problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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A $10$ digit number is called interesting if its digits are distinct and is divisible by $11111$. Then find the number of interesting numbers. | 1. **Divisibility by 11111 and 9:**
- A 10-digit number \( n \) is interesting if it is divisible by 11111 and has distinct digits.
- Since \( 11111 = 41 \times 271 \), and \( \gcd(9, 11111) = 1 \), \( n \) must be divisible by both 9 and 11111.
- For \( n \) to be divisible by 9, the sum of its digits must be... | 3456 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$ ; let $G$ be the centroid of $\Delta{ABC}$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta{ABC}$. | 1. **Given Information and Definitions:**
- Let \( \triangle ABC \) be a non-equilateral triangle with integer sides.
- \( D \) and \( E \) are the midpoints of \( BC \) and \( CA \) respectively.
- \( G \) is the centroid of \( \triangle ABC \).
- \( D, C, E, G \) are concyclic.
2. **Concyclic Points and ... | 37 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $N=6+66+666+....+666..66$, where there are hundred $6's$ in the last term in the sum. How many times does the digit $7$ occur in the number $N$ | To solve the problem, we need to determine how many times the digit $7$ appears in the number $N$, where $N$ is the sum of the sequence $6 + 66 + 666 + \ldots + 666\ldots66$ with the last term containing 100 sixes.
1. **Understanding the Pattern:**
Let's start by examining smaller cases to identify a pattern. We de... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the number of ways in which one can color the squares of a $4\times 4$ chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares? | To solve the problem of coloring a $4 \times 4$ chessboard with red (R) and blue (B) such that each row and each column has exactly two red squares and two blue squares, we can use combinatorial methods and casework.
1. **Case 1: The first two columns share no two colors in the same row.**
- We need to choose 2 po... | 90 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $? | 1. Given that $\Delta ABC$ is an acute-angled triangle and $H$ is its orthocenter. The centroids of the triangles $\Delta HBC$, $\Delta HCA$, and $\Delta HAB$ are $G_1$, $G_2$, and $G_3$ respectively.
2. We need to find the relationship between the area of $\Delta G_1G_2G_3$ and the area of $\Delta ABC$.
3. For any po... | 63 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest positive integer $n$ such that $$\frac{a^2}{\frac{b}{29} + \frac{c}{31}}+\frac{b^2}{\frac{c}{29} + \frac{a}{31}}+\frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a+b+c)$$holds for all positive real numbers $a,b,c$. | 1. We start with the given inequality:
\[
\frac{a^2}{\frac{b}{29} + \frac{c}{31}} + \frac{b^2}{\frac{c}{29} + \frac{a}{31}} + \frac{c^2}{\frac{a}{29} + \frac{b}{31}} \ge n(a+b+c)
\]
for all positive real numbers \(a, b, c\).
2. To simplify the left-hand side, we use the Titu's Lemma (or Engel's form of the... | 14 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $ | To solve the problem, we need to find the value of the expression:
\[ \sum_{1 \le i < j \le 10, \, i+j \text{ odd}} (i+j) - \sum_{1 \le i < j \le 10, \, i+j \text{ even}} (i+j) \]
We will break this down into several steps:
1. **Define the function \( f(k) \):**
Let \( f(k) \) represent the number of pairs \((i, ... | 55 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A book is published in three volumes, the pages being numbered from $1$ onwards. The page numbers are continued from the first volume to the second volume to the third. The number of pages in the second volume is $50$ more than that in the first volume, and the number pages in the third volume is one and a half times t... | 1. Let the number of pages in the first volume be \( x \).
2. The number of pages in the second volume is \( x + 50 \).
3. The number of pages in the third volume is \( \frac{3}{2}(x + 50) \).
4. The first page of the first volume is page number 1.
5. The first page of the second volume is page number \( x + 1 \).
6. ... | 17 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Consider all $6$-digit numbers of the form $abccba$ where $b$ is odd. Determine the number of all such $6$-digit numbers that are divisible by $7$. | 1. Consider a 6-digit number of the form $abccba$. This number can be expressed as:
\[
\overline{abccba} = 100000a + 10000b + 1000c + 100c + 10b + a = 100001a + 10010b + 1100c
\]
2. We need to determine when this number is divisible by 7. We will use modular arithmetic to simplify the expression modulo 7:
... | 70 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation. | 1. **Convert the given equation to a polynomial in base \( b \):**
The given equation is \( 166 \times 56 = 8590 \). We need to express each number in terms of base \( b \).
- \( 166_b = 1b^2 + 6b + 6 \)
- \( 56_b = 5b + 6 \)
- \( 8590_b = 8b^3 + 5b^2 + 9b \)
2. **Form the polynomial equation:**
\[
... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a trapezium in which $AB //CD$ and $AD \perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC = 36$ and $QB = 49$, find $PQ$. | 1. Given that $ABCD$ is a trapezium with $AB \parallel CD$ and $AD \perp AB$, and it has an incircle touching $AB$ at $Q$ and $CD$ at $P$.
2. We are given $PC = 36$ and $QB = 49$.
3. Since $ABCD$ has an incircle, it is a tangential quadrilateral. This means that the sum of the lengths of the opposite sides are equal: $... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ? | 1. We start with the given equations:
\[
a + b - c = 1
\]
\[
a^2 + b^2 - c^2 = -1
\]
2. From the first equation, we can express \(c\) in terms of \(a\) and \(b\):
\[
c = a + b - 1
\]
3. Substitute \(c = a + b - 1\) into the second equation:
\[
a^2 + b^2 - (a + b - 1)^2 = -1
\]
4. ... | 18 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ? | 1. We are given that \(2a - b\), \(a - 2b\), and \(a + b\) are all distinct squares. Let's denote these squares as \(x^2\), \(y^2\), and \(z^2\) respectively, where \(x\), \(y\), and \(z\) are distinct natural numbers.
2. Notice that:
\[
(a - 2b) + (a + b) = 2a - b
\]
This implies that \(y^2 + z^2 = x^2... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$ then what is the value of $a+b+c$ ? | 1. Given the equation \(4abc = (a+3)(b+3)(c+3)\) and the conditions \(a, b, c \ge 4\) are integers, not all equal, we need to find the value of \(a + b + c\).
2. Without loss of generality (WLOG), assume \(a \ge b \ge c \ge 4\). We will first consider the case where \(c \ge 6\).
3. If \(c \ge 6\), then we can analyze... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the sum of all possible positive integers $n, $ the product of whose digits equals $n^2 -15n -27$. | 1. We need to determine the sum of all possible positive integers \( n \) such that the product of their digits equals \( n^2 - 15n - 27 \).
