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Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$.
Determine the... | 1. Let \( N = \overline{abc} \) be a 3-digit number with distinct non-zero digits \( a, b, \) and \( c \). The six permutations of \( N \) are \( \{ \overline{abc}, \overline{acb}, \overline{bac}, \overline{bca}, \overline{cab}, \overline{cba} \} \).
2. The sum of these permutations can be written as:
\[
100a + ... | 629 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$39$ students participated in a math competition. The exam consisted of $6$ problems and each problem was worth $1$ point for a correct solution and $0$ points for an incorrect solution. For any $3$ students, there is at most $1$ problem that was not solved by any of the three. Let $B$ be the sum of all of the scores o... | 1. **Define the sets and conditions:**
Let \( S_i \) be the set of students that did not solve problem \( i \). The given condition states that for any 3 students, there is at most 1 problem that none of them solved. This can be translated into the condition that for any two problems \( i \) and \( j \), the interse... | 165 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a $12\times 12$ grid, colour each unit square with either black or white, such that there is at least one black unit square in any $3\times 4$ and $4\times 3$ rectangle bounded by the grid lines. Determine, with proof, the minimum number of black unit squares. | 1. **Divide the Grid into Rectangles**:
- Consider the $12 \times 12$ grid. We can divide this grid into non-overlapping $3 \times 4$ rectangles. Since the grid is $12 \times 12$, we can fit $12$ such $3 \times 4$ rectangles (3 rows and 4 columns of $3 \times 4$ rectangles).
2. **Minimum Number of Black Squares in ... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of distinct right-angled triangles such that its three sides are of integral lengths, and its area is $999$ times of its perimeter.
(Congruent triangles are considered identical.) | To determine the number of distinct right-angled triangles with integral side lengths such that the area is 999 times the perimeter, we start by using the properties of Pythagorean triples.
1. **Express the sides of the triangle:**
The sides of a right-angled triangle with integral lengths can be expressed as:
\... | 90 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $k$ such that for any positive integer $n$, $2^{(k-1)n+1}$ does not divide $\frac{(kn)!}{n!}$. | 1. **Define the valuation function and use Legendre's formula:**
Let \( v_p(x) \) denote the highest power of a prime \( p \) that divides \( x \). According to Legendre's formula, for a prime \( p \),
\[
v_p(n!) = \frac{n - S_p(n)}{p-1}
\]
where \( S_p(n) \) is the sum of the digits of \( n \) when wri... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Each of the numbers $1$ up to and including $2014$ has to be coloured; half of them have to be coloured red the other half blue. Then you consider the number $k$ of positive integers that are expressible as the sum of a red and a blue number. Determine the maximum value of $k$ that can be obtained. | 1. **Initial Setup and Constraints**:
- We need to color each of the numbers from \(1\) to \(2014\) such that half are red and half are blue.
- We are interested in the number \(k\) of positive integers that can be expressed as the sum of a red and a blue number.
- The smallest possible sum of a red and a blue... | 4023 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. Every square in a $n \times n$-square grid is either white or black.
How many such colourings exist, if every $2 \times 2$-square consists of exactly two white and two black squares?
The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from ea... | To solve this problem, we need to determine the number of valid colorings of an \( n \times n \) grid such that every \( 2 \times 2 \) sub-grid contains exactly two white and two black squares. Let's break down the solution step by step.
1. **Understanding the Constraints**:
- Each \( 2 \times 2 \) sub-grid must ha... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum? | 1. Let \( a_1, a_2, \ldots, a_{10} \) be a permutation of \( 1, 2, \ldots, 10 \) where \( a_1, a_2, \ldots, a_5 \) are on the vertices and the other 5 numbers are on the sides. We need to find the smallest possible value of the sum \( S \) such that the sum of the numbers on every side of the pentagon is the same.
2. ... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$. | To find the least positive integer \( n \) such that there exists a polynomial
\[ P(x) = a_{2n}x^{2n} + a_{2n-1}x^{2n-1} + \dots + a_1x + a_0 \]
with real coefficients satisfying:
1. \( 2014 \leq a_i \leq 2015 \) for \( i = 0, 1, \dots, 2n \),
2. There is a real number \( \xi \) such that \( P(\xi) = 0 \),
we proceed... | 2014 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The fraction $\tfrac1{2015}$ has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where $a$, $b$, and $c$ are integers with $0\leq a<5$ and $0\leq b<13$. Find $a+b$. | 1. We start with the given equation:
\[
\frac{1}{2015} = \frac{a}{5} + \frac{b}{13} + \frac{c}{31}
\]
To combine the fractions on the right-hand side, we need a common denominator, which is \(2015\) (since \(2015 = 5 \times 13 \times 31\)).
