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What is the smallest positive integer $m$ such that $15! \, m$ can be expressed in more than one way as a product of $16$ distinct positive integers, up to order? | To find the smallest positive integer \( m \) such that \( 15! \cdot m \) can be expressed in more than one way as a product of 16 distinct positive integers, we need to explore the factorization of \( 15! \cdot m \).
1. **Calculate \( 15! \):**
\[
15! = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 ... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $O$ be the set of odd numbers between 0 and 100. Let $T$ be the set of subsets of $O$ of size $25$. For any finite subset of integers $S$, let $P(S)$ be the product of the elements of $S$. Define $n=\textstyle{\sum_{S \in T}} P(S)$. If you divide $n$ by 17, what is the remainder? | 1. **Identify the set \( O \) and the set \( T \):**
- \( O \) is the set of odd numbers between 0 and 100, i.e., \( O = \{1, 3, 5, \ldots, 99\} \).
- \( T \) is the set of subsets of \( O \) of size 25.
2. **Define \( n \) as the sum of the products of elements of each subset \( S \in T \):**
- \( n = \sum_{... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_n = n(2n+1)$. Evaluate
\[
\biggl | \sum_{1 \le j < k \le 36} \sin\bigl( \frac{\pi}{6}(a_k-a_j) \bigr) \biggr |.
\] | To solve the given problem, we need to evaluate the expression:
\[
\left| \sum_{1 \le j < k \le 36} \sin\left( \frac{\pi}{6}(a_k-a_j) \right) \right|
\]
where \(a_n = n(2n+1)\).
1. **Simplify the Argument of the Sine Function:**
For any \(u \in \mathbb{Z}\), modulo \(2\pi\), we have:
\[
\frac{\pi}{6} \left( a... | 18 | Calculus | other | Yes | Yes | aops_forum | false |
[b]p1.[/b] A man is walking due east at $2$ m.p.h. and to him the wind appears to be blowing from the north. On doubling his speed to $4$ m.p.h. and still walking due east, the wind appears to be blowing from the nortl^eas^. What is the speed of the wind (assumed to have a constant velocity)?
[b]p2.[/b] Prove that an... | To show that the common difference of three prime numbers, each greater than \(3\), in arithmetic progression is a multiple of \(6\), we proceed as follows:
1. **Representation of Primes Greater than 3**:
Any prime number greater than \(3\) can be written in the form \(6k \pm 1\) for some integer \(k\). This is bec... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Two trains, $A$ and $B$, travel between cities $P$ and $Q$. On one occasion $A$ started from $P$ and $B$ from $Q$ at the same time and when they met $A$ had travelled $120$ miles more than $B$. It took $A$ four $(4)$ hours to complete the trip to $Q$ and B nine $(9)$ hours to reach $P$. Assuming each train t... | 1. Let \( D \) be the distance between cities \( P \) and \( Q \).
2. Let \( V_A \) be the speed of train \( A \) and \( V_B \) be the speed of train \( B \).
3. Given that train \( A \) takes 4 hours to travel from \( P \) to \( Q \), we have:
\[
V_A = \frac{D}{4}
\]
4. Given that train \( B \) takes 9 hours ... | 864 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Solve the system of equations
$$xy = 2x + 3y$$
$$yz = 2y + 3z$$
$$zx =2z+3x$$
[b]p2.[/b] For any integer $k$ greater than $1$ and any positive integer $n$ , prove that $n^k$ is the sum of $n$ consecutive odd integers.
[b]p3.[/b] Determine all pairs of real numbers, $x_1$, $x_2$ with $|x_1|\le 1$ and ... | To find the smallest positive integer having exactly 100 different positive divisors, we need to consider the prime factorization of the number. The number of divisors of a number \( n \) with prime factorization \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is given by:
\[
(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)
\]
We ... | 45360 | Other | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Find the largest integer which is a factor of all numbers of the form $n(n +1)(n + 2)$ where $n$ is any positive integer with unit digit $4$. Prove your claims.
[b]p2.[/b] Each pair of the towns $A, B, C, D$ is joined by a single one way road. See example. Show that for any such arrangement, a salesman can... | 1. To find the largest integer which is a factor of all numbers of the form \( n(n + 1)(n + 2) \) where \( n \) is any positive integer with unit digit \( 4 \), we need to analyze the divisibility properties of the product \( n(n + 1)(n + 2) \).
- **Divisibility by 2**: Since \( n \) has a unit digit of 4, \( n \) ... | 120 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.
[b]p2.[/b] Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$, each one of which is also tangent to two sides of ... | 1. We need to find the maximum number of distinct positive integers that sum to 100. Let's denote these integers as \(a_1, a_2, \ldots, a_k\) where \(a_1 < a_2 < \cdots < a_k\).
2. The sum of the first \(k\) positive integers is given by the formula:
\[
S_k = \frac{k(k+1)}{2}
\]
3. We need to find the largest ... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Consider a right triangle with legs of lengths $a$ and $b$ and hypotenuse of length $c$ such that the perimeter of the right triangle is numerically (ignoring units) equal to its area. Prove that there is only one possible value of $a + b - c$, and determine that value.
[b]p2.[/b] Last August, Jennifer McL... | 1. Given a right triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), we know the perimeter is \(a + b + c\) and the area is \(\frac{1}{2}ab\).
2. According to the problem, the perimeter is numerically equal to the area:
\[
a + b + c = \frac{1}{2}ab
\]
3. Rearrange the equation to isol... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Two players play the following game. On the lowest left square of an $8 \times 8$ chessboard there is a rook (castle). The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second layer is also allowed to move the rook up or to the right by an arbitrary numbe... | To find the area of the shape formed by rearranging the four equal arcs of a circle of radius 1, we can use the following steps:
1. **Identify the Shape and Symmetry**:
- The shape is formed by rearranging four equal arcs of a circle of radius 1.
- By symmetry, the area of the figure not contained in the square ... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were on each bus?
[b]p2.[/b... | 1. To determine the number of seats on each bus, we need to find a common divisor of 138 and 115. This common divisor must be greater than 1 since each bus has more than one seat.
2. First, we find the prime factorization of 138 and 115:
\[
138 = 2 \times 3 \times 23
\]
\[
115 = 5 \times 23
\]
3. Th... | 23 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family?
[b]p2.[/b] Solve each of the following problems.
(1) Find a pair of numbers with a sum of $11$ and a product of $24$.
(2) Find a pair of numbers with a sum of ... | 1. Let \( x \) be the number of boys and \( y \) be the number of girls in the family. According to the problem, the boy has as many sisters as brothers, and the sister has twice as many brothers as sisters.
- For the boy: The number of brothers is \( x - 1 \) (since he does not count himself), and the number of si... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] There are $5$ weights of masses $1,2,3,5$, and $10$ grams. One of the weights is counterfeit (its weight is different from what is written, it is unknown if the weight is heavier or lighter). How to find the counterfeit weight using simple balance scales only twice?