2. First, consider the case where \( n \) is a one-digit number. For one-digit numbers, \( n \) ranges from 1 to 9. However, for these values, \( n^2 - 15n - 27 \) will be negati... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen? | 1. We are given that the number of ways to select two cups without a handle and three cups with a handle is exactly 1200. We need to find the maximum possible number of cups in the kitchen, denoted as \(a + b\), where \(a\) is the number of cups without handles and \(b\) is the number of cups with handles.
2. The numbe... | 29 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$? | 1. We are given that \(a + b\) is a root of the quadratic equation \(x^2 + ax + b = 0\). Therefore, we can substitute \(x = a + b\) into the equation:
\[
(a + b)^2 + a(a + b) + b = 0
\]
Expanding and simplifying, we get:
\[
a^2 + 2ab + b^2 + a^2 + ab + b = 0
\]
\[
2a^2 + 3ab + b^2 + b = 0
... | 81 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If $x = cos 1^o cos 2^o cos 3^o...cos 89^o$ and $y = cos 2^o cos 6^o cos 10^o...cos 86^o$, then what is the integer nearest to $\frac27 \log_2 \frac{y}{x}$ ? | 1. **Identify the expressions for \( x \) and \( y \):**
\[
x = \cos 1^\circ \cos 2^\circ \cos 3^\circ \cdots \cos 89^\circ
\]
\[
y = \cos 2^\circ \cos 6^\circ \cos 10^\circ \cdots \cos 86^\circ
\]
2. **Express \( y \) in terms of sine using the identity \(\cos \theta = \sin (90^\circ - \theta)\):**
... | 13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of $8$-tuples $(\epsilon_1, \epsilon_2,...,\epsilon_8)$ such that $\epsilon_1, \epsilon_2, ..., 8 \in \{1,-1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 +...+ 8\epsilon_8$ is a multiple of $3$. | 1. We need to determine the number of $8$-tuples $(\epsilon_1, \epsilon_2, \ldots, \epsilon_8)$ such that $\epsilon_i \in \{1, -1\}$ and $\epsilon_1 + 2\epsilon_2 + 3\epsilon_3 + \cdots + 8\epsilon_8$ is a multiple of $3$.
2. Consider the polynomial:
\[
P(x) = \left(x + \frac{1}{x}\right)\left(x^2 + \frac{1}{x^2... | 88 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$? | 1. **Understanding the Problem:**
We need to find the number of triangles with integer angles that are not similar to each other. The sum of the angles in any triangle is \(180^\circ\). We need to count the number of distinct sets of angles \((a, b, c)\) where \(a, b, c\) are integers, \(a \leq b \leq c\), and \(a +... | 45 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $k$ is said to be [i]good [/i] if there exists a partition of $ \{1, 2, 3,..., 20\}$ into disjoint proper subsets such that the sum of the numbers in each subset of the partition is $k$. How many [i]good [/i] numbers are there? | 1. **Sum of the Set**: First, we calculate the sum of the set \(\{1, 2, 3, \ldots, 20\}\). This is an arithmetic series with the first term \(a = 1\) and the last term \(l = 20\), and the number of terms \(n = 20\). The sum \(S\) of the first \(n\) natural numbers is given by:
\[
S = \frac{n}{2} (a + l) = \frac{2... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common? | To solve the problem of choosing 60 unit squares from an $11 \times 11$ chessboard such that no two chosen squares have a side in common, we need to consider the structure of the chessboard and the constraints given.
1. **Total Number of Squares:**
The total number of squares on an $11 \times 11$ chessboard is:
... | 62 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N$. | To solve the problem, we need to distribute 8 different chocolates among 3 children such that each child gets at least one chocolate, and no two children get the same number of chocolates. We will consider the two possible distributions that satisfy these conditions.
1. **Distribution 1: 1, 2, and 5 chocolates**
- ... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ such that we can write numbers $1,2,\dots ,n$ in a 18*18 board such that:
i)each number appears at least once
ii)In each row or column,there are no two numbers having difference 0 or 1 | To solve this problem, we need to find the smallest positive integer \( n \) such that the numbers \( 1, 2, \dots, n \) can be arranged on an \( 18 \times 18 \) board with the following conditions:
1. Each number appears at least once.
2. In each row or column, there are no two numbers having a difference of 0 or 1.
L... | 37 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$,
$$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$
[i]Proposed by Morteza Saghafian[/i] | 1. **Understanding the Problem:**
We need to find the maximum possible value of \( k \) for which there exist distinct real numbers \( x_1, x_2, \ldots, x_k \) greater than 1 such that for all \( 1 \leq i, j \leq k \), the equation \( x_i^{\lfloor x_j \rfloor} = x_j^{\lfloor x_i \rfloor} \) holds.
2. **Rewriting th... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property? | Let \( abcd \) be a four-digit number, with \( 1 \le a \le 9 \) and \( 0 \le b, c, d \le 9 \). We need to satisfy two conditions:
1. \( 1000a + 100b + 10c + d \) is divisible by 27.
2. The digit sum \( a + b + c + d \) is also divisible by 27.
Since \( a, b, c, d \) are digits, the maximum digit sum is \( 9 + 9 + 9 + ... | 75 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given set $S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}$.Let $a$ and $n$ natural numbers such that $a+2^k\in S$ for all $k=1,2,3,...,n$.Find the greatest value of $n$. | 1. **Understanding the Set \( S \)**:
The set \( S \) is defined as \( S = \{ xy(x + y) \mid x, y \in \mathbb{N} \} \). We need to determine the possible values of \( xy(x + y) \mod 9 \).
2. **Possible Values of \( xy(x + y) \mod 9 \)**:
We observe that for any natural numbers \( x \) and \( y \), the product \(... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a quadratic function which satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$.
For two distinct real numbers $a,b$, if $f(a)=f(b)$, then $f(a^2-6b-1)=f(b^2+8)$. | 1. Let \( f(x) = c(x-h)^2 + d \) where \( c, h, d \) are constants. This is the general form of a quadratic function.
2. Given the condition: if \( f(a) = f(b) \) for distinct real numbers \( a \) and \( b \), then \( f(a^2 - 6b - 1) = f(b^2 + 8) \).
3. Since \( f(a) = f(b) \) and \( a \neq b \), it implies that \( a... | 13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integer $N$ which has not less than $4$ positive divisors, such that the sum of squares of the $4$ smallest positive divisors of $N$ is equal to $N$. | 1. **Identify the conditions:**
- \( N \) must have at least 4 positive divisors.
- The sum of the squares of the 4 smallest positive divisors of \( N \) must equal \( N \).
2. **Check if \( N \) can be odd:**
- If \( N \) is odd, all its divisors are odd.
- The sum of the squares of 4 odd numbers is even.... | 130 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There is a wooden $3 \times 3 \times 3$ cube and 18 rectangular $3 \times 1$ paper strips. Each strip has two dotted lines dividing it into three unit squares. The full surface of the cube is covered with the given strips, flat or bent. Each flat strip is on one face of the cube. Each bent strip (bent at one of its... | **
- Let's consider the placement of flat strips and bent strips:
- Place 4 flat strips on 4 faces of the cube, each covering 3 unit squares.
