from the properties of exponents, where $\sqrt[k]{x} = x^{\frac{1}{k}}$ and the chain rule for expo... |
Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid? | 750 | 1. **Identify the Components of the Trapezoid**:
Given that $ABCD$ is a trapezoid with $AB$ and $CD$ as the parallel sides, and the altitude (height) from $AB$ to $CD$ is $12$. The lengths of the sides are $AD = 15$, $AB = 50$, $BC = 20$.
2. **Draw Altitudes and Form Right Triangles**:
By drawing altitudes from ... |
If $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=6$ has area 6, then $ab=$ | 3 | 1. **Identify the intercepts**:
The equation of the line is given by $ax + by = 6$. To find the $x$-intercept, set $y = 0$:
\[
ax = 6 \implies x = \frac{6}{a}.
\]
Similarly, to find the $y$-intercept, set $x = 0$:
\[
by = 6 \implies y = \frac{6}{b}.
\]
2. **Calculate the area of the triangle**... |
A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list? | 6 | 1. **Define Variables:**
Let the integers in the list be $a_1, a_2, a_3, a_4, a_5$ such that $a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5$. Given that the mode and median are both $8$, we know $a_3 = 8$. Since the mode is $8$, at least one other number must be $8$. Without loss of generality, let $a_2 = 8$.
2. **Use th... |
The first four terms of an arithmetic sequence are $p$, $9$, $3p-q$, and $3p+q$. What is the $2010^{\text{th}}$ term of this sequence? | 8041 | 1. **Identify the common difference**:
Given the terms of the sequence are $p$, $9$, $3p-q$, and $3p+q$, we know that the common difference $d$ between consecutive terms can be calculated as:
\[
d = (3p+q) - (3p-q) = 2q
\]
2. **Set up equations for $p$ and $q$**:
Since $p$, $9$, $3p-q$, and $3p+q$ are ... |
How many ordered triples of integers $(a,b,c)$ satisfy $|a+b|+c = 19$ and $ab+|c| = 97$? | 12 | 1. **Symmetry and Reduction of Cases**:
Without loss of generality (WLOG), assume $a \geq 0$ and $a \geq b$. This assumption is valid because if $(a, b, c)$ is a solution, then $(-a, -b, c)$, $(b, a, c)$, and $(-b, -a, c)$ are also solutions due to the symmetry in the equations. If $a = b$, then $|a+b| = |2a| = 2a$,... |
How many three-digit numbers have at least one $2$ and at least one $3$? | 52 | To solve this problem, we will use the principle of inclusion-exclusion and case analysis to count the number of three-digit numbers that contain at least one $2$ and at least one $3$.
#### Total three-digit numbers:
A three-digit number ranges from $100$ to $999$. Thus, there are $999 - 100 + 1 = 900$ three-digit num... |
Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? | 8 | 1. **Calculate the area of the smaller circle (red area):**
The diameter of the smaller circle is $1$ inch, so its radius is $\frac{1}{2}$ inch. The area of a circle is given by the formula $A = \pi r^2$. Therefore, the area of the smaller circle is:
\[
A_{\text{red}} = \pi \left(\frac{1}{2}\right)^2 = \frac{\... |
As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour, and Emerson skates at a constant rate of ... | 15 | 1. **Determine the relative speed of Emily and Emerson**: Emily rides at $12$ mph and Emerson skates at $8$ mph. Since they are moving in the same direction, the relative speed of Emily with respect to Emerson is:
\[
12 \text{ mph} - 8 \text{ mph} = 4 \text{ mph}
\]
2. **Calculate the time taken to cover the ... |
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two... | 240 | 1. **Assigning angles in arithmetic progression**: Since the internal angles of quadrilateral $ABCD$ form an arithmetic progression, we denote them as $a$, $a+d$, $a+2d$, and $a+3d$. The sum of the internal angles of any quadrilateral is $360^\circ$, so:
\[
a + (a+d) + (a+2d) + (a+3d) = 360^\circ
\]
Simplif... |
The value of $\left(256\right)^{.16}\left(256\right)^{.09}$ is: | 4 | 1. **Identify the operation and simplify the expression**: We are given the expression $\left(256\right)^{0.16}\left(256\right)^{0.09}$. Using the property of exponents that states when multiplying like bases you add the exponents, we can simplify this expression:
\[
256^{0.16} \cdot 256^{0.09} = 256^{0.16+0.09} ... |
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? | 23 | 1. **Understanding the Display Format**: The digital watch displays time in a 12-hour format with AM and PM, showing hours and minutes. The hours can range from 01 to 12, and the minutes from 00 to 59.