[b]p2.[/b] There are $998$ candies and c... | To solve the problem of inserting arithmetic operations and brackets into the sequence \(2222222222\) to get the number \(999\), we need to carefully place the operations and brackets to achieve the desired result. Let's break down the given solution step-by-step:
1. **Group the digits and insert division:**
\[
... | 999 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Solve: $INK + INK + INK + INK + INK + INK = PEN$
($INK$ and $PEN$ are $3$-digit numbers, and different letters stand for different digits).
[b]p2. [/b]Two people play a game. They put $3$ piles of matches on the table:
the first one contains $1$ match, the second one $3$ matches, and the third one $4$ matc... | **p4.** Money in Wonderland comes in $\$5$ and $\$7$ bills. What is the smallest amount of money you need to buy a slice of pizza that costs $\$1$ and get back your change in full? (The pizza man has plenty of $\$5$ and $\$7$ bills.) For example, having $\$7$ won't do, since the pizza man can only give you $\$5$ back.
... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are a family of $5$ siblings. They have a pile of at least $2$ candies and are trying to split them up
amongst themselves. If the $2$ oldest siblings share the candy equally, they will have $1$ piece of candy left over.
If the $3$ oldest siblings share the candy equally, they will also have $1$ piece of candy lef... | 1. We start by translating the problem statement into mathematical congruences. The problem states that:
- If the 2 oldest siblings share the candy equally, they will have 1 piece left over. This can be written as:
\[
N \equiv 1 \pmod{2}
\]
- If the 3 oldest siblings share the candy equally, they w... | 31 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define $f(x)$ as $\frac{x^2-x-2}{x^2+x-6}$. $f(f(f(f(1))))$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p,q$. Find $10p+q$. | 1. First, we simplify the function \( f(x) \). Given:
\[
f(x) = \frac{x^2 - x - 2}{x^2 + x - 6}
\]
We factorize the numerator and the denominator:
\[
x^2 - x - 2 = (x + 1)(x - 2)
\]
\[
x^2 + x - 6 = (x + 3)(x - 2)
\]
Therefore, the function simplifies to:
\[
f(x) = \frac{(x + 1)(x... | 211 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks, then notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$. | 1. **Understanding the Problem:**
- We have a $420 \times 420$ square grid.
- The grid is tiled completely with $1 \times 2$ blocks.
- We need to find the total number of blocks that the two diagonals of the grid pass through.
2. **Analyzing the Diagonals:**
- Each diagonal of a $420 \times 420$ grid passe... | 2517 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$, denote $P(x)$ as the probability that the length of $AB$ is less than $x$. Find the minimum possible integer value of $x$ such that $\text{P}(x) > \frac{2}{3}$. | 1. **Fixing Point \( A \)**:
- Consider a circle with radius \( 100 \) and fix a point \( A \) on the circle.
- We need to find the probability that the length of \( AB \) is less than \( x \) for a randomly chosen point \( B \) on the circle.
2. **Using Law of Cosines**:
- Let \( O \) be the center of the ci... | 174 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Jason and Jared take turns placing blocks within a game board with dimensions $3 \times 300$, with Jason going first, such that no two blocks can overlap. The player who cannot place a block within the boundaries loses. Jason can only place $2 \times 100$ blocks, and Jared can only place $2 \times n$ blocks where $n$ i... | 1. **Understanding the Problem:**
- Jason and Jared are playing a game on a \(3 \times 300\) board.
- Jason places \(2 \times 100\) blocks.
- Jared places \(2 \times n\) blocks where \(n > 3\).
- The player who cannot place a block loses.
- We need to find the smallest \(n\) such that Jason can guarantee... | 51 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Derek fills a square $10$ by $10$ grid with $50$ $1$s and $50$ $2$s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N.$ Find the minimum possible value of $N.$ | 1. **Representation of the Products:**
- Derek fills a \(10 \times 10\) grid with \(50\) ones and \(50\) twos.
- The product of the numbers in each row can be represented as \(2^{a_i}\), where \(a_i\) is the number of twos in the \(i\)-th row.
- Similarly, the product of the numbers in each column can be repre... | 640 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The function $g\left(x\right)$ is defined as $\sqrt{\dfrac{x}{2}}$ for all positive $x$.
$ $\\
$$g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right)$$
$ $\\
can be expressed as $\cos(b)$ using degrees, where $0^\circ < b < 90^\circ$ and $b = p/q$ for some relatively prime posit... | 1. **Define the function \( g(x) \):**
\[
g(x) = \sqrt{\frac{x}{2}}
\]
This function is defined for all positive \( x \).
2. **Evaluate \( g\left(\frac{1}{2}\right) \):**
\[
g\left(\frac{1}{2}\right) = \sqrt{\frac{\frac{1}{2}}{2}} = \sqrt{\frac{1}{4}} = \frac{1}{2}
\]
3. **Recognize the relations... | 19 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to arrange the numbers $1$ through $8$ into a $2$ by $4$ grid such that the sum of the numbers in each of the two rows are all multiples of $6,$ and the sum of the numbers in each of the four columns are all multiples of $3$? | 1. **Identify the constraints and properties:**
- We need to arrange the numbers \(1\) through \(8\) in a \(2 \times 4\) grid.
- The sum of the numbers in each row must be a multiple of \(6\).
- The sum of the numbers in each column must be a multiple of \(3\).
2. **Analyze the numbers modulo \(3\):**
- Th... | 288 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In equilateral triangle $XYZ$ with side length $10$, define points $A, B$ on $XY,$ points $C, D$ on $YZ,$ and points $E, F$ on $ZX$ such that $ABDE$ and $ACEF$ are rectangles, $XA<XB,$ $YC<YD,$ and $ZE<ZF$. The area of hexagon $ABCDEF$ can be written as $\sqrt{x}$ for some positive integer $x$. Find $x$. | 1. **Define Variables and Points:**
Let \( XA = YB = x \). Since \( XYZ \) is an equilateral triangle with side length 10, we can use the properties of 30-60-90 triangles to find the coordinates of points \( A, B, C, D, E, \) and \( F \).
2. **Determine Coordinates:**
- \( A \) and \( B \) are on \( XY \):
... | 768 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x) = x^3 + 8x^2 - x + 3$ and let the roots of $P$ be $a, b,$ and $c.$ The roots of a monic polynomial $Q(x)$ are $ab - c^2, ac - b^2, bc - a^2.$ Find $Q(-1).$ | 1. **Identify the roots of \( P(x) \):**
Given \( P(x) = x^3 + 8x^2 - x + 3 \), let the roots be \( a, b, c \). By Vieta's formulas, we have:
\[
a + b + c = -8,
\]
\[
ab + bc + ca = -1,
\]
\[
abc = -3.
