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The function $g$, with domain and real numbers, fulfills the following:
$\bullet$ $g (x) \le x$, for all real $x$
$\bullet$ $g (x + y) \le g (x) + g (y)$ for all real $x,y$
Find $g (1990)$. | 1. **Given Conditions:**
- \( g(x) \le x \) for all real \( x \).
- \( g(x + y) \le g(x) + g(y) \) for all real \( x, y \).
2. **Setting \( x = 0 \):**
- From the first condition, \( g(0) \le 0 \).
- From the second condition, \( g(0 + 0) \le g(0) + g(0) \), which simplifies to \( g(0) \le 2g(0) \). This i... | 1990 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ be a number such that $x +\frac{1}{x}=-1$. Determine the value of $x^{1994} +\frac{1}{x^{1994}}$. | 1. Let \( x \) be a number such that \( x + \frac{1}{x} = -1 \). We need to determine the value of \( x^{1994} + \frac{1}{x^{1994}} \).
2. Define \( a_n = x^n + \frac{1}{x^n} \). We aim to find a recurrence relation for \( a_n \).
3. Using the given equation \( x + \frac{1}{x} = -1 \), we can derive the recurrence re... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Pedrito's lucky number is $34117$. His friend Ramanujan points out that $34117 = 166^2 + 81^2 = 159^2 + 94^2$ and $166-159 = 7$, $94- 81 = 13$. Since his lucky number is large, Pedrito decides to find a smaller one, but that satisfies the same properties, that is, write in two different ways as the sum of squares of po... | To solve the problem, we need to find the smallest number \( N \) that can be expressed as the sum of squares of two pairs of positive integers, such that the difference between the first integers in each pair is 7, and the difference between the second integers in each pair is 13.
We start with the system of equatio... | 545 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a function defined on the set of positive integers , and with values in the same set, which satisfies:
$\bullet$ $f (n + f (n)) = 1$ for all $n\ge 1$.
$\bullet$ $f (1998) = 2$
Find the lowest possible value of the sum $f (1) + f (2) +... + f (1999)$, and find the formula of $f$ for which this minimum is sati... | 1. **Understanding the given conditions:**
- We are given a function \( f \) defined on the set of positive integers with values in the same set.
- The function satisfies \( f(n + f(n)) = 1 \) for all \( n \geq 1 \).
- We are also given \( f(1998) = 2 \).
2. **Analyzing the function:**
- From the condition... | 1997003 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$.... | 1. **Define the function \( q(N) \)**: The function \( q(N) \) represents the sum of the digits of \( N \). For example, if \( N = 128 \), then \( q(128) = 1 + 2 + 8 = 11 \).
2. **Understand the problem**: We need to repeatedly apply the function \( q \) to \( 2^{2004} \) until we reach a single digit number. This pro... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle. | 1. Given that quadrilateral $ABCD$ is inscribed in a circle, we know that $|DA| = |BC| = 2$ and $|AB| = 4$. Let $E$ be the intersection point of lines $BC$ and $DA$. We are also given that $\angle AEB = 60^\circ$ and $|CD| < |AB|$.
2. Since $ABCD$ is a cyclic quadrilateral, we can use the property that opposite angles... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of different numbers of the form $\left\lfloor\frac{i^2}{2015} \right\rfloor$, with $i = 1,2, ..., 2015$. | 1. We need to find the number of different values of the expression $\left\lfloor \frac{i^2}{2015} \right\rfloor$ for $i = 1, 2, \ldots, 2015$.
2. First, consider the range of $i^2$ for $i$ in the given range. The smallest value of $i^2$ is $1^2 = 1$ and the largest value is $2015^2$.
3. We need to evaluate the floor f... | 2016 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all postitive integers n such that
$$\left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor=n^2$$
where $\lfloor x \rfloor$ represents the largest integer less than the real number $x$. | To find all positive integers \( n \) such that
\[ \left\lfloor \frac{n}{2} \right\rfloor \cdot \left\lfloor \frac{n}{3} \right\rfloor \cdot \left\lfloor \frac{n}{4} \right\rfloor = n^2, \]
we will proceed step-by-step.
1. **Initial Observation:**
We start by noting that for any integer \( n \), the floor function... | 24 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A square of $3 \times 3$ is subdivided into 9 small squares of $1 \times 1$. It is desired to distribute the nine digits $1, 2, . . . , 9$ in each small square of $1 \times 1$, a number in each small square. Find the number of different distributions that can be formed in such a way that the difference of the digits in... | To solve the problem of distributing the digits \(1, 2, \ldots, 9\) in a \(3 \times 3\) grid such that the difference between digits in adjacent cells is at most 3, we need to consider the constraints and systematically count the valid configurations.
1. **Initial Constraints**:
- The digit \(1\) cannot be placed i... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f (n) $ be a function that fulfills the following properties:
$\bullet$ For each natural $ n $, $ f (n) $ is an integer greater than or equal to $ 0 $.
$\bullet$ $f (n) = 2010 $, if $ n $ ends in $ 7 $. For example, $ f (137) = 2010 $.
$\bullet$ If $ a $ is a divisor of $ b $, then: $ f \left(\frac {b} {a} \... | 1. **Understanding the properties of the function \( f(n) \):**
- \( f(n) \) is an integer greater than or equal to 0 for each natural \( n \).
- \( f(n) = 2010 \) if \( n \) ends in 7.
- If \( a \) is a divisor of \( b \), then \( f\left(\frac{b}{a}\right) = |f(b) - f(a)| \).
2. **Analyzing the given number ... | 2010 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are real numbers $a,b$ and $c$ and a positive number $\lambda$ such that $f(x)=x^3+ax^2+bx+c$ has three real roots $x_1, x_2$ and $x_3$ satisfying
$(1) x_2-x_1=\lambda$
$(2) x_3>\frac{1}{2}(x_1+x_2)$.
Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$ | 1. Given the polynomial \( f(x) = x^3 + ax^2 + bx + c \) with roots \( x_1, x_2, x_3 \), we know from Vieta's formulas:
\[
x_1 + x_2 + x_3 = -a,
\]
\[
x_1x_2 + x_2x_3 + x_3x_1 = b,
\]
\[
x_1x_2x_3 = -c.
