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In a circle, let $AB$ and $BC$ be chords , with $AB =\sqrt3, BC =3\sqrt3, \angle ABC =60^o$. Find the length of the circle chord that divides angle $ \angle ABC$ in half. | 1. **Identify the given elements and apply the angle bisector theorem:**
- Given: \(AB = \sqrt{3}\), \(BC = 3\sqrt{3}\), \(\angle ABC = 60^\circ\).
- Let the angle bisector of \(\angle ABC\) intersect \(AC\) at \(D\).
- By the angle bisector theorem, \(\frac{AD}{DC} = \frac{AB}{BC} = \frac{\sqrt{3}}{3\sqrt{3}}... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Diagonals of trapezium $ABCD$ are mutually perpendicular and the midline of the trapezium is $5$. Find the length of the segment that connects the midpoints of the bases of the trapezium. | 1. Let $ABCD$ be a trapezium with $AB \parallel CD$ and diagonals $AC$ and $BD$ intersecting at point $O$ such that $AC \perp BD$.
2. Let $E$ and $G$ be the midpoints of $AB$ and $CD$ respectively. The line segment $EG$ is the midline of the trapezium, and it is given that $EG = 5$.
3. Let $F$ and $H$ be the midpoints ... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Inside the rectangle $ABCD$ is taken a point $M$ such that $\angle BMC + \angle AMD = 180^o$. Determine the sum of the angles $BCM$ and $DAM$. | 1. **Reflect point \( M \) over the midsegment of rectangle \( ABCD \) to get point \( M' \).**
- The midsegment of a rectangle is the line segment that connects the midpoints of two opposite sides. In this case, we can consider the midsegment connecting the midpoints of \( AB \) and \( CD \).
2. **Let \(\angle AMD... | 90 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The kid cut out of grid paper with the side of the cell $1$ rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle.
- My rectangle ... - kid sobbed. - There is somet... | Let's denote the sides of the original rectangle as \(a\) and \(b\), and the side length of the square that Carlson cut out as \(x\).
1. **Calculate the area and perimeter of the original rectangle:**
- The area of the original rectangle is \(ab\).
- The perimeter of the original rectangle is \(2(a + b)\).
2. *... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A right triangle when rotating around a large leg forms a cone with a volume of $100\pi$. Calculate the length of the path that passes through each vertex of the triangle at rotation of $180^o$ around the point of intersection of its bisectors, if the sum of the diameters of the circles, inscribed in the triangle and c... | 1. **Identify the given conditions and variables:**
- The volume of the cone formed by rotating the right triangle around its larger leg is \(100\pi\).
- The sum of the diameters of the inscribed and circumscribed circles of the triangle is 17.
- We need to calculate the length of the path that passes through ... | 30 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices of three being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a f... | 1. **Total number of ways to choose 3 knights out of 25:**
\[
\binom{25}{3} = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = 2300
\]
2. **Calculate the number of ways to choose 3 knights such that at least two are adjacent:**
- **Case 1: All three knights are adjacent.**
- There are 25 possible ... | 113 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For $\{1, 2, 3, \dots, n\}$ and each of its nonempty subsets a unique [b]alternating sum[/b] is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. (For example, the alternating sum for $\{1, 2, 4, 6, 9\}$ is $... | 1. **Understanding the Problem:**
We need to find the sum of all alternating sums for the set $\{1, 2, 3, \dots, 7\}$. The alternating sum for a subset is defined by arranging the numbers in decreasing order and then alternately adding and subtracting them.
2. **Analyzing the Alternating Sum:**
Let's denote the ... | 448 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there? | 1. **Identify the structure of the number:**
We are looking for four-digit numbers that start with 1 and have exactly two identical digits. Let's denote the four-digit number as \(1abc\), where \(a, b, c\) are digits and one of them is repeated.
2. **Determine the possible positions for the repeated digit:**
The... | 432 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The integer $n$ is the smallest positive multiple of 15 such that every digit of $n$ is either 8 or 0. Compute $\frac{n}{15}$. | 1. **Determine the conditions for \( n \):**
- \( n \) must be a multiple of 15.
- Every digit of \( n \) must be either 8 or 0.
2. **Break down the conditions for \( n \) to be a multiple of 15:**
- \( n \) must be a multiple of both 3 and 5.
- For \( n \) to be a multiple of 5, the last digit must be 0 (... | 592 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Three circles, each of radius 3, are drawn with centers at $(14,92)$, $(17,76)$, and $(19,84)$. A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absol... | 1. **Identify the centers of the circles and their radii:**
- The centers of the circles are at \((14, 92)\), \((17, 76)\), and \((19, 84)\).
- Each circle has a radius of 3.
2. **Determine the condition for the line:**
- The line must pass through \((17, 76)\) and divide the total area of the parts of the th... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$. | 1. We start with the given function definition:
\[
f(n) = \begin{cases}
n - 3 & \text{if } n \ge 1000 \\
f(f(n + 5)) & \text{if } n < 1000
\end{cases}
\]
We need to find \( f(84) \).
2. Since \( 84 < 1000 \), we use the second case of the function definition:
\[
f(84) = f(f(89))
\]
3. ... | 98 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is 15 $\text{cm}^2$ and the area of face $ABD$ is 12 $\text{cm}^2$. These two faces meet each other at a $30^\circ$ angle. Find the volume of the tetrahedron in $\text{cm}^3$. | 1. **Identify the given information:**
- Edge \( AB \) has length 3 cm.
