problem stringlengths 11 5.15k | answer stringlengths 1 42 | solution stringlengths 0 8.75k |
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The sides $PQ$ and $PR$ of triangle $PQR$ are respectively of lengths $4$ inches, and $7$ inches. The median $PM$ is $3\frac{1}{2}$ inches. Then $QR$, in inches, is: | 9 | 1. **Using the Median Formula**: The formula for the length of a median in a triangle, which connects a vertex to the midpoint of the opposite side, is given by:
\[
PM = \frac{1}{2}\sqrt{2PQ^2 + 2PR^2 - QR^2}
\]
where $PQ$, $PR$, and $QR$ are the lengths of the sides of the triangle, and $PM$ is the median ... |
The mean of three numbers is $10$ more than the least of the numbers and $15$ less than the greatest. The median of the three numbers is $5$. What is their sum? | 30 | 1. Let the three numbers be $x$, $y$, and $z$. Arrange them in ascending order, so $x \leq y \leq z$. Given that the median of the three numbers is $5$, we have $y = 5$.
2. The problem states that the mean of the numbers is $10$ more than the least of the numbers and $15$ less than the greatest. This translates to the... |
The $y$-intercepts, $P$ and $Q$, of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of $\triangle APQ$? | 60 | 1. **Identify the properties of the lines**: Given that the lines are perpendicular and intersect at $A(6,8)$, we can denote the equations of the lines as $y = m_1x + b_1$ and $y = m_2x + b_2$. Since the lines are perpendicular, their slopes satisfy $m_1 \cdot m_2 = -1$.
2. **Determine the $y$-intercepts**: The proble... |
Let ($a_1$, $a_2$, ... $a_{10}$) be a list of the first 10 positive integers such that for each $2 \le i \le 10$ either $a_i + 1$ or $a_i - 1$ or both appear somewhere before $a_i$ in the list. How many such lists are there? | 512 | To solve this problem, we need to understand the constraints on the sequence and how they affect the arrangement of the numbers. The key constraint is that for each $2 \leq i \leq 10$, either $a_i + 1$ or $a_i - 1$ (or both) must appear before $a_i$ in the list. This constraint significantly restricts the order in whic... |
Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What is $m+n$? | 18 | 1. **Assign Variables and Use the Angle Bisector Theorem**:
Let $AC = x$. By the Angle Bisector Theorem, we have:
\[
\frac{AB}{AC} = \frac{BD}{CD} \implies \frac{10}{x} = \frac{BD}{3}
\]
Solving for $BD$, we get:
\[
BD = \frac{10}{x} \times 3 = \frac{30}{x}
\]
2. **Apply the Triangle Inequality... |
Joe has a collection of $23$ coins, consisting of $5$-cent coins, $10$-cent coins, and $25$-cent coins. He has $3$ more $10$-cent coins than $5$-cent coins, and the total value of his collection is $320$ cents. How many more $25$-cent coins does Joe have than $5$-cent coins? | 2 | Let $n$, $d$, and $q$ represent the number of $5$-cent coins, $10$-cent coins, and $25$-cent coins in Joe's collection, respectively. We are given the following equations based on the problem statement:
\[
n + d + q = 23 \quad \text{(1)}
\]
\[
5n + 10d + 25q = 320 \quad \text{(2)}
\]
\[
d = n + 3 \quad \text{(3)}
\]
F... |
In the expansion of $(a + b)^n$ there are $n + 1$ dissimilar terms. The number of dissimilar terms in the expansion of $(a + b + c)^{10}$ is: | 66 | To find the number of dissimilar terms in the expansion of $(a + b + c)^{10}$, we can use the concept of combinations to determine the number of distinct terms generated by the expansion.
1. **Understanding the Expansion**:
The expression $(a + b + c)^{10}$ can be expanded using the multinomial theorem. Each term i... |
Set $u_0 = \frac{1}{4}$, and for $k \ge 0$ let $u_{k+1}$ be determined by the recurrence
\[u_{k+1} = 2u_k - 2u_k^2.\]This sequence tends to a limit; call it $L$. What is the least value of $k$ such that
\[|u_k-L| \le \frac{1}{2^{1000}}?\] | 10 | 1. **Define the sequence and transformation**: Given the recurrence relation:
\[
u_{k+1} = 2u_k - 2u_k^2
\]
We start with $u_0 = \frac{1}{4}$. To simplify the recurrence relation, we perform a transformation by defining $v_k = 2u_k - 1$. Substituting $u_k = \frac{v_k + 1}{2}$ into the recurrence relation, w... |
Let $a_1, a_2, \dots, a_k$ be a finite arithmetic sequence with $a_4 + a_7 + a_{10} = 17$ and $a_4 + a_5 + \dots + a_{13} + a_{14} = 77$. If $a_k = 13$, then $k = $ | 18 | 1. **Identify the common difference and terms in terms of $a_7$:**
Given the arithmetic sequence $a_1, a_2, \ldots, a_k$, we know that $a_n = a_1 + (n-1)d$ where $d$ is the common difference. For the terms $a_4, a_7, a_{10}$, we can express them in terms of $a_7$:
\[
a_4 = a_7 - 3d, \quad a_7 = a_7, \quad a_{1... |
What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$? | 970 | To find the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$, we start by simplifying the expression.
1. **Expand $(\sqrt{3}+\sqrt{2})^6$ using the binomial theorem:**
\[
(\sqrt{3}+\sqrt{2})^6 = \sum_{k=0}^6 \binom{6}{k} (\sqrt{3})^{6-k} (\sqrt{2})^k
\]
Here, $\binom{6}{k}$ is the binomial coefficient,... |
Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips? | 5 | 1. **Identify the chips in each bag:**
- Bag A contains odd numbers: 1, 3, 5.
