problem stringlengths 11 5.15k | answer stringlengths 1 42 | solution stringlengths 0 8.75k |
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At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000... | 100 | 1. **Define Variables:**
Let $a$ be the number of sets of twins, $b$ be the number of sets of triplets, and $c$ be the number of sets of quadruplets.
2. **Set Up Equations:**
From the problem, we have the following relationships:
- There are four times as many sets of triplets as sets of quadruplets: $b = 4c$... |
$\frac{1000^2}{252^2-248^2}$ equals | 500 | 1. **Identify the expression and apply the difference of squares formula**:
The given expression is $\frac{1000^2}{252^2-248^2}$. We can simplify the denominator using the difference of squares formula, which states that $a^2 - b^2 = (a-b)(a+b)$.
2. **Apply the formula to the denominator**:
Here, $a = 252$ a... |
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? | 13 | 1. **Analyze the distribution of file sizes:**
- There are 3 files of 0.8 MB each.
- There are 12 files of 0.7 MB each.
- The remaining files are 15 files of 0.4 MB each (since 3 + 12 = 15 files are accounted for, and there are 30 files in total).
2. **Optimize the storage of 0.8 MB files:**
- Each 0.8 MB ... |
How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit? | 60 | We are tasked with finding the number of three-digit numbers that satisfy the following conditions:
1. The number is not divisible by $5$.
2. The sum of the digits is less than $20$.
3. The first and third digits are the same.
Let's denote such a number as $\overline{xyx}$, where $x$ and $y$ are its digits. The number... |
If $r$ is the remainder when each of the numbers $1059$, $1417$, and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals | 15 | 1. We start with the given congruences:
\[
1059 \equiv r \pmod{d} \quad \text{(i)}
\]
\[
1417 \equiv r \pmod{d} \quad \text{(ii)}
\]
\[
2312 \equiv r \pmod{d} \quad \text{(iii)}
\]
2. Subtract congruence (i) from (ii) to eliminate $r$:
\[
1417 - 1059 \equiv r - r \pmod{d} \implies 358 ... |
Let $B$ be a right rectangular prism (box) with edges lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,... | 19 | To solve the problem, we analyze the volume of $S(r)$ by decomposing it into different geometric regions as described in the problem statement. We then calculate the coefficients $a$, $b$, $c$, and $d$ in the volume formula $ar^3 + br^2 + cr + d$.
1. **Region 1: The Rectangular Prism Itself**
- The volume of the re... |
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? | 170 | 1. **Understanding the Problem:**
- We have 6 individuals: Adam, Benin, Chiang, Deshawn, Esther, and Fiona.
- Each individual has the same number of internet friends within this group.
- The number of friends each person has ranges from 1 to 4, as no one can be friends with all 5 others (since some are not fri... |
Let $1$; $4$; $\ldots$ and $9$; $16$; $\ldots$ be two arithmetic progressions. The set $S$ is the union of the first $2004$ terms of each sequence. How many distinct numbers are in $S$? | 3722 | 1. **Identify the sequences and their general terms**:
- The first sequence is $1, 4, 7, \ldots$ which is an arithmetic progression with the first term $a_1 = 1$ and common difference $d_1 = 3$. The $k$-th term of this sequence can be expressed as $a_k = 3k + 1$ for $k \geq 0$.
- The second sequence is $9, 16, 23... |
Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is | 16 | To solve this problem, we need to count all the triangles formed by the lines in the given figure. The figure consists of a square, its diagonals, and segments joining the midpoints of opposite sides. Let's analyze the figure and count the triangles of different sizes.
1. **Identify the vertices and lines:**
- The ... |
The first three terms of a geometric progression are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. Find the fourth term. | 1 | 1. **Identify the sequence and rewrite terms with fractional exponents:**
The given terms are $\sqrt{2}, \sqrt[3]{2}, \sqrt[6]{2}$. We can rewrite these terms using fractional exponents:
\[
\sqrt{2} = 2^{1/2}, \quad \sqrt[3]{2} = 2^{1/3}, \quad \sqrt[6]{2} = 2^{1/6}.
\]
2. **Calculate the common ratio of t... |
Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents? | 64 | 1. **Calculate the number of students who can sing, dance, and act**:
- Students who can sing: $100 - 42 = 58$
- Students who can dance: $100 - 65 = 35$
- Students who can act: $100 - 29 = 71$
2. **Sum the number of students who can sing, dance, or act**:
- Total (ignoring overlaps): $58 + 35 + 71 = 164$
... |
Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$% discount as children. The two members of the oldest generation receive a $25\%$ discount as senior citizens. The two members of the middle generation receive no discount. Gra... | 36 | 1. **Calculate the cost of a regular ticket**:
Given that a senior ticket (with a 25% discount) costs $6.00, we need to find the original price of the ticket before the discount. The discount implies that seniors pay 75% (or $\frac{3}{4}$) of the original price. Let $x$ be the cost of a regular ticket. Then:
\[... |
How many ways are there to paint each of the integers $2, 3, \cdots , 9$ either red, green, or blue so that each number has a different color from each of its proper divisors? | 432 | To solve this problem, we need to consider the constraints imposed by the requirement that each number must have a different color from each of its proper divisors. We start by identifying the divisors of each number from $2$ to $9$:
- $2$: No proper divisors in the list.
- $3$: No proper divisors in the list.