2. **Maximizing the Hour Digits**:
- The hours are displayed as either 01, 02, ..., 12. We need to find the hour ... |
If $\frac{2+3+4}{3}=\frac{1990+1991+1992}{N}$, then $N=$ | 1991 | 1. **Calculate the left-hand side of the equation:**
\[
\frac{2+3+4}{3} = \frac{9}{3} = 3
\]
2. **Set up the equation:**
Given that
\[
\frac{2+3+4}{3} = \frac{1990+1991+1992}{N}
\]
substituting the value from step 1, we have:
\[
3 = \frac{1990+1991+1992}{N}
\]
3. **Calculate the sum i... |
From a group of boys and girls, 15 girls leave. There are then left two boys for each girl. After this 45 boys leave. There are then 5 girls for each boy. The number of girls in the beginning was: | 40 | 1. **Define Variables:**
Let $b$ represent the number of boys and $g$ represent the number of girls initially.
2. **Translate the Problem into Equations:**
- After 15 girls leave, the number of girls becomes $g - 15$. According to the problem, there are then two boys for each girl, so we have the equation:
... |
The vertex of the parabola $y = x^2 - 8x + c$ will be a point on the $x$-axis if the value of $c$ is: | 16 | 1. **Identify the Vertex Formula**: The vertex form of a parabola given by $y = ax^2 + bx + c$ is $(h, k)$ where $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$. For the given parabola $y = x^2 - 8x + c$, we have $a = 1$ and $b = -8$.
2. **Calculate the x-coordinate of the Vertex**:
\[
h = -\frac{-8}{2 \times ... |
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score of $90$, and that $65\%$ of the votes cas... | 300 | 1. **Define Variables:**
Let $x$ be the total number of votes cast. Since $65\%$ of the votes are likes and $35\%$ are dislikes, we can denote the number of like votes as $0.65x$ and the number of dislike votes as $0.35x$.
2. **Set Up the Equation:**
Each like vote increases the score by $1$ and each dislike vot... |
Using a table of a certain height, two identical blocks of wood are placed as shown in Figure 1. Length $r$ is found to be $32$ inches. After rearranging the blocks as in Figure 2, length $s$ is found to be $28$ inches. How high is the table? | 30 | Let's denote the height of the table as $h$, the length of each wood block as $l$, and the width of each wood block as $w$. All measurements are in inches.
From the problem statement, we have two configurations:
1. In Figure 1, the blocks are arranged such that the total length from the end of one block to the end of ... |
Twenty percent less than 60 is one-third more than what number? | 36 | 1. **Calculate 20% less than 60**:
To find 20% less than a number, we subtract 20% of the number from the number itself.
\[
60 - 0.20 \times 60 = 60 - 12 = 48
\]
Alternatively, we can calculate this as 80% of 60:
\[
0.80 \times 60 = 48
\]
or using fractions:
\[
\frac{4}{5} \times 60 =... |
Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100? | 20201 | 1. **Identify the pattern**: We observe that the number of unit squares in figures $0$, $1$, $2$, and $3$ are $1$, $5$, $13$, and $25$ respectively. We need to find a formula that describes this sequence.
2. **Recognize the sequence type**: The sequence appears to be quadratic because the differences between consecuti... |
Let $n$ be a positive integer and $d$ be a digit such that the value of the numeral $32d$ in base $n$ equals $263$, and the value of the numeral $324$ in base $n$ equals the value of the numeral $11d1$ in base six. What is $n + d$? | 11 | 1. **Convert $\underline{32d}$ in base $n$ to decimal:**
The numeral $\underline{32d}$ in base $n$ can be expressed in decimal as:
\[
3n^2 + 2n + d
\]
Given that this equals 263, we have the equation:
\[
3n^2 + 2n + d = 263
\]
2. **Convert $\underline{324}$ in base $n$ to decimal:**
The ... |
Suppose $d$ is a digit. For how many values of $d$ is $2.00d5 > 2.005$? | 5 | To solve this problem, we need to compare the number $2.00d5$ with $2.005$ for each possible digit $d$ (where $d$ can be any digit from $0$ to $9$).