- Bag B contains even numbers: 2, 4, 6.
2. **Understand the nature of sums:**
- The sum of an odd number (from Bag A) and an even number (from Bag B) is always odd.
3. **Calculate the possible sums:**
- The smallest sum is obtai... |
When $x^9-x$ is factored as completely as possible into polynomials and monomials with integral coefficients, the number of factors is: | 5 | 1. **Factor out the greatest common factor**:
The polynomial $x^9 - x$ can be rewritten by factoring out the common factor of $x$, giving:
\[
x^9 - x = x(x^8 - 1)
\]
2. **Factorize $x^8 - 1$ using difference of squares**:
The expression $x^8 - 1$ can be factored using the difference of squares formula,... |
Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? | 45 | 1. **Identify the given information and the goal:** We are given a rectangle $ABCD$ with $AB = 6$ and $BC = 3$. We need to find the degree measure of $\angle AMD$ given that $\angle AMD = \angle CMD$.
2. **Assign variables and use the Pythagorean Theorem:** Let $AM = x$. Then, $MB = 6 - x$. Since $D$ and $C$ are on th... |
In $\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$? | 61 | 1. **Assign Variables:**
Let $x$ represent $CX$, and let $y$ represent $BX$. Since the circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$, we have $AB = AX = 86$.
2. **Apply Stewart's Theorem:**
Stewart's Theorem states that for a point $X$ on side $BC$ of $\triangle ABC$, th... |
For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$? | 14 | 1. **Identify the form of the terms**: We are interested in terms that include all four variables $a, b, c, d$ each raised to some positive power in the expansion of $(a+b+c+d+1)^N$. This means we are looking for terms of the form $a^x b^y c^z d^w 1^t$ where $x, y, z, w > 0$ and $x + y + z + w + t = N$.
2. **Adjust th... |
The data set $[6, 19, 33, 33, 39, 41, 41, 43, 51, 57]$ has median $Q_2 = 40$, first quartile $Q_1 = 33$, and third quartile $Q_3 = 43$. An outlier in a data set is a value that is more than $1.5$ times the interquartile range below the first quartle ($Q_1$) or more than $1.5$ times the interquartile range above the thi... | 1 | 1. **Identify the Interquartile Range (IQR):**
The interquartile range (IQR) is calculated as $Q_3 - Q_1$. Given $Q_1 = 33$ and $Q_3 = 43$, we have:
\[
\text{IQR} = Q_3 - Q_1 = 43 - 33 = 10.
\]
2. **Calculate the Outlier Thresholds:**
Outliers are defined as values that are more than $1.5 \times \text{I... |
For the positive integer $n$, let $\langle n\rangle$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\langle 4\rangle=1+2=3$ and $\langle 12 \rangle =1+2+3+4+6=16$. What is $\langle\langle\langle 6\rangle\rangle\rangle$? | 6 | To solve for $\langle\langle\langle 6\rangle\rangle\rangle$, we need to first determine $\langle 6\rangle$, then $\langle \langle 6 \rangle \rangle$, and finally $\langle \langle \langle 6 \rangle \rangle \rangle$.
1. **Calculate $\langle 6 \rangle$:**
The divisors of $6$ are $1, 2, 3, 6$. The sum of all divisors e... |
The limit of $\frac {x^2-1}{x-1}$ as $x$ approaches $1$ as a limit is: | 2 | To find the limit of the function $\frac{x^2-1}{x-1}$ as $x$ approaches 1, we start by simplifying the expression.
1. **Factorize the Numerator**:
The numerator $x^2 - 1$ is a difference of squares, which can be factored as:
\[
x^2 - 1 = (x + 1)(x - 1)
\]
2. **Simplify the Expression**:
The expression... |
How many different four-digit numbers can be formed by rearranging the four digits in $2004$? | 6 | To find the number of different four-digit numbers that can be formed by rearranging the digits in $2004$, we need to consider the repetitions of the digits and the restrictions on the arrangement.
1. **Identify the digits and their repetitions**: The digits in $2004$ are $2$, $0$, $0$, and $4$. Here, the digit $0$ is... |
On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? | 13 | The AMC 8 scoring system is structured as follows:
- Each correct answer awards 1 point.
- Each incorrect answer awards 0 points.
- Questions left unanswered also award 0 points.
Given the problem statement:
- Billy answers 13 questions correctly.
- Billy answers 7 questions incorrectly.
- Billy does not answer 5 ques... |
One of the sides of a triangle is divided into segments of $6$ and $8$ units by the point of tangency of the inscribed circle. If the radius of the circle is $4$, then the length of the shortest side is | 12 | 1. **Identify the segments created by the inscribed circle**: Let's label the triangle as $\triangle ABC$ with the inscribed circle touching $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$ respectively. Given that one side is divided into segments of $6$ and $8$ units by the point of tangency, let's assume this side i... |
Estimate the year in which the population of Nisos will be approximately 6,000. | 2075 |
We are given that the population of Nisos triples every 25 years, starting from a population of 200 in the year 2000. We need to estimate the year when the population will be approximately 6000.
#### Solution 1: Using a tabular approach
1. **Start with the initial population and year:**
- Year: 2000, Population: 2... |
Let n be the number of real values of $p$ for which the roots of
$x^2-px+p=0$
are equal. Then n equals: | 2 | 1. **Identify the condition for equal roots:** For a quadratic equation $Ax^2 + Bx + C = 0$ to have equal roots, the discriminant must be zero. The discriminant is given by $\Delta = B^2 - 4AC$.