\]
2. We are given the conditions:
\[
x_2 - x_1 = \lambda,
\]
\[
x_3 > \fr... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimet... | 1. We start with the given condition:
\[
\left\{\frac{3^\ell}{10^4}\right\} = \left\{\frac{3^m}{10^4}\right\} = \left\{\frac{3^n}{10^4}\right\}
\]
This implies:
\[
3^\ell \equiv 3^m \equiv 3^n \pmod{10^4}
\]
2. To solve this, we need to understand the order of 3 modulo \(10^4\). We use the Chinese... | 3003 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of $\sum_{k=1}^{200} f(k)$. | To solve the problem, we need to evaluate the sum \(\sum_{k=1}^{200} f(k)\) where the function \(f(n)\) is defined as follows:
\[ f(n) = \begin{cases}
0, & \text{if } n \text{ is the square of an integer} \\
\left\lfloor \frac{1}{\{\sqrt{n}\}} \right\rfloor, & \text{if } n \text{ is not the square of an integer}
\end{... | 629 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $A = {x|5x-a \le 0}$, $B = {x|6x-b > 0}$, $a,b \in \mathbb{N}$, and $A \cap B \cap \mathbb{N} = {2,3,4}$. The number of such pairs $(a,b)$ is
${ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}} 42\qquad $ | 1. **Determine the range for \(a\):**
- Given \(A = \{x \mid 5x - a \leq 0\}\), we can rewrite this as \(5x \leq a\).
- For \(x \in \{2, 3, 4\}\), we have:
\[
5 \cdot 2 \leq a \implies 10 \leq a
\]
\[
5 \cdot 3 \leq a \implies 15 \leq a
\]
\[
5 \cdot 4 \leq a \implies 20 \l... | 25 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
In a $7\times 8$ chessboard, $56$ stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed. | To solve this problem, we need to ensure that no five stones are adjacent horizontally, vertically, or diagonally on a $7 \times 8$ chessboard. We start with 56 stones and need to determine the minimal number of stones to remove to meet this condition.
1. **Initial Setup and Constraints**:
- The chessboard has dime... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each square of a $33\times 33$ square grid is colored in one of the three colors: red, yellow or blue, such that the numbers of squares in each color are the same. If two squares sharing a common edge are in different colors, call that common edge a separating edge. Find the minimal number of separating edges in the gr... | 1. **Define the problem and variables:**
- We have a $33 \times 33$ grid where each square is colored either red, yellow, or blue.
- The number of squares in each color is the same.
- A separating edge is defined as an edge shared by two squares of different colors.
- We need to find the minimal number of s... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $k$ with the following property: if each cell of a $100\times 100$ grid is dyed with one color and the number of cells of each color is not more than $104$, then there is a $k\times1$ or $1\times k$ rectangle that contains cells of at least three different colors. | To find the smallest positive integer \( k \) such that any \( 100 \times 100 \) grid dyed with colors, where no color appears in more than 104 cells, contains a \( k \times 1 \) or \( 1 \times k \) rectangle with at least three different colors, we need to analyze the distribution of colors and the constraints given.
... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a=1+10^{-4}$. Consider some $2023\times 2023$ matrix with each entry a real in $[1,a]$. Let $x_i$ be the sum of the elements of the $i$-th row and $y_i$ be the sum of the elements of the $i$-th column for each integer $i\in [1,n]$. Find the maximum possible value of $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2... | 1. Let \( A \) be a \( 2023 \times 2023 \) matrix with each entry \( a_{ij} \) such that \( 1 \leq a_{ij} \leq a \), where \( a = 1 + 10^{-4} \).
2. Define \( x_i \) as the sum of the elements in the \( i \)-th row:
\[
x_i = \sum_{j=1}^{2023} a_{ij}
\]
Similarly, define \( y_i \) as the sum of the elements... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Compute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$. | 1. Let \( n \) be the smallest positive integer that satisfies the given conditions.
2. The number \( n \) is 3 more than a multiple of 5. This can be written as:
\[
n = 5k + 3 \quad \text{for some integer } k.
\]
3. The number \( n \) is also twice a multiple of 6. This can be written as:
\[
n = 2 \cdot... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers. | To determine the number of integers between \(1\) and \(100\) that have an odd number of factors, we need to understand the conditions under which a number has an odd number of factors.
1. **Understanding Factors**:
- A number \(n\) has an odd number of factors if and only if \(n\) is a perfect square. This is beca... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords. | 1. **Understanding the Problem**: We need to find the number of ways to draw chords between pairs of 7 points on a circle such that no two chords intersect and each point belongs to at most one chord. This is a classic problem related to Motzkin numbers.
2. **Motzkin Numbers**: The Motzkin number $M(n)$ counts the num... | 127 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two math students play a game with $k$ sticks. Alternating turns, each one chooses a number from the set $\{1,3,4\}$ and removes exactly that number of sticks from the pile (so if the pile only has $2$ sticks remaining the next player must take $1$). The winner is the player who takes the last stick. For $1\leq k\leq10... | 1. **Initial Conditions**:
- For \( k = 1 \), the first player can take the last stick and win. Thus, \( l_1 = T \).
- For \( k = 2 \), the first player can only take 1 stick, leaving 1 stick for the second player, who will then win. Thus, \( l_2 = F \).
- For \( k = 3 \), the first player can take all 3 stic... | 71 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Nick has a $3\times3$ grid and wants to color each square in the grid one of three colors such that no two squares that are adjacent horizontally or vertically are the same color. Compute the number of distinct grids that Nick can create. | 1. **Define the problem and colors:**
We need to color a $3 \times 3$ grid using three colors (red, blue, and green) such that no two adjacent squares (horizontally or vertically) have the same color.
2. **Fix the middle square:**
Assume without loss of generality that the middle square is red. This assumption s... | 174 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the remainder when $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$ is divided by $43$. Your answer should be an integer between $0$ and $42$. | 1. Let \( p = 43 \). We need to find the remainder when \( (1^2+1)(2^2+1)(3^2+1)\dots(42^2+1) \) is divided by \( 43 \).