- This leaves 2 faces uncovered by flat strips.
- Use bent strips to cover the remaining unit squares, ensuring that each bent strip covers 2 adjacent faces.
7... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$? | 1. **Identify the set \( S \)**: The set \( S \) consists of all positive integers from 1 through 1000 that are not perfect squares. First, we need to determine the perfect squares within this range.
- The smallest perfect square is \( 1^2 = 1 \).
- The largest perfect square less than or equal to 1000 is \( 31^2... | 333 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$, with value 13.) Each positive digit is equally likely, each ar... | 1. **Determine the expected value of a single digit:**
Each single-digit positive integer (1 through 9) is equally likely. Therefore, the expected value \( E[X] \) of a single digit \( X \) is the average of these digits:
\[
E[X] = \frac{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9}{9} = \frac{45}{9} = 5
\]
2. **Iden... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A mustache is created by taking the set of points $(x, y)$ in the $xy$-coordinate plane that satisfy $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$ and $-24 \le x \le 24$. What is the area of the mustache? | To find the area of the mustache, we need to calculate the area between the curves \( y = 4 + 4 \cos\left(\frac{\pi x}{24}\right) \) and \( y = 6 + 6 \cos\left(\frac{\pi x}{24}\right) \) over the interval \( -24 \le x \le 24 \).
1. **Determine the difference between the upper and lower curves:**
\[
\Delta y = (6... | 96 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
How many 3-term geometric sequences $a$, $b$, $c$ are there where $a$, $b$, and $c$ are positive integers with $a < b < c$ and $c = 8000$? | 1. Given that \(a, b, c\) form a geometric sequence with \(a < b < c\) and \(c = 8000\), we can express the terms of the sequence as:
\[
a = \frac{c}{r^2}, \quad b = \frac{c}{r}, \quad c = 8000
\]
where \(r\) is the common ratio and \(r > 1\).
2. Since \(c = 8000\), we have:
\[
a = \frac{8000}{r^2}, ... | 39 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$. Let $c$ be the coefficient of $x^{48}$ in $f(x)$. When $c$ is divided by 101, what is the remainder? (The remainder is an integer between 0 and 100.) | 1. The polynomial \( f(x) \) is given by:
\[
f(x) = \prod_{k=1}^{50} (x - (2k-1))
\]
This is a polynomial of degree 50 with roots at the first 50 odd numbers: \(1, 3, 5, \ldots, 99\).
2. We need to find the coefficient \( c \) of \( x^{48} \) in \( f(x) \). This coefficient is the sum of the products of th... | 60 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$,... | 1. **Reflecting the point over the $x$-axis:**
- The initial point is \( X = (126, 21) \).
- Reflecting \( X \) over the \( x \)-axis gives \( Y = (126, -21) \).
2. **Reflecting the first quadrant over the \( x \)-axis:**
- We want the ray to hit the \( x \)-axis, then the line \( y = x \), and finally return... | 111 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider the sum
\[
S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \, .
\]
Determine $\lfloor S_{4901} \rfloor$. Recall that if $x$ is a real number, then $\lfloor x \rfloor$ (the [i]floor[/i] of $x$) is the greatest integer that is less than or equal to $x$.
| 1. **Estimate the sum using an integral approximation:**
We start by approximating the sum \( S_n = \sum_{k=1}^n \frac{1}{\sqrt{2k-1}} \) using an integral. The idea is to approximate the sum by the integral of a similar function. Specifically, we use:
\[
\sum_{k=1}^n \frac{1}{\sqrt{2k-1}} \approx \int_1^n \f... | 98 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
[b]Problem Section #1
a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and y. Find the greatest possible value of: $x + y$.
[color=red]NOTE: There is a high chance that this problems was copied.[/color] | Let \( a, b, c, d \) be the four numbers in the set. The six pairwise sums of these numbers are given as \( 189, 320, 287, 264, x, \) and \( y \). We need to find the greatest possible value of \( x + y \).
We consider two cases:
1. **Case 1: \( x = a + b \) and \( y = c + d \) (no common number)**
- The remaining... | 761 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]Problem Section #4
b) Let $A$ be a unit square. What is the largest area of a triangle whose vertices lie on the perimeter of
$A$? Justify your answer. | 1. **Positioning the Square and Vertices:**
- Place the unit square \( A \) on the Cartesian plane with vertices at \((0,0)\), \((1,0)\), \((0,1)\), and \((1,1)\).
- Let the vertices of the triangle be \( a = (1,0) \), \( b = (0,y) \), and \( c = (x,1) \) where \( x, y \in [0,1] \).
2. **Area of the Triangle:**
... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The area of a circle (in square inches) is numerically larger than its circumference (in inches). What is the smallest possible integral area of the circle, in square inches?
[i]Proposed by James Lin[/i] | 1. Let \( r \) be the radius of the circle. The area \( A \) of the circle is given by:
\[
A = \pi r^2
\]
The circumference \( C \) of the circle is given by:
\[
C = 2\pi r
\]
2. According to the problem, the area of the circle is numerically larger than its circumference:
\[
\pi r^2 > 2\pi ... | 29 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ordered quadruples $(a,b,c,d)$ of distinct positive integers such that $\displaystyle \binom{\binom{a}{b}}{\binom{c}{d}}=21$.
[i]Proposed by Luke Robitaille[/i] | To solve the problem, we need to find the number of ordered quadruples \((a, b, c, d)\) of distinct positive integers such that \(\binom{\binom{a}{b}}{\binom{c}{d}} = 21\).
First, we note that \(21\) can be expressed as a binomial coefficient in the following ways:
\[ 21 = \binom{7}{2} = \binom{7}{5} = \binom{21}{1} =... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Lunasa, Merlin, and Lyrica are performing in a concert. Each of them will perform two different solos, and each pair of them will perform a duet, for nine distinct pieces in total. Since the performances are very demanding, no one is allowed to perform in two pieces in a row. In how many different ways can the pieces b... | 1. **Identify the pieces and constraints:**
- There are 9 distinct pieces: 6 solos (2 for each of Lunasa, Merlin, and Lyrica) and 3 duets (one for each pair: Lunasa-Merlin, Lunasa-Lyrica, and Merlin-Lyrica).
- No one is allowed to perform in two pieces in a row.