\]
2. **Form the polynomial \( Q(x) \):**
The roots of \( Q(x) \) are \( ab - c^2, a... | 1536 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There exist positive integers $N, M$ such that $N$'s remainders modulo the four integers $6, 36,$ $216,$ and $M$ form an increasing nonzero geometric sequence in that order. Find the smallest possible value of $M$. | 1. **Define the remainders:**
Let \( r_6 \), \( r_{36} \), \( r_{216} \), and \( r_M \) be the remainders of \( N \) when divided by \( 6 \), \( 36 \), \( 216 \), and \( M \) respectively. These remainders form an increasing nonzero geometric sequence.
2. **Geometric sequence properties:**
Since \( r_6 \), \( r_... | 2001 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A point $X$ exactly $\sqrt{2}-\frac{\sqrt{6}}{3}$ away from the origin is chosen randomly. A point $Y$ less than $4$ away from the origin is chosen randomly. The probability that a point $Z$ less than $2$ away from the origin exists such that $\triangle XYZ$ is an equilateral triangle can be expressed as $\frac{a\pi + ... | 1. **Fixing Point \(X\)**:
Let \(O\) be the origin. Due to symmetry, we can fix \(X\) at some arbitrary point exactly \(\sqrt{2} - \frac{\sqrt{6}}{3}\) away from the origin without loss of generality.
2. **Understanding the Problem**:
The problem is asking for the probability that a rotation around \(X\) by \(... | 34 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5,$ $b_1 = 7,$
$$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$
and
$$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$
$ $ \\
How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the ... | 1. We start with the given sequences \(a_n\) and \(b_n\) defined by:
\[
a_1 = 5, \quad b_1 = 7
\]
and the recurrence relations:
\[
a_{n+1} = \frac{\sqrt{(a_n + b_n - 1)^2 + (a_n - b_n + 1)^2}}{2}
\]
\[
b_{n+1} = \frac{\sqrt{(a_n + b_n + 1)^2 + (a_n - b_n - 1)^2}}{2}
\]
2. We need to deter... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$. | 1. **Determine the side length of the squares:**
The perimeter of each square is given as \(8\). Since the perimeter \(P\) of a square with side length \(s\) is \(P = 4s\), we have:
\[
4s = 8 \implies s = 2
\]
2. **Set up the equations for the dimensions of the rectangle:**
Let \(x\) and \(y\) be the di... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The sum
$$\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}$$
$ $ \\
can be expressed as a rational number $N$. Find the last 3 digits of $2021! \cdot N$. | 1. **Expressing the General Term:**
For any positive integer \( n \), we can express the term \(\frac{n^2 - 2}{n!}\) as follows:
\[
\frac{n^2 - 2}{n!} = \frac{n(n-1) + n - 2}{n!} = \frac{n(n-1)}{n!} + \frac{n-2}{n!}
\]
Simplifying further:
\[
\frac{n^2 - 2}{n!} = \frac{1}{(n-2)!} + \frac{1}{(n-1)!}... | 977 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In square $ABCD$ with $AB = 10$, point $P, Q$ are chosen on side $CD$ and $AD$ respectively such that $BQ \perp AP,$ and $R$ lies on $CD$ such that $RQ \parallel PA.$ $BC$ and $AP$ intersect at $X,$ and $XQ$ intersects the circumcircle of $PQD$ at $Y$. Given that $\angle PYR = 105^{\circ},$ $AQ$ can be expressed in sim... | 1. **Identify the given information and setup the problem:**
- We have a square \(ABCD\) with side length \(AB = 10\).
- Points \(P\) and \(Q\) are chosen on sides \(CD\) and \(AD\) respectively such that \(BQ \perp AP\).
- Point \(R\) lies on \(CD\) such that \(RQ \parallel PA\).
- \(BC\) and \(AP\) inters... | 23 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The infinite sequence of integers $a_1, a_2, \cdots $ is defined recursively as follows: $a_1 = 3$, $a_2 = 7$, and $a_n$ equals the alternating sum
$$a_1 - 2a_2 + 3a_3 - 4a_4 + \cdots (-1)^n \cdot (n-1)a_{n-1}$$
for all $n > 2$. Let $a_x$ be the smallest positive multiple of $1090$ appearing in this sequence. Find t... | 1. **Define the sequence and initial conditions:**
The sequence \( \{a_n\} \) is defined recursively with initial conditions:
\[
a_1 = 3, \quad a_2 = 7
\]
For \( n > 2 \), the sequence is defined by:
\[
a_n = a_1 - 2a_2 + 3a_3 - 4a_4 + \cdots + (-1)^n \cdot (n-1)a_{n-1}
\]
2. **Analyze the recu... | 51 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered triples of integers $(a,b,c)$ such that $$a^2 + b^2 + c^2 - ab - bc - ca - 1 \le 4042b - 2021a - 2021c - 2021^2$$ and $|a|, |b|, |c| \le 2021.$ | 1. **Rearrange the given inequality:**
We start with the given inequality:
\[
a^2 + b^2 + c^2 - ab - bc - ca - 1 \le 4042b - 2021a - 2021c - 2021^2
\]
Rearrange the terms to group all terms involving \(a\), \(b\), and \(c\) on one side:
\[
a^2 + b^2 + c^2 - ab - bc - ca - 4042b + 2021a + 2021c + ... | 14152 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers $a,b,c$ exist such that $a+b+c+1$, $a^2+b^2+c^2 +1$, $a^3+b^3+c^3+1,$ and $a^4+b^4+c^4+7459$ are all multiples of $p$ for some prime $p$. Find the sum of all possible values of $p$ less than $1000$. | 1. **Initial Setup and Simplification:**
We are given that \(a, b, c\) are positive integers such that:
\[
a + b + c + 1 \equiv 0 \pmod{p}
\]
\[
a^2 + b^2 + c^2 + 1 \equiv 0 \pmod{p}
\]
\[
a^3 + b^3 + c^3 + 1 \equiv 0 \pmod{p}
\]
\[
a^4 + b^4 + c^4 + 7459 \equiv 0 \pmod{p}
\]
W... | 59 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The integers from $1$ through $9$ inclusive, are placed in the squares of a $3 \times 3$ grid. Each square contains a different integer. The product of the integers in the first and second rows are $60$ and $96$ respectively. Find the sum of the integers in the third row.
[i]Proposed by bissue [/i] | 1. We are given a \(3 \times 3\) grid with integers from 1 to 9, each appearing exactly once. The product of the integers in the first row is 60, and the product of the integers in the second row is 96. We need to find the sum of the integers in the third row.
2. First, calculate the product of all integers from 1 to ... | 17 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a room, each person is an painter and/or a musician. $2$ percent of the painters are musicians, and $5$ percent of the musicians are painters. Only one person is both an painter and a musician. How many people are in the room?
[i]Proposed by Evan Chang[/i] | 1. Let \( P \) be the number of painters and \( M \) be the number of musicians. We are given that 2% of the painters are musicians and 5% of the musicians are painters. Additionally, we know that there is exactly one person who is both a painter and a musician.
2. Since 2% of the painters are musicians, we can write:... | 69 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Evan has $10$ cards numbered $1$ through $10$. He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$, the remainder is $1$. Find the maximum number of cards he could have chose.