- $4$: ... |
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? | 100 | 1. **Understanding the Offer**: The store offers one free window for every four purchased. This means for every five windows considered (four purchased + one free), the cost is for four windows only.
2. **Calculating Individual Savings**:
- **Dave's Purchase**: Dave needs 7 windows.
- Without the offer, Dave w... |
How many ordered pairs $(m,n)$ of positive integers, with $m \ge n$, have the property that their squares differ by $96$? | 4 | To find the number of ordered pairs $(m,n)$ of positive integers such that $m \ge n$ and $m^2 - n^2 = 96$, we start by factoring the expression for the difference of squares:
\[ m^2 - n^2 = (m+n)(m-n) = 96. \]
We need to find all pairs of factors of $96$ that can be expressed as $(m+n, m-n)$. Since $m$ and $n$ are in... |
Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score? | 40 |
#### Detailed Steps:
1. **Understanding the Problem:**
We are given the scores of three students as $70, 80,$ and $90$. We also know that the average score of all four students is $70$. We need to find the score of the fourth student, denoted as $r$.
2. **Setting Up the Equation:**
The average of four scores i... |
The number of positive integers less than $1000$ divisible by neither $5$ nor $7$ is: | 686 | 1. **Count the integers divisible by 5:**
The number of positive integers less than $1000$ that are divisible by $5$ can be calculated using the floor function:
\[
\left\lfloor \frac{999}{5} \right\rfloor = 199
\]
This is because $999$ is the largest number less than $1000$, and dividing it by $5$ and ta... |
Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read $1/5$ of the pages plus $12$ more, and on the second day she read $1/4$ of the remaining pages plus $15$ pages. On the third day she read $1/3$ of the remaining pages plus $18$ pages. She then realized that there w... | 240 | Let $x$ be the total number of pages in the book.
1. **Reading on the first day:**
Hui reads $\frac{1}{5}x + 12$ pages. The remaining pages are:
\[
x - \left(\frac{1}{5}x + 12\right) = \frac{4}{5}x - 12
\]
2. **Reading on the second day:**
From the remaining pages, Hui reads $\frac{1}{4}(\frac{4}{5}x -... |
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product
\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are count... | 112 | To find $D(96)$, we need to consider all possible ways to write $96$ as a product of integers greater than $1$, where the order of factors matters. We start by noting the prime factorization of $96$:
\[ 96 = 2^5 \cdot 3 \]
We will consider different cases based on the number of factors in each product.
#### Case 1: D... |
Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? | 3120 | To solve this problem, we need to consider the constraints given by the diagonals of the pentagon. Each diagonal connects two vertices, and the vertices at the ends of each diagonal must have different colors. We will analyze the possible colorings by considering different cases based on the color assignments of the ve... |
John scores 93 on this year's AHSME. Had the old scoring system still been in effect, he would score only 84 for the same answers.
How many questions does he leave unanswered? | 9 | Let $c$, $w$, and $u$ be the number of correct, wrong, and unanswered questions respectively. We are given three pieces of information:
1. Under the old scoring system, John's score is $84$. The old scoring system awards $30$ points initially, $4$ points for each correct answer, subtracts $1$ point for each wrong answ... |
A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture? | 8 | 1. **Identify the total width and picture width**: The wall is $19$ feet wide and the picture is $3$ feet wide.
2. **Set up the equation**: The picture is centered, so the distances from each end of the wall to the nearest edge of the picture are equal. Let $x$ be the distance from the end of the wall to the nearest e... |
A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for $x$ are possible? | 5 | To determine the possible values of $x$, the cost of each ticket, we need to consider the total amounts spent by the 9th and 10th graders and find a common ticket price that divides both totals.
1. **Identify the total amounts spent by each grade:**
- 9th graders: $48$
- 10th graders: $64$
2. **Find the greates... |
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
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path p2=(0,1)--(1,1)--(1,0);
path p3=(2,0)--(2,1)--(3,1);
path p4=(3,2)--(2... | 56 | 1. **Identify the basic unit of tiling**: The tiling pattern consists of a large square divided into 9 smaller squares, each with side length $a$. The large square thus has a side length of $3a$.
2. **Calculate the area of the large square**: The area of the large square is $(3a)^2 = 9a^2$.
3. **Identify the areas co... |
The base three representation of $x$ is $12112211122211112222$. The first digit (on the left) of the base nine representation of $x$ is | 5 | To solve this problem, we need to convert the base three number $12112211122211112222_3$ to its base nine representation and find the first digit of that representation.
#### Step 1: Convert from base 3 to base 10
The number in base 3 is $12112211122211112222_3$. We convert it to base 10 by evaluating each digit's con... |
The average cost of a long-distance call in the USA in $1985$ was
$41$ cents per minute, and the average cost of a long-distance
call in the USA in $2005$ was $7$ cents per minute. Find the
approximate percent decrease in the cost per minute of a long-
distance call. | 80 | 1. **Identify the original and new costs**: The original cost in 1985 was $41$ cents per minute, and the new cost in 2005 was $7$ cents per minute.
2. **Calculate the amount of decrease**:
\[
\text{Amount of decrease} = 41 - 7 = 34 \text{ cents per minute}
\]
3. **Calculate the percent decrease**: The perce... |
At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many ha... | 441 | 1. **Count the total number of twins and triplets:**
- There are $9$ sets of twins, with $2$ twins per set, giving a total of $9 \times 2 = 18$ twins.