- The area of face \( ABC \) is 15 \(\text{cm}^2\).
- The area of face \( ABD \) is 12 \(\text{cm}^2\).
- The angle between faces \( ABC \) and \( ABD \) is \( 30^\circ \).
2. **Calculate the height of face \( ABD \):**
- The area o... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have determined this. What was Mary's score? (Recall that... | 1. We start by understanding the scoring formula for the AHSME:
\[
s = 30 + 4c - w
\]
where \( s \) is the score, \( c \) is the number of correct answers, and \( w \) is the number of wrong answers. Note that the total number of questions is 30, so \( c + w + u = 30 \), where \( u \) is the number of unan... | 119 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A gardener plants three maple trees, four oak trees, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\frac{m}{n}$ in lowest terms be the probability that no two birch trees are next to one another. Find $m + n$. | 1. **Calculate the total number of permutations of the trees:**
The gardener plants 3 maple trees, 4 oak trees, and 5 birch trees, making a total of 12 trees. The total number of unique permutations of these trees is given by the multinomial coefficient:
\[
\frac{12!}{3!4!5!}
\]
Calculating this, we get:... | 106 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$. If $x = 0$ is a root of $f(x) = 0$, what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$? | 1. **Given Functional Equations:**
\[
f(2 + x) = f(2 - x) \quad \text{and} \quad f(7 + x) = f(7 - x)
\]
These equations imply that \( f \) is symmetric about \( x = 2 \) and \( x = 7 \).
2. **Transforming the Equations:**
Let's denote:
\[
a(x) = 4 - x \quad \text{and} \quad b(x) = 14 - x
\]
... | 401 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The pages of a book are numbered 1 through $n$. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice? | 1. The sum of the first $n$ natural numbers is given by the formula:
\[
S = \frac{n(n+1)}{2}
\]
We are given that the incorrect sum is 1986, which means one of the page numbers was added twice.
2. To find the correct $n$, we need to find the largest $n$ such that:
\[
\frac{n(n+1)}{2} \leq 1986
\]
... | 33 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\text{th}}$ term of this sequence. | 1. **Identify the sequence in base 3:**
The given sequence consists of numbers that are either powers of 3 or sums of distinct powers of 3. The first few terms are \(1, 3, 4, 9, 10, 12, 13, \ldots\). When these numbers are written in base 3, they become \(1_3, 10_3, 11_3, 100_3, 101_3, 110_3, 111_3, \ldots\).
2. **... | 981 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to $S$? | 1. First, we need to determine the proper divisors of \( 1000000 \). We start by expressing \( 1000000 \) in its prime factorized form:
\[
1000000 = 10^6 = (2 \times 5)^6 = 2^6 \times 5^6
\]
The number of divisors of \( 1000000 \) is given by \((6+1)(6+1) = 49\). Since we are interested in proper divisors, ... | 141 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$. | 1. **Identify the given elements and the goal:**
- We are given a triangle \( \triangle ABC \) with sides \( AB = 425 \), \( BC = 450 \), and \( AC = 510 \).
- An interior point \( P \) is drawn, and segments through \( P \) parallel to the sides of the triangle are of equal length \( d \).
- We need to find t... | 306 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let triangle $ABC$ be a right triangle in the xy-plane with a right angle at $C$. Given that the length of the hypotenuse $AB$ is 60, and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4$ respectively, find the area of triangle $ABC$. | 1. **Determine the angle between the lines \( y = x + 3 \) and \( y = 2x + 4 \):**
Let the acute angle the red line makes with the \( x \)-axis be \( \alpha \) and the acute angle the blue line makes with the \( x \)-axis be \( \beta \). Then, we know that:
\[
\tan \alpha = 1 \quad \text{and} \quad \tan \beta... | 400 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
By a proper divisor of a natural number we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? | To solve the problem, we need to identify the "nice" numbers and then find the sum of the first ten such numbers. A "nice" number is defined as a natural number greater than 1 that is equal to the product of its distinct proper divisors.
1. **Understanding the Definition of Proper Divisors:**
- Proper divisors of a... | 604 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$? | 1. **Identify the centers and radii of the spheres:**
- Sphere 1: Center at $(-2, -10, 5)$ with radius $19$.
- Sphere 2: Center at $(12, 8, -16)$ with radius $87$.
2. **Calculate the distance between the centers of the two spheres using the distance formula:**
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (... | 137 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$. | 1. Given the equation \( y^2 + 3x^2 y^2 = 30x^2 + 517 \), we start by letting \( a = x^2 \) and \( b = y^2 \). This transforms the equation into:
\[
b + 3ab = 30a + 517
\]
2. We can use Simon's Favorite Factoring Trick to factor the equation. First, we rearrange the terms:
\[
b + 3ab - 30a = 517
\]
... | 588 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers. | 1. Let the \( k \) consecutive positive integers be \( a, a+1, a+2, \ldots, a+(k-1) \). The sum of these integers can be expressed as:
\[
a + (a+1) + (a+2) + \cdots + (a+k-1)
\]
This is an arithmetic series with the first term \( a \) and the last term \( a+(k-1) \).
2. The sum of an arithmetic series is g... | 486 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more ``bubble passes''. A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the t... | 1. **Understanding the Problem:**
We need to determine the probability that the number initially at position \( r_{20} \) will end up at position \( r_{30} \) after one bubble pass through a sequence of 40 distinct real numbers.