2. Consider the polynomial \( f(x) = (x^2 + 1^2)(x^2 + 2^2) \cdots (x^2 + (p-1)^2) \).
3. For any \( a \) coprime to \( p \), the set \( \{a, 2a, \dots, (p-1)a\} \) is the same as \( \{1, 2, \dots,... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$. | To compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$, we use Euler's Totient Function, $\phi(n)$. The function $\phi(n)$ counts the number of integers up to $n$ that are relatively prime to $n$.
1. **Prime Factorization of 2014**:
\[
2014 = 2 \times 19 \ti... | 4648 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A robot is standing on the bottom left vertex $(0,0)$ of a $5\times5$ grid, and wants to go to $(5,5)$, only moving to the right $(a,b)\mapsto(a+1,b)$ or upward $(a,b)\mapsto(a,b+1)$. However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path $(a,b)\mapsto(a-1,b+1)$. As the grid is... | 1. **Define the problem and constraints:**
- The robot starts at \((0,0)\) and wants to reach \((5,5)\) on a \(5 \times 5\) grid.
- The robot can move right \((a,b) \mapsto (a+1,b)\), up \((a,b) \mapsto (a,b+1)\), or diagonally up-left \((a,b) \mapsto (a-1,b+1)\).
- The robot takes the diagonal path exactly on... | 1650 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given taht $RI=3$, $CE=8$, and $ER=7$, compute $IC$. | 1. Let $X$, $Y$, $Z$, and $W$ be the points of tangency on sides $RI$, $IC$, $CE$, and $ER$ respectively.
2. By the property of tangents from a point to a circle, we have:
- $RX = RW$
- $IY = IX$
- $CZ = CY$
- $EW = EZ$
3. Let $RX = RW = a$, $IY = IX = b$, $CZ = CY = c$, and $EW = EZ = d$.
4. The lengths of... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider all right triangles with integer side lengths that form an arithmetic sequence. Compute the $2014$th smallest perimeter of all such right triangles. | 1. We are given a right triangle with integer side lengths that form an arithmetic sequence. Let the side lengths be \(a\), \(b\), and \(c\) such that \(a < b < c\). Since they form an arithmetic sequence, we can write:
\[
b = a + d \quad \text{and} \quad c = a + 2d
\]
for some positive integer \(d\).
2. S... | 24168 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$. | 1. **Apply the Angle Bisector Theorem**: The Angle Bisector Theorem states that the angle bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides. For triangle $ABC$ with $BD$ as the angle bisector of $\angle ABC$, we have:
\[
\frac{AB}{AD} = \frac{BC... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$. | 1. **Identify the primes less than or equal to 17:**
The primes less than or equal to 17 are:
\[
2, 3, 5, 7, 11, 13, 17
\]
There are 7 such primes.
2. **Consider the number 1:**
The number 1 is a special case because it is not divisible by any prime number. Therefore, it can be included in our set w... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the last two digits of $\tbinom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g. if the last two digits are $05$, just write $5$.) | 1. To find the last two digits of $\binom{200}{100}$, we need to compute $\binom{200}{100} \mod 100$. We will use the Chinese Remainder Theorem (CRT) to combine results modulo 4 and modulo 25, since $100 = 4 \times 25$.
2. First, we compute $\binom{200}{100} \mod 4$. Using Lucas' Theorem, which states that for a prime... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Screws are sold in packs of $10$ and $12$. Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$? | To solve this problem, we need to find the smallest number of screws, \( k \), that can be bought in two different ways using packs of 10 and 12 screws. This means we need to find the smallest \( k \) such that \( k \) can be expressed as both \( 10x \) and \( 12y \) for some integers \( x \) and \( y \), where \( x \)... | 60 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, we have that $AB=AC$, $BC=16$, and that the area of $\triangle ABC$ is $120$. Compute the length of $AB$. | 1. Given that \( \triangle ABC \) is isosceles with \( AB = AC \) and \( BC = 16 \), we can drop a perpendicular from \( A \) to \( BC \) at point \( D \), which bisects \( BC \). Therefore, \( BD = DC = \frac{BC}{2} = \frac{16}{2} = 8 \).
2. The area of \( \triangle ABC \) is given as \( 120 \). The area of a triangl... | 17 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve. | 1. We need to determine the maximum number of problems \( N \) that Ben can solve in 60 minutes, given that the time taken to solve the \( i \)-th problem is \( i \) minutes.
2. The total time taken to solve the first \( N \) problems is given by the sum of the first \( N \) natural numbers:
\[
\sum_{i=1}^N i = 1... | 10 | Other | math-word-problem | Yes | Yes | aops_forum | false |
George and two of his friends go to a famous jiaozi restaurant, which serves only two kinds of jiaozi: pork jiaozi, and vegetable jiaozi. Each person orders exactly $15$ jiaozi. How many different ways could the three of them order? Two ways of ordering are different if one person orders a different number of pork jiao... | 1. Each person orders exactly 15 jiaozi, which can be either pork or vegetable jiaozi.
2. Let \( x_i \) be the number of pork jiaozi ordered by the \( i \)-th person, where \( i = 1, 2, 3 \).
3. Since each person orders exactly 15 jiaozi, the number of vegetable jiaozi ordered by the \( i \)-th person is \( 15 - x_i \)... | 4096 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid.
[center]<see attached>[/center]
Given that Mr. Ambulando wants ... | 1. **Identify the problem and constraints:**
- Mr. Ambulando starts at the intersection of $5^{\text{th}}$ and $\text{A St}$.
- He needs to reach the intersection of $1^{\text{st}}$ and $\text{F St}$.
- He must avoid the intersection of $4^{\text{th}}$ and $\text{B St}$.
- He wants to take the shortest path... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible? | 1. **Identify the constraints**: Antoine (A), Benoît (B), and Claude (C) cannot sit next to each other. This means that no two of A, B, and C can be adjacent in the seating arrangement.
2. **Determine possible seatings for A, B, and C**: To ensure that A, B, and C are not adjacent, we need to place them in such a way ... | 144 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$. | To solve this problem, we need to find all possible geometric sequences of length 3 where each term is a positive integer no larger than 10. A geometric sequence is defined by the property that the ratio between consecutive terms is constant. Let's denote the terms of the sequence by \(a, ar, ar^2\), where \(a\) is the... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers. | 1. **Identify the digits that are not prime:**
- The digits that are not prime are \(1, 4, 6, 8, 9\).