- There are $6$ sets of triplets, with $3$ triplets per set, giving a total of $6 \times 3 = 18$ triplets.
2. **Calculate handshakes among twins:**
- Each twin... |
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 repeated digits of $9$, increasing by one $9$ at each step.
2. **Express each number as a sum**: Each number in the set can be expressed as a sum of $9$s multiplied by powers of $10... |
$\diamondsuit$ and $\Delta$ are whole numbers and $\diamondsuit \times \Delta =36$. The largest possible value of $\diamondsuit + \Delta$ is | 37 | To find the largest possible value of $\diamondsuit + \Delta$ given that $\diamondsuit \times \Delta = 36$, we need to consider all pairs of whole numbers $(\diamondsuit, \Delta)$ that multiply to 36. We will then calculate the sum $\diamondsuit + \Delta$ for each pair and determine the maximum.
1. **Factorize 36**: W... |
What is the difference between the sum of the first $2003$ even counting numbers and the sum of the first $2003$ odd counting numbers? | 2003 | 1. **Identify the sequences**:
- The first $2003$ odd counting numbers form the sequence $O = 1, 3, 5, \ldots, 4005$.
- The first $2003$ even counting numbers can be considered in two cases:
- Including $0$: $E_1 = 0, 2, 4, \ldots, 4004$
- Excluding $0$: $E_2 = 2, 4, 6, \ldots, 4006$
2. **Calculate th... |
In the base ten number system the number $526$ means $5 \times 10^2+2 \times 10 + 6$. In the Land of Mathesis, however, numbers are written in the base $r$. Jones purchases an automobile there for $440$ monetary units (abbreviated m.u). He gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The ba... | 8 | 1. **Understanding the Problem**: In the Land of Mathesis, numbers are written in base $r$. The problem states that Jones buys an automobile for $440$ m.u. in base $r$, pays with a $1000$ m.u. bill in base $r$, and receives $340$ m.u. in base $r$ as change. We need to find the base $r$.
2. **Setting Up the Equation**:... |
A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? | 15 | Let's denote the large rectangle as $ABCD$, with $AB$ and $CD$ being the length and $BC$ and $AD$ being the width. Suppose two segments parallel to $AB$ and $BC$ partition $ABCD$ into four smaller rectangles: $PQRS$, $PQTU$, $UVWX$, and $RSWX$. Given the areas of three of these rectangles, we need to find the area of t... |
The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams? | 88 | 1. **Calculate the number of games within the conference:**
- There are 8 teams in the conference.
- Each team plays every other team twice (once at home and once away).
- The number of ways to choose 2 teams from 8 is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the tota... |
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior a... | 100 | 1. **Understanding the Geometry of the Keystone Arch**:
The keystone arch is composed of $9$ congruent isosceles trapezoids. These trapezoids are arranged such that their non-parallel sides (legs) meet at a common point when extended. The bottom sides of the end trapezoids are horizontal.
2. **Intersection of Exten... |
Laura added two three-digit positive integers. All six digits in these numbers are different. Laura's sum is a three-digit number $S$. What is the smallest possible value for the sum of the digits of $S$? | 4 | 1. **Define the problem**: We need to find the smallest possible value for the sum of the digits of $S$, where $S$ is the sum of two three-digit numbers $a$ and $b$. Each digit among $a$ and $b$ is unique.
2. **Set constraints on $a$ and $b$**: Since $a$ and $b$ are three-digit numbers and all digits are different, we... |
For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$? | 16 | 1. **Understanding the repeating base-$k$ representation**: Given that $0.\overline{23}_k = 0.232323..._k$, we interpret this as the infinite series:
\[
0.232323..._k = \frac{2}{k} + \frac{3}{k^2} + \frac{2}{k^3} + \frac{3}{k^4} + \cdots
\]
2. **Converting the series into a single fraction**: We can split the... |
A positive number is mistakenly divided by $6$ instead of being multiplied by $6.$ Based on the correct answer, the error thus committed, to the nearest percent, is | 97 | Let the positive number be $x$. According to the problem, the number $x$ should have been multiplied by $6$, which would give the correct result as $6x$. However, it was mistakenly divided by $6$, giving the erroneous result $\frac{x}{6}$.
To find the error committed, we calculate the absolute difference between the c... |
The domain of the function $f(x)=\log_{\frac{1}{2}}(\log_4(\log_{\frac{1}{4}}(\log_{16}(\log_{\frac{1}{16}}x))))$ is an interval of length $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 271 | To find the domain of the function
\[ f(x) = \log_{\frac{1}{2}}\left(\log_4\left(\log_{\frac{1}{4}}\left(\log_{16}\left(\log_{\frac{1}{16}}x\right)\right)\right)\right), \]
we need to ensure that each logarithm in the nested expression is defined and positive, as the logarithm function is only defined for positive arg... |
A rising number, such as $34689$, is a positive integer each digit of which is larger than each of the digits to its left. There are $\binom{9}{5} = 126$ five-digit rising numbers. When these numbers are arranged from smallest to largest, the $97^{\text{th}}$ number in the list does not contain the digit | 5 | 1. **Understanding the Problem**: We need to find the $97^{\text{th}}$ five-digit rising number and identify which digit from the given options it does not contain.
2. **Counting Rising Numbers Starting with '1'**:
- A five-digit rising number starting with '1' can be formed by choosing 4 more digits from 2 to 9.