2. **Label the pieces:**
- Let \( S_1, S_2 \) b... | 384 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 7, BC = 5,$ and $CA = 6$. Let $D$ be a variable point on segment $BC$, and let the perpendicular bisector of $AD$ meet segments $AC, AB$ at $E, F,$ respectively. It is given that there is a point $P$ inside $\triangle ABC$ such that $\frac{AP}{PC} = \frac{AE}{EC}$ and $\frac{AP}{PB} =... | 1. **Define the circumcircle and key points:**
Let $\Omega$ be the circumcircle of $\triangle ABC$ centered at $O$. Define points $E_1$ and $F_1$ such that $(A, C; E, E_1) = (A, B; F, F_1) = -1$. Let $U$ and $V$ be the midpoints of $EE_1$ and $FF_1$, respectively.
2. **Define circles $\omega_E$ and $\omega_F$:**
... | 240124 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with incenter $I$. Let $P$ and $Q$ be points such that $IP\perp AC$, $IQ\perp AB$, and $IA\perp PQ$. Assume that $BP$ and $CQ$ intersect at the point $R\neq A$ on the circumcircle of $ABC$ such that $AR\parallel BC$. Given that $\angle B-\angle C=36^\circ$, the value of $\cos A$ can be expressed... | 1. **Define the problem and setup:**
Let \( \triangle ABC \) be a triangle with incenter \( I \). Points \( P \) and \( Q \) are such that \( IP \perp AC \), \( IQ \perp AB \), and \( IA \perp PQ \). Assume \( BP \) and \( CQ \) intersect at point \( R \neq A \) on the circumcircle of \( \triangle ABC \) such that \... | 1570 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $q<50$ be a prime number. Call a sequence of polynomials $P_0(x), P_1(x), P_2(x), ..., P_{q^2}(x)$ [i]tasty[/i] if it satisfies the following conditions:
[list]
[*] $P_i$ has degree $i$ for each $i$ (where we consider constant polynomials, including the $0$ polynomial, to have degree $0$)
[*] The coefficients of ... | 1. **Initial Setup and Transformations**:
- We start by performing transformations \( Q(x) \rightarrow a Q\left(\frac{x-c}{a}\right) + c \) to simplify the problem. This allows us to assume without loss of generality that \( P_0(x) = 0 \) and \( P_2(x) \) is monic. This implies all \( P_i(x) \) (except \( P_0(x) \))... | 30416 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=x^2+x$ for all real $x$. There exist positive integers $m$ and $n$, and distinct nonzero real numbers $y$ and $z$, such that $f(y)=f(z)=m+\sqrt{n}$ and $f(\frac{1}{y})+f(\frac{1}{z})=\frac{1}{10}$. Compute $100m+n$.
[i]Proposed by Luke Robitaille[/i] | 1. Given the function \( f(x) = x^2 + x \), we need to find positive integers \( m \) and \( n \), and distinct nonzero real numbers \( y \) and \( z \) such that \( f(y) = f(z) = m + \sqrt{n} \) and \( f\left(\frac{1}{y}\right) + f\left(\frac{1}{z}\right) = \frac{1}{10} \).
2. Since \( f(y) = f(z) \) and \( y \neq z ... | 1735 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$. Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$. Find $100p+q$.
[i]Proposed by Michael Ren[/i] | 1. Given the equations \(a^3 - 3a + 1 = 0\), \(b^3 - 3b + 1 = 0\), and \(c^3 - 3c + 1 = 0\), we need to find the value of \(\frac{1}{a^2 + b} + \frac{1}{b^2 + c} + \frac{1}{c^2 + a}\).
2. Notice that the polynomial \(x^3 - 3x + 1 = 0\) has roots \(a\), \(b\), and \(c\). We can use trigonometric identities to solve for... | 301 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Farmer James has three types of cows on his farm. A cow with zero legs is called a $\textit{ground beef}$, a cow with one leg is called a $\textit{steak}$, and a cow with two legs is called a $\textit{lean beef}$. Farmer James counts a total of $20$ cows and $18$ legs on his farm. How many more $\textit{ground beef}$s ... | 1. Let \( x \) be the number of ground beefs (cows with 0 legs), \( y \) be the number of steaks (cows with 1 leg), and \( z \) be the number of lean beefs (cows with 2 legs).
2. We are given two equations based on the problem statement:
\[
x + y + z = 20 \quad \text{(total number of cows)}
\]
\[
0x + 1y... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0,1,2,\ldots, 2018$, and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$, and the mouse has at least $k$ slices of cheese remaining, then t... | **
Since the recurrence relation and the initial conditions suggest that \( T_k \) converges to a constant value, we conclude that:
\[
T_{2018} = 2019
\]
The final answer is \( \boxed{2019} \). | 2019 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says:
"Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers.''
How many possibilities are there ... | **
- We need to determine the possible sets of truth-tellers that satisfy the conditions for all \(i\).
6. **Simplifying the Problem:**
- Consider the last person in line (the 100th person). They say that more than half of the 1 person (themselves) is a truth-teller. This is trivially true if they are a truth-te... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than $1.999$.
[i]Proposed by James Lin | 1. **Rewrite the polynomial using the geometric series sum formula:**
The given polynomial is:
\[
P(x) = x^n - x^{n-1} - x^{n-2} - \cdots - x - 1
\]
Notice that the sum of the geometric series \(1 + x + x^2 + \cdots + x^{n-1}\) is:
\[
\frac{x^n - 1}{x - 1}
\]
Therefore, we can rewrite the pol... | 10 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In a rectangular $57\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?
[i]Proposed by James Lin | To solve this problem, we need to determine the smallest number \( k \) such that in a \( 57 \times 57 \) grid, there must exist a rectangle with its four vertices at the centers of distinct black cells.
1. **Understanding the Problem:**
- We have a \( 57 \times 57 \) grid.
- We need to find the smallest \( k \)... | 457 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be a polynomial of degree at most $2018$ such that $P(i)=\binom{2018}i$ for all integer $i$ such that $0\le i\le 2018$. Find the largest nonnegative integer $n$ such that $2^n\mid P(2020)$.
[i]Proposed by Michael Ren | 1. **Understanding the Problem:**
We are given a polynomial \( P(x) \) of degree at most 2018 such that \( P(i) = \binom{2018}{i} \) for all integers \( i \) where \( 0 \leq i \leq 2018 \). We need to find the largest nonnegative integer \( n \) such that \( 2^n \) divides \( P(2020) \).
2. **Using Finite Differenc... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 9001$ be a prime number and let $\mathbb{Z}/p\mathbb{Z}$ denote the additive group of integers modulo $p$. Furthermore, if $A, B \subset \mathbb{Z}/p\mathbb{Z}$, then denote $A+B = \{a+b \pmod{p} | a \in A, b \in B \}.$ Let $s_1, s_2, \dots, s_8$ are positive integers that are at least $2$. Yang the Sheep not... | 1. We are given that \( p = 9001 \) is a prime number and we are working in the additive group \(\mathbb{Z}/p\mathbb{Z}\).