2. **Identify the two-digit numbers ending in non-prime digits:**
- Since we are looking for two-digit numbers that are prime but not primeable, we need to consider numbers ending in \(1\) or \(9\) (since these ... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A bitstring of length $\ell$ is a sequence of $\ell$ $0$'s or $1$'s in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$'s or $1$'s? | 1. **Count the bitstrings with exactly 2014 consecutive 0's:**
- There is only one such bitstring: \(000\ldots000\) (2014 zeros).
2. **Count the bitstrings with exactly 2013 consecutive 0's:**
- There are two such bitstrings:
- \(000\ldots0001\) (2013 zeros followed by a 1)
- \(100\ldots0000\) (1 follo... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Apples cost $2$ dollars. Bananas cost $3$ dollars. Oranges cost $5$ dollars. Compute the number of distinct baskets of fruit such that there are $100$ pieces of fruit and the basket costs $300$ dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit... | To solve the problem, we need to find the number of distinct baskets of fruit such that there are $100$ pieces of fruit and the basket costs $300$ dollars. We are given the following costs for each type of fruit:
- Apples cost $2$ dollars each.
- Bananas cost $3$ dollars each.
- Oranges cost $5$ dollars each.
Let:
- $... | 34 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A college math class has $N$ teaching assistants. It takes the teaching assistants $5$ hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only $4$ hours to grade. Assuming everyone did the same amount of work, compute the number of hours it w... | 1. Let \( W \) be the total amount of work required to grade all the homework assignments.
2. If there are \( N \) teaching assistants, and it takes them 5 hours to grade all the homework, then the work done by each teaching assistant per hour is:
\[
\frac{W}{N \cdot 5}
\]
3. When another teaching assistant jo... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=\sum_{i=1}^{2014}|x-i|$. Compute the length of the longest interval $[a,b]$ such that $f(x)$ is constant on that interval. | 1. Consider the function \( f(x) = \sum_{i=1}^{2014} |x - i| \). This function represents the sum of the absolute differences between \( x \) and each integer from 1 to 2014.
2. To understand where \( f(x) \) is constant, we need to analyze the behavior of \( f(x) \) as \( x \) varies. The absolute value function \( |x... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2+kx+2014$ and $x^2+kx-2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous. | 1. To determine if a positive integer \( k \) is \( 2014 \)-ambiguous, we need to check if both quadratics \( x^2 + kx + 2014 \) and \( x^2 + kx - 2014 \) have integer roots.
2. For the quadratic \( x^2 + kx + 2014 \) to have integer roots, its discriminant must be a perfect square. The discriminant of \( x^2 + kx + 2... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be the roots of the quadratic $x^2-7x+c$. Given that $a^2+b^2=17$, compute $c$. | 1. Let \( a \) and \( b \) be the roots of the quadratic equation \( x^2 - 7x + c = 0 \). By Vieta's formulas, we know:
\[
a + b = 7 \quad \text{and} \quad ab = c
\]
2. We are given that \( a^2 + b^2 = 17 \). We can use the identity for the sum of squares of the roots:
\[
a^2 + b^2 = (a + b)^2 - 2ab
... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$. Suppose that $a_1$, $a_3$, and $a_6$ also form a geometric sequence. Compute $a_1$. | 1. Let \( a = a_1 \). Since \( a_1, a_2, a_3, \dots, a_6 \) is an arithmetic sequence with common difference \( 3 \), we can express the terms as follows:
\[
a_1 = a, \quad a_2 = a + 3, \quad a_3 = a + 6, \quad a_4 = a + 9, \quad a_5 = a + 12, \quad a_6 = a + 15
\]
2. Given that \( a_1, a_3, a_6 \) form a geo... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=(x-a)^3$. If the sum of all $x$ satisfying $f(x)=f(x-a)$ is $42$, find $a$. | 1. Let \( f(x) = (x-a)^3 \). We need to find the value of \( a \) such that the sum of all \( x \) satisfying \( f(x) = f(x-a) \) is 42.
2. Define \( g(x) = f(x) - f(x-a) \). Notice that if \( x \) satisfies \( f(x) = f(x-a) \), then \( x \) is also a root of \( g(x) \).
3. Calculate \( g(x) \):
\[
g(x) = (x-a)^3... | 14 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
There is a rectangular field that measures $20\text{m}$ by $15\text{m}$. Xiaoyu the butterfly is sitting at the perimeter of the field on one of the $20\text{m}$ sides such that he is $6\text{m}$ from a corner. He flies in a straight line to another point on the perimeter. His flying path splits the field into two part... | 1. **Identify the problem and given data:**
- The rectangular field measures $20\text{m}$ by $15\text{m}$.
- Xiaoyu is sitting on one of the $20\text{m}$ sides, $6\text{m}$ from a corner.
- He flies in a straight line to another point on the perimeter, splitting the field into two equal areas.
2. **Calculate ... | 17 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In trapezoid $ABCD$ with $AD\parallel BC$, $AB=6$, $AD=9$, and $BD=12$. If $\angle ABD=\angle DCB$, find the perimeter of the trapezoid. | 1. Given the trapezoid \(ABCD\) with \(AD \parallel BC\), we know that \(AB = 6\), \(AD = 9\), and \(BD = 12\). We are also given that \(\angle ABD = \angle DCB\).
2. Since \(\angle ABD = \angle DCB\) and \(\angle ADB = \angle DBC\) (because \(AD \parallel BC\)), triangles \(\triangle ABD\) and \(\triangle DCB\) are s... | 39 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a rectangle $ABCD$, two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$, $CX$ intersects $HF$ and $Y$, $DY$ intersects $EG$ at $Z$. Given that $AH=4$, $HD=6$, $AE=4$, and $EB=5$, find the area of quadrilateral $HXYZ$. | 1. **Identify the given dimensions and points:**
- Rectangle \(ABCD\) with \(AH = 4\), \(HD = 6\), \(AE = 4\), and \(EB = 5\).
- Points \(E\) and \(G\) divide \(AB\) and \(CD\) respectively.
- Points \(F\) and \(H\) divide \(AD\) and \(BC\) respectively.