... |
Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the $6$ faces of the cube equal to $2$? | 20 | To solve this problem, we need to ensure that each face of the cube has exactly two edges labeled with $1$ and two edges labeled with $0$. We will analyze the possible configurations that satisfy this condition.
#### Step 1: Understanding the Cube's Faces and Edges
A cube has 6 faces and 12 edges. Each face shares an ... |
Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding $4$ more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed in miles per hour? | 22 | 1. **Identify the initial and final odometer readings**: Barney's initial odometer reading is $1441$ and after riding for some time, the final reading is $1661$.
2. **Calculate the total distance traveled**:
\[
\text{Distance} = \text{Final odometer reading} - \text{Initial odometer reading} = 1661 - 1441 = 220... |
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one? | 18 | 1. **Initial Set and Operation Definition**: We start with a set of tiles numbered from 1 to 100. The operation, denoted as $P(x)$, involves removing all tiles numbered with a perfect square and renumbering the remaining tiles consecutively starting with 1.
2. **Understanding Perfect Squares**: The perfect squares bet... |
How many terms are in the arithmetic sequence $13$, $16$, $19$, $\dotsc$, $70$, $73$? | 21 | 1. **Identify the first term and common difference**: The given sequence is $13, 16, 19, \dots, 70, 73$. The first term ($a$) is $13$ and the common difference ($d$) can be calculated as $16 - 13 = 3$.
2. **Formulate the general term**: The general term of an arithmetic sequence can be expressed as $a_n = a + (n-1)d$.... |
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
Suppose $PQ=6$ and... | 81 | 1. **Identify the given information and setup:**
- $ABCD$ is a parallelogram with area $15$.
- $PQ = 6$ and $RS = 8$ are the lengths of the projections of $A$ and $C$ onto $BD$, and $B$ and $D$ onto $AC$, respectively.
- We need to find $d^2$, where $d$ is the length of diagonal $BD$.
2. **Use the properties ... |
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:
$\circ$ Roger's cookies are rectangles:
$\circ$ Paul's... | 24 | To solve this problem, we need to determine the number of cookies Trisha can make from the same amount of dough as Art, given that all cookies have the same thickness and thus the same volume per unit area.
1. **Calculate the area of one of Art's cookies**:
Art's cookies are trapezoids. The formula for the area of ... |
Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose... | 17 | To determine the number of rods that can be chosen as the fourth rod to form a quadrilateral with the rods of lengths $3$ cm, $7$ cm, and $15$ cm, we need to ensure that the sum of the lengths of any three sides must be greater than the length of the fourth side. This is a direct application of the triangle inequality ... |
A regular octahedron has side length $1$. A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area $\frac {a\sqrt {b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively pr... | 14 |
To solve this problem, we need to understand the geometry of the octahedron and how the plane intersects it. An octahedron can be thought of as two square pyramids base-to-base. Each face of the octahedron is an equilateral triangle.
#### Step 1: Understanding the Intersection
The plane is parallel to two opposite fa... |
If one side of a triangle is $12$ inches and the opposite angle is $30^{\circ}$, then the diameter of the circumscribed circle is: | 24 | 1. **Identify the Known Values:**
- One side of the triangle (let's call it $a$) is given as $12$ inches.
- The angle opposite to this side ($\angle A$) is $30^\circ$.
2. **Apply the Extended Law of Sines:**
- The Extended Law of Sines states that for any triangle, the diameter $D$ of the circumscribed circle... |
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\frac{4}{5}$ as $N$ grows large. What is th... | 12 | 1. **Understanding the Problem:**
We are given a line of balls consisting of one red ball and $N$ green balls, where $N$ is a multiple of 5. We need to find the probability $P(N)$ that at least $\frac{3}{5}$ of the green balls are on the same side of the red ball.
2. **Initial Observations:**
- For $N=5$, all gr... |
Terri produces a sequence of positive integers by following three rules. She starts with a positive integer, then applies the appropriate rule to the result, and continues in this fashion.
Rule 1: If the integer is less than 10, multiply it by 9.
Rule 2: If the integer is even and greater than 9, divide it by 2.
Ru... | 27 | To solve the problem, we first need to understand the sequence generated by the rules provided and then determine the $98^\text{th}$ term of the sequence starting with 98.
1. **Starting the sequence**: We begin with 98 and apply the rules:
- **98** is even and greater than 9, so by Rule 2, we divide it by 2: $98 / ... |
$A$ can do a piece of work in $9$ days. $B$ is $50\%$ more efficient than $A$. The number of days it takes $B$ to do the same piece of work is: | 6 | 1. **Calculate $A$'s work rate**: If $A$ can complete the work in $9$ days, then $A$'s work rate is $\frac{1}{9}$ of the work per day.
2. **Determine $B$'s work rate**: Since $B$ is $50\%$ more efficient than $A$, $B$'s work rate is $1.5$ times that of $A$. Therefore, $B$'s work rate is $1.5 \times \frac{1}{9} = \frac... |
Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get? | 380 | 1. **Identify the operation**: Ahn performs the operation $2(200 - n)$, where $n$ is a two-digit integer.
2. **Determine the range of $n$**: Since $n$ is a two-digit integer, $10 \leq n \leq 99$.