2. We need to find the minimum possible value of \( s_8 \) such that for any sets \( T_1, T_2, \dots, T_8 \subset \mathbb{Z}/p\mathbb{Z} \) with \( |T_i| = s_i \) for \( 1 \le i \le 8 \), the sum \... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn parallel to one pair of sides and $n-1$ lines are drawn parallel to the other pair of sides (with each line distinct and intersecting the interior of the rectangle), thus dividing the rectangle into an $m\times n$... | 1. **Determine \( C_{m,n} \):**
- For \( m = 1 \), there is only one possibility, and it always works. Thus, \( C_{1,n} = 1 \).
- For \( m = 2 \), we claim that \( C_{2,n} = 2^{m-1} \cdot m \). This is because there must be one rectangle known in each column and two rectangles known in one row. There are \( m \) ... | 1289 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Hen Hao randomly selects two distinct squares on a standard $8\times 8$ chessboard. Given that the two squares touch (at either a vertex or a side), the probability that the two squares are the same color can be expressed in the form $\frac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
[i]Prop... | 1. **Identify the total number of squares on the chessboard:**
A standard $8 \times 8$ chessboard has $64$ squares.
2. **Determine the number of pairs of squares that touch at a side (S):**
- Each row has $7$ internal vertical edges, and there are $8$ rows, so there are $8 \times 7 = 56$ vertical edges.
- Eac... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $k$ be a positive integer. In the coordinate plane, circle $\omega$ has positive integer radius and is tangent to both axes. Suppose that $\omega$ passes through $(1,1000+k)$. Compute the smallest possible value of $k$.
[i]Proposed by Luke Robitaille | 1. Since the circle $\omega$ is tangent to both axes, its center must be at $(h, h)$ where $h$ is the radius of the circle. This is because the distance from the center to each axis must be equal to the radius.
2. The equation of the circle with center $(h, h)$ and radius $h$ is:
\[
(x - h)^2 + (y - h)^2 = h^2
... | 58 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathbb{N}$ denote the set of positive integers. For how many positive integers $k\le 2018$ do there exist a function $f: \mathbb{N}\to \mathbb{N}$ such that $f(f(n))=2n$ for all $n\in \mathbb{N}$ and $f(k)=2018$?
[i]Proposed by James Lin | 1. **Understanding the Problem:**
We need to find the number of positive integers \( k \leq 2018 \) for which there exists a function \( f: \mathbb{N} \to \mathbb{N} \) such that \( f(f(n)) = 2n \) for all \( n \in \mathbb{N} \) and \( f(k) = 2018 \).
2. **Analyzing the Function:**
Given \( f(f(n)) = 2n \), we c... | 1512 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $n=2^{2018}$ and let $S=\{1,2,\ldots,n\}$. For subsets $S_1,S_2,\ldots,S_n\subseteq S$, we call an ordered pair $(i,j)$ [i]murine[/i] if and only if $\{i,j\}$ is a subset of at least one of $S_i, S_j$. Then, a sequence of subsets $(S_1,\ldots, S_n)$ of $S$ is called [i]tasty[/i] if and only if:
1) For all $i$, $i\... | 1. **Define the problem in terms of graph theory:**
Consider a directed graph where you draw an edge from \(i\) to \(j\) if \(j \in S_i\). The given conditions imply that if \(i \to j\) and \(j \to k\), then \(i \to k\) as well. This is because the union of the sets \(S_j\) for \(j \in S_i\) must be equal to \(S_i\)... | 2018 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Leonhard has five cards. Each card has a nonnegative integer written on it, and any two cards show relatively prime numbers. Compute the smallest possible value of the sum of the numbers on Leonhard's cards.
Note: Two integers are relatively prime if no positive integer other than $1$ divides both numbers.
[i]Propose... | 1. **Understanding the problem**: We need to find the smallest possible sum of five nonnegative integers such that any two of them are relatively prime. Two numbers are relatively prime if their greatest common divisor (gcd) is 1.
2. **Considering the smallest nonnegative integers**: The smallest nonnegative integer i... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Compute the largest integer that can be expressed in the form $3^{x(3-x)}$ for some real number $x$.
[i]Proposed by James Lin | 1. We start by considering the expression \(3^{x(3-x)}\). To maximize this expression, we need to first maximize the exponent \(x(3-x)\).
2. The function \(f(x) = x(3-x)\) is a quadratic function. To find its maximum value, we can complete the square or use the vertex formula for a parabola. The standard form of a qua... | 11 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ways to erase 24 letters from the string ``OMOMO$\cdots$OMO'' (with length 27), such that the three remaining letters are O, M and O in that order. Note that the order in which they are erased does not matter.
[i]Proposed by Yannick Yao | 1. **Select the position of the $M$**:
- There are 13 $M$'s in the string. Let's denote the position of the selected $M$ as $k$, where $k$ ranges from 1 to 13.
2. **Count the $O$'s before and after the selected $M$**:
- Before the $k$-th $M$, there are $k$ $O$'s.
- After the $k$-th $M$, there are $14 - k$ $O... | 455 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $(p_1, p_2, \dots) = (2, 3, \dots)$ be the list of all prime numbers, and $(c_1, c_2, \dots) = (4, 6, \dots)$ be the list of all composite numbers, both in increasing order. Compute the sum of all positive integers $n$ such that $|p_n - c_n| < 3$.
[i]Proposed by Brandon Wang[/i] | 1. **Identify the sequences of prime and composite numbers:**
- The sequence of prime numbers \( (p_1, p_2, \ldots) \) is \( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, \ldots \).
- The sequence of composite numbers \( (c_1, c_2, \ldots) \) is \( 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, \ldots \).
2. **Check the condition \(... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Patchouli is taking an exam with $k > 1$ parts, numbered Part $1, 2, \dots, k$. It is known that for $i = 1, 2, \dots, k$, Part $i$ contains $i$ multiple choice questions, each of which has $(i+1)$ answer choices. It is known that if she guesses randomly on every single question, the probability that she gets exactly o... | 1. **Calculate the probability of getting no questions correct:**
For each part \(i\) (where \(i = 1, 2, \dots, k\)), there are \(i\) questions, each with \(i+1\) answer choices. The probability of getting a single question wrong is \(\frac{i}{i+1}\). Therefore, the probability of getting all \(i\) questions wrong ... | 2037171 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Three non-collinear lattice points $A,B,C$ lie on the plane $1+3x+5y+7z=0$. The minimal possible area of triangle $ABC$ can be expressed as $\frac{\sqrt{m}}{n}$ where $m,n$ are positive integers such that there does not exists a prime $p$ dividing $n$ with $p^2$ dividing $m$. Compute $100m+n$.
[i]Proposed by Yannick Y... | 1. Let \( A = (x_1, y_1, z_1) \), \( B = (x_2, y_2, z_2) \), and \( C = (x_3, y_3, z_3) \) be three lattice points on the plane \( 1 + 3x + 5y + 7z = 0 \). We aim to find the minimal possible area of triangle \( \triangle ABC \).