2. **Apply the condition to the given equation:** The given quadratic equation is $x^2 - px + p = 0$. Here, $A = 1$, $B = -p... |
A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils ... | 11 | 1. **Define Variables:**
Let $s$ be the number of students that bought pencils, $c$ be the cost of each pencil in cents, and $n$ be the number of pencils each student bought.
2. **Analyze Given Information:**
- A majority of the 30 students bought pencils, so $s > \frac{30}{2} = 15$.
- Each student bought mor... |
Sides $AB$, $BC$, and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4$, $5$, and $20$, respectively.
If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B = \frac{3}{5}$, then side $AD$ has length
A polygon is called “simple” if it is not self intersecting. | 25 | 1. **Given Information and Angle Relationships**:
- Sides $AB = 4$, $BC = 5$, and $CD = 20$.
- Angles $B$ and $C$ are obtuse.
- $\sin C = -\cos B = \frac{3}{5}$.
Since $B$ and $C$ are obtuse, we have:
\[
\sin(180^\circ - C) = \sin C = \frac{3}{5} \quad \text{and} \quad \cos(180^\circ - B) = -\cos B =... |
In the expansion of $\left(a - \dfrac{1}{\sqrt{a}}\right)^7$ the coefficient of $a^{-\frac{1}{2}}$ is: | -21 | 1. **Identify the general term in the binomial expansion**:
The binomial expansion of $\left(a-\dfrac{1}{\sqrt{a}}\right)^7$ can be expressed using the Binomial Theorem, which states that:
\[
(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k
\]
Here, $x = a$ and $y = -\dfrac{1}{\sqrt{a}}$. Thus, the gene... |
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? | 70 | 1. **Identify the fraction of pink and red flowers**:
Given that six tenths (or $\frac{6}{10}$) of the flowers are pink, we can simplify this fraction to $\frac{3}{5}$. Consequently, the remaining flowers must be red, which is $\frac{2}{5}$ of the total flowers (since the total must sum to 1, or $\frac{5}{5}$).
2.... |
If the value of $20$ quarters and $10$ dimes equals the value of $10$ quarters and $n$ dimes, then $n=$ | 35 | 1. **Set up the equation based on the problem statement:**
The problem states that the value of $20$ quarters and $10$ dimes is equal to the value of $10$ quarters and $n$ dimes. We know that the value of a quarter is $25$ cents and the value of a dime is $10$ cents. Therefore, we can set up the equation:
\[
... |
How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}$? | 13 | 1. **Identify the set and its properties**: The set given is $\{1, 4, 7, 10, 13, 16, 19\}$. This set is an arithmetic sequence where each term increases by 3 from the previous term.
2. **Determine the range of possible sums**: We need to find the sums of three distinct elements from the set. The smallest sum occurs wh... |
Given the progression $10^{\frac{1}{11}}, 10^{\frac{2}{11}}, 10^{\frac{3}{11}}, 10^{\frac{4}{11}},\dots , 10^{\frac{n}{11}}$. The least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$ is | 11 | 1. **Identify the General Term and Product of the Sequence**:
The given sequence is $10^{\frac{1}{11}}, 10^{\frac{2}{11}}, 10^{\frac{3}{11}}, \dots, 10^{\frac{n}{11}}$. The product of the first $n$ terms of this sequence can be expressed as:
\[
10^{\frac{1}{11}} \cdot 10^{\frac{2}{11}} \cdot 10^{\frac{3}{11}... |
Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian? | 150 | 1. **Define Variables:**
Let $d$ be the distance Ian drove, $t$ be the time Ian drove, and $s$ be Ian's speed. Thus, Ian's equation is:
\[
d = s \cdot t
\]
This is equation (1).
2. **Set Up Han's Equation:**
Han drove 1 hour longer than Ian and at a speed 5 mph faster. Han also drove 70 miles more th... |
The number of real values of $x$ satisfying the equation $2^{2x^2 - 7x + 5} = 1$ is: | 2 | 1. **Identify the Equation Form**: The given equation is $2^{2x^2 - 7x + 5} = 1$. We know that any number raised to the power of zero equals one, i.e., $a^0 = 1$ for any $a \neq 0$. Therefore, we need $2x^2 - 7x + 5 = 0$.
2. **Solve the Quadratic Equation**: The quadratic equation is $2x^2 - 7x + 5 = 0$. We can attemp... |
Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,
\[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k... | 1341 | To solve this problem, we need to understand the behavior of the sequence $\{a_k\}_{k=1}^{2011}$ and determine when $a_k = b_k$, where $\{b_k\}_{k=1}^{2011}$ is the sequence $\{a_k\}_{k=1}^{2011}$ rearranged in decreasing order.
#### Step 1: Analyze the sequence $\{a_k\}_{k=1}^{2011}$
The sequence is defined recursive... |
Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers $d$, $e$, and $10d+e$, where $d$ and $e$ are single digits. What is the sum of the digits of $n$? | 12 | 1. **Identify the possible values for $d$ and $e$:**
- Since $d$ and $e$ are single-digit prime numbers, the possible values for $d$ are $\{2, 3, 5, 7\}$.
- Since $e$ is also a single-digit prime number and must be the units digit of the prime number $10d + e$, the possible values for $e$ are $\{3, 7\}$ (as these... |
The number of triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations
$ab+bc=44$
$ac+bc=23$
is | 2 | 1. **Factor the second equation**: Given $ac + bc = 23$, we can factor out $c$:
\[
c(a + b) = 23
\]
Since $23$ is a prime number, the possible values for $c$ are $1$ and $23$.