[i]Proposed by Evan Chang [/i] | 1. **Identify the cards that cannot be chosen:**
- The cards are numbered from $1$ to $10$.
- We need the product of the chosen cards to leave a remainder of $1$ when divided by $3$.
- Any card that is a multiple of $3$ will make the product divisible by $3$. Therefore, we cannot choose cards numbered $3$, $6$... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Calvin makes a number. He starts with $1$, and on each move, he multiplies his current number by $3$, then adds $5$. After $10$ moves, find the sum of the digits (in base $10$) when Calvin's resulting number is expressed in base $9$.
[i]Proposed by Calvin Wang [/i] | 1. Let's denote the number Calvin makes after $n$ moves as $a_n$. Initially, $a_0 = 1$.
2. The recurrence relation for the sequence is given by:
\[
a_{n+1} = 3a_n + 5
\]
3. We need to find the value of $a_{10}$ after 10 moves. To do this, we will solve the recurrence relation.
4. First, let's find the general... | 21 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$. Find $a + b$.
[i]Proposed by Isaac Chen[/i] | 1. **Define the problem and variables:**
- Isaac earns a point whenever a particular face appears for the \(2n+1\)th time.
- We need to find the expected number of flips to get 10 points.
2. **Understand the process:**
- Each point is earned when a particular face (say heads) appears for the \(2n+1\)th time.
... | 201 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$12$ people stand in a row. Each person is given a red shirt or a blue shirt. Every minute, exactly one pair of two people with the same color currently standing next to each other in the row leave. After $6$ minutes, everyone has left. How many ways could the shirts have been assigned initially?
[i]Proposed by Evan C... | To solve this problem, we need to determine the number of ways to assign red and blue shirts to 12 people such that every minute, exactly one pair of two people with the same color standing next to each other leaves, and after 6 minutes, everyone has left.
We will use combinatorial methods and casework based on the n... | 837 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $ABC$. If $XA = \tfrac ab$, where $a$ and $b$ are coprime positive integers, find $a + b$.
[i]Proposed by Alexander Wang[/i] | 1. **Identify the point \( X \)**:
- \( X \) is the point such that \( XB = XC \) and \( XA \) is tangent to the circumcircle of \( \triangle ABC \). This implies that \( X \) is the excenter opposite \( A \).
2. **Use the Law of Cosines to find \( \cos \angle BAC \)**:
\[
\cos \angle BAC = \frac{b^2 + c^2 - ... | 61 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Katelyn is building an integer (in base $10$). She begins with $9$. Each step, she appends a randomly chosen digit from $0$ to $9$ inclusive to the right end of her current integer. She stops immediately when the current integer is $0$ or $1$ (mod $11$). The probability that the final integer ends up being $0$ (mod $11... | 1. **Initial Setup and Problem Understanding**:
- Katelyn starts with the integer \(9\).
- At each step, she appends a randomly chosen digit from \(0\) to \(9\) to the right end of her current integer.
- She stops when the current integer is \(0\) or \(1\) modulo \(11\).
- We need to find the probability th... | 31 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $1 = x_{1} < x_{2} < \dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2} \dots x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that
$$n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}.$$
[i]Proposed by Justin Lee[/i] | 1. **Determine the nature of \( n \):**
- We need to find the smallest \( n \) such that \( n = x_1^2 + x_2^2 + x_3^2 + x_4^2 \), where \( x_1, x_2, x_3, x_4 \) are the divisors of \( n \).
- Since \( x_1 = 1 \) and \( x_2 = 2 \) (as \( n \) is even), we have:
\[
n = 1^2 + 2^2 + x_3^2 + x_4^2 = 1 + 4 + ... | 130 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$ABCD$ is a rhombus where $\angle BAD = 60^\circ$. Point $E$ lies on minor arc $\widehat{AD}$ of the circumcircle of $ABD$, and $F$ is the intersection of $AC$ and the circle circumcircle of $EDC$. If $AF = 4$ and the circumcircle of $EDC$ has radius $14$, find the squared area of $ABCD$.
[i]Proposed by Vivian Loh [/i... | 1. **Identify the properties of the rhombus and the given angles:**
- Since \(ABCD\) is a rhombus, all sides are equal, i.e., \(AB = BC = CD = DA\).
- Given \(\angle BAD = 60^\circ\), we know that \(\angle BCD = 60^\circ\) because opposite angles in a rhombus are equal.
2. **Circumcircle and cyclic quadrilateral... | 2916 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off of each of the mirrors once and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\tfrac{AB... | 1. **Define the problem and setup the geometry:**
We are given an isosceles right triangle \( \triangle ABC \) with \( AB = BC \). The walls \( BC \) and \( CA \) are mirrors. A laser is shot from \( A \), hits each mirror once, and lands at a point \( X \) on \( AB \). We need to find the ratio \( \frac{CA}{AY} \) ... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$ with $AB = 10$, $BC = 12$, and $AC = 14$, let $E$ and $F$ be the midpoints of $AB$ and $AC$. If a circle passing through $B$ and $C$ is tangent to the circumcircle of $AEF$ at point $X \ne A$, find $AX$.
[i]Proposed by Vivian Loh [/i] | 1. **Identify the midpoints**:
Let \( E \) and \( F \) be the midpoints of \( AB \) and \( AC \) respectively. Therefore, we have:
\[
E = \left( \frac{A + B}{2} \right) \quad \text{and} \quad F = \left( \frac{A + C}{2} \right)
\]
2. **Calculate the lengths of \( AE \) and \( AF \)**:
Since \( E \) and \... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define mutually externally tangent circles $\omega_1$, $\omega_2$, and $\omega_3$. Let $\omega_1$ and $\omega_2$ be tangent at $P$. The common external tangents of $\omega_1$ and $\omega_2$ meet at $Q$. Let $O$ be the center of $\omega_3$. If $QP = 420$ and $QO = 427$, find the radius of $\omega_3$.
[i]Proposed by Tan... | 1. **Define the problem setup:**
- Let $\omega_1$, $\omega_2$, and $\omega_3$ be mutually externally tangent circles.
- $\omega_1$ and $\omega_2$ are tangent at point $P$.
- The common external tangents of $\omega_1$ and $\omega_2$ meet at point $Q$.
- Let $O$ be the center of $\omega_3$.
- Given $QP = 4... | 77 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let \[\mathcal{S} = \sum_{i=1}^{\infty}\left(\prod_{j=1}^i \dfrac{3j - 2}{12j}\right).\] Then $(\mathcal{S} + 1)^3 = \tfrac mn$ with $m$ and $n$ coprime positive integers. Find $10m + n$.
[i]Proposed by Justin Lee and Evan Chang[/i] | 1. We start with the given series:
\[
\mathcal{S} = \sum_{i=1}^{\infty}\left(\prod_{j=1}^i \dfrac{3j - 2}{12j}\right).
\]
We need to simplify the product inside the sum.