- \(BH\) intersects \(EG\) at \(X\).
- \(CX\) inte... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tourname... | 1. **Calculate the number of IDs where the letters are the same:**
- There are 26 possible uppercase letters.
- The letters must be the same, so there are \(26\) choices for the letters.
- There are 100 possible combinations of two digits (from 00 to 99).
- Therefore, the number of such IDs is:
\[
... | 9100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a positive integer $x>1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil\tfrac{n}{2}\rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $X$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$... | 1. **Identify the smallest value of \( x \) such that \( f(x) \) is a multiple of \( x \):**
- Given \( x = 24 \), the divisors of \( 24 \) are \(\{1, 2, 3, 4, 6, 8, 12, 24\}\).
- The number of divisors \( n = 8 \).
- The smallest \(\lceil \frac{n}{2} \rceil = \lceil \frac{8}{2} \rceil = 4\) divisors are \(\{1... | 31 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Four men are each given a unique number from $1$ to $4$, and four women are each given a unique number from $1$ to $4$. How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to ea... | 1. **Labeling and Initial Constraints**:
- Denote the men as \( A, B, C, D \) and the women as \( a, b, c, d \).
- Each man and woman pair share the same number: \( A \) and \( a \) have number 1, \( B \) and \( b \) have number 2, and so on.
- The arrangement must alternate between men and women, and no two p... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$? | 1. We need to find the largest integer \( n \) such that \( n \) is divisible by every integer less than \( \sqrt[3]{n} \). Let's denote \( k = \sqrt[3]{n} \). This means \( n = k^3 \).
2. For \( n \) to be divisible by every integer less than \( k \), \( n \) must be divisible by the least common multiple (LCM) of all... | 420 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different pa... | 1. **Identify the problem**: We need to find the number of Hamiltonian cycles in a cube graph, where each vertex represents a friend, and each edge represents a possible path between friends. A Hamiltonian cycle visits each vertex exactly once and returns to the starting vertex.
2. **Vertices and edges of a cube**: A ... | 96 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Rachel has $3$ children, all of which are at least $2$ years old. The ages of the children are all pairwise relatively prime, but Rachel’s age is a multiple of each of her children’s ages. What is Rachel’s minimum possible age? | 1. **Identify the conditions:**
- Rachel has 3 children.
- Each child is at least 2 years old.
- The ages of the children are pairwise relatively prime.
- Rachel's age is a multiple of each of her children's ages.
2. **Select the ages of the children:**
- To minimize Rachel's age, we need to select the ... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Meena owns a bottle cap collection. While on a vacation, she finds a large number of bottle caps, increasing her collection size by $40\%$. Later on her same vacation, she decides that she does not like some of the bottle caps, so she gives away $20\%$ of her current collection. Suppose that Meena owns $21$ more bottle... | 1. Let \( x \) be the number of bottle caps that Meena owned before her vacation.
2. During her vacation, Meena increases her collection by \( 40\% \). Therefore, the number of bottle caps after the increase is:
\[
x + 0.4x = 1.4x
\]
3. Later, she gives away \( 20\% \) of her current collection. The number of ... | 175 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Consider a unit circle with center $O$. Let $P$ be a point outside the circle such that the two line segments passing through $P$ and tangent to the circle form an angle of $60^\circ$. Compute the length of $OP$. | 1. Let \( O \) be the center of the unit circle, and let \( P \) be a point outside the circle such that the two line segments passing through \( P \) and tangent to the circle form an angle of \( 60^\circ \). Let \( A \) and \( B \) be the points of tangency of the two line segments from \( P \) to the circle.
2. Sin... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A circle $A$ is circumscribed about a unit square and a circle $B$ is inscribed inside the same unit square. Compute the ratio of the area of $A$ to the area of $B$. | 1. **Determine the radius of circle \( B \):**
- Circle \( B \) is inscribed inside the unit square.
- The side length of the unit square is 1.
- The radius of circle \( B \) is half the side length of the square.
\[
r_B = \frac{1}{2}
\]
- The area of circle \( B \) is:
\[
\text{Area}_B = \pi... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How many purple... | 1. Let \( r, p, g \) be the number of red, purple, and green cars Lynnelle has respectively. Then Moor has \( 27-r, 27-p, 27-g \) red, purple, and green cars respectively.
2. Given that the number of red cars Lynnelle has is the same as the number of green cars Moor has, we have:
\[
r = 27 - g
\]
Thus, we c... | 22 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The rectangular faces of rectangular prism $A$ have perimeters $12$, $16$, and $24$. The rectangular faces of rectangular prism $B$ have perimeters $12$, $16$, and $20$. Let $V_A$ denote the volume of $A$ and $V_B$ denote the volume of $B$. Find $V_A-V_B$. | 1. Let \( x, y, z \) be the dimensions of rectangular prism \( A \). The perimeters of the faces are given by:
\[
2(x+y) = 12, \quad 2(x+z) = 16, \quad 2(y+z) = 24
\]
Dividing each equation by 2, we get:
\[
x + y = 6, \quad x + z = 8, \quad y + z = 12
\]
2. To find \( x, y, z \), we add the three ... | -13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A three-digit number $x$ in base $10$ has a units-digit of $6$. When $x$ is written is base $9$, the second digit of the number is $4$, and the first and third digit are equal in value. Compute $x$ in base $10$. | 1. Let \( x \) be a three-digit number in base 10, written as \( x = 100A + 10B + 6 \), where \( A \) and \( B \) are digits.
2. When \( x \) is written in base 9, it is given as \( x_9 = C4C \), where \( C \) is a digit in base 9.
3. Convert \( x_9 = C4C \) to base 10:
\[
x = C \cdot 9^2 + 4 \cdot 9 + C = 81C + ... | 446 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to write $91$ as the sum of at least $2$ consecutive positive integers? | 1. Suppose we write \( 91 \) as the sum of the consecutive integers from \( a \) to \( b \), inclusive. This gives us the equation:
\[
\frac{(b+a)}{2}(b-a+1) = 91
\]
because the sum of a sequence is equal to the mean of the sequence multiplied by the number of terms.