2. Define vectors \( \vec{AB} \) and \( \vec{AC} \) as follows:
\[
\vec{AB} = (x_2 ... | 8302 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the largest possible number of distinct real solutions for $x$ to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where $a$, $b$, and $c$ are real numbers.
[i]Proposed by Tristan Shin | 1. **Understanding the Polynomial and Its Degree:**
The given polynomial is of degree 6:
\[
P(x) = x^6 + ax^5 + 60x^4 - 159x^3 + 240x^2 + bx + c
\]
A polynomial of degree 6 can have at most 6 real roots.
2. **Application of Newton's Inequalities:**
Newton's inequalities provide conditions on the coef... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$, and define the $A$-[i]ntipodes[/i] to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 ... | 1. **Understanding the Problem:**
- We are given a triangle \(ABC\) with vertices at the center of masses of three houses.
- We define \(N = 2017\).
- We need to find the number of ordered triples \((\ell_A, \ell_B, \ell_C)\) of concurrent qevians through \(A\), \(B\), and \(C\).
2. **Defining Antipodes:**
... | 6049 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
On Lineland there are 2018 bus stations numbered 1 through 2018 from left to right. A self-driving bus that can carry at most $N$ passengers starts from station 1 and drives all the way to station 2018, while making a stop at each bus station. Each passenger that gets on the bus at station $i$ will get off at station $... | **
- We need to find the minimum \( N \) such that every possible good group is covered.
- Consider the worst-case scenario where each passenger covers a unique pair \((i, j)\). The total number of such pairs is given by the binomial coefficient \( \binom{2018}{2} \).
4. **Calculating the Minimum \( N \):**
-... | 1009 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Jay has a $24\times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e. it switches lights that are on to off and lights that are off to on. Jay toggles each of the $48$ rows and columns exactly once, s... | 1. **Understanding the Problem:**
Jay has a $24 \times 24$ grid of lights, all initially off. Each of the 48 rows and columns has a switch that toggles all the lights in that row or column. Jay toggles each of the 48 rows and columns exactly once in a random order. We need to compute the expected value of the total ... | 9408 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Ann and Drew have purchased a mysterious slot machine; each time it is spun, it chooses a random positive integer such that $k$ is chosen with probability $2^{-k}$ for every positive integer $k$, and then it outputs $k$ tokens. Let $N$ be a fixed integer. Ann and Drew alternate turns spinning the machine, with Ann goin... | 1. **Problem Translation**: We need to determine the value of \( N \) such that the probability of Ann reaching \( N \) tokens before Drew reaches \( M = 2^{2018} \) tokens is \( \frac{1}{2} \). This can be visualized as a particle moving on a grid, starting at \((0, 0)\), moving up or right with equal probability, and... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let an ordered pair of positive integers $(m, n)$ be called [i]regimented[/i] if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $\left(2016^{2016}, N\right)$ is regimented. Compute the largest positive i... | 1. We start by understanding the definition of a regimented pair \((m, n)\). For \((2016^{2016}, N)\) to be regimented, the number of positive integer divisors of \(2016^{2016}\) and \(N^k\) must be the same for all nonnegative integers \(k\).
2. First, we factorize \(2016\):
\[
2016 = 2^5 \cdot 3^2 \cdot 7
\... | 10086 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that a sequence $a_0, a_1, \ldots$ of real numbers is defined by $a_0=1$ and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for $n\geq1$. There is a positive real number $r$ such that \[a_0+a_1r+... | 1. Define the generating functions \( f(x) = \sum_{n=0}^{\infty} a_{2n} x^n \) and \( g(x) = \sum_{n=0}^{\infty} a_{2n+1} x^n \).
2. From the given recurrence relations, we have:
\[
f(x)^2 = g(x)
\]
and
\[
g(x)^2 = \frac{f(x) - 1}{x}
\]
3. We are interested in the generating function \( h(x) = f(x^... | 1923 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 101$ and let $S$ be the set of $p$-tuples $(a_1, a_2, \dots, a_p) \in \mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \to \{0, 1, \dots, p-1\}$ such that
[list]
[*] $f(a + b) + f(a - b) \equiv 2\big(f(a) + f(b)\big) \pmod{p}$ for all $a, b \in S$, and
[*] $f(a) = f(b)$ whenever all co... | 1. **Understanding the problem:**
We need to find the number of functions \( f: S \to \{0, 1, \dots, p-1\} \) that satisfy the given conditions, where \( p = 101 \) and \( S \) is the set of \( p \)-tuples of integers. The conditions are:
- \( f(a + b) + f(a - b) \equiv 2(f(a) + f(b)) \pmod{p} \) for all \( a, b ... | 5152 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given two positive integers $x,y$, we define $z=x\,\oplus\,y$ to be the bitwise XOR sum of $x$ and $y$; that is, $z$ has a $1$ in its binary representation at exactly the place values where $x,y$ have differing binary representations. It is known that $\oplus$ is both associative and commutative. For example, $20 \oplu... | 1. **Understanding the Problem:**
- We are given a set \( S = \{1, 2, \dots, 2018\} \).
- For each subset \( S \subseteq \{1, 2, \dots, 2018\} \), we need to compute \( f(S) \), which is the XOR of all elements in \( S \).
- We then compute \( g(S) \), which is the number of divisors of \( f(S) \) that are at ... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $k,n$ with $k\leq n$, we say that a $k$-tuple $\left(a_1,a_2,\ldots,a_k\right)$ of positive integers is [i]tasty[/i] if
[list]
[*] there exists a $k$-element subset $S$ of $[n]$ and a bijection $f:[k]\to S$ with $a_x\leq f\left(x\right)$ for each $x\in [k]$,
[*] $a_x=a_y$ for some distinct $x,y\in... | 1. **Understanding the Problem:**
We need to find the least possible number of tasty tuples for some positive integer \( n \) such that there are more than 2018 tasty tuples as \( k \) ranges through \( 2, 3, \ldots, n \).