3. **Maximize the expression**: To maximize $2(200 - n)$, we need to minimize $n$ because subtracting a smaller number fro... |
How many positive integers $b$ have the property that $\log_{b} 729$ is a positive integer? | 4 | 1. **Express the equation in exponential form**: Given $\log_b 729 = n$, we can rewrite this as $b^n = 729$.
2. **Factorize 729**: We know that $729 = 3^6$. Therefore, we can express $b^n = 3^6$.
3. **Determine possible values of $b$**: Since $b^n = 3^6$, $b$ must be a form of $3^k$ where $k$ is a divisor of 6. The d... |
Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$? | 8 |
1. **Identify the form of the sequences**: Given that $\left(a_n\right)$ and $\left(b_n\right)$ are arithmetic sequences with integer terms and $a_1 = b_1 = 1$, we can express the $n$-th terms of these sequences as:
\[
a_n = 1 + (n-1)x \quad \text{and} \quad b_n = 1 + (n-1)y
\]
where $x$ and $y$ are the co... |
Al and Barb start their new jobs on the same day. Al's schedule is 3 work-days followed by 1 rest-day. Barb's schedule is 7 work-days followed by 3 rest-days. On how many of their first 1000 days do both have rest-days on the same day? | 100 | 1. **Identify the Cycles**: Al's schedule is a cycle of 4 days (3 work-days followed by 1 rest-day), and Barb's schedule is a cycle of 10 days (7 work-days followed by 3 rest-days).
2. **Calculate the Least Common Multiple (LCM)**: To find when their schedules align completely, we calculate the LCM of their individual... |
The fraction
\[\frac{\left(3^{2008}\right)^2-\left(3^{2006}\right)^2}{\left(3^{2007}\right)^2-\left(3^{2005}\right)^2}\]
simplifies to what? | 9 | 1. **Rewrite the expression using the difference of squares formula**:
\[
\frac{(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2} = \frac{(3^{2008} + 3^{2006})(3^{2008} - 3^{2006})}{(3^{2007} + 3^{2005})(3^{2007} - 3^{2005})}
\]
2. **Simplify each term**:
- For the numerator:
\[
3^{2008} ... |
Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: | 1 | 1. **Understanding the angle formula for equiangular polygons**: For any equiangular polygon with $n$ sides, each interior angle is given by the formula:
\[
\text{Angle} = \frac{180(n-2)}{n} = 180 - \frac{360}{n}
\]
This formula arises from the fact that the sum of the interior angles of an $n$-sided polygo... |
A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed? | 6 | 1. **Determine the number of cans needed to feed 15 children**:
Given that one can of soup can feed 5 children, we calculate the number of cans needed to feed 15 children as follows:
\[
\text{Number of cans} = \frac{15 \text{ children}}{5 \text{ children per can}} = 3 \text{ cans}
\]
2. **Calculate the re... |
Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. ... | 4 | 1. **Calculate the total cost of the pizza:**
The pizza is plain and costs $8$ dollars. Adding anchovies to half of it costs an additional $2$ dollars. Therefore, the total cost of the pizza is:
\[
8 + 2 = 10 \text{ dollars}
\]
2. **Determine the cost per slice:**
The pizza has $8$ slices, so the co... |
Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ ar... | 752 | 1. **Understanding the relationship between $f$ and $g$:**
Given that $g(x) = -f(100 - x)$, we can infer that the graph of $g$ is a $180^\circ$ rotation of the graph of $f$ around the point $(50, 0)$. This is because replacing $x$ with $100 - x$ reflects the graph across the line $x = 50$, and the negative sign refl... |
The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? | 90 | 1. **Given Information**: The mean, median, and mode of the data set $60, 100, x, 40, 50, 200, 90$ are all equal to $x$.
2. **Calculating the Mean**:
The mean of the data set is given by:
\[
\text{Mean} = \frac{60 + 100 + x + 40 + 50 + 200 + 90}{7} = \frac{540 + x}{7}
\]
Since the mean is equal to $x$, ... |
A cart rolls down a hill, travelling $5$ inches the first second and accelerating so that during each successive $1$-second time interval, it travels $7$ inches more than during the previous $1$-second interval. The cart takes $30$ seconds to reach the bottom of the hill. How far, in inches, does it travel? | 3195 | 1. **Identify the sequence type**: The problem describes a sequence where the distance traveled each second increases by a constant amount. This is an arithmetic sequence.
2. **Determine the first term and common difference**:
- The first term $a_1 = 5$ inches (distance traveled in the first second).
- The comm... |
What is the value of the following expression?
\[ \frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)} \] | 1 | 1. **Identify the expression and apply the difference of squares formula:**
The given expression is:
\[
\frac{100^2-7^2}{70^2-11^2} \cdot \frac{(70-11)(70+11)}{(100-7)(100+7)}
\]
We can apply the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$, to both the numerator and the denominator of the fir... |
Ten women sit in $10$ seats in a line. All of the $10$ get up and then reseat themselves using all $10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated? | 89 | To solve this problem, we will use a recursive approach to determine the number of ways the women can be reseated under the given constraints. We define $S_n$ as the number of ways $n$ women can be reseated in $n$ seats such that each woman sits in her original seat or in a seat adjacent to her original seat.
#### Ste... |
Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD}, BC=CD=43$, and $\overline{AD}\perp\overline{BD}$. Let $O$ be the intersection of the diagonals $\overline{AC}$ and $\overline{BD}$, and let $P$ be the midpoint of $\overline{BD}$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt{n}$, wh... | 194 | 1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\overline{AB}\parallel\overline{CD}$ and $BC=CD=43$. Also, $\overline{AD}\perp\overline{BD}$, which implies that $\triangle ABD$ is a right triangle.