2. Removing fractions by multiplying bo... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_n$ be a sequence with $a_0=1$ and defined recursively by
$$a_{n+1}=\begin{cases}a_n+2&\text{if }n\text{ is even},\\2a_n&\text{if }n\text{ is odd.}\end{cases}$$
What are the last two digits of $a_{2015}$? | 1. We start by defining the sequence \(a_n\) with \(a_0 = 1\) and the recursive relation:
\[
a_{n+1} = \begin{cases}
a_n + 2 & \text{if } n \text{ is even}, \\
2a_n & \text{if } n \text{ is odd}.
\end{cases}
\]
2. To simplify the problem, we introduce a new sequence \(b_n\) defined by:
\[
b_n ... | 72 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $D$ is a point on $AB$ between $A$ and $B$, $E$ is a point on $AC$ between $A$ and $C$, and $F$ is a point on $BC$ between $B$ and $C$ such that $AF$, $BE$, and $CD$ all meet inside $\triangle ABC$ at a point $G$. Given that the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and... | 1. Given the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and the area of $\triangle ACD$ is $10$.
2. We need to find the area of $\triangle ABF$.
First, let's analyze the given areas and their implications:
- The area of $\triangle ABE$ is $5$, so the area of $\triangle BCE$ is $15 - 5 = 10$.
... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider $13$ marbles that are labeled with positive integers such that the product of all $13$ integers is $360$. Moor randomly picks up $5$ marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain? | 1. **Factorization of 360**:
We start by factorizing 360 into its prime factors:
\[
360 = 2^3 \cdot 3^2 \cdot 5
\]
This means that any integer factor of 360 can be expressed as \(2^a \cdot 3^b \cdot 5^c\) where \(0 \leq a \leq 3\), \(0 \leq b \leq 2\), and \(0 \leq c \leq 1\).
2. **Counting the Factors*... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $10$ mathematics teachers gather at a circular table with $25$ seats to discuss the upcoming mathematics competition. Each teacher is assigned a unique integer ID number between $1$ and $10$, and the teachers arrange themselves in such a way that teachers with consecutive ID numbers are not separated by an... | 1. **Understanding the Problem:**
We need to arrange 10 teachers with unique IDs from 1 to 10 at a circular table with 25 seats. The arrangement must ensure that:
- Teachers with consecutive IDs are not separated by any other teacher.
- Each pair of teachers is separated by at least one empty seat.
- Arrang... | 10010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of digits is $2015!$. Your score will be given by $\max\{\lfloor125(\min\{\tfrac{A}{C},\tfrac{C}{A}\}-\tfrac{1}{5})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer. | To find the number of digits in \(2015!\), we use the formula for the number of digits of a factorial, which is given by:
\[
\text{Number of digits of } n! = \left\lfloor \log_{10}(n!) \right\rfloor + 1
\]
Using Stirling's approximation for \(n!\):
\[
n! \approx \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n
\]
Taking t... | 5787 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of natural numbers $1\leq n\leq10^6$ such that the least prime divisor of $n$ is $17$. Your score will be given by $\lfloor26\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor$, where $A$ is your answer and $C$ is the actual answer. | To solve the problem of finding the number of natural numbers \(1 \leq n \leq 10^6\) such that the least prime divisor of \(n\) is \(17\), we can use the Principle of Inclusion-Exclusion (PIE) and some number theory.
1. **Counting Multiples of 17**:
First, we count the number of multiples of \(17\) in the range \(1... | 11323 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of positive integers less than $10^6$ that can be written as the sum of two perfect squares. Compute the number of elements in $S$. Your score will be given by $\max\{\lfloor75(\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}-\tfrac{2}{3})\rfloor,0\}$, where $A$ is your answer and $C$ is the actual answer. | To solve the problem, we need to count the number of positive integers less than $10^6$ that can be written as the sum of two perfect squares. We will use the following steps:
1. **Identify the relevant theorem**:
According to Fermat's theorem on sums of two squares, a positive integer $n$ can be expressed as the s... | 215907 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A blue square of side length $10$ is laid on top of a coordinate grid with corners at $(0,0)$, $(0,10)$, $(10,0)$, and $(10,10)$. Red squares of side length $2$ are randomly placed on top of the grid, changing the color of a $2\times2$ square section red. Each red square when placed lies completely within the blue squa... | 1. **Understanding the Problem:**
- We have a blue square of side length \(10\) laid on a coordinate grid with corners at \((0,0)\), \((0,10)\), \((10,0)\), and \((10,10)\).
- Red squares of side length \(2\) are randomly placed on the grid, changing the color of a \(2 \times 2\) square section to red.
- Each ... | 201 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3\leq n\leq8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have? | 1. Let \( x \) be the number of lychees that Jonah currently has. According to the problem, for all \( n \) where \( 3 \leq n \leq 8 \), when the lychees are distributed evenly into \( n \) groups, \( n-1 \) lychees remain. This can be mathematically expressed as:
\[
x \equiv n-1 \pmod{n} \quad \text{for} \quad n... | 839 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Laurie loves multiplying numbers in her head. One day she decides to multiply two $2$-digit numbers $x$ and $y$ such that $x\leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of rem... | 1. Let \( x = 10A + B \) and \( y = 10C + D \), where \( A, B, C, D \) are digits from 1 to 9 (since 0 would make the number not a 2-digit number).
2. According to the problem, Laurie remembers the digits in the opposite order, so she remembers \( x \) as \( 10B + A \) and \( y \) as \( 10D + C \).
3. The condition giv... | 28 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An ant is walking on the edges of an icosahedron of side length $1$. Compute the length of the longest path that the ant can take if it never travels over the same edge twice, but is allowed to revisit vertices.
[center]<see attached>[/center] | 1. **Understanding the structure of the icosahedron**:
- An icosahedron has 12 vertices and 30 edges.
- Each vertex is connected to 5 other vertices (degree 5).
2. **Constraints on the ant's path**:
- The ant cannot travel over the same edge twice.
- The ant can revisit vertices.