2. **Conditions for Tasty Tuples:**
- There exists a \( k \)-element subset \( S \) of \([... | 4606 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 2027$ be the smallest prime greater than $2018$, and let $P(X) = X^{2031}+X^{2030}+X^{2029}-X^5-10X^4-10X^3+2018X^2$. Let $\mathrm{GF}(p)$ be the integers modulo $p$, and let $\mathrm{GF}(p)(X)$ be the set of rational functions with coefficients in $\mathrm{GF}(p)$ (so that all coefficients are taken modulo $p... | 1. **Understanding the Problem:**
We are given a polynomial \( P(X) \) and a function \( D \) defined on the field of rational functions over \(\mathrm{GF}(p)\), where \( p = 2027 \). The function \( D \) satisfies a specific property for rational functions and reduces the degree of polynomials. We need to determine... | 4114810 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For integers $0 \le m,n \le 2^{2017}-1$, let $\alpha(m,n)$ be the number of nonnegative integers $k$ for which $\left\lfloor m/2^k \right\rfloor$ and $\left\lfloor n/2^k \right\rfloor$ are both odd integers. Consider a $2^{2017} \times 2^{2017}$ matrix $M$ whose $(i,j)$th entry (for $1 \le i, j \le 2^{2017}$) is \[ (-... | 1. **Understanding $\alpha(m, n)$:**
- $\alpha(m, n)$ counts the number of nonnegative integers $k$ for which $\left\lfloor \frac{m}{2^k} \right\rfloor$ and $\left\lfloor \frac{n}{2^k} \right\rfloor$ are both odd integers.
- This is equivalent to counting the number of positions where both $m$ and $n$ have a 1 in... | 1382 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $7/9$. How many players took part at the tournament? | 1. Let \( g_1, g_2, \ldots, g_{2x} \) be the girls and \( b_1, b_2, \ldots, b_x \) be the boys. The total number of players is \( 3x \) since there are \( 2x \) girls and \( x \) boys.
2. Each player plays against every other player exactly once. Therefore, the total number of matches played is given by the binomial co... | 33 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A natural number is a [i]factorion[/i] if it is the sum of the factorials of each of its decimal digits. For example, $145$ is a factorion because $145 = 1! + 4! + 5!$.
Find every 3-digit number which is a factorion. | To find every 3-digit number which is a factorion, we need to check if the number is equal to the sum of the factorials of its digits. Let's denote the 3-digit number as \( N = \overline{abc} \), where \( a, b, \) and \( c \) are its digits.
1. **Determine the range of digits:**
- Since \( N \) is a 3-digit number,... | 145 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times $111$ appears on this sequence? | To determine how many times the number \(111\) appears in the sequence \((a_n)\), we need to analyze the sequence generation rules and how they can produce the number \(111\).
Given:
- \(a_1 = 1\)
- \(a_{2n} = a_n + 1\)
- \(a_{2n+1} = 10a_n\)
We need to find all \(n\) such that \(a_n = 111\).
1. **Initial Step**:
... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations? | 1. **Labeling the Coins**: We start by labeling the heads/tails with zeros/ones to get a cyclic binary sequence. This helps in analyzing the problem using binary sequences.
2. **Case 1: \( n \) is odd**:
- **Invariance Modulo 2**: When \( n \) is odd, the sum of the numbers (heads as 0 and tails as 1) is invariant... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $n$? | To solve this problem, we need to understand the constraints and how they affect the placement of arrows in the $10 \times 10$ grid. Let's break down the problem step by step.
1. **Understanding the Arrow Placement:**
Each arrow is drawn from one corner of a cell to the opposite corner. For example, an arrow in cel... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a regular cube with side length $2$. Let $A$ and $B$ be $2$ vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1$, $A_2$, $\ldots$, $A_k$ so that $A_1=A$, $A_k=B$ and for any $i = 1,\ldots, k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$. Find the minimum value ... | 1. **Identify the vertices of the cube:**
- Let the vertices of the cube be labeled as follows:
\[
(0,0,0), (2,0,0), (0,2,0), (0,0,2), (2,2,0), (2,0,2), (0,2,2), (2,2,2)
\]
- The vertices \(A\) and \(B\) that are furthest apart are \((0,0,0)\) and \((2,2,2)\), respectively. The distance between the... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P. | 1. Define \( a_n \) as the number of ways to color a \( 2 \times n \) grid such that in any \( 2 \times 2 \) block, at least one of the four squares is white.
2. We establish the recurrence relation for \( a_n \) by considering the last column of the grid:
- If the last column is not composed of two black squares, t... | 1009 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer $n$ whose prime factors are all greater than $18$, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$. | 1. We start with the equation \( n = a^3 + b^3 \). Using the identity for the sum of cubes, we have:
\[
n = a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Since all prime factors of \( n \) must be greater than 18, \( a + b \) must be at least 19.
2. To minimize \( n \), we should minimize \( a^3 + b^3 \) while en... | 1843 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares. | 1. We start by noting that \(2x + 1\) is a perfect square. Let \(2x + 1 = k^2\) for some integer \(k\). Therefore, we have:
\[
2x + 1 = k^2 \implies 2x = k^2 - 1 \implies x = \frac{k^2 - 1}{2}
\]
Since \(x\) is a positive integer, \(k^2 - 1\) must be even, which implies \(k\) must be odd.
2. Next, we need ... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $n$ stone piles each consisting of $2018$ stones. The weight of each stone is equal to one of the numbers $1, 2, 3, ...25$ and the total weights of any two piles are different. It is given that if we choose any two piles and remove the heaviest and lightest stones from each of these piles then the pile which ... | 1. **Claim**: The maximum possible value of \( n \) is \( \boxed{12} \).
2. **Construction of Piles**:
- Let pile \( k \) include:
- \( 1 \) stone weighing \( 2k \),
- \( k + 2004 \) stones weighing \( 24 \),
- \( 13 - k \) stones weighing \( 25 \),
- for \( k = 1, 2, \ldots, 12 \).
3. **Verif... | 12 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For integers $a, b$, call the lattice point with coordinates $(a,b)$ [b]basic[/b] if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$... | 1. **Prove that any point \((x, y)\) is connected to exactly one of the points \((1, 1), (1, 0), (0, 1), (-1, 0)\) or \((0, -1)\):**
Consider a basic point \((x, y)\) where \(\gcd(x, y) = 1\). We need to show that this point is connected to one of the five specified points.
- If \(x = 1\) and \(y = 1\), then \... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$a_0, a_1, \ldots, a_{100}$ and $b_1, b_2,\ldots, b_{100}$ are sequences of real numbers, for which the property holds: for all $n=0, 1, \ldots, 99$, either
$$a_{n+1}=\frac{a_n}{2} \quad \text{and} \quad b_{n+1}=\frac{1}{2}-a_n,$$
or
$$a_{n+1}=2a_n^2 \quad \text{and} \quad b_{n+1}=a_n.$$
Given $a_{100}\leq a_0$, what i... | 1. **Initial Assumptions and Setup:**
We are given two sequences \(a_0, a_1, \ldots, a_{100}\) and \(b_1, b_2, \ldots, b_{100}\) with specific recursive properties. For each \(n = 0, 1, \ldots, 99\), either:
\[
a_{n+1} = \frac{a_n}{2} \quad \text{and} \quad b_{n+1} = \frac{1}{2} - a_n,
\]
or
\[
a_{... | 50 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $(x_n)$ is defined as follows:
$$x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3}$$
for all $n\geq 1$.
a. Prove that $(x_n)$ has a finite limit and find that limit.
b. For every $n\geq 1$, prove that
$$n\leq x_1+x_2+\dots +x_n\leq n+1.$$ | ### Part (a): Prove that $(x_n)$ has a finite limit and find that limit.