1. **Expression of $2.00d5$:**
The number $2.00d5$ can be expressed as $2.00d5 = 2 + 0.00d + 0.0005$. Here, $0.00d$ represents the contribution of the digit $d$ in t... |
Square corners, 5 units on a side, are removed from a $20$ unit by $30$ unit rectangular sheet of cardboard. The sides are then folded to form an open box. The surface area, in square units, of the interior of the box is | 500 | 1. **Identify the dimensions of the original cardboard sheet**: The original sheet is $20$ units by $30$ units.
2. **Calculate the area of the original sheet**:
\[
\text{Area}_{\text{original}} = 20 \times 30 = 600 \text{ square units}
\]
3. **Determine the size of the squares removed**: Each square removed... |
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How m... | 23 | 1. **Understanding the problem setup**: Each high school sends 3 students, and each student has a unique score. Andrea's score is the median of all scores, and she has the highest score on her team. Her teammates Beth and Carla placed 37th and 64th, respectively.
2. **Determining Andrea's rank**: Since Andrea's score ... |
Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$, and the ratio of blue to green marbles in Jar $2$ is $8:1$. There are $95$ green marbles in all. How many more blue marbles are in Jar $1$ than in Jar $2$? | 5 | 1. **Set up the ratios for each jar:**
For Jar $1$, the ratio of blue to green marbles is $9:1$. This means for every $10$ marbles in Jar $1$, $9$ are blue and $1$ is green.
For Jar $2$, the ratio of blue to green marbles is $8:1$. This means for every $9$ marbles in Jar $2$, $8$ are blue and $1$ is green.
2... |
There is a unique positive integer $n$ such that $\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}.$ What is the sum of the digits of $n?$ | 13 | 1. **Rewrite the equation using logarithmic identities:**
Given the equation \(\log_2{(\log_{16}{n})} = \log_4{(\log_4{n})}\), we use the identity \(\log_{a^b} c = \frac{1}{b} \log_a c\) to simplify the bases:
\[
\log_2{(\log_{2^4}{n})} = \log_{2^2}{(\log_{2^2}{n})}
\]
which simplifies to:
\[
\log_... |
A man has $2.73 in pennies, nickels, dimes, quarters and half dollars. If he has an equal number of coins of each kind, then the total number of coins he has is | 15 | Let $x$ represent the number of each type of coin the man has. Since he has pennies, nickels, dimes, quarters, and half dollars, and he has an equal number of each, we can set up the following equation to represent the total value of his coins in cents:
- Pennies contribute $1x$ cents.
- Nickels contribute $5x$ cents.... |
An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if th... | 191 | To solve this problem, we need to determine the maximum number of cables that can be used such that every employee can communicate with each other, under the constraint that cables can only connect a brand A computer to a brand B computer.
#### Step 1: Understand the problem constraints
- There are 30 employees: 20 wi... |
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c, a \neq 0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}$.) | 4 | We are given a quadratic polynomial $ax^2 + bx + c = 0$ with real coefficients, and we need to find the number of such polynomials where the set of roots equals the set of coefficients $\{a, b, c\}$. We will consider different cases based on the possible equalities among $a$, $b$, and $c$.
#### Case 1: $a = b = r$ and... |
Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is $47.$ What is the digit represented by $... | 5 | 1. **Identify the sums along each line**: Given the points $A, B, C, D, E, F$, each representing a unique digit from 1 to 6, and the lines connecting them, we can write the sums for each line as:
- Line through $A, B, C$: $A + B + C$
- Line through $A, E, F$: $A + E + F$
- Line through $C, D, E$: $C + D + E$
... |
The perimeter of an equilateral triangle exceeds the perimeter of a square by $1989 \text{ cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d \text{ cm}$. The square has perimeter greater than 0. How many positive integers are NOT possible value for $d$? | 663 | 1. **Set up the equations based on the problem statement:**
Let $t$ be the length of each side of the equilateral triangle, and $s$ be the length of each side of the square. The perimeter of the triangle is $3t$ and the perimeter of the square is $4s$. According to the problem, the perimeter of the triangle exceeds ... |
If $S=1!+2!+3!+\cdots +99!$, then the units' digit in the value of S is: | 3 | 1. **Identify the relevant factorials**: We need to find the units' digit of the sum $S = 1! + 2! + 3! + \cdots + 99!$. We start by observing the units' digits of the first few factorials:
- $1! = 1$
- $2! = 2$
- $3! = 6$
- $4! = 24$
- $5! = 120$
- $6! = 720$
- and so on.