2. **Key Observations:**
- During a bubble pass, each element is compared with its s... | 931 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $|x_i| < 1$ for $i = 1, 2, \dots, n$. Suppose further that
\[ |x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|. \]
What is the smallest possible value of $n$? | 1. We are given that \( |x_i| < 1 \) for \( i = 1, 2, \dots, n \) and
\[
|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.
\]
We need to find the smallest possible value of \( n \).
2. First, consider the inequality \( |x_i| < 1 \). This implies that the sum of the absolute values of \( x_i ... | 20 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The function $f$, defined on the set of ordered pairs of positive integers, satisfies the following properties:
\begin{eqnarray*} f(x,x) &=& x, \\ f(x,y) &=& f(y,x), \quad \text{and} \\ (x + y) f(x,y) &=& yf(x,x + y). \end{eqnarray*}
Calculate $f(14,52)$. | To solve for \( f(14,52) \), we will use the given properties of the function \( f \):
1. \( f(x,x) = x \)
2. \( f(x,y) = f(y,x) \)
3. \( (x + y) f(x,y) = y f(x,x + y) \)
We will use these properties step-by-step to find \( f(14,52) \).
1. **Using the symmetry property**:
\[
f(14,52) = f(52,14)
\]
This a... | 364 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile and types it. There are nine letters to be typed during the day, and the boss deli... | 1. Let's analyze the problem step by step. The boss delivers the letters in the order 1, 2, 3, 4, 5, 6, 7, 8, 9, and the secretary types them in a Last-In-First-Out (LIFO) manner, meaning the most recently delivered letter is on top of the pile and will be typed first.
2. We know that letter 8 has already been typed. ... | 256 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$. | 1. Let \( p \) be the probability of getting heads in a single flip. The probability of getting tails is \( 1 - p \).
2. The probability of getting exactly one head in five flips is given by the binomial probability formula:
\[
P(X = 1) = \binom{5}{1} p^1 (1-p)^4 = 5p(1-p)^4
\]
3. The probability of getting ... | 283 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[ \begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array} \] Find the value of \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100... | 1. Define the function \( f(x) \) as follows:
\[
f(x) = \sum_{k=1}^7 (x+k)^2 x_k
\]
This function is a quadratic polynomial in \( x \).
2. Given the values:
\[
f(1) = 1, \quad f(2) = 12, \quad f(3) = 123
\]
We need to find \( f(4) \).
3. Calculate the first differences:
\[
f(2) - f(1) = ... | 334 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer $ n$ such that \[133^5 \plus{} 110^5 \plus{} 84^5 \plus{} 27^5 \equal{} n^5.\] Find the value of $ n$. | 1. **Units Digit Analysis:**
- We start by analyzing the units digits of the numbers involved. The units digits of \(133\), \(110\), \(84\), and \(27\) are \(3\), \(0\), \(4\), and \(7\) respectively.
- The units digit of any number \(k\) and \(k^5\) are the same. Therefore, the units digit of \(n\) must be the s... | 144 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ P_1$ be a regular $ r$-gon and $ P_2$ be a regular $ s$-gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$. What's the largest possible value of $ s$? | 1. The interior angle of a regular $r$-gon is given by:
\[
\text{Interior angle of } P_1 = \frac{180(r-2)}{r}
\]
2. The interior angle of a regular $s$-gon is given by:
\[
\text{Interior angle of } P_2 = \frac{180(s-2)}{s}
\]
3. According to the problem, each interior angle of $P_1$ is $\frac{59}... | 117 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that $25\%$ of these fi... | 1. **Determine the number of tagged fish remaining in the lake on September 1:**
- Initially, 60 fish were tagged on May 1.
- Given that 25% of the tagged fish had died or emigrated, 75% of the tagged fish remain.
\[
\text{Remaining tagged fish} = 60 \times \left(\frac{3}{4}\right) = 60 \times 0.75 = 45
... | 630 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$. | 1. **Identify the vertices and the sides of the triangle:**
The vertices of the triangle are \( P = (-8, 5) \), \( Q = (-15, -19) \), and \( R = (1, -7) \).
2. **Calculate the lengths of the sides using the distance formula:**
\[
PQ = \sqrt{(-15 + 8)^2 + (-19 - 5)^2} = \sqrt{(-7)^2 + (-24)^2} = \sqrt{49 + 576... | 89 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a shooting match, eight clay targets are arranged in two hanging columns of three targets each and one column of two targets. A marksman is to break all the targets according to the following rules:
1) The marksman first chooses a column from which a target is to be broken.
2) The marksman must then break the ... | 1. **Define the problem in terms of sequences:**
- Let $X$ represent shooting a target from the first column.
- Let $Y$ represent shooting a target from the second column.
- Let $Z$ represent shooting a target from the third column.
2. **Understand the constraints:**
- Each column has a fixed number of tar... | 560 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A fair coin is to be tossed $10$ times. Let $i/j$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j$. | 1. **Define the problem in terms of sequences:**
We need to find the number of sequences of length 10 consisting of heads (H) and tails (T) such that no two heads are consecutive.
2. **Use dynamic programming to count valid sequences:**
Let \( a_n \) be the number of valid sequences of length \( n \) ending in T... | 73 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Someone observed that $6! = 8 \cdot 9 \cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers. | To find the largest positive integer \( n \) for which \( n! \) can be expressed as the product of \( n - 3 \) consecutive positive integers, we need to set up the equation and solve for \( n \).