2. Rewrite the right-hand side with the common denominator:
\[
... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:
[list]
[*] t... | To solve the problem, we need to find the number of ordered pairs \((a, b) \in \{1, 2, \ldots, 35\}^2\) such that \(ax + b\) is a quadratic residue modulo \(x^2 + 1\) and 35. This means there exists a polynomial \(f(x)\) with integer coefficients such that either of the following equivalent conditions holds:
1. There ... | 144 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $N=30^{2015}$. Find the number of ordered 4-tuples of integers $(A,B,C,D)\in\{1,2,\ldots,N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3+Bn^2+2Cn+D$ is divisible by $N$. | 1. Let \( N = 30^{2015} \). We need to find the number of ordered 4-tuples \((A, B, C, D) \in \{1, 2, \ldots, N\}^4\) such that for every integer \( n \), the polynomial \( An^3 + Bn^2 + 2Cn + D \) is divisible by \( N \).
2. First, consider the case when \( n = 0 \). Substituting \( n = 0 \) into the polynomial, we g... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The number $125$ can be written as a sum of some pairwise coprime integers larger than $1$. Determine the largest number of terms that the sum may have. | 1. **Define the problem and constraints:**
We need to express \(125\) as a sum of pairwise coprime integers greater than \(1\). We aim to find the maximum number of terms in this sum.
2. **Initial observations:**
- Since the integers are pairwise coprime, there can be at most one even number in the sum.
- The... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Beto plays the following game with his computer: initially the computer randomly picks $30$ integers from $1$ to $2015$, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some if the numbers written on the chalkboard, and subtracts $k$ from each ... | 1. **Initial Setup and Problem Restatement**:
- Beto's game involves reducing 30 integers, each between 1 and 2015, to zero.
- On each turn, Beto can choose a positive integer \( k \) and subtract \( k \) from any subset of the numbers, provided the result is non-negative.
- The goal is to determine the minimu... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $10$-digit number is called a $\textit{cute}$ number if its digits belong to the set $\{1,2,3\}$ and the difference of every pair of consecutive digits is $1$.
a) Find the total number of cute numbers.
b) Prove that the sum of all cute numbers is divisibel by $1408$. | 1. Define \( t_n \) as the total number of cute numbers with \( n \) digits.
2. Let \( a_n, b_n, c_n \) denote the number of cute numbers ending with \( 1, 2, 3 \) respectively. Thus, \( t_n = a_n + b_n + c_n \).
3. Consider the cases based on the last digit of the number:
- If the number ends with \( 1 \) or \( 3 \... | 64 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Every cell of a $3\times 3$ board is coloured either by red or blue. Find the number of all colorings in which there are no $2\times 2$ squares in which all cells are red. | To solve the problem, we need to count the number of ways to color a $3 \times 3$ board such that no $2 \times 2$ sub-square is entirely red. We will consider two cases based on the color of the center cell.
1. **Case 1: The center cell is blue.**
- If the center cell is blue, the remaining 8 cells can be colored e... | 417 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are four basketball players $A,B,C,D$. Initially the ball is with $A$. The ball is always passed from one person to a different person.
In how many ways can the ball come back to $A$ after $\textbf{seven}$ moves? (for example $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A\rightarrow B\rightarrow C\righta... | 1. **Define the problem and initial conditions:**
We need to find the number of ways the ball can return to player \( A \) after 7 moves. Let \( a_n \) represent the number of ways the ball can be with \( A \) after \( n \) passes. The initial conditions are \( a_0 = 0 \) and \( a_1 = 1 \).
2. **Formulate the recur... | 1094 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite? | 1. **Calculate the total number of ways to choose 3 objects from 32 objects:**
\[
\binom{32}{3} = \frac{32 \times 31 \times 30}{3 \times 2 \times 1} = 4960
\]
2. **Subtract the number of ways in which two of the chosen objects are adjacent:**
- There are 32 pairs of adjacent objects.
- For each pair, th... | 3648 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $$N=2^5+2^{5^2}+2^{5^3}+\cdots{}+2^{5^{2015}}.$$Wrotten in the usual decimal form, find the last two digits of the number $N$.
| To find the last two digits of \( N = 2^5 + 2^{5^2} + 2^{5^3} + \cdots + 2^{5^{2015}} \), we need to compute \( N \mod 100 \).
1. **Compute \( N \mod 4 \):**
\[
2^5 \equiv 0 \pmod{4}, \quad 2^{5^2} \equiv 0 \pmod{4}, \quad \ldots, \quad 2^{5^{2015}} \equiv 0 \pmod{4}
\]
Since each term in the sum is divisi... | 80 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\gamma_1, \gamma_2,\gamma_3 $ be three circles of unit radius which touch each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed by the common tangents be $\triangle XYZ$ . Find the length of each side of $\triangle XYZ$ | 1. **Identify the centers and radii of the circles:**
Let \( \gamma_1, \gamma_2, \gamma_3 \) be three circles of unit radius that touch each other externally. Let \( O_1, O_2, O_3 \) be the centers of these circles. Since the circles touch each other externally, the distance between any two centers is \( 2 \) units ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For the all $(m,n,k)$ positive integer triples such that $|m^k-n!| \le n$ find the maximum value of $\frac{n}{m}$
[i]Proposed by Melih Üçer[/i] | 1. **Verification of the initial claim:**
We need to verify that the triples \((m, n, k) = (1, 2, k)\) satisfy the inequality \(|m^k - n!| \le n\) for all possible values of \(k\).