2. **Analyze the possible values of $c$**:
- If $c = 23$, then $a + b = \frac{23}{23} = 1$. However, $a$ and $b$ are positive in... |
Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? | 2 | Let $b$ be the number of bagels Jane bought, and $m$ be the number of muffins. Since Jane buys these items over a five-day workweek, we have the equation:
\[ b + m = 5 \]
The cost of a bagel is $75$ cents and the cost of a muffin is $50$ cents. The total cost for the week, in cents, is given by:
\[ 75b + 50m \]
We kn... |
Let $n$ be a $5$-digit number, and let $q$ and $r$ be the quotient and the remainder, respectively, when $n$ is divided by $100$. For how many values of $n$ is $q+r$ divisible by $11$? | 8181 |
1. **Understanding the Problem:**
Let $n$ be a 5-digit number. When $n$ is divided by 100, $q$ is the quotient and $r$ is the remainder. We need to find how many values of $n$ make $q+r$ divisible by 11.
2. **Expressing $n$ in terms of $q$ and $r$:**
Since $q$ is the quotient and $r$ is the remainder when $n$ i... |
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$ | 8 | 1. **Identify the Incorrect Sum**: First, we add the given numbers without changing any digits:
- $742586 + 829430 = 1572016$
- The provided sum is $1212016$.
2. **Analyze the Incorrectness**: The provided sum $1212016$ differs significantly from the actual sum $1572016$. We need to change one digit $d$ to anoth... |
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$? | 12 | 1. **Count the number of factors of 10 in $90!$:**
The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:
\[
\left\lfloor \frac{90}{5} \right\rfloor + \left\lf... |
Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? | 120 | 1. **Understanding the problem**: We are given that Brenda received $36$ votes, which represents $\frac{3}{10}$ of the total votes in the election.
2. **Calculating the total number of votes**:
- Since $36$ votes is $\frac{3}{10}$ of the total votes, we can find the total number of votes by setting up the equation:... |
For how many positive integers $n$ less than or equal to $24$ is $n!$ evenly divisible by $1 + 2 + \cdots + n?$ | 16 | 1. **Sum of the first $n$ positive integers**: The sum of the first $n$ positive integers is given by the formula:
\[
1 + 2 + \cdots + n = \frac{n(n+1)}{2}
\]
This formula can be derived by pairing terms from the beginning and end of the sequence, each pair summing to $n+1$, and noting there are $\frac{n}{2... |
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order)... | 80 |
To solve this problem, we need to ensure that the average after each score is entered is an integer. This means that the sum of the scores entered at each step must be divisible by the number of scores entered up to that step.
Given scores in ascending order: $71, 76, 80, 82, 91$.
#### Step 1: Calculate the total su... |
Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle? | 125 | 1. **Understanding the Martian Measurement**: A full circle on Mars is measured as $500$ clerts.
2. **Relating Martian Measurement to Earth Measurement**: On Earth, a full circle is $360^\circ$. A right angle, which is $90^\circ$, is $\frac{1}{4}$ of a full circle.
3. **Calculating Clerts in a Right Angle**: Since a ... |
Quadrilateral $ABCD$ is inscribed in a circle with $\angle BAC=70^{\circ}, \angle ADB=40^{\circ}, AD=4,$ and $BC=6$. What is $AC$? | 6 | 1. **Identify Angles Subtended by the Same Arc**:
Since $\angle ADB$ and $\angle ACB$ are both subtended by the same arc $AB$ in the circle, by the Inscribed Angle Theorem, we have:
\[
\angle ACB = \angle ADB = 40^\circ.
\]
2. **Calculate $\angle ABC$ in $\triangle ABC$**:
We know that the sum of an... |
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.
- To find the maximum... |
For positive integers $n$, denote $D(n)$ by the number of pairs of different adjacent digits in the binary (base two) representation of $n$. For example, $D(3) = D(11_{2}) = 0$, $D(21) = D(10101_{2}) = 4$, and $D(97) = D(1100001_{2}) = 2$. For how many positive integers less than or equal to $97$ does $D(n) = 2$? | 26 | To solve for the number of positive integers less than or equal to $97$ for which $D(n) = 2$, we analyze the binary representations of numbers and count those with exactly two transitions between adjacent digits (from 0 to 1 or from 1 to 0).
#### Case Analysis:
For $D(n) = 2$, the binary representation of $n$ must hav... |
How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$? | 2 | To solve the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ over the interval $[0,\pi]$, we analyze the behavior of both sides of the equation.
#### Step 1: Evaluate the functions at critical points
We start by evaluating the functions at the endpoints and the midpoint of the in... |
How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017}=\log_b(a^{2017})?$ | 597 | 1. **Rewrite the given equation using logarithmic identities:**
The given equation is $(\log_b a)^{2017} = \log_b(a^{2017})$. By the power rule of logarithms, $\log_b(a^{2017}) = 2017 \log_b a$. Thus, the equation simplifies to:
\[
(\log_b a)^{2017} = 2017 \log_b a
\]
2. **Introduce a substitution:**
Le... |
The region consisting of all points in three-dimensional space within 3 units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $\textit{AB}$? | 20 | 1. **Understanding the Geometry**: The region described is a cylinder with hemispheres capping both ends. The radius of the cylinder and hemispheres is $3$ units.