1. **Set up the equation:**
We need to express \( n! \) as the product of \( n - 3 \) consecutive integers. Let these in... | 23 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $T = \{9^k : k \ \text{is an integer}, 0 \le k \le 4000\}$. Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit? | 1. **Understanding the Problem:**
We need to determine how many elements of the set \( T = \{9^k : k \ \text{is an integer}, 0 \le k \le 4000\} \) have 9 as their leftmost digit. We are given that \( 9^{4000} \) has 3817 digits and its first digit is 9.
2. **Number of Digits in \( 9^k \):**
The number of digits ... | 184 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ ... | 1. We start by identifying the term \( A_k = \binom{1000}{k}(0.2)^k \) in the binomial expansion of \( (1+0.2)^{1000} \). We need to find the value of \( k \) for which \( A_k \) is the largest.
2. Let \( n \) be the value of \( k \) such that \( A_n \) is the largest. This implies that \( A_n > A_{n-1} \) and \( A_n ... | 166 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find $A^2$, where $A$ is the sum of the absolute values of all roots of the following equation: \begin{eqnarray*}x &=& \sqrt{19} + \frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{{\displaystyle \sqrt{19}+\frac{91}{x}}}}}}}}}\end{eqnarray*} | 1. Let \( f(x) = \sqrt{19} + \frac{91}{x} \). We need to solve the equation \( x = f(x) \), which translates to:
\[
x = \sqrt{19} + \frac{91}{x}
\]
2. Multiply both sides by \( x \) to clear the fraction:
\[
x^2 = x \sqrt{19} + 91
\]
3. Rearrange the equation to standard quadratic form:
\[
x^2 -... | 383 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$. | 1. **Geometric Interpretation**: Consider \( n \) right triangles joined at their vertices, with bases \( a_1, a_2, \ldots, a_n \) and heights \( 1, 3, \ldots, 2n-1 \). The sum of their hypotenuses is the value of \( S_n \). The minimum value of \( S_n \) is the length of the straight line connecting the bottom vertex ... | 12 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with t... | 1. Let \( a \) be the number of red socks and \( b \) be the number of blue socks. The total number of socks is \( a + b \leq 1991 \).
2. The problem states that the probability of selecting two socks of the same color (both red or both blue) is exactly \( \frac{1}{2} \). This can be expressed as:
\[
\frac{\bino... | 990 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two three-letter strings, $aaa$ and $bbb$, are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an $a$ when it should have been a $b$, or as a $b$ when it should be an $a$. However, whether a given lette... | 1. **Define the problem and notation:**
- We are given two three-letter strings, \(aaa\) and \(bbb\), transmitted electronically.
- Each letter has a \( \frac{1}{3} \) chance of being received incorrectly.
- Let \( S_a \) be the string received when \( aaa \) is transmitted.
- Let \( S_b \) be the string re... | 532 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Rhombus $PQRS$ is inscribed in rectangle $ABCD$ so that vertices $P$, $Q$, $R$, and $S$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB=15$, $BQ=20$, $PR=30$, and $QS=40$. Let $m/n$, in lowest terms, denote the perimeter of $ABCD$. ... | 1. **Identify the center of the rhombus:**
Let \( O \) be the intersection of diagonals \( PR \) and \( QS \). Since \( O \) is the center of the rhombus, it is also the midpoint of both diagonals.
2. **Determine the lengths of the diagonals:**
Given \( PR = 30 \) and \( QS = 40 \), the half-lengths of the diago... | 677 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$. | 1. **Label the vertices and identify the diagonals:**
- Let the vertices of the hexagon be labeled \( A, B, C, D, E, F \) in clockwise order.
- The side lengths are given as \( AB = 31 \) and \( BC = CD = DE = EF = FA = 81 \).
- We need to find the lengths of the diagonals \( AC, AD, \) and \( AE \).
2. **Use... | 384 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron. | 1. **Identify the given information:**
- The angle between faces $ABC$ and $BCD$ is $30^\circ$.
- The area of face $ABC$ is $120$.
- The area of face $BCD$ is $80$.
- The length of edge $BC$ is $10$.
2. **Calculate the height of triangle $BCD$ relative to side $BC$:**
- The area of triangle $BCD$ is giv... | 320 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l$, the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$, and the resulting line is reflected in $l_2... | 1. **Define the angles:**
Let \( a = \frac{\pi}{70} \) and \( b = \frac{\pi}{54} \) be the angles that lines \( l_1 \) and \( l_2 \) make with the positive x-axis, respectively. Let \( x \) be the angle that line \( l \) makes with the positive x-axis. Given that \( l \) is the line \( y = \frac{19}{92}x \), we can ... | 945 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC$, and $AB$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O$, and that \[\frac{AO}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92,\] find \[\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}.\] | 1. Let \( \frac{AO}{OA'} = k_1 \), \( \frac{BO}{OB'} = k_2 \), and \( \frac{CO}{OC'} = k_3 \). We are given that:
\[
k_1 + k_2 + k_3 = 92
\]
2. By the properties of cevians in a triangle, we know that:
\[
\frac{1}{k_1 + 1} + \frac{1}{k_2 + 1} + \frac{1}{k_3 + 1} = 1
\]
3. To find \( k_1 k_2 k_3 \), we... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick o... | 1. **Define the problem and set up the notation:**
- We have two sets of numbers: $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$.
- These numbers are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$.