For \(m = 1\), \(n = 2\), and any \(k\):
\[
|1^k - 2!| = |1 - 2| = 1 \le 2
\]
This confirms that the inequality holds fo... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
In a country consisting of $2015$ cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies. | 1. **Graph Representation and Problem Restatement**:
- Represent the problem using graph theory. We need to find the minimal number of colors to color the edges of the complete graph \( K_{2015} \) such that every triangle (or 3-cycle) has edges of different colors.
2. **Lower Bound on the Number of Colors**:
- ... | 2015 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]quadratic[/i] number is a real root of the equations $ax^2 + bx + c = 0$ where $|a|,|b|,|c|\in\{1,2,\ldots,10\}$. Find the smallest positive integer $n$ for which at least one of the intervals$$\left(n-\dfrac{1}{3}, n\right)\quad \text{and}\quad\left(n, n+\dfrac{1}{3}\right)$$does not contain any quadratic number. | 1. **Understanding the problem**: We need to find the smallest positive integer \( n \) such that at least one of the intervals \( \left(n-\dfrac{1}{3}, n\right) \) and \( \left(n, n+\dfrac{1}{3}\right) \) does not contain any quadratic number. A quadratic number is a real root of the equation \( ax^2 + bx + c = 0 \) w... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be [i]critical[/i] if for each $i,j\in S$ at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$ and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is colored red. Find the maximal possible number of... | To solve this problem, we need to find the maximal number of red points in a critical coloring of the plane points with coordinates in the set \( S = \{0, 1, \ldots, 99\} \). A coloring is critical if for each \( i, j \in S \), at least one of the four points \((i, j), (i + 1, j), (i, j + 1), (i + 1, j + 1)\) is colore... | 5000 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ such that if we color in red $n$ arbitrary vertices of the cube , there will be a vertex of the cube which has the three vertices adjacent to it colored in red. | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( n \) such that if we color \( n \) arbitrary vertices of a cube in red, there will be at least one vertex of the cube which has all three of its adjacent vertices colored in red.
2. **Analyzing the Cube:**
A cube has 8 vertices an... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c>0$ such that $a \geq bc^2$ , $b \geq ca^2$ and $c \geq ab^2$ . Find the maximum value that the expression :
$$E=abc(a-bc^2)(b-ca^2)(c-ab^2)$$ can acheive. | Given the conditions \(a \geq bc^2\), \(b \geq ca^2\), and \(c \geq ab^2\), we need to find the maximum value of the expression:
\[ E = abc(a - bc^2)(b - ca^2)(c - ab^2). \]
1. **Analyzing the conditions:**
- \(a \geq bc^2\)
- \(b \geq ca^2\)
- \(c \geq ab^2\)
2. **Considering the equality case:**
Let's a... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$? | 1. **Restate the problem in a more manageable form:**
We need to prove that the number \(1\) can be represented as a sum of a finite number \(n\) of real numbers, each less than \(1\), and each containing only the digits \(0\) and \(7\) in their decimal representation. We also need to find the least possible value o... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this oper... | 1. **Understanding the Problem:**
- We have an 8-by-8 grid, divided into 64 unit squares.
- Each square is either black or white.
- The number of black squares is even.
- We can flip the colors of two adjacent squares (1-by-2 or 2-by-1 rectangle) in one step.
- We need to find the maximum number of steps... | 32 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In how many different ways can 900 be expressed as the product of two (possibly equal) positive integers? Regard $m \cdot n$ and $n \cdot m$ as the same product. | 1. First, we need to find the prime factorization of 900. We have:
\[
900 = 2^2 \cdot 3^2 \cdot 5^2
\]
2. The number of divisors of 900 can be calculated using the formula for the number of divisors of a number \( n = p_1^{e_1} \cdot p_2^{e_2} \cdot \ldots \cdot p_k^{e_k} \), which is:
\[
(e_1 + 1)(e_2 +... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ and $y$ be real numbers such that
\[
2 < \frac{x - y}{x + y} < 5.
\]
If $\frac{x}{y}$ is an integer, what is its value? | 1. Let $\frac{x}{y} = m$, where $m$ is an integer. Then we can rewrite the given inequality in terms of $m$:
\[
2 < \frac{x - y}{x + y} < 5.
\]
Substituting $x = my$ into the inequality, we get:
\[
2 < \frac{my - y}{my + y} < 5.
\]
2. Simplify the fraction:
\[
\frac{my - y}{my + y} = \frac{y(... | -2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
How many distinct positive integers can be expressed in the form $ABCD - DCBA$, where $ABCD$ and $DCBA$ are 4-digit positive integers? (Here $A$, $B$, $C$ and $D$ are digits, possibly equal.)