2. **Volume of Hemispheres**: Each hemisphere has a radius of $3$ units. The volume $V$ of a sphere is given by $V = \frac{4}{3}\pi r^3$. Therefore, the vol... |
The square
$\begin{tabular}{|c|c|c|} \hline 50 & \textit{b} & \textit{c} \\ \hline \textit{d} & \textit{e} & \textit{f} \\ \hline \textit{g} & \textit{h} & 2 \\ \hline \end{tabular}$
is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries ... | 35 | 1. **Identify the products of rows, columns, and diagonals:** Given the multiplicative magic square, we know that the product of the numbers in each row, column, and diagonal must be the same. Let's denote this common product by $P$. Thus, we have:
- $50 \cdot b \cdot c = P$
- $d \cdot e \cdot f = P$
- $g \cdo... |
A $2$ by $2$ square is divided into four $1$ by $1$ squares. Each of the small squares is to be painted either green or red. In how many different ways can the painting be accomplished so that no green square shares its top or right side with any red square? There may be as few as zero or as many as four small green... | 6 | To solve this problem, we need to consider all possible configurations of green and red squares in the $2 \times 2$ grid such that no green square shares its top or right side with a red square. We will analyze each case based on the number of green squares.
#### Case 1: No green squares
- All squares are red.
- **Num... |
(1901 + 1902 + 1903 + \cdots + 1993) - (101 + 102 + 103 + \cdots + 193) = | 167400 | 1. **Identify the pattern in the sequences**:
The problem involves two sums of arithmetic sequences. The first sequence starts at 1901 and ends at 1993, and the second sequence starts at 101 and ends at 193.
2. **Recognize the relationship between the sequences**:
Each term in the first sequence can be expresse... |
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? | 12 | 1. **Identify the possible placements for $R_5$ and $B_5$:**
- Since $R_5$ divides $B_5$ and $B_5 = 5$ is the only blue card divisible by $R_5 = 5$, $R_5$ must be adjacent to $B_5$.
- $B_5$ cannot be adjacent to any other red card except $R_1$ and $R_5$ because $5$ is not divisible by $2, 3,$ or $4$.
- Thus, $... |
How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$? | 6 | 1. **Use the Pythagorean Theorem**: Given a right triangle with legs $a$ and $b$ and hypotenuse $b+1$, we apply the Pythagorean theorem:
\[
a^2 + b^2 = (b+1)^2
\]
Expanding the square on the right side, we get:
\[
a^2 + b^2 = b^2 + 2b + 1
\]
Simplifying this, we find:
\[
a^2 = 2b + 1
\]... |
How many $3$-digit positive integers have digits whose product equals $24$? | 21 | To find how many 3-digit positive integers have digits whose product equals 24, we need to consider all possible sets of three digits (where each digit is between 1 and 9) that multiply together to give 24. We also need to consider the different permutations of these digits.
1. **Identify possible sets of digits:**
... |
Bricklayer Brenda would take $9$ hours to build a chimney alone, and bricklayer Brandon would take $10$ hours to build it alone. When they work together they talk a lot, and their combined output is decreased by $10$ bricks per hour. Working together, they build the chimney in $5$ hours. How many bricks are in the chim... | 900 | 1. **Define Variables:**
Let $h$ be the total number of bricks in the chimney.
2. **Individual Rates:**
- Brenda's rate of laying bricks is $\frac{h}{9}$ bricks per hour.
- Brandon's rate of laying bricks is $\frac{h}{10}$ bricks per hour.
3. **Combined Rate with Decreased Output:**
When Brenda and Brando... |
For all non-zero real numbers $x$ and $y$ such that $x-y=xy$, $\frac{1}{x}-\frac{1}{y}$ equals | -1 | 1. **Start with the given equation and manipulate it:**
Given that \(x - y = xy\), we can rearrange this equation to:
\[
x - y - xy = 0
\]
Adding 1 to both sides, we get:
\[
x - y - xy + 1 = 1
\]
Factoring the left-hand side, we have:
\[
(x-1)(y+1) = 1
\]
2. **Analyze the expression... |
Jeremy's father drives him to school in rush hour traffic in 20 minutes. One day there is no traffic, so his father can drive him 18 miles per hour faster and gets him to school in 12 minutes. How far in miles is it to school? | 9 |
Let's denote the distance to school as $d$ miles and the usual speed during rush hour as $v$ miles per hour.
1. **Convert time to hours**:
- The time taken in rush hour traffic is 20 minutes, which is $\frac{20}{60} = \frac{1}{3}$ hours.
- The time taken without traffic is 12 minutes, which is $\frac{12}{60} ... |
Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer $S$. For example, if $N = 749$, Bernardo writes the numbers $1... | 25 | 1. **Understanding the Problem**: Bernardo writes the base-5 and base-6 representations of a three-digit number $N$ on a blackboard. LeRoy, treating these numbers as base-10 integers, adds them to get $S$. We need to find how many such $N$ exist such that the two rightmost digits of $S$ are the same as those of $2N$.
... |
Find the sum of the squares of all real numbers satisfying the equation $x^{256} - 256^{32} = 0$. | 8 | 1. **Identify the form of the equation**: The given equation is $x^{256} - 256^{32} = 0$. We can rewrite this as $x^{256} = 256^{32}$.
2. **Simplify the equation**: Notice that $256 = 2^8$, so $256^{32} = (2^8)^{32} = 2^{256}$. Thus, the equation becomes $x^{256} = 2^{256}$.
3. **Roots of the equation**: The equation... |
The number $2013$ is expressed in the form $2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$? | 2 | 1. **Prime Factorization of 2013**:
The prime factorization of $2013$ is $2013 = 3 \times 11 \times 61$.