- We need to find the probability $p$ that a brick of dimensions $a_1 \times a_2 \ti... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P ... | 1. **Using Euler's Formula**: Euler's formula for a convex polyhedron states that:
\[
V - E + F = 2
\]
Given that the polyhedron has \( F = 32 \) faces, we can rewrite Euler's formula as:
\[
V - E + 32 = 2 \implies V - E = -30
\]
2. **Relating Edges and Vertices**: Each vertex is joined by \( T + ... | 250 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and... | 1. **Understanding the problem:**
- Kenny and Jenny are walking on parallel paths 200 feet apart.
- Kenny walks at 3 feet per second, and Jenny walks at 1 foot per second.
- A circular building with a diameter of 100 feet is centered midway between the paths.
- At the moment the building first blocks their ... | 163 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a point $P$ on a triangular piece of paper $ABC,$ consider the creases that are formed in the paper when $A, B,$ and $C$ are folded onto $P.$ Let us call $P$ a fold point of $\triangle ABC$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=72,$ and... | 1. **Determine the length of \(BC\):**
Given \(AB = 36\), \(AC = 72\), and \(\angle B = 90^\circ\), we can use the Pythagorean theorem to find \(BC\):
\[
BC = \sqrt{AB^2 + AC^2} = \sqrt{36^2 + 72^2} = \sqrt{1296 + 5184} = \sqrt{6480} = 36\sqrt{3}
\]
2. **Identify the creases and their properties:**
The ... | 597 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The graphs of the equations \[ y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k, \] are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.$ These 63 lines cut part of the plane into equilateral triangles of side $2/\sqrt{3}.$ How many such triangles are formed? | 1. **Understanding the Problem:**
We are given three sets of lines:
\[
y = k, \quad y = \sqrt{3}x + 2k, \quad y = -\sqrt{3}x + 2k
\]
where \( k \) ranges from \(-10\) to \(10\). These lines form equilateral triangles with side length \(\frac{2}{\sqrt{3}}\).
2. **Finding Intersections:**
To form equil... | 660 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there? | To solve the problem, we need to find the number of ordered pairs \((a, b)\) such that the system of equations
\[
\begin{cases}
ax + by = 1 \\
x^2 + y^2 = 50
\end{cases}
\]
has at least one solution where \((x, y)\) are integers.
1. **Identify integer solutions to \(x^2 + y^2 = 50\):**
We need to find all pairs \((... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues $4''$ or $10''$ or $19''$ to the total height of the tower. How many differnt tower heights can be achieved using all 94 of the bricks? | To determine the number of different tower heights that can be achieved using all 94 bricks, we need to consider the possible contributions to the height from each brick. Each brick can contribute either $4''$, $10''$, or $19''$ to the total height.
1. **Formulate the problem in terms of variables:**
Let:
- $x$... | 465 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field... | 1. **Define the side length of the squares:**
Let the side length of the squares be \( a \). The field is to be partitioned into congruent square plots, so the number of squares along the length (52 meters) and the width (24 meters) must be integers. Let \( n \) be the number of squares along the length and \( m \) ... | 78 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a positive integer $n$, let $p(n)$ be the product of the non-zero digits of $n$. (If $n$ has only one digits, then $p(n)$ is equal to that digit.) Let \[ S=p(1)+p(2)+p(3)+\cdots+p(999). \] What is the largest prime factor of $S$? | 1. **Understanding the Problem:**
We need to find the sum \( S \) of the products of the non-zero digits of all numbers from 1 to 999. For a number \( n \), \( p(n) \) is defined as the product of its non-zero digits.
2. **Breaking Down the Problem:**
We can consider the numbers from 1 to 999 as three-digit numb... | 11567 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Circles of radius 3 and 6 are externally tangent to each other and are internally tangent to a circle of radius 9. The circle of radius 9 has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. | 1. **Identify the centers and radii of the circles:**
- Let \( O_1 \) be the center of the circle with radius 3.
- Let \( O_2 \) be the center of the circle with radius 6.
- Let \( O_3 \) be the center of the circle with radius 9.
2. **Determine the distances between the centers:**
- Since the circles with... | 72 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $p$ be the probability that the object reaches $(2,2)$ in six or fewer steps. Given that $p$ can be written in the form $m/n,$ where $m$ and $n... | 1. **Understanding the problem**: We need to find the probability \( p \) that an object starting at \((0,0)\) reaches \((2,2)\) in six or fewer steps. Each step is either left, right, up, or down, and all four directions are equally likely.
2. **Formulating the problem**: To reach \((2,2)\), the object must take exac... | 67 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$ | 1. **Identify the roots and their properties:**
Given that the polynomial \(x^4 + ax^3 + bx^2 + cx + d = 0\) has four non-real roots, and the roots must come in conjugate pairs. Let the roots be \( \alpha, \overline{\alpha}, \beta, \overline{\beta} \).
2. **Given conditions:**
- The product of two of the roots i... | 26 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$? | 1. **Determine the prime factorization of \( n \):**
\[
n = 2^{31} \cdot 3^{19}
\]
2. **Calculate \( n^2 \):**
\[
n^2 = (2^{31} \cdot 3^{19})^2 = 2^{62} \cdot 3^{38}
\]
3. **Find the total number of divisors of \( n^2 \):**
The number of divisors of a number \( a^m \cdot b^n \) is given by \((m+1... | 589 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $(1+\sin t)(1+\cos t)=5/4$ and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$ | 1. Given the equations:
\[
(1+\sin t)(1+\cos t) = \frac{5}{4}
\]
and
\[
(1-\sin t)(1-\cos t) = \frac{m}{n} - \sqrt{k},
\]
we need to find the values of \(k\), \(m\), and \(n\) such that \(k + m + n\) is maximized.