2. **Observation about un... |
Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch? | 20 | 1. **Identify the dimensions of the index card**: Menkara has a $4 \times 6$ index card. The two possible dimensions are 4 inches and 6 inches.
2. **Calculate the initial area of the card**:
\[
\text{Area} = \text{Length} \times \text{Width} = 4 \times 6 = 24 \text{ square inches}
\]
3. **Scenario when one ... |
If it is known that $\log_2(a)+\log_2(b) \ge 6$, then the least value that can be taken on by $a+b$ is: | 16 | 1. **Use the logarithm property of addition**:
Given $\log_2(a) + \log_2(b) \geq 6$, we can apply the logarithmic property that states $\log_b(x) + \log_b(y) = \log_b(xy)$ for any base $b$. Thus,
\[
\log_2(a) + \log_2(b) = \log_2(ab).
\]
Therefore, we have:
\[
\log_2(ab) \geq 6.
\]
2. **Expone... |
In a given arithmetic sequence the first term is $2$, the last term is $29$, and the sum of all the terms is $155$. The common difference is: | 3 | Let's denote the first term of the arithmetic sequence as $a = 2$, the common difference as $d$, and the number of terms as $n$. The last term, which is also the $n$-th term, is given as $29$. The sum of all terms is $155$.
1. **Expression for the $n$-th term:**
The $n$-th term of an arithmetic sequence can be expr... |
According to the standard convention for exponentiation,
\[2^{2^{2^{2}}} = 2^{(2^{(2^2)})} = 2^{16} = 65536.\]
If the order in which the exponentiations are performed is changed, how many other values are possible? | 1 | To solve this problem, we need to evaluate the expression $2^{2^{2^{2}}}$ with different parenthesizations and determine how many distinct values can be obtained.
The expression $2^{2^{2^{2}}}$ can be parenthesized in the following ways:
1. $2^{(2^{(2^2)})}$
2. $2^{((2^2)^2)}$
3. $((2^2)^2)^2$
4. $(2^{(2^2)})^2$
5. $(... |
A wooden cube $n$ units on a side is painted red on all six faces and then cut into $n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $n$? | 4 | 1. **Understanding the problem**: We start with a cube of side length $n$, which is painted on all six faces. This cube is then cut into $n^3$ smaller unit cubes. Each unit cube has 6 faces. We need to find $n$ such that exactly one-fourth of the total number of faces of all the unit cubes are red.
2. **Total number o... |
Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain? | 280 | 1. **Calculate the volume of the original rectangular fort**:
The fort is built using the dimensions given as $12$ feet in length, $10$ feet in width, and $5$ feet in height. The volume of a rectangular prism is calculated by multiplying its length, width, and height. Therefore, the volume of the original fort is:
... |
John is walking east at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up to John? | 30 | 1. **Determine the relative speed of Bob with respect to John**: Since both are moving east, we subtract John's speed from Bob's speed to find the rate at which the distance between them is closing.
\[
\text{Relative speed} = 5 \text{ mph (Bob's speed)} - 3 \text{ mph (John's speed)} = 2 \text{ mph}
\]
2. **C... |
Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$? | 120 | 1. **Identify the length of each side of the rhombus**: Given that the perimeter of rhombus $ABCD$ is $52$ meters, and knowing that a rhombus has four sides of equal length, we calculate the length of each side as follows:
\[
\text{Side length} = \frac{\text{Perimeter}}{4} = \frac{52}{4} = 13 \text{ meters}
\]... |
The number of the distinct solutions to the equation $|x-|2x+1||=3$ is | 2 | To solve the equation $|x - |2x + 1|| = 3$, we need to consider the different cases that arise from the absolute values.
#### Step 1: Analyze the inner absolute value
The expression $|2x + 1|$ can be simplified by considering two cases:
- **Case 1:** $2x + 1 \geq 0 \Rightarrow x \geq -\frac{1}{2}$
- Here, $|2x + 1| ... |
After Euclid High School's last basketball game, it was determined that $\frac{1}{4}$ of the team's points were scored by Alexa and $\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the total number of points scored by the other $... | 11 | Let $x$ be the total number of points scored by the team. According to the problem, the points can be broken down as follows:
- Alexa scored $\frac{1}{4}x$ points.
- Brittany scored $\frac{2}{7}x$ points.
- Chelsea scored $15$ points.
- The other $7$ team members scored $y$ points in total.
The total points scored by ... |
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