2. **Form of the Expression**:
We need to express $2013$ in the form $\frac{a_1!a_2!\cdots a_m!}{b_1!b_2!\cdots b_n!}$, where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positiv... |
The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? | 70 | 1. **Identify the highest and lowest prices:** From the problem, we know that the highest price of gasoline was $17 in Month 1, and the lowest price was $10 in Month 3.
2. **Calculate the percentage increase from the lowest to the highest price:** To find by what percent the highest price ($17) is more than the lowest... |
Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing? | 7 | 1. **Identify the number of songs each girl sang**:
- Hanna sang 7 songs.
- Mary sang 4 songs.
- Alina and Tina sang more than Mary but less than Hanna, so they sang either 5 or 6 songs each.
2. **Set up the equation for total songs**:
- Let $a$ be the number of songs Alina sang and $t$ be the number of s... |
Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is
[asy]
draw((2.7,3.99)--(0,3)--(0,0));
draw((3.7,3.99)--(1,3)--(1,0));
draw((4.7,3.99)--(2,3)--(2,0));
draw((5.7,3.99)--(3,3)--(3,0));
draw((0,0)--(3,0)--(5.7,0.99));
draw((0,1)--(3,1... | 54 | 1. **Calculate the original surface area of the cube**:
The original cube has dimensions $3\text{ cm} \times 3\text{ cm} \times 3\text{ cm}$. Each face of the cube is a square with an area of $3^2 = 9\text{ cm}^2$. Since a cube has 6 faces, the total surface area of the cube is:
\[
6 \times 9 = 54\text{ cm}^2... |
Find the smallest integer $n$ such that $(x^2+y^2+z^2)^2 \le n(x^4+y^4+z^4)$ for all real numbers $x,y$, and $z$. | 3 | To find the smallest integer $n$ such that $(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)$ holds for all real numbers $x, y, z$, we can utilize the Cauchy-Schwarz inequality in its sum of squares form.
1. **Applying Cauchy-Schwarz Inequality:**
The Cauchy-Schwarz inequality states that for any real numbers $a_1, a_2, a_3, b_1,... |
Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively.
A third tangent intersects segment $AB$ in $P$ and $AC$ in $R$, and touches the circle at $Q$. If $AB=20$, then the perimeter of $\triangle APR$ is | 40 | 1. **Identify Tangent Properties**: From the problem, we know that $AB$ and $AC$ are tangents from point $A$ to the circle, touching the circle at points $B$ and $C$ respectively. By the tangent properties, $AB = AC$.
2. **Tangent Lengths**: Since $AB = 20$, we also have $AC = 20$.
3. **Third Tangent Properties**: Th... |
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $y=p(x)$ and $y=q(x)$, each with leading coefficient 1? | 3 | 1. **Identify the Problem**: We need to find the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $y=p(x)$ and $y=q(x)$, where both have leading coefficients of 1.
2. **Set Up the Equation for Intersections**: The points of intersection of the graphs of $p(x)$ ... |
A sign at the fish market says, "50% off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars? | 10 | 1. **Understanding the problem**: The problem states that half-pound packages of fish are being sold for $3 per package after a 50% discount. We need to find the regular price for a full pound of fish.
2. **Setting up the equation**: Let $x$ be the regular price for a full pound of fish. Since the fish is currently be... |
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.)
The number pick... | 1 | Let's denote the number picked by person $i$ as $a_i$. According to the problem, each person announces the average of the numbers picked by their two immediate neighbors. Therefore, if person $i$ announces $i$, the equation relating the numbers picked by their neighbors is:
\[
\frac{a_{i-1} + a_{i+1}}{2} = i
\]
which s... |
Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0,1,2\}$. How many equilateral triangles have all their vertices in $S$? | 80 |
#### Step 1: Understanding the Set $S$
The set $S$ consists of all points $(x,y,z)$ where $x, y, z \in \{0,1,2\}$. This forms a $3 \times 3 \times 3$ grid of points in three-dimensional space.
#### Step 2: Possible Distances Between Points
Using the three-dimensional distance formula, the distance $d$ between any two... |
Call a positive real number special if it has a decimal representation that consists entirely of digits $0$ and $7$. For example, $\frac{700}{99}= 7.\overline{07}= 7.070707\cdots$ and $77.007$ are special numbers. What is the smallest $n$ such that $1$ can be written as a sum of $n$ special numbers? | 8 | 1. **Understanding the Problem**: We need to find the smallest number of special numbers (numbers with only the digits $0$ and $7$ in their decimal representation) that can sum up to $1$.
2. **Relating to Super-Special Numbers**: We introduce the concept of super-special numbers, which are numbers with only $0$s and $... |
For how many three-digit whole numbers does the sum of the digits equal $25$? | 6 | 1. **Identify the range of digits**: Since we are dealing with three-digit numbers, the digits range from $1$ to $9$ for the hundreds place and $0$ to $9$ for the tens and units places.
2. **Set up the equation for the sum of the digits**: Let the digits of the number be $a$, $b$, and $c$. We need to find the number o... |
The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=4$ is: | 2 | 1. **Identify the curves**: The given equations are $x^2 + 4y^2 = 1$ and $4x^2 + y^2 = 4$. These represent an ellipse and another ellipse respectively.