Clarification: $A$ and $D$ can't be zero (because otherwise $ABCD$ or $DCBA$ wouldn't be a true 4-digit integer). | To determine how many distinct positive integers can be expressed in the form \(ABCD - DCBA\), where \(ABCD\) and \(DCBA\) are 4-digit positive integers, we start by expressing the numbers in terms of their digits.
1. **Express \(ABCD\) and \(DCBA\) in terms of digits:**
\[
ABCD = 1000A + 100B + 10C + D
\]
... | 161 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In baseball, a player's [i]batting average[/i] is the number of hits divided by the number of at bats, rounded to three decimal places. Danielle's batting average is $.399$. What is the fewest number of at bats that Danielle could have? | 1. Let \( x \) be the number of hits and \( y \) be the number of at bats. Danielle's batting average is given by:
\[
\frac{x}{y} = 0.399
\]
Since the batting average is rounded to three decimal places, we have:
\[
0.3985 < \frac{x}{y} < 0.3995
\]
2. To eliminate the fraction, multiply all parts o... | 138 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$. Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points. Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior o... | 1. **Identify the coordinates of the points and the distance from the origin:**
The $2^n$ points in $n$-dimensional space have coordinates $(\pm 1, \pm 1, \ldots, \pm 1)$. Each of these points is at a distance of $\sqrt{n}$ from the origin, as calculated by the Euclidean distance formula:
\[
\text{Distance} = ... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Say that a rational number is [i]special[/i] if its decimal expansion is of the form $0.\overline{abcdef}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are digits (possibly equal) that include each of the digits $2$, $0$, $1$, and $5$ at least once (in some order). How many special rational numbers are there?
| To solve the problem, we need to count the number of 6-digit sequences where each of the digits 2, 0, 1, and 5 appears at least once. We will use combinatorial methods to count the different cases based on the frequency of the digits.
1. **Case 1: Each of the digits 2, 0, 1, and 5 appears exactly once.**
- We need ... | 22080 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]permutation[/i] of a finite set is a one-to-one function from the set onto itself. A [i]cycle[/i] in a permutation $P$ is a nonempty sequence of distinct items $x_1$, $\ldots\,$, $x_n$ such that $P(x_1) = x_2$, $P(x_2) = x_3$, $\ldots\,$, $P(x_n) = x_1$. Note that we allow the 1-cycle $x_1$ where $P(x_1) = x_1$ ... | To find the number of bi-cyclic permutations of the 7-element set formed by the letters of "PROBLEM", we need to count the permutations that split the set into exactly two disjoint cycles.
1. **Identify the possible cycle structures:**
- Since we need exactly two disjoint cycles, the possible cycle structures for ... | 1554 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. When the leftmost digit of (the standard base 10 representation of) $n$ is shifted to the rightmost position (the units position), the result is $n/3$. Find the smallest possible value of the sum of the digits of $n$. | 1. Let \( n \) be a positive integer with \( d \) digits. When the leftmost digit of \( n \) is shifted to the rightmost position, the result is \( \frac{n}{3} \). We need to find the smallest possible value of the sum of the digits of \( n \).
2. Let \( n \) be represented as \( n = a_1a_2a_3 \ldots a_d \) in base 10... | 126 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that ... | 1. Define \( e_i \) as the expected number of moves to get all four numbers in the sequence given that we have currently seen \( i \) distinct values. We need to find \( e_0 \).
2. We start with the following equations:
\[
e_0 = e_1 + 1
\]
\[
e_1 = \frac{1}{3}e_1 + \frac{2}{3}e_2 + 1
\]
\[
e_2 ... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$. | 1. **List the primes in increasing order**: Let the primes be listed as \( p_1, p_2, p_3, \ldots \). We need to show that for any \( n \), the terms \( a_k \) with \( k \in (p_n, p_{n+1}] \) permute the integers in \( (p_n, p_{n+1}] \).
2. **Base case**: For \( n = 1 \), we have \( p_1 = 2 \). The set \( \{a_1, a_2\} ... | 1 | Number Theory | proof | Yes | Yes | aops_forum | false |
Let $\{x_n\}$ be a Van Der Corput series,that is,if the binary representation of $n$ is $\sum a_{i}2^{i}$ then $x_n=\sum a_i2^{-i-1}$.Let $V$ be the set of points on the plane that have the form $(n,x_n)$.Let $G$ be the graph with vertex set $V$ that is connecting any two points $(p,q)$ if there is a rectangle $R$ whic... | 1. **Understanding the Van Der Corput Series**:
The Van Der Corput series $\{x_n\}$ is defined such that if the binary representation of $n$ is $\sum a_i 2^i$, then $x_n = \sum a_i 2^{-i-1}$. This means that $x_n$ is obtained by reversing the binary digits of $n$ and placing them after the binary point.