2. **Intersection and midpoint properties**: Let $O$ be the intersection of the d... |
Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last ... | 7 |
To solve this problem, we need to determine the smallest initial number $N$ such that Bernardo wins the game. We will analyze the sequence of operations and the conditions under which Bernardo wins.
1. **Sequence of Operations**:
- Bernardo receives $N$, doubles it: $2N$
- Silvia receives $2N$, adds 50: $2N + 5... |
Points $A = (3,9)$, $B = (1,1)$, $C = (5,3)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D$? | 10 | 1. **Identify the midpoints of sides of the quadrilateral**:
- The midpoint of $\overline{AB}$ is calculated as:
\[
\left(\frac{3+1}{2}, \frac{9+1}{2}\right) = (2, 5)
\]
- The midpoint of $\overline{BC}$ is calculated as:
\[
\left(\frac{1+5}{2}, \frac{1+3}{2}\right) = (3, 2)
\]
2. ... |
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE$? | 170 | 1. **Identify the given information and setup**: We know that $ABCD$ is a square, and $E$ lies on the opposite half-plane determined by line $CD$ from point $A$. The angle $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ such that $DE = DF$.
2. **Extend $\overline{AD}$ to point $G$**: By extending $\overlin... |
How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$? | 2 | To find the number of integer pairs $(m,n)$ that satisfy the equation $m+n=mn$, we can start by rearranging the equation:
\[ m+n = mn \]
Subtracting $m+n$ from both sides gives:
\[ mn - m - n = 0 \]
Adding 1 to both sides, we can factorize the left-hand side:
\[ mn - m - n + 1 = 1 \]
\[ (m-1)(n-1) = 1 \]
Now, we... |
Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$? | 21 | 1. **Convert 1000 to Hexadecimal**:
The decimal number 1000 can be converted to hexadecimal. We find that $1000_{10} = 3E8_{16}$. This conversion is done by dividing 1000 by 16 repeatedly and considering the remainders:
- $1000 \div 16 = 62$ remainder $8$
- $62 \div 16 = 3$ remainder $14$ (which is 'E' in hex... |
For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ not in lowest terms? | 86 | 1. **Identify the condition for $\frac{N^2+7}{N+4}$ not being in lowest terms**: We need to find the number of integers $N$ such that $\gcd(N^2+7, N+4) > 1$.
2. **Express $N$ in terms of a divisor $m$ of $N+4**: Assume $m$ is a divisor of $N+4$. Then, $N+4 \equiv 0 \pmod{m}$, which implies $N \equiv -4 \pmod{m}$.
3. ... |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$? | 6 | 1. **Given Information and Assumptions**:
- Convex quadrilateral $ABCD$ with $AB = 9$, $CD = 12$, and diagonals $AC = 14$.
- Diagonals $AC$ and $BD$ intersect at $E$.
- $\triangle AED$ and $\triangle BEC$ have equal areas.
2. **Using Equal Areas to Infer Similarity**:
- Since $\triangle AED$ and $\triangle... |
The expression $\frac{2021}{2020} - \frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$ in which $p$ and $q$ are positive integers whose greatest common divisor is 1. What is $p?$ | 2021 | 1. Let $a = 2020$. We start by rewriting the given expression using $a$:
\[
\frac{2021}{2020} - \frac{2020}{2021} = \frac{a+1}{a} - \frac{a}{a+1}
\]
2. To combine these fractions, find a common denominator, which is $a(a+1)$:
\[
\frac{(a+1)^2 - a^2}{a(a+1)}
\]
3. Simplify the numerator:
\[
(a+... |
For real numbers $w$ and $z$, $\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014.$ What is $\frac{w+z}{w-z}$? | -2014 | 1. Start with the given equation:
\[
\frac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014
\]
2. Multiply both sides by $\left(\frac{1}{w} - \frac{1}{z}\right)$ to eliminate the denominator:
\[
\frac{1}{w} + \frac{1}{z} = 2014 \left(\frac{1}{w} - \frac{1}{z}\right)
\]
3. Expand the righ... |
Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leav... | 5 | 1. **Set up the problem**: Michael walks at $5$ feet per second and trash pails are located every $200$ feet. The garbage truck travels at $10$ feet per second and stops for $30$ seconds at each pail. Michael sees the truck leaving a pail $200$ feet ahead as he passes a pail.
2. **Define positions**: Let $M(t)$ be Mic... |
A teacher tells the class,
"Think of a number, add 1 to it, and double the result. Give the answer to your partner. Partner, subtract 1 from the number you are given and double the result to get your answer."
Ben thinks of $6$, and gives his answer to Sue. What should Sue's answer be? | 26 | 1. **Ben's Calculation:**
- Ben thinks of a number, which is $6$.
- He adds $1$ to this number: $6 + 1 = 7$.
- He then doubles the result: $7 \times 2 = 14$.
- Ben gives the number $14$ to Sue.
2. **Sue's Calculation:**
- Sue receives the number $14$.
- She subtracts $1$ from this number: $14 - 1 = 1... |
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle? | 8 | 1. **Understanding the Problem**: Mary divides a circle into 12 sectors with central angles that are integers and form an arithmetic sequence. The sum of these angles must be $360^\circ$ since they complete a circle.