2. Consider the product:
\[
\prod_{j=1}^i \frac{3j-2}{12j}.
\]
We can separate the numerator and the denominator:
\[
\p... | 43 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \tfrac pq$ for coprime positive integers $p$, $q$, find $p + q$.
[i]Pro... | 1. **Define the Tangent Lines:**
Let the three tangent lines be described by \( y = mx + b \). When we substitute this equation into \( x = y^2 + a \) and \( y = x^2 + a \), each of the resulting quadratics must have a unique solution in \( x \), i.e., each discriminant must equal \( 0 \).
2. **Substitute into \( x... | 91 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $N$ is [i]apt[/i] if for each integer $0 < k < 1009$, there exists exactly one divisor of $N$ with a remainder of $k$ when divided by $1009$. For a prime $p$, suppose there exists an [i]apt[/i] positive integer $N$ where $\tfrac Np$ is an integer but $\tfrac N{p^2}$ is not. Find the number of possibl... | 1. **Understanding the Problem:**
We need to find the number of possible remainders when a prime \( p \) is divided by \( 1009 \) such that there exists an *apt* positive integer \( N \) where \( \frac{N}{p} \) is an integer but \( \frac{N}{p^2} \) is not. An integer \( N \) is *apt* if for each integer \( 0 < k < 1... | 946 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A $39$-tuple of real numbers $(x_1,x_2,\ldots x_{39})$ satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
The ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots x_{39})$ satisfying the condition is $\tfrac ab$ for coprime positive integers ... | 1. **Modeling the Problem:**
We start by modeling each \( x_i \) as a vector \( \vec{v_i} = (\cos(x_i), \sin(x_i)) \) with unit magnitude. Given the conditions:
\[
2\sum_{i=1}^{39} \sin(x_i) = -34 \quad \text{and} \quad \sum_{i=1}^{39} \cos(x_i) = -34,
\]
we can rewrite these as:
\[
\sum_{i=1}^{39}... | 37 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $K > 0$ be an integer. An integer $k \in [0,K]$ is randomly chosen. A sequence of integers is defined starting on $k$ and ending on $0$, where each nonzero term $t$ is followed by $t$ minus the largest Lucas number not exceeding $t$.
The probability that $4$, $5$, or $6$ is in this sequence approaches $\tfrac{a -... | 1. **Define the Lucas and Fibonacci sequences:**
- Lucas numbers: \( L_0 = 2, L_1 = 1, L_2 = 3, L_3 = 4, L_4 = 7, \ldots \)
- Fibonacci numbers: \( F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, \ldots \)
2. **Identify the problem:**
- We need to find the probability that the sequence starting from a randomly c... | 31 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $n$ where the mean and median of $\{20, 42, 69, n\}$ are both integers.
[i]Proposed by bissue[/i] | 1. **Determine the condition for the mean to be an integer:**
The mean of the set \(\{20, 42, 69, n\}\) is given by:
\[
\frac{20 + 42 + 69 + n}{4} = \frac{131 + n}{4}
\]
For this to be an integer, \(131 + n\) must be divisible by 4. Therefore:
\[
131 + n \equiv 0 \pmod{4}
\]
Simplifying, we g... | 45 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Parabolas $P_1, P_2$ share a focus at $(20,22)$ and their directrices are the $x$ and $y$ axes respectively. They intersect at two points $X,Y.$ Find $XY^2.$
[i]Proposed by Evan Chang[/i] | 1. **Identify the properties of the parabolas:**
- The focus of both parabolas is at \((20, 22)\).
- The directrix of \(P_1\) is the \(x\)-axis, and the directrix of \(P_2\) is the \(y\)-axis.
2. **Determine the equations of the parabolas:**
- For \(P_1\) (directrix is the \(x\)-axis):
The general form o... | 3520 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $p=2^{16}+1$ is a prime, find the maximum possible number of elements in a set $S$ of positive integers less than $p$ so no two distinct $a,b$ in $S$ satisfy $$a^2\equiv b\pmod{p}.$$ | 1. **Define the problem in terms of a generator:**
Given \( p = 2^{16} + 1 \) is a prime, we need to find the maximum number of elements in a set \( S \) of positive integers less than \( p \) such that no two distinct \( a, b \in S \) satisfy \( a^2 \equiv b \pmod{p} \).
2. **Use a generator \( g \) modulo \( p \)... | 43691 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
John has cut out these two polygons made out of unit squares. He joins them to each other to form a larger polygon (but they can't overlap). Find the smallest possible perimeter this larger polygon can have. He can rotate and reflect the cut out polygons. | 1. **Identify the polygons and their properties:**
- Each polygon is made out of unit squares.
- The total area of the two polygons combined is 16 unit squares.
2. **Determine the smallest rectangular box that can contain the polygons:**
- The area of the box must be at least 16 square units.
- The smalles... | 18 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Two cars travel along a circular track $n$ miles long, starting at the same point. One car travels $25$ miles along the track in some direction. The other car travels $3$ miles along the track in some direction. Then the two cars are once again at the same point along the track. If $n$ is a positive integer, find the s... | 1. **Determine the conditions for the cars to meet again:**
- The cars travel along a circular track of length \( n \) miles.
- One car travels 25 miles, and the other car travels 3 miles.
- They meet again at the same point on the track.
2. **Case 1: Cars travel in the same direction:**
- The distance bet... | 89 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Five identical circles are placed in a line inside a larger one as shown. If the shown chord has length $16,$ find the radius of the large circle. | 1. **Identify the given information and variables:**
- Let \( r \) be the radius of the small circles.
- The chord length given is \( 16 \).
2. **Understand the geometric configuration:**
- Five identical small circles are placed in a line inside a larger circle.
- The centers of the small circles are coll... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
An ant lies on each corner of a $20 \times 23$ rectangle. Each second, each ant independently and randomly chooses to move one unit vertically or horizontally away from its corner. After $10$ seconds, find the expected area of the convex quadrilateral whose vertices are the positions of the ants. | 1. **Understanding the Problem:**
- We have a $20 \times 23$ rectangle with ants at each corner.
- Each ant moves randomly either vertically or horizontally each second for 10 seconds.
- We need to find the expected area of the convex quadrilateral formed by the ants after 10 seconds.
2. **Analyzing the Movem... | 130 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Ryan uses $91$ puzzle pieces to make a rectangle. Each of them is identical to one of the tiles shown. Given that pieces can be flipped or rotated, find the number of pieces that are red in the puzzle. (He is not allowed to join two ``flat sides'' together.) | 1. **Understanding the Problem:**
- Ryan uses 91 puzzle pieces to form a rectangle.
- Each piece can be flipped or rotated.
- The puzzle pieces have "in" and "out" thingys on their edges.
- The red pieces are indifferent to the "in" and "out" thingys.