2. **Substitute variables**: Let $a = x^2$ and $b = y^2$. This substitution transforms the equations into:
\[
a + 4b = 1 \quad \text{(1)}
\]
\[
4a + b =... |
The angles in a particular triangle are in arithmetic progression, and the side lengths are $4, 5, x$. The sum of the possible values of x equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is $a+b+c$? | 36 | 1. **Identify the angles in the triangle**: Since the angles are in arithmetic progression and their sum is $180^\circ$, let the angles be $\alpha - d$, $\alpha$, and $\alpha + d$. Solving $\alpha - d + \alpha + \alpha + d = 180^\circ$ gives $3\alpha = 180^\circ$, so $\alpha = 60^\circ$. Thus, the angles are $60^\circ ... |
Starting with some gold coins and some empty treasure chests, I tried to put $9$ gold coins in each treasure chest, but that left $2$ treasure chests empty. So instead I put $6$ gold coins in each treasure chest, but then I had $3$ gold coins left over. How many gold coins did I have? | 45 | Let $n$ be the number of gold coins and $c$ be the number of treasure chests.
1. **Analyzing the first condition:**
- If we try to put $9$ gold coins in each chest, but $2$ chests remain empty, then the number of chests that actually contain coins is $c - 2$. The total number of coins is then $9(c - 2)$.
- This ... |
A speaker talked for sixty minutes to a full auditorium. Twenty percent of the audience heard the entire talk and ten percent slept through the entire talk. Half of the remainder heard one third of the talk and the other half heard two thirds of the talk. What was the average number of minutes of the talk heard by memb... | 33 | 1. **Assumptions and Initial Setup**:
Let's assume there are $100$ people in the audience for simplicity of calculation. The total duration of the talk is $60$ minutes.
2. **Distribution of Audience**:
- **20% heard the entire talk**: This is $20\%$ of $100$ people, which equals $20$ people. Each of these heard ... |
In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of... | 5 | 1. **Calculate the total number of games**: In a round-robin tournament with 6 teams, each team plays against every other team exactly once. The total number of games played is given by the combination formula $\binom{n}{2}$, where $n$ is the number of teams. Thus, the total number of games is:
\[
\binom{6}{2} = ... |
$2.46 \times 8.163 \times (5.17 + 4.829)$ is closest to | 200 | To solve the problem, we will approximate the values and calculate the product.
1. **Approximate the values:**
- $2.46 \approx 2.5$
- $8.163 \approx 8$
- $5.17 + 4.829 = 10$ (This is exact since $5.17 + 4.829$ is designed to sum to $10$).
2. **Calculate the product using the approximations:**
- Multiply t... |
Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\circ}$. Thus the three polygons form a new polygon with $A$ as an interior point. What is the largest... | 21 | 1. **Identify the Problem Requirements:**
We need to find three non-overlapping regular polygons, at least two of which are congruent, that meet at a point $A$ such that the sum of their interior angles at $A$ is $360^\circ$. The sides of each polygon are of length $1$. We aim to maximize the perimeter of the new po... |
The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum? | 18 | 1. **Identify the total sum of numbers on the cube**: Each number from $1$ to $8$ is placed on a vertex of the cube. The sum of these numbers is $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8$. Using the formula for the sum of the first $n$ natural numbers, $\frac{n(n+1)}{2}$, where $n = 8$, we get:
\[
\frac{8 \times 9}{2} = 36
... |
Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can have? | 8072 | 1. **Define Variables:**
Let $p$ be the number of pennies Ricardo has. Since the rest of the coins are nickels, he has $2020 - p$ nickels.
2. **Constraints:**
Ricardo has at least one penny and at least one nickel. Therefore, we have the constraints:
\[
p \geq 1 \quad \text{and} \quad 2020 - p \geq 1
\]... |
Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren? | 146 | 1. **Identify the problem and apply the Principle of Inclusion-Exclusion (PIE):**
We need to find the number of days Mrs. Sanders does not receive a call from any of her grandchildren. Each grandchild calls her every 3, 4, and 5 days respectively. We use PIE to count the days she receives at least one call and subtr... |
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled? | 1462 |
To solve this problem, we need to determine how many miles the car has actually traveled when the odometer reads 002005, given that it skips every occurrence of the digit 4.
#### Method 1: Counting Skipped Numbers
1. **Counting Numbers with Digit 4**:
- **Hundreds Place**: There are 200 numbers from 0000 to 1999 ... |
For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ | 10 | To solve this problem, we need to understand the function $f_j(n)$ and its behavior as $j$ increases. We start by analyzing the function $f_1(n)$, which is defined as twice the number of positive integer divisors of $n$. We then recursively apply $f_1$ to its own outputs to determine $f_j(n)$ for $j \geq 2$.
#### Step... |
What is the value of $\sqrt{(3-2\sqrt{3})^2}+\sqrt{(3+2\sqrt{3})^2}$? | 6 | 1. **Define the expression and simplify each square root:**
Let $x = \sqrt{(3-2\sqrt{3})^2} + \sqrt{(3+2\sqrt{3})^2}$.
Since squaring a real number and then taking the square root gives the absolute value of the original number, we have:
\[
\sqrt{(3-2\sqrt{3})^2} = |3-2\sqrt{3}|
\]
\[
\sqrt{(3+2\s... |
Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is: | 48 | 1. Let the two numbers be $x$ and $y$. According to the problem, their difference, sum, and product are in the ratio $1:7:24$. This can be expressed as:
\[
x - y : x + y : xy = 1 : 7 : 24
\]
This implies:
\[
x+y = 7(x-y) \quad \text{and} \quad xy = 24(x-y)
\]
2. From the first equation $x+y = 7(x-... |
The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. The number of males in... | 10 | 1. **Calculate the total number of females in either band or orchestra**:
- The total number of females in the band is $100$.
- The total number of females in the orchestra is $80$.