2. **Defini... | 4 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
A function $f$ from the positive integers to the nonnegative integers is defined recursively by $f(1) = 0$ and $f(n+1) = 2^{f(n)}$ for every positive integer $n$. What is the smallest $n$ such that $f(n)$ exceeds the number of atoms in the observable universe (approximately $10^{80}$)?
[i]Proposed by Evan Chen[/i] | 1. We start with the given recursive function \( f \) defined as:
\[
f(1) = 0
\]
\[
f(n+1) = 2^{f(n)}
\]
2. We need to find the smallest \( n \) such that \( f(n) \) exceeds \( 10^{80} \). First, we estimate \( 10^{80} \) in terms of powers of 2:
\[
10^{80} \approx 2^{240}
\]
This approxi... | 7 | Other | math-word-problem | Yes | Yes | aops_forum | false |
There exists a unique strictly increasing arithmetic sequence $\{a_i\}_{i=1}^{100}$ of positive integers such that \[a_1+a_4+a_9+\cdots+a_{100}=\text{1000},\] where the summation runs over all terms of the form $a_{i^2}$ for $1\leq i\leq 10$. Find $a_{50}$.
[i]Proposed by David Altizio and Tony Kim[/i] | 1. Let \( a_1 = a \) and \( a_{k+1} - a_k = d \). This means the sequence is an arithmetic sequence with the first term \( a \) and common difference \( d \).
2. We are given the condition:
\[
a_1 + a_4 + a_9 + \cdots + a_{100} = 1000
\]
where the summation runs over all terms of the form \( a_{i^2} \) for... | 123 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt 3$. We construct four semicircles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ whose diameters are the segments $AB$, $BC$, $CD$, $DA$. It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4... | 1. **Define the coordinates of the rectangle:**
Let \( A = (0, 0) \), \( B = (0, 6) \), \( C = (6\sqrt{3}, 6) \), and \( D = (6\sqrt{3}, 0) \). The sides of the rectangle are \( AB = 6 \) and \( BC = 6\sqrt{3} \).
2. **Write the equations of the semicircles:**
- Semicircle \(\omega_1\) with diameter \(AB\):
... | 243 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a $4\times 4$ grid of unit squares, five squares are chosen at random. The probability that no two chosen squares share a side is $\tfrac mn$ for positive relatively prime integers $m$ and $n$. Find $m+n$.
[i]Proposed by David Altizio[/i] | To solve this problem, we need to calculate the probability that no two chosen squares in a \(4 \times 4\) grid share a side. We will use combinatorial methods and casework to find the number of valid configurations and then divide by the total number of ways to choose 5 squares from 16.
1. **Total number of ways to c... | 2309 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $1\leq k\leq 99$ such that there exist positive integers $a$ and $b$ with the property that \[x^{100}-ax^k+b=(x^2-2x+1)P(x)\] for some polynomial $P$ with integer coefficients.
[i]Proposed by David Altizio[/i] | 1. Notice that \(1\) is a double root of \(x^2 - 2x + 1\). This can be rewritten as \((x-1)^2\).
2. Given that \(x^2 - 2x + 1\) is a divisor of \(x^{100} - ax^k + b\), it follows that \((x-1)^2\) must also be a factor of \(x^{100} - ax^k + b\).
3. For \((x-1)^2\) to be a factor of \(x^{100} - ax^k + b\), \(x-1\) must b... | 117 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.
[i]Proposed by Evan Chen[/i] | To solve the problem, we need to compute the number of subsets \( S \) of \( \{0, 1, \dots, 14\} \) such that for each \( n = 0, 1, \dots, 6 \), either \( n \) is in \( S \) or both \( 2n+1 \) and \( 2n+2 \) are in \( S \).
1. **Understanding the Condition:**
- For each \( n \) in \( \{0, 1, \dots, 6\} \), either \... | 2306 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Omega_1$ and $\Omega_2$ be two circles in the plane. Suppose the common external tangent to $\Omega_1$ and $\Omega_2$ has length $2017$ while their common internal tangent has length $2009$. Find the product of the radii of $\Omega_1$ and $\Omega_2$.
[i]Proposed by David Altizio[/i] | 1. Let the radii of the circles \(\Omega_1\) and \(\Omega_2\) be \(r_1\) and \(r_2\) respectively. Let the distance between the centers of the circles be \(d\).
2. The length of the common external tangent between two circles is given by:
\[
\sqrt{d^2 - (r_1 + r_2)^2} = 2017
\]
Squaring both sides, we get:... | 8052 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $O$, $A$, $B$, and $C$ be points in space such that $\angle AOB=60^{\circ}$, $\angle BOC=90^{\circ}$, and $\angle COA=120^{\circ}$. Let $\theta$ be the acute angle between planes $AOB$ and $AOC$. Given that $\cos^2\theta=\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed... | 1. **Define the problem in terms of geometry and vectors:**
- We are given four points \(O\), \(A\), \(B\), and \(C\) in space with specific angles between them: \(\angle AOB = 60^\circ\), \(\angle BOC = 90^\circ\), and \(\angle COA = 120^\circ\).