2. **Setting Up the Equations**: Let $a_1$ be the first term and $d$ be the common difference of the a... |
Qiang drives $15$ miles at an average speed of $30$ miles per hour. How many additional miles will he have to drive at $55$ miles per hour to average $50$ miles per hour for the entire trip? | 110 | 1. **Determine the time for the first part of the trip**: Qiang drives 15 miles at an average speed of 30 miles per hour. The time taken for this part of the trip is calculated by the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{15 \text{ miles}}{30 \text{ mph}} = \frac{1}{2} \text{ hour... |
Makarla attended two meetings during her $9$-hour work day. The first meeting took $45$ minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? | 25 | 1. **Convert the work day into minutes**:
Makarla's work day is $9$ hours long. Since there are $60$ minutes in an hour, the total number of minutes in her work day is:
\[
9 \times 60 = 540 \text{ minutes}
\]
2. **Calculate the total time spent in meetings**:
- The duration of the first meeting is $45$... |
Let $f(x)=x^2+3x+2$ and let $S$ be the set of integers $\{0, 1, 2, \dots , 25 \}$.
The number of members $s$ of $S$ such that $f(s)$ has remainder zero when divided by $6$ is: | 17 | 1. **Understanding the Polynomial Function**:
Given the polynomial function $f(x) = x^2 + 3x + 2$. We need to find the number of integers $s$ in the set $S = \{0, 1, 2, \dots, 25\}$ such that $f(s)$ is divisible by $6$.
2. **Periodicity of Polynomial Functions Modulo $6$**:
We observe that for any polynomial $f... |
A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? | 2 | To solve this problem, we need to determine how many days of the week could be the first day of a month with 31 days such that the number of Mondays and Wednesdays in that month are equal.
1. **Understanding the Weekday Distribution:**
A month with 31 days spans exactly 4 weeks and 3 extra days. This means that eac... |
In a tournament there are six teams that play each other twice. A team earns $3$ points for a win, $1$ point for a draw, and $0$ points for a loss. After all the games have been played it turns out that the top three teams earned the same number of total points. What is the greatest possible number of total points for ... | 24 | 1. **Understanding the Tournament Setup**: In this tournament, there are six teams, and each team plays every other team twice. This results in a total of $\binom{6}{2} \times 2 = 15 \times 2 = 30$ games played in the tournament.
2. **Scoring System**: Each team earns 3 points for a win, 1 point for a draw, and 0 poin... |
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? | 7 | Let the three numbers be $a$, $b$, and $c$, where $a < b < c$. We are given the sums of these numbers taken in pairs:
1. $a + b = 12$
2. $a + c = 17$
3. $b + c = 19$
We aim to find the value of $b$, the middle number. To do this, we can manipulate the given equations to isolate $b$. We start by adding the first and th... |
Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in ... | 22 | Let $x$ be the cost of Anna's dinner before tax and tip.
1. **Calculate the tax:** The tax rate is 10%, so the tax amount is $\frac{10}{100}x = 0.1x$.
2. **Calculate the tip:** The tip is 15% of the pre-tax meal cost, so the tip amount is $\frac{15}{100}x = 0.15x$.
3. **Total cost:** The total cost of the meal inclu... |
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? | 75 | 1. **Identify the areas of sub-triangles**:
- The area of $\triangle ABC$ is given as $120$.
- Since $D$ and $E$ are midpoints of $AB$ and $AC$ respectively, $AD = DB = 25$ and $AE = EC = 5$.
- The area of $\triangle ADE$ can be calculated using the formula for the area of a triangle, $\frac{1}{2} \cdot \text... |
How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)? | 37 | To find the number of positive integer divisors of $201^9$ that are perfect squares or perfect cubes (or both), we start by factoring $201$:
\[ 201 = 3 \times 67. \]
Thus,
\[ 201^9 = (3 \times 67)^9 = 3^9 \times 67^9. \]
A divisor of $201^9$ can be written as $3^a \times 67^b$ where $0 \leq a, b \leq 9$.
#### Step 1... |
The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this? | 665 | To find the total duration of the tennis match in minutes, we need to convert the hours into minutes and then add the remaining minutes.
1. **Convert hours to minutes**:
- There are 60 minutes in one hour.