3. **Analyzing the vertic... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a given positive integer $m$, the series
$$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$
evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$. | 1. We start with the given series:
\[
\sum_{k=1, k \neq m}^{\infty} \frac{1}{(k+m)(k-m)}
\]
We can use partial fraction decomposition to rewrite the summand:
\[
\frac{1}{(k+m)(k-m)} = \frac{A}{k+m} + \frac{B}{k-m}
\]
Solving for \(A\) and \(B\), we get:
\[
1 = A(k-m) + B(k+m)
\]
Sett... | 7 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop? | 1. Let \( C \) be the number of cats, \( D \) be the number of ducks, and \( P \) be the number of parrots. We are given that \( C = 56 \).
2. Let \( T \) be the total number of animals in the pet shop. We know that \( \frac{1}{12} \) of the animals are ducks:
\[
D = \frac{1}{12}T
\]
3. We are also given tha... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The largest factor of $n$ not equal to $n$ is $35$. Compute the largest possible value of $n$. | 1. We are given that the largest factor of \( n \) not equal to \( n \) is \( 35 \). This implies that \( 35 \) is a divisor of \( n \).
2. The prime factorization of \( 35 \) is \( 35 = 5 \times 7 \). Therefore, \( n \) must be a multiple of \( 35 \).
3. To find the largest possible value of \( n \), we need to consid... | 35 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the area of the trapezoid $ABCD$ with right angles $BAD$ and $ADC$ and side lengths of $AB=3$, $BC=5$, and $CD=7$. | 1. **Identify the shape and given dimensions:**
- The trapezoid \(ABCD\) has right angles at \(BAD\) and \(ADC\).
- Given side lengths are \(AB = 3\), \(BC = 5\), and \(CD = 7\).
2. **Determine the height of the trapezoid:**
- Since \(BAD\) and \(ADC\) are right angles, \(AB\) and \(CD\) are perpendicular to ... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=ax^3+bx^2+cx+d$ be some cubic polynomial. Given that $f(1)=20$ and $f(-1)=16$, what is $b+d$? | 1. We start with the given polynomial \( f(x) = ax^3 + bx^2 + cx + d \).
2. We are given two conditions:
\[
f(1) = 20 \quad \text{and} \quad f(-1) = 16
\]
3. Substituting \( x = 1 \) into the polynomial, we get:
\[
f(1) = a(1)^3 + b(1)^2 + c(1) + d = a + b + c + d = 20 \quad \text{(Equation 1)}
\]
4. ... | 18 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping? | To solve this problem, we need to cover a $5 \times 5$ square using the minimum number of smaller squares of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$, and $4 \times 4$. We must ensure that no two squares overlap.
1. **Calculate the total area to be covered:**
The area of the $5 \times 5$ square is:
\[
5... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist? | 1. We need to determine how many possible values of \( n \) exist such that Moor and his \( n \) friends can split 2016 candies equally. This means that \( n+1 \) must be a divisor of 2016.
2. First, we find the prime factorization of 2016:
\[
2016 = 2^5 \times 3^2 \times 7
\]
3. The number of divisors of a nu... | 35 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Eric has $2$ boxes of apples, with the first box containing red and yellow apples and the second box containing green apples. Eric observes that the red apples make up $\tfrac{1}{2}$ of the apples in the first box. He then moves all of the red apples to the second box, and observes that the red apples now make up $\tfr... | 1. Let the number of red apples be \( r \), yellow apples be \( y \), and green apples be \( g \).
2. Since the red apples make up \(\frac{1}{2}\) of the apples in the first box, and the only other color in the first box is yellow, we have:
\[
r = y
\]
3. After moving all the red apples to the second box, Eric... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the maximum possible value for the sum of the squares of the roots of $x^4+ax^3+bx^2+cx+d$ where $a$, $b$, $c$, and $d$ are $2$, $0$, $1$, and $7$ in some order? | 1. Let the roots of the polynomial \(x^4 + ax^3 + bx^2 + cx + d\) be \(p, q, r, s\).
2. We need to find the maximum possible value for the sum of the squares of the roots, \(p^2 + q^2 + r^2 + s^2\).
3. Using the identity for the sum of squares of the roots, we have:
\[
p^2 + q^2 + r^2 + s^2 = (p + q + r + s)^2 - ... | 49 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If $a$, $6$, and $b$, in that order, form an arithmetic sequence, compute $a+b$. | 1. Let the common difference of the arithmetic sequence be \( d \).
2. In an arithmetic sequence, the terms are given by:
- The first term: \( a \)
- The second term: \( a + d \)
- The third term: \( a + 2d \)
3. Given that the second term is 6, we have:
\[
a + d = 6
\]
4. The third term is given by:
... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$. What is the minimum possible value of $a+b$? | 1. Start with the given equation:
\[
ab - 7a - 11b + 13 = 0
\]
2. Use Simon's Favorite Factoring Trick (SFFT) to factor the equation. First, rearrange the terms to facilitate factoring:
\[
ab - 7a - 11b + 13 = 0 \implies ab - 7a - 11b = -13
\]
3. Add and subtract 77 to complete the factorization:
... | 34 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many $6$-digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit. | To determine how many $6$-digit positive integers have their digits in nondecreasing order from left to right, we can use the combinatorial method known as "stars and bars."
1. **Define the problem in terms of combinatorics:**
We need to find the number of ways to distribute $6$ digits (from $1$ to $9$) such that t... | 3003 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sum
$$\sum_{n=0}^{2016\cdot2017^2}2018^n$$
can be represented uniquely in the form $\sum_{i=0}^{\infty}a_i\cdot2017^i$ for nonnegative integers $a_i$ less than $2017$. Compute $a_0+a_1$. | To solve the problem, we need to represent the sum \( \sum_{n=0}^{2016 \cdot 2017^2} 2018^n \) in the form \( \sum_{i=0}^{\infty} a_i \cdot 2017^i \) where \( 0 \leq a_i < 2017 \). We are asked to compute \( a_0 + a_1 \).
1. **Understanding the Sum**:
The given sum is a geometric series with the first term \( a = 1... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of positive integers $n\leq1330$ for which $\tbinom{2n}{n}$ is not divisible by $11$. | To solve the problem of finding the number of positive integers \( n \leq 1330 \) for which \( \binom{2n}{n} \) is not divisible by 11, we need to use properties of binomial coefficients and divisibility rules.