2. First, expand the given product:
\[
(1+\sin t)(1+\cos t) = 1 + \sin t + ... | 27 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer? | 1. We need to find the largest positive integer \( n \) such that \( n \) cannot be expressed as the sum of a positive integral multiple of 42 and a positive composite integer. In other words, we need \( n \) such that \( n = 42k + c \) where \( k \) is a positive integer and \( c \) is a positive composite integer.
2... | 215 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $... | 1. **Identify the problem constraints and setup:**
We are given a right rectangular prism \( P \) with sides of integral lengths \( a, b, c \) such that \( a \leq b \leq c \). A plane parallel to one of the faces of \( P \) cuts \( P \) into two smaller prisms, one of which is similar to \( P \). We are given \( b =... | 40 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$. | 1. Define a "good" string as one that contains neither 5 consecutive heads nor 2 consecutive tails. We seek the probability that repeatedly flipping a coin generates a good string directly followed by 5 consecutive heads.
2. Let \( t_k \) denote the number of good strings of length \( k \) ( \( k \ge 0 \) ). Define \(... | 37 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible b... | 1. **Understanding the Problem:**
We need to find the length of the longest proper sequence of dominos from the set \( D_{40} \), where each domino is an ordered pair of distinct positive integers \((i, j)\) with \(1 \leq i, j \leq 40\). The sequence must satisfy:
- The first coordinate of each domino after the f... | 39 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square cen... | 1. **Understanding the Problem:**
- A wooden cube with edge length 1 cm is placed on a horizontal surface.
- A point source of light is positioned \( x \) cm directly above an upper vertex of the cube.
- The shadow cast by the cube on the horizontal surface, excluding the area directly beneath the cube, is 48 ... | 166 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker ... | To solve this problem, we need to carefully track the student's movements and the lockers he opens. Let's break down the process step-by-step.
1. **First Pass (Forward):**
- The student starts at locker 1 and opens it.
- He then skips one locker and opens the next one, continuing this pattern until he reaches th... | 854 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For each permutation $ a_1, a_2, a_3, \ldots,a_{10}$ of the integers $ 1,2,3,\ldots,10,$ form the sum
\[ |a_1 \minus{} a_2| \plus{} |a_3 \minus{} a_4| \plus{} |a_5 \minus{} a_6| \plus{} |a_7 \minus{} a_8| \plus{} |a_9 \minus{} a_{10}|.\]
The average value of all such sums can be written in the form $ p/q,$ where $ p$... | 1. **Understanding the Problem:**
We need to find the average value of the sum
\[
|a_1 - a_2| + |a_3 - a_4| + |a_5 - a_6| + |a_7 - a_8| + |a_9 - a_{10}|
\]
over all permutations of the integers \(1, 2, 3, \ldots, 10\).
2. **Symmetry and Expected Value:**
Due to symmetry, the expected value of each t... | 58 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$ | 1. Let $\angle DBA = \angle BDC = x$ and $\angle CAB = \angle DBC = \angle ADB = \angle ACD = 2x$. This is given in the problem statement.
2. Since $ABCD$ is a parallelogram, opposite sides are equal, so $BC = AD = Z$.
3. Pick point $P$ on $DB$ such that $CP = BC = Z$. Notice that $\triangle BPC$ is isosceles, so $\ang... | 777 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$. | 1. **Define the function and properties:**
Given the function \( f(x) = \frac{ax + b}{cx + d} \) with properties \( f(19) = 19 \), \( f(97) = 97 \), and \( f(f(x)) = x \) for all \( x \neq -\frac{d}{c} \).
2. **Consider the function \( h(x) = f(x) - x \):**
Since \( f(19) = 19 \) and \( f(97) = 97 \), the equati... | 58 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many of the integers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers? | 1. **Understanding the problem**: We need to determine how many integers between 1 and 1000 can be expressed as the difference of squares of two nonnegative integers.
2. **Expressing the difference of squares**: The difference of squares of two integers \(a\) and \(b\) can be written as:
\[
a^2 - b^2 = (a+b)(a-... | 750 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. **Counting the Rectangles:**
- To count the total number of rectangles in an $8 \times 8$ checkerboard, we need to choose 2 out of the 9 horizontal lines and 2 out of the 9 vertical lines.
- The number of ways to choose 2 lines from 9 is given by the binomial coefficient $\binom{9}{2}$.
- Therefore, the to... | 125 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\frac 27.$ What is the number of p... | To solve this problem, we need to find the number of possible values for \( r \) such that \( r \) is closest to \(\frac{2}{7}\) when expressed as a four-place decimal \(0.abcd\).
1. **Convert \(\frac{2}{7}\) to a decimal:**
\[
\frac{2}{7} \approx 0.285714285714\ldots
\]
The decimal representation of \(\fr... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of t... | To solve this problem, we need to count the number of complementary three-card sets from a deck of 27 cards, where each card is uniquely defined by a combination of shape, color, and shade. We will consider two main cases: when the cards share no properties and when the cards share at least one property.
1. **Case 1: ... | 702 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S$ be the set of points in the Cartesian plane that satisfy
\[ \Big|\big|{|x| \minus{} 2}\big| \minus{} 1\Big| \plus{} \Big|\big|{|y| \minus{} 2}\big| \minus{} 1\Big| \equal{} 1.