- We need to find the acute angle \(\theta\) between the planes \(... | 101 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$ be positive integers and $p$ be a prime number. Assume that \[ a^n(b+c)+b^n(a+c)+c^n(a+b)\equiv 8\pmod{p} \] for each nonnegative integer $n$. Let $m$ be the remainder when $a^p+b^p+c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$.
[... | Given the problem, we need to find the maximum possible value of \( k \) when \( m^p \) is divided by \( p^4 \). Let's break down the solution step-by-step.
1. **Initial Condition Analysis:**
The given condition is:
\[
a^n(b+c) + b^n(a+c) + c^n(a+b) \equiv 8 \pmod{p}
\]
for each nonnegative integer \( n... | 399 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$?
[i]Based on a proposal by Neelabh Deka[/i] | 1. **Determine the first day of year \( N+1 \):**
- Given that the last day of year \( N \) is a Tuesday, the first day of year \( N+1 \) will be the next day, which is a Wednesday.
2. **Identify if year \( N+1 \) is a leap year:**
- Since the first day of year \( N+2 \) is a Friday, we need to check if \( N+1 \... | 23 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square with side length $100$. Denote by $M$ the midpoint of $AB$. Point $P$ is selected inside the square so that $MP = 50$ and $PC = 100$. Compute $AP^2$.
[i]Based on a proposal by Amogh Gaitonde[/i] | 1. **Identify the given information and setup the problem:**
- Square \(ABCD\) with side length \(100\).
- \(M\) is the midpoint of \(AB\), so \(AM = MB = 50\).
- Point \(P\) inside the square such that \(MP = 50\) and \(PC = 100\).
2. **Analyze the geometric properties:**
- Since \(M\) is the midpoint of ... | 2000 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.)
[i]Based on a proposal by Yannick Yao[/i] | 1. **Understanding the constraints:**
- $N$ is a 5-digit number in base 10.
- $N$ contains no digits greater than 3, so each digit of $N$ can be 0, 1, 2, or 3.
- The leading digit of $N$ cannot be zero.
- $\gcd(N, 15) = 1$ and $\gcd(N, 20) = 1$.
2. **Analyzing the gcd conditions:**
- $\gcd(N, 15) = 1$ i... | 256 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of positive integers $a \le 250$ for which the set $\{a+1, a+2, \dots, a+1000\}$ contains
$\bullet$ Exactly $333$ multiples of $3$,
$\bullet$ Exactly $142$ multiples of $7$, and
$\bullet$ Exactly $91$ multiples of $11$.
[i]Based on a proposal by Rajiv Movva[/i] | To determine the number of positive integers \( a \le 250 \) for which the set \(\{a+1, a+2, \dots, a+1000\}\) contains exactly \(333\) multiples of \(3\), \(142\) multiples of \(7\), and \(91\) multiples of \(11\), we need to analyze the conditions given.
1. **Condition for multiples of 3:**
- The set \(\{a+1, a+2... | 20 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of ways a series of $+$ and $-$ signs can be inserted between the numbers $0,1,2,\cdots, 12$ such that the value of the resulting expression is divisible by 5.
[i]Proposed by Matthew Lerner-Brecher[/i] | 1. **Define the Laurent Polynomial:**
Consider the Laurent polynomial:
\[
f(x) = \prod_{k=1}^{12}\left(x^k + \frac{1}{x^k}\right)
\]
This polynomial represents the insertion of $+$ and $-$ signs between the numbers $0, 1, 2, \ldots, 12$.
2. **Primitive 5th Root of Unity:**
Let $\omega$ be a primitive... | 816 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For an integer $30 \le k \le 70$, let $M$ be the maximum possible value of \[ \frac{A}{\gcd(A,B)} \quad \text{where } A = \dbinom{100}{k} \text{ and } B = \dbinom{100}{k+3}. \] Find the remainder when $M$ is divided by $1000$.
[i]Based on a proposal by Michael Tang[/i] | To solve the problem, we need to find the maximum possible value of \(\frac{A}{\gcd(A,B)}\) where \(A = \binom{100}{k}\) and \(B = \binom{100}{k+3}\) for \(30 \le k \le 70\). We will then find the remainder when this maximum value is divided by 1000.
1. **Express \(A\) and \(B\) in terms of factorials:**
\[
A = ... | 664 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$, $k=1,2,...,7$. Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$. What is the most probable sum he can get? | 1. **Identify the numbers on the cards:**
The numbers on the cards are given by \(2^{k-1}\) for \(k = 1, 2, \ldots, 7\). Therefore, the numbers on the cards are:
\[
2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 = 1, 2, 4, 8, 16, 32, 64
\]
2. **Determine the sum of all cards:**
The sum of all the numbers on the card... | 127 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
| 1. Let's first observe the pattern in the sequence. The sequence starts as follows:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, \ldots\]
We notice that every third number is repeated. Specifically, the numbers at positions \(3k+1\) (where \(k\) is a non-negative integer) are not repeated, while the numbe... | 1344 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Consider all sums that add up to $2015$. In each sum, the addends are consecutive positive integers, and all sums have less than $10$ addends. How many such sums are there? | To solve the problem, we need to find all sets of consecutive positive integers that sum to 2015, with the number of addends being less than 10. Let's denote the first integer in the sequence by \( a \) and the number of addends by \( n \).