1. **Understanding the Problem:**
We need to determine when \( \binom{2n}{n} \) is not divisible by 11. A... | 295 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all integers $0\le a \le124$ so that $a^3-2$ is a multiple of $125$. | To find the sum of all integers \(0 \le a \le 124\) such that \(a^3 - 2\) is a multiple of \(125\), we need to solve the congruence \(a^3 \equiv 2 \pmod{125}\).
1. **Solve \(a^3 \equiv 2 \pmod{5}\):**
- We need to find \(a\) such that \(a^3 \equiv 2 \pmod{5}\).
- Testing values \(a = 0, 1, 2, 3, 4\):
\[
... | 265 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many primes between $2$ and $2^{30}$ are $1$ more than a multiple of $2017$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max(0,25-15|\ln\tfrac{A}{C}|)$. | 1. **Understanding the problem**: We need to find the number of prime numbers between \(2\) and \(2^{30}\) that are \(1\) more than a multiple of \(2017\). This can be expressed as finding primes \(p\) such that \(p \equiv 1 \pmod{2017}\).
2. **Using Dirichlet's theorem on arithmetic progressions**: Dirichlet's theore... | 25560 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Moor has $3$ different shirts, labeled $T, E,$ and $A$. Across $5$ days, the only days Moor can wear shirt $T$ are days $2$ and $5$. How many different sequences of shirts can Moor wear across these $5$ days? | To solve this problem, we will use casework to count the number of valid sequences of shirts Moor can wear across the 5 days, given the constraint that shirt $T$ can only be worn on days 2 and 5.
1. **Case 1: Moor does not wear shirt $T$ at all.**
- Since Moor cannot wear shirt $T$ on any day, he must wear either s... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$? | 1. **Understanding the problem**: We need to find the last two digits of \(2018^{2018}\) when written in base 7. This is equivalent to finding \(2018^{2018} \mod 49\) because \(49 = 7^2\).
2. **Simplifying the base**: First, we reduce \(2018\) modulo \(49\):
\[
2018 \div 49 = 41 \quad \text{remainder} \quad 9 \i... | 44 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Alice’s age in years is twice Eve’s age in years. In $10$ years, Eve will be as old as Alice is now. Compute Alice’s age in years now. | 1. Denote Alice's age as \( a \) and Eve's age as \( b \).
2. From the problem statement, we have the following system of equations:
\[
a = 2b \quad \text{(Alice's age is twice Eve's age)}
\]
\[
a = b + 10 \quad \text{(In 10 years, Eve will be as old as Alice is now)}
\]
3. We can equate the two expre... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The integers $a, b,$ and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$. What is the maximum possible value of $a + b + c$? | 1. Let the first term of the geometric sequence be \( a \) and the common ratio be \( r \). Therefore, the terms of the sequence are \( a \), \( ar \), and \( ar^2 \).
2. Given that the product of the terms is \( abc = 216 \), we can write:
\[
a \cdot ar \cdot ar^2 = a^3 r^3 = 216
\]
3. Taking the cube root of... | 43 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $x\ge0$, $y\ge0$, $x+2y\le6$, and $2x+y\le6$, compute the maximum possible value of $x+y$. | To find the maximum possible value of \(x + y\) given the constraints \(x \ge 0\), \(y \ge 0\), \(x + 2y \le 6\), and \(2x + y \le 6\), we can follow these steps:
1. **Graph the inequalities**:
- The inequality \(x \ge 0\) represents the region to the right of the y-axis.
- The inequality \(y \ge 0\) represents ... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $P_1,P_2,\dots,P_{720}$ denote the integers whose digits are a permutation of $123456$, arranged in ascending order (so $P_1=123456$, $P_2=123465$, and $P_{720}=654321$). What is $P_{144}$? | To find \( P_{144} \), we need to understand the permutations of the digits \( 1, 2, 3, 4, 5, 6 \) and how they are ordered.
1. **Calculate the total number of permutations:**
The total number of permutations of 6 digits is given by \( 6! \):
\[
6! = 720
\]
This means there are 720 permutations in total... | 216543 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Benny the Bear has $100$ rabbits in his rabbit farm. He observes that $53$ rabbits are spotted, and $73$ rabbits are blue-eyed. Compute the minimum number of rabbits that are both spotted and blue-eyed. | 1. Let \( A \) be the set of spotted rabbits and \( B \) be the set of blue-eyed rabbits. We are given:
\[
|A| = 53, \quad |B| = 73, \quad \text{and} \quad |A \cup B| \leq 100.
\]
2. We need to find the minimum number of rabbits that are both spotted and blue-eyed, i.e., \( |A \cap B| \).
3. Using the princi... | 26 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For a real number $x,$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to $x,$ and let $\{x\} = x -\lfloor x\rfloor$ denote the fractional part of $x.$ The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2+\{\alpha\}=21$$ can be expressed in the form $$\frac{\sqrt{a}-\sqrt{b}}{... | To solve the given problem, we need to find all real numbers $\alpha$ that satisfy the equation:
\[
\alpha^2 + \{\alpha\} = 21
\]
where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $\{x\} = x - \lfloor x \rfloor$ denotes the fractional part of $x$.
1. **Range of $\alpha^2$**:
Sinc... | 169 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Assume that $ A_1, A_2, \ldots, A_8$ are eight points taken arbitrarily on a plane. For a directed line $ l$ taken arbitrarily on the plane, assume that projections of $ A_1, A_2, \ldots, A_8$ on the line are $ P_1, P_2, \ldots, P_8$ respectively. If the eight projections are pairwise disjoint, they can be arranged as ... | 1. **Understanding the Problem:**
We are given eight points \( A_1, A_2, \ldots, A_8 \) on a plane. For any directed line \( l \) on the plane, the projections of these points on \( l \) are \( P_1, P_2, \ldots, P_8 \). If these projections are pairwise disjoint, they can be arranged in a specific order along \( l \... | 56 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say... | 1. **Define the problem and variables:**
- There are 47 students in a classroom with seats arranged in a 6 rows $\times$ 8 columns grid.
- Each seat is denoted by $(i, j)$ where $1 \leq i \leq 6$ and $1 \leq j \leq 8$.
- The position value of a student moving from seat $(i, j)$ to $(m, n)$ is defined as $a + b... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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