- We need to find the number of red pieces in the puz... | 79 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Initially five variables are defined: $a_1=1, a_2=0, a_3=0, a_4=0, a_5=0.$ On a turn, Evan can choose an integer $2 \le i \le 5.$ Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2,$ then now $a_1=1, a_2=0+1=1, a_3=0, a_4=0, a_5=0.$ Find the minimum number of turns Evan ne... | 1. **Initialization and Strategy**:
- We start with the variables \(a_1 = 1, a_2 = 0, a_3 = 0, a_4 = 0, a_5 = 0\).
- Evan can choose an integer \(2 \le i \le 5\) and add \(a_{i-1}\) to \(a_i\).
- The goal is to make \(a_5\) exceed \(1,000,000\) with the minimum number of turns.
2. **Optimal Strategy**:
- I... | 127 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
James the naked mole rat is hopping on the number line. He starts at $0$ and jumps exactly $2^{n}$ either forward or backward at random at time $n$ seconds, his first jump being at time $n = 0$. What is the expected number of jumps James takes before he is on a number that exceeds $8$? | 1. **Understanding the Problem:**
James starts at position \(0\) on the number line and makes jumps of \(2^n\) either forward or backward at time \(n\) seconds. We need to find the expected number of jumps before James lands on a number greater than \(8\).
2. **Analyzing the States:**
At time \(n = 3\), the poss... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1$, $a_2$, $\cdots$ be a sequence such that $a_1=a_2=\frac 15$, and for $n \ge 3$, $$a_n=\frac{a_{n-1}+a_{n-2}}{1+a_{n-1}a_{n-2}}.$$ Find the smallest integer $n$ such that $a_n>1-5^{-2022}$. | 1. **Define the sequence and initial conditions:**
Given the sequence \(a_1, a_2, \cdots\) with initial conditions \(a_1 = a_2 = \frac{1}{5}\) and the recurrence relation for \(n \geq 3\):
\[
a_n = \frac{a_{n-1} + a_{n-2}}{1 + a_{n-1}a_{n-2}}
\]
2. **Transform the sequence into a polynomial form:**
Defi... | 21 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Kevin writes a nonempty subset of $S = \{ 1, 2, \dots 41 \}$ on a board. Each day, Evan takes the set last written on the board and decreases each integer in it by $1.$ He calls the result $R.$ If $R$ does not contain $0$ he writes $R$ on the board. If $R$ contains $0$ he writes the set containing all elements of $S$ n... | 1. **Understanding the Problem:**
Kevin writes a nonempty subset of \( S = \{ 1, 2, \dots, 41 \} \) on a board. Each day, Evan takes the set last written on the board and decreases each integer in it by \( 1 \). He calls the result \( R \). If \( R \) does not contain \( 0 \), he writes \( R \) on the board. If \( R... | 94 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be two distinct points on line $AC$ ($A, P_1, C, P_2$ appear in that order on the line) and $Q_1$ and $Q_2$ be two distinct points on line $AB$ ($A, Q_1, B, Q_2$ appear in that order on the line) such that $BQ_1 = P_1Q_1 = P_1C... | To find the distance between the circumcenters of $\triangle BP_1P_2$ and $\triangle CQ_1Q_2$, we will follow the steps outlined in the solution sketch and provide detailed calculations and justifications.
1. **Identify the points and their properties:**
- Given $AB = 7$, $AC = 8$, and $BC = 3$.
- Points $P_1$ a... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define the minimum real $C$ where for any reals $0 = a_0 < a_{1} < \dots < a_{1000}$ then $$\min_{0 \le k \le 1000} (a_{k}^2 + (1000-k)^2) \le C(a_1+ \dots + a_{1000})$$ holds. Find $\lfloor 100C \rfloor.$ | 1. We need to find the minimum real \( C \) such that for any sequence of reals \( 0 = a_0 < a_1 < \dots < a_{1000} \), the inequality
\[
\min_{0 \le k \le 1000} (a_k^2 + (1000-k)^2) \le C(a_1 + \dots + a_{1000})
\]
holds.
2. Define the function \( f(a_0, \dots, a_{1000}) = \frac{\min(a_k^2 + (1000-k)^2)}... | 127 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Define the Fibonacci numbers such that $F_{1} = F_{2} = 1,$ $F_{k} = F_{k-1} + F_{k-2}$ for $k > 2.$ For large positive integers $n,$ the expression (containing $n$ nested square roots)
$$\sqrt{2023 F^{2}_{2^{1}} + \sqrt{2023 F^{2}_{2^{2}} + \sqrt{2023 F_{2^{3}}^{2} \dots + \sqrt{2023 F^{2}_{2^{n}} }}}}$$
approaches... | 1. **Define the Fibonacci sequence and the given expression:**
The Fibonacci sequence is defined as:
\[
F_1 = F_2 = 1, \quad F_k = F_{k-1} + F_{k-2} \text{ for } k > 2.
\]
We need to evaluate the nested square root expression:
\[
\sqrt{2023 F_{2^1}^2 + \sqrt{2023 F_{2^2}^2 + \sqrt{2023 F_{2^3}^2 + ... | 8102 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken?
$\emph{(... | 1. **Understanding the Problem:**
- The fly starts on the second hand at noon.
- The fly switches to another hand whenever the hand it is on crosses with another hand.
- We need to determine the total number of revolutions the fly makes by midnight.
2. **Analyzing the Hands of the Clock:**
- The second han... | 245 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $x$ such that $$|x^2-x-6|$$ has exactly $4$ positive integer divisors.
[i]Proposed by Evan Chang (squareman), USA[/i] | 1. Define the function \( f(x) = |x^2 - x - 6| \). We need to find the sum of all positive integers \( x \) such that \( f(x) \) has exactly 4 positive integer divisors.
2. First, evaluate \( f(x) \) for small values of \( x \):
\[
f(1) = |1^2 - 1 - 6| = |1 - 1 - 6| = | - 6| = 6
\]
\[
f(2) = |2^2 - 2 - ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Choose a permutation of$ \{1,2, ..., 20\}$ at random. Let $m$ be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of $2^m$.
[i]Proposed by Evan Chang (squareman), USA[/i] | To find the expected value of \(2^m\) where \(m\) is the number of elements in a permutation of \(\{1, 2, \ldots, 20\}\) that are larger than all previous elements, we can use the concept of indicator random variables and linearity of expectation.
1. **Define the Indicator Random Variables:**
Let \(X_i\) be an indi... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[i]B stands for Begginners , A stands for Advanced[/i]
[b]B1.[/b] What is the last digit of $2022^2 + 2202^2$?
[b]B2 / A1.[/b] Find the area of the shaded region!
[img]https://cdn.artofproblemsolving.com/attachments/d/4/dd49bfe56ba77e8eec9f91220725ced15f61d8.png[/img]
[b]B3 / A2.[/b] If $\Psi (n^2 + k) = n - 2k$, ... | To solve for \( g(12) \) given the function \( f(x) = x(x+1) \) and the equation \( f(g(x)) = 9x^2 + 3x \), we need to find \( g(x) \) such that \( f(g(x)) = g(x)(g(x) + 1) = 9x^2 + 3x \).