\]
If a model of $ S$ were built from wire of negligible thickness, then the total length of wire required would be $ a\sqrt {b},$ w... | 1. **Understanding the given equation:**
\[
\Big|\big|{|x| - 2}\big| - 1\Big| + \Big|\big|{|y| - 2}\big| - 1\Big| = 1
\]
We need to analyze the behavior of the function \( f(a) = \Big|\big||a| - 2\big| - 1\Big| \).
2. **Analyzing \( f(a) \):**
- For \( 0 \leq a \leq 1 \):
\[
f(a) = \Big|\big|a... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Let $m/n$ be the probability that $\sqrt{2+\sqrt{3}}\le |v+w|,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. **Identify the roots of the equation**: The roots of the equation \(z^{1997} - 1 = 0\) are the 1997th roots of unity. These roots are given by \(z_k = e^{2\pi i k / 1997}\) for \(k = 0, 1, 2, \ldots, 1996\).
2. **Express the roots in terms of cosine and sine**: Let \(v = e^{2\pi i n / 1997}\) and \(w = e^{2\pi i m ... | 2312047 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The sides of rectangle $ABCD$ have lengths 10 and 11. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any... | 1. **Place the rectangle on the complex plane:**
Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 11$. Place the rectangle on the complex plane with $A$ at the origin $(0,0)$, $B$ at $(0,10)$, $C$ at $(11,10)$, and $D$ at $(11,0)$.
2. **Position the equilateral triangle:**
To maximize the area of the equilater... | 554 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6, 8^8,$ and $k$? | 1. We start by noting that the least common multiple (LCM) of the numbers \(6^6\), \(8^8\), and \(k\) must be \(12^{12}\). We need to find the values of \(k\) such that this condition holds.
2. First, we express each number in terms of its prime factorization:
\[
6^6 = (2 \cdot 3)^6 = 2^6 \cdot 3^6
\]
\[
... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x\le 2y\le 60$ and $y\le 2x\le 60.$ | 1. **Identify the constraints and initial pairs**:
We need to find the number of ordered pairs \((x, y)\) of positive integers that satisfy:
\[
x \leq 2y \leq 60 \quad \text{and} \quad y \leq 2x \leq 60.
\]
First, consider pairs of the form \((a, a)\). For these pairs, both conditions are automatically s... | 480 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The graph of $y^2+2xy+40|x|=400$ partitions the plane into several regions. What is the area of the bounded region? | 1. **Case 1: \( x \geq 0 \)**
- When \( x \geq 0 \), \( |x| = x \). The equation becomes:
\[
y^2 + 2xy + 40x = 400
\]
- Rearrange the equation:
\[
y^2 + 2xy + 40x - 400 = 0
\]
- Factor the equation:
\[
(y + 20)(y - 20) = 2x(y + 20)
\]
- This gives two cases:
... | 800 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Nine tiles are numbered $1, 2, 3, \ldots, 9,$ respectively. Each of three players randomly selects and keeps three of the tile, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. **Understanding the Problem:**
We need to find the probability that all three players obtain an odd sum when each player randomly selects three tiles from a set of nine tiles numbered \(1, 2, 3, \ldots, 9\).
2. **Condition for Odd Sum:**
For the sum of three numbers to be odd, one or all three of those numbe... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Except for the first two terms, each term of the sequence $1000, x, 1000-x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encounted. What positive integer $x$ produces a sequence of maximum length? | 1. **Define the sequence and initial terms:**
The sequence is given as \(1000, x, 1000-x, \ldots\). Each term is obtained by subtracting the preceding term from the one before that.
2. **Compute the first few terms:**
\[
\begin{aligned}
a_1 &= 1000, \\
a_2 &= x, \\
a_3 &= 1000 - x, \\
a_4 &= x - (... | 618 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ mintues. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m=a-b\sqrt{c},$ where $a, b,$ and $c$ are posi... | 1. **Define the problem in terms of probability and geometry:**
- Let \( X \) and \( Y \) be the arrival times of the two mathematicians, measured in minutes after 9 a.m. Thus, \( X \) and \( Y \) are uniformly distributed over the interval \([0, 60]\).
- The condition that one arrives while the other is in the c... | 197 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. **Identify the vertices of the parallelogram:**
The vertices of the parallelogram are given as \((10, 45)\), \((10, 114)\), \((28, 153)\), and \((28, 84)\).
2. **Find the center (midpoint) of the parallelogram:**
The center of the parallelogram can be found by taking the midpoint of the diagonals. The diagona... | 118 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $x$, let $S(x)$ be the sum of the digits of $x$, and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values $T(x)$ do not exceed 1999? | 1. **Understanding the Problem:**
We need to find how many values \( T(x) \) do not exceed 1999, where \( T(x) = |S(x+2) - S(x)| \) and \( S(x) \) is the sum of the digits of \( x \).
2. **Analyzing Edge Cases:**
We start by examining the edge cases where \( x \) is close to a power of 10. This is because the su... | 222 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There is a set of 1000 switches, each of which has four positions, called $A, B, C,$ and $D.$ When the position of any switch changes, it is only from $A$ to $B,$ from $B$ to $C,$ from $C$ to $D,$ or from $D$ to $A.$ Initially each switch is in position $A.$ The switches are labeled with the 1000 different integers ... | 1. **Understanding the Problem:**
- We have 1000 switches, each labeled with \(2^x 3^y 5^z\) where \(x, y, z \in \{0, 1, \ldots, 9\}\).
- Each switch can be in one of four positions: \(A, B, C,\) or \(D\).
- Initially, all switches are in position \(A\).