- Therefore, for 11 hours, the total minutes are:
\[
11 \text{ hours} \times 60 \text{ minutes/h... |
In base $R_1$ the expanded fraction $F_1$ becomes $.373737\cdots$, and the expanded fraction $F_2$ becomes $.737373\cdots$. In base $R_2$ fraction $F_1$, when expanded, becomes $.252525\cdots$, while the fraction $F_2$ becomes $.525252\cdots$. The sum of $R_1$ and $R_2$, each written in the base ten, is: | 19 | 1. **Express $F_1$ and $F_2$ in base $R_1$ and $R_2$:**
- For $F_1$ in base $R_1$, we have:
\[
F_1 = 0.373737\ldots_{R_1}
\]
Multiplying by $R_1^2$:
\[
R_1^2 F_1 = 37.373737\ldots_{R_1}
\]
Subtracting the original from this:
\[
R_1^2 F_1 - F_1 = 37_{R_1} \implies (R_... |
A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $R$ is the ratio of the lesser part to the greater part, then the value of
\[R^{\left(R^{(R^2+R^{-1})}+R^{-1}\right)}+R^{-1}\]is | 2 | 1. **Identify the Relationship**: Given that a line segment is divided into two parts, $w$ (lesser part) and $l$ (greater part), such that the ratio of the lesser part to the greater part is the same as the ratio of the greater part to the whole segment. This can be expressed as:
\[
\frac{w}{l} = \frac{l}{w+l}
... |
How many ordered pairs $(a, b)$ of positive integers satisfy the equation
\[a \cdot b + 63 = 20 \cdot \text{lcm}(a, b) + 12 \cdot \text{gcd}(a,b),\] where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple? | 2 | 1. **Expressing $a$ and $b$ in terms of $\text{lcm}$ and $\text{gcd}$**:
Let $x = \text{lcm}(a, b)$, and $y = \text{gcd}(a, b)$. We know that $a \cdot b = \text{lcm}(a, b) \cdot \text{gcd}(a, b) = x \cdot y$. Substituting this into the given equation, we have:
\[
x \cdot y + 63 = 20x + 12y
\]
Rearrangi... |
Using the letters $A$, $M$, $O$, $S$, and $U$, we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $USAMO$ occupies position | 115 | To find the position of the word "USAMO" in the alphabetical order of all possible permutations of the letters $A$, $M$, $O$, $S$, and $U$, we can proceed as follows:
1. **Count the permutations starting with each letter before 'U':**
- The letters before 'U' in alphabetical order are $A$, $M$, $O$, and $S$.
- F... |
The number of ounces of water needed to reduce $9$ ounces of shaving lotion containing $50$ % alcohol to a lotion containing $30$ % alcohol is: | 6 | 1. **Identify the amount of alcohol in the original lotion**:
The original shaving lotion is 9 ounces with 50% alcohol. Therefore, the amount of alcohol in the lotion is:
\[
\frac{50}{100} \times 9 = \frac{9}{2} \text{ ounces}
\]
2. **Set up the equation for the final alcohol concentration**:
Let $N... |
Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class's mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all... | 76 | 1. **Identify the given information:**
- Mean score of the morning class, $M = 84$.
- Mean score of the afternoon class, $A = 70$.
- Ratio of the number of students in the morning class to the afternoon class, $\frac{m}{a} = \frac{3}{4}$.
2. **Express the number of students in the morning class in terms of th... |
Convex quadrilateral $ABCD$ has $AB=3$, $BC=4$, $CD=13$, $AD=12$, and $\angle ABC=90^{\circ}$, as shown. What is the area of the quadrilateral? | 36 | 1. **Identify the right triangle**: Given $\angle ABC = 90^\circ$ and the sides $AB = 3$ and $BC = 4$, we can apply the Pythagorean theorem to find $AC$:
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.
\]
2. **Check if $\triangle CAD$ is a right triangle**: We know $AC = 5$, $AD... |
Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cd... | 19 | To solve the problem, we need to calculate the sum of the reciprocals of all positive integers that have no prime factors other than $2$, $3$, or $5$. These integers can be expressed in the form $2^a 3^b 5^c$ where $a$, $b$, and $c$ are non-negative integers.
1. **Express the sum as a product of geometric series**:
... |
For how many integers $n$ between $1$ and $50$, inclusive, is $\frac{(n^2-1)!}{(n!)^n}$ an integer? | 34 | 1. **Understanding the Expression**: We start by analyzing the expression \[\frac{(n^2-1)!}{(n!)^n}.\] We need to determine for how many integers $n$ between $1$ and $50$, inclusive, this expression is an integer.
2. **Rewriting the Expression**: We can rewrite the expression as:
\[\frac{(n^2-1)!}{(n!)^n} = \frac{(... |
$\frac{(3!)!}{3!} = $ | 120 | 1. **Calculate $3!$:**
\[
3! = 3 \times 2 \times 1 = 6
\]
2. **Calculate $(3!)!$:**
\[
(3!)! = (6)!
\]
3. **Expand $(6)!$:**
\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1
\]
4. **Simplify the expression $\frac{(3!)!}{3!}$:**
\[
\frac{(3!)!}{3!} = \frac{6!}{6}
\]
\[
... |
Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)... | 45 | 1. **Understanding the problem setup**: Each of the 2010 boxes contains 1 red marble and an increasing number of white marbles as we move from the first to the last box. Isabella draws marbles one by one from each box starting from the first box and stops when she draws a red marble.
2. **Calculating the probability o... |
How many distinguishable arrangements are there of $1$ brown tile, $1$ purple tile, $2$ green tiles, and $3$ yellow tiles in a row from left to right? (Tiles of the same color are indistinguishable.) | 420 | 1. **Identify the Total Number of Tiles and Their Colors**:
- We have 1 brown tile (B), 1 purple tile (P), 2 green tiles (G), and 3 yellow tiles (Y).
2. **Set Up the Problem as a Permutation of Indistinguishable Objects**:
- We need to arrange these tiles in a row. Since tiles of the same color are indistinguish... |
A line passes through $A\ (1,1)$ and $B\ (100,1000)$. How many other points with integer coordinates are on the line and strictly between $A$ and $B$? | 8 | 1. **Identify the slope of the line passing through points $A$ and $B$:**
Given points $A(1,1)$ and $B(100,1000)$, the slope of the line can be calculated using the formula:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1000 - 1}{100 - 1} = \frac{999}{99}
\]
Simplifying the fraction, we get:
\... |
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