1. Start with the given equation:
\[
g(x)(g(x) + 1) = 9x^2 + 3x
\]
2. Let \( g(x) = y \). Then the equation becomes:
... | 36 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In eight years Henry will be three times the age that Sally was last year. Twenty five years ago their ages added to $83$. How old is Henry now? | 1. Let Henry's current age be \( h \) and Sally's current age be \( s \).
2. According to the problem, in eight years, Henry will be three times the age that Sally was last year. This can be written as:
\[
h + 8 = 3(s - 1)
\]
3. Simplify the equation:
\[
h + 8 = 3s - 3
\]
\[
h = 3s - 11
\]
... | 97 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest number that could be the date of the first Saturday after the second Monday following the second Thursday of a month? | 1. Let's start by identifying the second Thursday of the month. If the month starts on a Thursday, the first Thursday is the 1st, and the second Thursday is the 8th.
2. Next, we need to find the second Monday following the second Thursday. Since the second Thursday is the 8th, the first Monday after the 8th is the 12th... | 17 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest $n$ such that every subset of $\{1, 2, 3, . . . , 2004 \}$ with $n$ elements contains at least two elements that are relatively prime. | 1. We need to find the smallest \( n \) such that every subset of \(\{1, 2, 3, \ldots, 2004\}\) with \( n \) elements contains at least two elements that are relatively prime.
2. Two numbers are relatively prime if their greatest common divisor (gcd) is 1.
3. Consider the set \(\{2, 4, 6, \ldots, 2004\}\). This set con... | 1003 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many gallons of a solution which is $15\%$ alcohol do we have to mix with a solution that is $35\%$ alcohol to make $250$ gallons of a solution that is $21\%$ alcohol? | 1. Let \( x \) be the number of gallons of the solution with \( 15\% \) alcohol.
2. Let \( y \) be the number of gallons of the solution with \( 35\% \) alcohol.
3. We need to mix these to get \( 250 \) gallons of a solution that is \( 21\% \) alcohol.
We can set up the following system of equations based on the given... | 175 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $r$ be a real number such that $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}}=2$. Find $r^3 - \frac{1}{r^3}$. | 1. **Define the variable and the given equation:**
Let \( x = \sqrt[3]{r} \). The given equation is:
\[
x - \frac{1}{x} = 2
\]
2. **Square both sides to eliminate the fraction:**
\[
\left( x - \frac{1}{x} \right)^2 = 2^2
\]
\[
x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 4
\]
\[... | 14 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A circle radius $320$ is tangent to the inside of a circle radius $1000$. The smaller circle is tangent to a diameter of the larger circle at a point $P$. How far is the point $P$ from the outside of the larger circle? | 1. Let \( O_1 \) be the center of the larger circle with radius \( 1000 \), and \( O_2 \) be the center of the smaller circle with radius \( 320 \). The smaller circle is tangent to the inside of the larger circle and to a diameter of the larger circle at point \( P \).
2. Since the smaller circle is tangent to the di... | 400 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can we form three teams of four players each from a group of $12$ participants? | 1. **Choose the first team:**
- We need to select 4 players out of 12 for the first team. The number of ways to do this is given by the binomial coefficient:
\[
\binom{12}{4}
\]
Calculating this, we get:
\[
\binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} = \frac{12 \times 11 \times ... | 5775 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $m$ and $n$ define function $f(m, n)$ by $f(1, 1) = 1$, $f(m+ 1, n) = f(m, n) +m$ and $f(m, n + 1) = f(m, n) - n$. Find the sum of all the values of $p$ such that $f(p, q) = 2004$ for some $q$. | 1. **Define the function and initial condition:**
The function \( f(m, n) \) is defined by:
\[
f(1, 1) = 1
\]
\[
f(m+1, n) = f(m, n) + m
\]
\[
f(m, n+1) = f(m, n) - n
\]
2. **Express \( f(p, q) \) in terms of sums:**
We need to find \( f(p, q) \). Starting from \( f(1, 1) = 1 \), we ca... | 3007 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ in. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac23$ . Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches. | 1. Given that $\triangle ABC$ is isosceles with $AB = AC = 16$ inches and $\angle A = 30^\circ$, we can use the properties of isosceles triangles and trigonometry to find the lengths of the sides and heights.
2. Since $\angle A = 30^\circ$, $\angle B = \angle C = 75^\circ$ because the sum of angles in a triangle is $1... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given that $(1 + \tan 1^{\circ})(1 + \tan 2^{\circ}) \ldots (1 + \tan 45^{\circ}) = 2^n$, find $n$. | 1. We start by considering the product \((1 + \tan 1^{\circ})(1 + \tan 2^{\circ}) \ldots (1 + \tan 45^{\circ})\).
2. Notice that for any pair of angles \(\alpha\) and \(\beta\) such that \(\alpha + \beta = 45^{\circ}\), we have:
\[
(1 + \tan \alpha)(1 + \tan \beta)
\]
Expanding this, we get:
\[
1 + \... | 23 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
This year February $29$ fell on a Sunday. In what year will February $29$ next fall on a Sunday? | 1. **Identify the starting point**: We are given that February 29 fell on a Sunday in the year 2004.
2. **Understand the leap year cycle**: Leap years occur every 4 years, except for years that are divisible by 100 but not by 400. Therefore, the leap years after 2004 are 2008, 2012, 2016, 2020, 2024, 2028, 2032, etc.... | 2032 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A rectangle has area $1100$. If the length is increased by ten percent and the width is
decreased by ten percent, what is the area of the new rectangle? | 1. Let the original length of the rectangle be \( l \) and the original width be \( w \). Given that the area of the rectangle is 1100, we have:
\[
l \cdot w = 1100
\]
2. When the length is increased by 10%, the new length becomes:
\[
l_{\text{new}} = 1.1l
\]
3. When the width is decreased by 10%, t... | 1089 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The number $2.5081081081081\ldots$ can be written as $\frac{m}{n}$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$. | 1. Let \( x = 2.5081081081081\ldots \). We can express \( x \) as \( x = 2.5\overline{081} \).
2. We can separate \( x \) into its integer and fractional parts:
\[
x = 2 + 0.5 + 0.00\overline{081}
\]
3. Let \( y = 0.00\overline{081} \). We need to express \( y \) as a fraction. Notice that \( y \) is a repea... | 86417 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers less that $200$ are relatively prime to either $15$ or $24$? | To find the number of positive integers less than $200$ that are relatively prime to either $15$ or $24$, we need to count the integers that are not divisible by $2$, $3$, or $5$. We will use the principle of inclusion-exclusion (PIE) to solve this problem.
1. **Count the multiples of $2$, $3$, and $5$ less than $200$... | 53 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
One rainy afternoon you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005$ th digit that you write? | 1. We need to determine the 2005th digit in the sequence where the number \( n \) is written \( n \) times for \( n = 1, 2, 3, \ldots, 99 \).