1. **Sum of Consecutive Integers**:
The sum of \( n \) consecutive integers... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers. | 1. **Define the problem and the set construction:**
We need to find the largest natural number \( m \) such that among any five 500-element subsets of \(\{1, 2, \dots, 1000\}\), there exist two sets whose intersection contains at least \( m \) numbers.
2. **Construct specific sets:**
Consider the sets \( S_i = \... | 200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be at triangle with incircle $\Gamma$. Let $\Gamma_1$, $\Gamma_2$, $\Gamma_3$ be three circles inside $\triangle ABC$ each of which is tangent to $\Gamma$ and two sides of the triangle and their radii are $1,4,9$. Find the radius of $\Gamma$. | 1. Let \( r \) be the radius of the incircle \(\Gamma\) of \(\triangle ABC\), and let \( r_1, r_2, r_3 \) be the radii of the circles \(\Gamma_1, \Gamma_2, \Gamma_3\) respectively, with \( r_1 = 1 \), \( r_2 = 4 \), and \( r_3 = 9 \).
2. The circles \(\Gamma_1, \Gamma_2, \Gamma_3\) are each tangent to the incircle \(\... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $ABC$ at $D \neq C $. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $AIC$ at $Y$ outside $\omega_1$. She... | 1. **Identify Key Points and Circles:**
- Let \( I \) be the incenter of \(\triangle ABC\).
- Given \( AB = 20 \), \( BC = 15 \), and \( BI = 12 \).
- Let \( CI \) intersect the circumcircle \(\omega_1\) of \(\triangle ABC\) at \( D \neq C \).
- Let \( I_a \) and \( I_b \) be the excenters opposite \( A \) ... | 13 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are real numbers $a, b, c, d$ such that for all $(x, y)$ satisfying $6y^2 = 2x^3 + 3x^2 + x$, if $x_1 = ax + b$ and $y_1 = cy + d$, then $y_1^2 = x_1^3 - 36x_1$. What is $a + b + c + d$? | 1. We start with the given equation \(6y^2 = 2x^3 + 3x^2 + x\). We need to transform this equation into the form \(y_1^2 = x_1^3 - 36x_1\) using the linear transformations \(x_1 = ax + b\) and \(y_1 = cy + d\).
2. First, let's rewrite the given equation in a factored form:
\[
6y^2 = 2x^3 + 3x^2 + x = x(2x^2 + 3x... | 90 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n = \frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?
| 1. We start with the given sequence definition:
\[
a_1 = 1, \quad a_2 = 2015, \quad a_n = \frac{na_{n-1}^2}{a_{n-1} + na_{n-2}} \quad \text{for} \quad n > 2
\]
2. To find the least \( k \) such that \( a_k < a_{k-1} \), we need to analyze the ratio \( \frac{a_n}{a_{n-1}} \):
\[
a_n = \frac{na_{n-1}^2}{a... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$. Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$. Over all such choices of $a, b, c$, what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\] | 1. **Symmetry of \( f(x, y) \)**:
We start by noting that the function \( f(x, y) = x^3 + (y-4)x^2 + (y^2-4y+4)x + (y^3-4y^2+4y) \) is symmetric in \( x \) and \( y \). This means \( f(x, y) = f(y, x) \).
2. **Given Condition**:
We are given that \( f(a, b) = f(b, c) = f(c, a) \). This implies that the polynomia... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
[/list]
A closed interval $I$ is ``boring'' if
[list]
[*]$I$ contains no ridiculous numbers, and
[*]There exists an interval $[b... | 1. **Identify the recursive definition of ridiculous numbers:**
- \(1\) is a ridiculous number.
- If \(a\) is a ridiculous number, then \(\sqrt{a}\) and \(1 + \sqrt{a}\) are also ridiculous numbers.
2. **Determine the least upper bound for ridiculous numbers:**
- Start with \(1\).
- Apply the recursive ru... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Since counting the numbers from 1 to 100 wasn't enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining $\zeta_{15} = e^{2\pi i/15}$ where $i = \sqrt{-1}$, the teacher wrote the 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k < 15$ on the board. Then, he to... | 1. **Define the problem and the initial setup:**
We are given 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k < 15$, where $\zeta_{15} = e^{2\pi i/15}$. We need to find the expected value of the last number remaining after repeatedly applying the operation $2ab - a - b + 1$.