- The number of females in both the band and orchestra is $60$.
Using the principle of inclusion-exclusion for the females... |
The shaded region formed by the two intersecting perpendicular rectangles, in square units, is | 38 | 1. **Identify the areas of the individual rectangles**:
- The first rectangle has dimensions $2 \times 10$. Therefore, its area is:
\[
2 \cdot 10 = 20 \text{ square units}
\]
- The second rectangle has dimensions $3 \times 8$. Therefore, its area is:
\[
3 \cdot 8 = 24 \text{ square units... |
Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km... | 65 | 1. **Calculate Minnie's Time:**
- **Uphill from A to B (10 km):** Minnie's speed uphill is $5$ kph. Time taken is:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{10 \text{ km}}{5 \text{ kph}} = 2 \text{ hours}
\]
- **Downhill from B to C (15 km):** Minnie's speed downhill is $30$ kph... |
For what value of $k$ does the equation $\frac{x-1}{x-2} = \frac{x-k}{x-6}$ have no solution for $x$? | 5 | 1. **Identify the domain**: The equation given is $\frac{x-1}{x-2} = \frac{x-k}{x-6}$. We must exclude values of $x$ that make the denominators zero, hence the domain is $\mathbb{R} \setminus \{2,6\}$.
2. **Cross-multiply to eliminate fractions**:
\[
(x-1)(x-6) = (x-k)(x-2)
\]
Expanding both sides, we get... |
When the mean, median, and mode of the list
\[10,2,5,2,4,2,x\]
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of $x$? | 20 | 1. **Calculate the Mean**:
The mean of the list $10, 2, 5, 2, 4, 2, x$ is calculated as follows:
\[
\text{Mean} = \frac{10 + 2 + 5 + 2 + 4 + 2 + x}{7} = \frac{25 + x}{7}.
\]
2. **Determine the Mode**:
The mode is the number that appears most frequently in the list. Here, the number $2$ appears three t... |
Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon... | 147 | 1. **Understanding the Problem:**
We have three identical square sheets of paper, each with side length $6$. The middle sheet is rotated $30^\circ$ clockwise, and the top sheet is rotated $60^\circ$ clockwise. We need to find the area of the resulting $24$-sided polygon.
2. **Breaking Down the Polygon:**
The pol... |
For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:
(1) two successive $15\%$ discounts
(2) three successive $10\%$ discounts
(3) a $25\%$ discount followed by a $5\%$ discount
What is the smallest possible positive integer value of $n$? | 29 | Let the original price be $x$. We need to find the smallest integer $n$ such that a single discount of $n\%$ is more advantageous than any of the given series of discounts.
1. **Calculating the effective discount for option (1):**
Two successive $15\%$ discounts:
\[
(1 - 0.15)(1 - 0.15)x = 0.85 \times 0.85x =... |
When the number $2^{1000}$ is divided by $13$, the remainder in the division is | 3 | 1. **Application of Fermat's Little Theorem**: Fermat's Little Theorem states that if $p$ is a prime number and $a$ is an integer not divisible by $p$, then $a^{p-1} \equiv 1 \pmod{p}$. Here, $p = 13$ and $a = 2$, so we have:
\[
2^{12} \equiv 1 \pmod{13}
\]
2. **Reduction of Exponent Modulo 12**: Since $2^{12... |
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family? | 6 | Let $m$ be the age of the mother.
Let $x$ be the number of children and $y$ be the average age of the children. The total age of the children is then $xy$.
We are given two key pieces of information:
1. The average age of the family (mother, father, and children) is $20$.
2. The average age of the mother and children... |
When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list? | 4 | Let $n$ be the number of integers in the original list and $m$ be the mean of these integers. The sum of the integers in the original list is then $mn$.
1. **Appending 15 to the list:**
When 15 is added to the list, the new list has $n+1$ integers, and the new mean is $m+2$. The sum of the integers in the new list ... |
Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1$. | 0 | We start by analyzing the given equation:
\[ m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1. \]
We can rewrite the left-hand side as:
\[ m^3 + 6m^2 + 5m = m(m^2 + 6m + 5) = m(m+1)(m+5). \]
This factorization is achieved by recognizing that $m^2 + 6m + 5$ can be factored as $(m+1)(m+5)$.
Now, the equation becomes:
\[ m(m+1)... |
Find a positive integral solution to the equation $\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}$. | 115 | 1. **Identify the series in the numerator and denominator:**
The numerator is the sum of the first $n$ odd numbers, which can be expressed as:
\[
1 + 3 + 5 + \dots + (2n-1)
\]
The denominator is the sum of the first $n$ even numbers, which can be expressed as:
\[
2 + 4 + 6 + \dots + 2n
\]
2. **... |
Point $P$ is outside circle $C$ on the plane. At most how many points on $C$ are $3$ cm from $P$? | 2 | 1. **Identify the Geometric Configuration**: We are given a circle $C$ and a point $P$ outside this circle. We need to find the maximum number of points on circle $C$ that are exactly $3$ cm away from point $P$.
2. **Construct a Circle Around $P$**: Consider a circle centered at $P$ with a radius of $3$ cm. This circl... |
$(2 \times 3 \times 4)\left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4}\right) = $ | 26 | 1. **Calculate the product of the integers:**
\[
2 \times 3 \times 4 = 24
\]
2. **Calculate the sum of the fractions:**
\[
\frac{1}{2} + \frac{1}{3} + \frac{1}{4}
\]
To add these fractions, find a common denominator. The least common multiple of 2, 3, and 4 is 12. Thus, convert each fraction:
... |
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