1. **Define a new sequence**:
Let \( y_n = x_n - 1 \). Then the sequence \( y_n \) is defined as:
\[
y_1 = 1, \quad y_{n+1} = \sqrt{y_n + 9} - \sqrt{y_n + 4} - 1
\]
2. **Simplify the recurrence relation**:
We can rewrite \( y_{n+1... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i] | 1. **Understanding the Problem:**
We need to find all positive integers \( n \) less than 300 such that \( n \) has one less positive divisor than \( n + 1 \). This means:
\[
d(n) = d(n+1) - 1
\]
where \( d(x) \) denotes the number of positive divisors of \( x \).
2. **Analyzing the Divisors:**
For \... | 687 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1\leq x < y \leq 26$ there is exactly one passenger that goes from the $x$-th stop to the $y$-th one. If every passenger want... | 1. **Understanding the Problem:**
- The train has 26 stops, numbered from 1 to 26.
- For each pair \((x, y)\) with \(1 \leq x < y \leq 26\), there is exactly one passenger traveling from stop \(x\) to stop \(y\).
- We need to find the minimum number of seats required to ensure every passenger has a seat during... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the last three digits of the sum of all the real values of $m$ such that the ellipse $x^2+xy+y^2=m$ intersects the hyperbola $xy=n$ only at its two vertices, as $n$ ranges over all non-zero integers $-81\le n \le 81$.
[i]Proposed by [b]AOPS12142015[/b][/i] | 1. Given the ellipse equation \(x^2 + xy + y^2 = m\) and the hyperbola equation \(xy = n\), we need to find the values of \(m\) such that the ellipse intersects the hyperbola only at its two vertices for all non-zero integers \(n\) in the range \(-81 \le n \le 81\).
2. From the hyperbola equation \(xy = n\), we can ex... | 704 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{P}$ be the set of all polynomials $p(x)=x^4+2x^2+mx+n$, where $m$ and $n$ range over the positive reals. There exists a unique $p(x) \in \mathcal{P}$ such that $p(x)$ has a real root, $m$ is minimized, and $p(1)=99$. Find $n$.
[i]Proposed by [b]AOPS12142015[/b][/i] | 1. **Identify the polynomial and its properties:**
Given the polynomial \( p(x) = x^4 + 2x^2 + mx + n \), where \( m \) and \( n \) are positive reals, we need to find \( n \) such that \( p(x) \) has a real root, \( m \) is minimized, and \( p(1) = 99 \).
2. **Analyze the polynomial for real roots:**
Since \( p... | 56 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A, B, C, D$ be points, in order, on a straight line such that $AB=BC=CD$. Let $E$ be a point closer to $B$ than $D$ such that $BE=EC=CD$ and let $F$ be the midpoint of $DE$. Let $AF$ intersect $EC$ at $G$ and let $BF$ intersect $EC$ at $H$. If $[BHC]+[GHF]=1$, then $AD^2 = \frac{a\sqrt{b}}{c}$ where $a,b,$ and $c... | 1. Let \( AB = BC = CD = BE = EC = x \). We need to find \( AD^2 \).
2. Since \( BE = EC = x \), triangle \( BEC \) is equilateral. Also, \( CD = x \), so \( BED \) is a 30-60-90 triangle.
3. Let \( F \) be the midpoint of \( DE \). Since \( DE = x \), \( DF = FE = \frac{x}{2} \).
4. We need to find the areas of tri... | 250 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1,2,3,4,5,6,7,8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positi... | 1. **Determine the number of possible subsets:**
We need to count the number of nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ that consist of consecutive integers. These subsets can be of lengths from 1 to 8.
- For length 1: There are 8 subsets.
- For length 2: There are 7 subsets.
- For length 3: There... | 421 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the set $\{1, 2, \dots, 2018\}$. For each subset $A$ of $N$ with exactly $1009$ elements, define $$f(A)=\sum\limits_{i \in A} i \sum\limits_{j \in N, j \notin A} j.$$If $\mathbb{E}[f(A)]$ is the expected value of $f(A)$ as $A$ ranges over all the possible subsets of $N$ with exactly $1009$ elements, find the... | 1. **Define the problem and notation:**
Let \( N = \{1, 2, \dots, 2018\} \). For each subset \( A \) of \( N \) with exactly \( 1009 \) elements, define
\[
f(A) = \sum_{i \in A} i \sum_{j \in N, j \notin A} j.
\]
We need to find the expected value \( \mathbb{E}[f(A)] \) and then determine the remainder ... | 441 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define a permutation of the set $\{1,2,3,...,n\}$ to be $\textit{sortable}$ if upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when $$\sum\limits_{i=1}^{2018} (-1)^i \cdo... | To solve the problem, we need to find the number of sortable permutations of the set $\{1, 2, 3, \ldots, n\}$, denoted as $f(n)$, and then compute the sum $\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)$ modulo $1000$.
1. **Understanding Sortable Permutations**:
A permutation of $\{1, 2, 3, \ldots, n\}$ is sortable if,... | 153 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be a triangle with $AB=6, BC=8, AC=10$, and let $D$ be a point such that if $I_A, I_B, I_C, I_D$ are the incenters of the triangles $BCD,$ $ ACD,$ $ ABD,$ $ ABC$, respectively, the lines $AI_A, BI_B, CI_C, DI_D$ are concurrent. If the volume of tetrahedron $ABCD$ is $\frac{15\sqrt{39}}{2}$, then the... | 1. **Given Data and Setup:**
- We have a triangle $\triangle ABC$ with sides $AB = 6$, $BC = 8$, and $AC = 10$.
- Point $D$ is such that the lines $AI_A, BI_B, CI_C, DI_D$ are concurrent, where $I_A, I_B, I_C, I_D$ are the incenters of triangles $BCD, ACD, ABD, ABC$ respectively.
- The volume of tetrahedron $A... | 49 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $n$ is said to be $m$-free if $n \leq m!$ and $\gcd(i,n)=1$ for each $i=1,2,...,m$. Let $\mathcal{S}_k$ denote the sum of the squares of all the $k$-free integers. Find the remainder when $\mathcal{S}_7-\mathcal{S}_6$ is divided by $1000$.
[i]Proposed by [b]FedeX333X[/b][/i] | 1. **Understanding the Problem:**
A positive integer \( n \) is said to be \( m \)-free if \( n \leq m! \) and \(\gcd(i, n) = 1\) for each \( i = 1, 2, \ldots, m \). We need to find the sum of the squares of all \( k \)-free integers, denoted as \(\mathcal{S}_k\), and then find the remainder when \(\mathcal{S}_7 - \... | 80 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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