2. First, we calculate the total number of digits written for each number \( n \):
- For single-digit numbers (1 to 9), each number \( n \) is written \( n \) times, contribu... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all integers $x$ satisfying $1 + 8x \le 358 - 2x \le 6x + 94$. | 1. We start with the compound inequality:
\[
1 + 8x \le 358 - 2x \le 6x + 94
\]
This can be split into two separate inequalities:
\[
1 + 8x \le 358 - 2x \quad \text{and} \quad 358 - 2x \le 6x + 94
\]
2. Solve the first inequality:
\[
1 + 8x \le 358 - 2x
\]
Add \(2x\) to both sides:
... | 102 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season? | 1. Let \( H \) be the number of hits Sandy had before this week, and \( A \) be the number of at-bats Sandy had before this week.
2. Given that Sandy's batting average a week ago was \( 0.360 \), we can write the equation:
\[
\frac{H}{A} = 0.360
\]
Therefore,
\[
H = 0.360A
\]
3. After five more at-... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of addition problems in which a two digit number is added to a second two digit number to give a two digit answer, such as in the three examples: \[\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}23\\42\\\hline 65\end{tabular}\,,\qquad\begin{tabular}{@{\hspace{3pt}}c@{\hspace{3pt}}}36\\36\\\hline 72\end... | 1. **Define the problem constraints:**
- We need to find the number of addition problems where two two-digit numbers sum to another two-digit number.
- Let the two-digit numbers be \(a\) and \(b\), and their sum \(a + b\) must also be a two-digit number.
2. **Determine the range of valid sums:**
- The smalles... | 3240 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find $n$ such that $n - 76$ and $n + 76$ are both cubes of positive integers. | 1. We need to find an integer \( n \) such that both \( n - 76 \) and \( n + 76 \) are cubes of positive integers.
2. Let \( n - 76 = a^3 \) and \( n + 76 = b^3 \) for some positive integers \( a \) and \( b \).
3. Subtracting the first equation from the second, we get:
\[
(n + 76) - (n - 76) = b^3 - a^3
\]
... | 140 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of different quadruples $(a, b, c, d)$ of positive integers such that $ab =cd = a + b + c + d - 3$. | 1. We start with the given system of equations:
\[
\begin{cases}
ab = a + b + c + d - 3 \\
cd = a + b + c + d - 3
\end{cases}
\]
Let \( k = a + b + c + d - 3 \). Then we have:
\[
ab = k \quad \text{and} \quad cd = k
\]
2. Adding these two equations, we get:
\[
ab + cd = 2k
\]
... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain? | 1. **Calculate the total number of unit cubes in the cubic block:**
The total number of unit cubes in an \( n \times n \times n \) cubic block is given by:
\[
n^3
\]
2. **Determine the number of unit cubes in the outer layer:**
The outer layer consists of all the unit cubes that are on the surface of th... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be a real number greater than $1$ such that $\frac{20a}{a^2+1} = \sqrt{2}$. Find $\frac{14a}{a^2 - 1}$. | 1. Given the equation:
\[
\frac{20a}{a^2 + 1} = \sqrt{2}
\]
We need to find the value of \(\frac{14a}{a^2 - 1}\).
2. First, solve for \(a\) from the given equation. Multiply both sides by \(a^2 + 1\):
\[
20a = \sqrt{2}(a^2 + 1)
\]
Rearrange the equation:
\[
20a = \sqrt{2}a^2 + \sqrt{2}
... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\] | 1. Given the polynomial:
\[
P(x) = x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x
\]
We need to determine the number of real roots.
2. The polynomial can be factored as:
\[
P(x) = (x^4 - 37x^3 - 2x^2 + 74x)(x^5 + 1)
\]
3. Further factorization of \(x^4 - 37x^3 - 2x^2 + 74x\):
\[
x^4 ... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The number $2.5081081081081 \ldots$ can be written as $m/n$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$. | 1. Let \( x = 2.5081081081081 \ldots \). We can express \( x \) as \( 2.5\overline{081} \).
2. We can decompose \( x \) into two parts: the integer part and the repeating decimal part. Thus, we write:
\[
x = 2.5 + 0.00\overline{081}
\]
3. Let \( y = 0.00\overline{081} \). We need to express \( y \) as a fracti... | 86417 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$. | 1. Start with the given inequality:
\[
4x(x - 5) \le 375
\]
2. Expand and rearrange the inequality to standard quadratic form:
\[
4x^2 - 20x \le 375
\]
\[
4x^2 - 20x - 375 \le 0
\]
3. Factor the quadratic expression:
\[
4x^2 - 20x - 375 = 0
\]
To factor this, we can use the quad... | 20 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A cubic block with dimensions $n$ by $n$ by $n$ is made up of a collection of $1$ by $1$ by $1$ unit cubes. What is the smallest value of $n$ so that if the outer two layers of unit cubes are removed from the block, more than half the original unit cubes will still remain? | To solve this problem, we need to determine the smallest value of \( n \) such that removing the outer two layers of unit cubes from an \( n \times n \times n \) cubic block leaves more than half of the original unit cubes.
1. **Calculate the total number of unit cubes in the original block:**
\[
\text{Total num... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the total length of the set of real numbers satisfying \[\frac{x^2 - 80x + 1500}{x^2 - 55x + 700} < 0.\] | 1. **Factor the given expression:**
\[
\frac{x^2 - 80x + 1500}{x^2 - 55x + 700}
\]
We need to factor both the numerator and the denominator.
- For the numerator \(x^2 - 80x + 1500\):
\[
x^2 - 80x + 1500 = (x - 30)(x - 50)
\]
- For the denominator \(x^2 - 55x + 700\):
\[
x^2 ... | 25 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
We want to paint some identically-sized cubes so that each face of each cube is painted a solid color and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce? | 1. **Determine the number of ways to choose 6 colors out of 7:**
Since we need to paint each cube with 6 different colors out of 7 available colors, we first need to choose which 6 colors to use. The number of ways to choose 6 colors out of 7 is given by the binomial coefficient:
\[
\binom{7}{6} = 7
\]
2. ... | 210 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had... | 1. **Case 1: The first candy is green.**
- **Subcase 1: The first candy was from Bag 1.**
- Probability of selecting Bag 1: \(\frac{1}{3}\)
- Probability of selecting a green candy from Bag 1: \(\frac{3}{4}\)
- Probability of selecting the second bag (Bag 2 or Bag 3): \(\frac{1}{2}\)
- Probabili... | 217 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $70$ foot pole stands vertically in a plane supported by three $490$ foot wires, all attached to the top of the pole, pulled taut, and anchored to three equally spaced points in the plane. How many feet apart are any two of those anchor points? | 1. **Understanding the Geometry**:
- We have a $70$ foot pole standing vertically.
- Three $490$ foot wires are attached to the top of the pole and anchored to three equally spaced points in the plane.
- These three points form an equilateral triangle on the plane.
2. **Analyzing the Right Triangle**:
- Ea... | 840 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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