- At each step \(i\), the \(i\)-th switch and all swi... | 500 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{T}$ be the set of ordered triples $(x,y,z)$ of nonnegative real numbers that lie in the plane $x+y+z=1.$ Let us say that $(x,y,z)$ supports $(a,b,c)$ when exactly two of the following are true: $x\ge a, y\ge b, z\ge c.$ Let $\mathcal{S}$ consist of those triples in $\mathcal{T}$ that support $\left(\fra... | 1. **Understanding the Problem:**
We need to find the ratio of the area of the set $\mathcal{S}$ to the area of the set $\mathcal{T}$, where:
- $\mathcal{T}$ is the set of ordered triples $(x, y, z)$ of nonnegative real numbers that lie in the plane $x + y + z = 1$.
- $\mathcal{S}$ consists of those triples in... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m$ and $n$ are relatively prime p... | 1. Given the function \( f(z) = (a + bi)z \), where \( a \) and \( b \) are positive numbers, and the property that the image of each point in the complex plane is equidistant from that point and the origin, we have:
\[
|f(z)| = |z - f(z)|
\]
This implies:
\[
|(a + bi)z| = |z - (a + bi)z|
\]
2. Si... | 259 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m$ and $n$ are r... | 1. **Calculate the total number of segments:**
Given 10 points in the plane, the number of segments that can be formed by joining pairs of these points is given by the binomial coefficient:
\[
\binom{10}{2} = \frac{10 \times 9}{2} = 45
\]
2. **Calculate the total number of ways to choose 4 segments:**
T... | 51345 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P,$ and its radius is 21. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle. | 1. **Identify the given information and set up the problem:**
- The radius of the inscribed circle (inradius) is \( r = 21 \).
- \( AP = 23 \) and \( PB = 27 \).
- The point of tangency on \( \overline{AC} \) is \( Q \), and let \( CQ = x \).
2. **Use the property of tangents from a point to a circle:**
- ... | 150 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $\log_2 n.$ | 1. **Determine the total number of games played:**
Since there are 40 teams and each team plays every other team exactly once, the total number of games played is given by the combination formula:
\[
\binom{40}{2} = \frac{40 \times 39}{2} = 780
\]
2. **Calculate the probability of a specific arrangement:**... | 742 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $u$ and $v$ be integers satisfying $0<v<u.$ Let $A=(u,v),$ let $B$ be the reflection of $A$ across the line $y=x,$ let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon $ABCDE$ is 451. Find... | 1. **Identify the coordinates of the points:**
- Point \( A = (u, v) \).
- Point \( B \) is the reflection of \( A \) across the line \( y = x \), so \( B = (v, u) \).
- Point \( C \) is the reflection of \( B \) across the y-axis, so \( C = (-v, u) \).
- Point \( D \) is the reflection of \( C \) across th... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positi... | 1. Let \( b_1 \) and \( w_1 \) be the number of black and white marbles in the first box, respectively. Similarly, let \( b_2 \) and \( w_2 \) be the number of black and white marbles in the second box, respectively. We know that:
\[
b_1 + w_1 + b_2 + w_2 = 25
\]
2. The probability that both marbles drawn are ... | 61 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz=1,$ $x+\frac{1}{z}=5,$ and $y+\frac{1}{x}=29.$ Then $z+\frac{1}{y}=\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ | 1. Given the equations:
\[
xyz = 1, \quad x + \frac{1}{z} = 5, \quad y + \frac{1}{x} = 29
\]
2. From the second equation, solve for \(x\):
\[
x + \frac{1}{z} = 5 \implies x = 5 - \frac{1}{z}
\]
3. From the third equation, solve for \(y\):
\[
y + \frac{1}{x} = 29 \implies y = 29 - \frac{1}{x}
... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the liquid is $m-n\sqrt[3]{p},$ wh... | 1. **Determine the volume of the cone and the volume of the liquid:**
- The volume \( V \) of a cone is given by:
\[
V = \frac{1}{3} \pi r^2 h
\]
where \( r \) is the radius of the base and \( h \) is the height.
- For the given cone, \( r = 5 \) inches and \( h = 12 \) inches:
\[
V_... | 52 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the firetruck within six minutes. The area of... | 1. **Determine the time constraints and convert to hours:**
The firetruck can travel for 6 minutes, which is equivalent to:
\[
\frac{6}{60} = 0.1 \text{ hours}
\]
2. **Set up the distance equations:**
Let \( x \) be the distance traveled on the highway and \( r \) be the distance traveled on the prairie... | 731 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from... | 1. **Define the problem recursively:**
Let \( a_n \) be the number of cards above the card labeled \( n-1 \) in a stack of \( n \) cards such that they will be laid out \( 1, 2, 3, \ldots, n \) from left to right.
2. **Base case:**
For \( n = 2 \), the stack must be \( (1, 2) \) from top to bottom. Therefore, \(... | 927 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A deck of forty cards consists of four 1's, four 2's,..., and four 10's. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form a pair, where $m$ and $n$ are relatively prim... | 1. **Initial Setup**: We start with a deck of 40 cards, consisting of four cards each of the numbers 1 through 10. A matching pair (two cards with the same number) is removed from the deck. This leaves us with 38 cards, with three cards of one number and four cards of each of the other nine numbers.
2. **Total Number ... | 760 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n.$ | 1. **Choosing 5 out of 8 rings:**
We need to choose 5 rings out of the 8 available rings. This can be done in $\binom{8}{5}$ ways. Using the binomial coefficient formula, we have:
\[
\binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56
\]
2. **Ordering... | 376 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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