2. **Understand the operation:**
... | 0 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be a polynomial with positive integer coefficients and degree 2015. Given that there exists some $\omega \in \mathbb{C}$ satisfying
$$\omega^{73} = 1\quad \text{and}$$
$$P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0,$$
what is the minimum possible value of $P... | 1. **Understanding the problem:**
We are given a polynomial \( P(x) \) with positive integer coefficients and degree 2015. We also know that there exists a complex number \( \omega \) such that \( \omega^{73} = 1 \). This means \( \omega \) is a 73rd root of unity. The problem states that the sum of the polynomial e... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n = 2^{2015} - 1$. For any integer $1 \le x < n$, let \[f_n(x) = \sum\limits_p s_p(n-x) + s_p(x) - s_p(n),\] where $s_q(k)$ denotes the sum of the digits of $k$ when written in base $q$ and the summation is over all primes $p$. Let $N$ be the number of values of $x$ such that $4 | f_n(x)$. What is the remainder wh... | 1. **Understanding the function \( f_n(x) \):**
\[
f_n(x) = \sum\limits_p s_p(n-x) + s_p(x) - s_p(n)
\]
where \( s_q(k) \) denotes the sum of the digits of \( k \) when written in base \( q \), and the summation is over all primes \( p \).
2. **Simplifying the expression modulo \( p-1 \):**
Note that \(... | 382 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$
[i]Note: A primitive root of $r$ of a prime $p$ is an integer $1 \le r < p$ such that the smallest positive integer $k$ such that $r^k \equiv 1... | 1. **Understanding the Problem:**
We need to find the number of positive integers \( k < 2015 \) for which \( f(k) \) is divisible by 73. Here, \( f(k) \) is the sum of the \( k \)-th powers of the primitive roots of 73.
2. **Primitive Roots and Orders:**
A primitive root \( r \) of a prime \( p \) is an intege... | 1841 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A word is an ordered, non-empty sequence of letters, such as $word$ or $wrod$. How many distinct $3$-letter words can be made from a subset of the letters $c, o, m, b, o$, where each letter in the list is used no more than the number of times it appears? | To determine the number of distinct 3-letter words that can be made from the subset of letters \( \{c, o, m, b, o\} \), we need to consider different cases based on the number of times the letter 'o' is used.
### Case 1: No 'o's are used
If no 'o's are used, we are left with the letters \( \{c, m, b\} \). We need to ... | 33 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Jonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ ... | 1. **Understanding the Problem:**
- Jonathan starts with 15 coins.
- He can make entries of 7, 8, or 9 coins.
- Each entry has a specific outcome:
- Entry of 7 coins: loses 4 coins (15 - 7 + 3 = 11 coins).
- Entry of 8 coins: gains 3 coins (15 - 8 + 11 = 18 coins).
- Entry of 9 coins: loses 5 co... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Princeton’s Math Club recently bought a stock for $\$2$ and sold it for $\$9$ thirteen days later. Given that the stock either increases or decreases by $\$1$ every day and never reached $\$0$, in how many possible ways could the stock have changed during those thirteen days?
| 1. **Determine the number of days the stock increases and decreases:**
- The stock starts at $2 and ends at $9, which is a net increase of $7.
- Since the stock either increases or decreases by $1 each day over 13 days, let \( x \) be the number of days the stock increases and \( y \) be the number of days it dec... | 273 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Andrew has $10$ balls in a bag, each a different color. He randomly picks a ball from the bag $4$ times, with replacement. The expected number of distinct colors among the balls he picks is $\tfrac{p}{q}$, where $\gcd(p, q) = 1$ and $p, q > 0$. What is $p + q$?
| 1. **Define the problem and the variables:**
- Andrew has 10 balls, each of a different color.
- He picks a ball 4 times with replacement.
- We need to find the expected number of distinct colors among the balls he picks.
2. **Calculate the probability that a specific color is not picked in one draw:**
- T... | 4439 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 2... | 1. **Calculate the probability that \(a_i - a_j > 266\):**
- Consider a random pair \((a_i, a_j)\). The total number of possible pairs \((a_i, a_j)\) is \(2015 \times 2014\) since \(a_i\) and \(a_j\) are distinct.
- We need to count the number of pairs \((a_i, a_j)\) such that \(a_i - a_j > 266\). For each \(a_j\... | 651222 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A number is [i]interesting [/i]if it is a $6$-digit integer that contains no zeros, its first $3$ digits are strictly increasing, and its last $3$ digits are non-increasing. What is the average of all interesting numbers? | 1. Let the 6-digit number be $\overline{abcdef}$, where each digit is non-zero, the first three digits $a, b, c$ are strictly increasing, and the last three digits $d, e, f$ are non-increasing.
2. We need to find the expected value of each digit. By symmetry, $\mathbb{E}(b) = \mathbb{E}(e) = 5$.
3. We also have the r... | 308253 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For her daughter’s $12\text{th}$ birthday, Ingrid decides to bake a dodecagon pie in celebration. Unfortunately, the store does not sell dodecagon shaped pie pans, so Ingrid bakes a circular pie first and then trims off the sides in a way such that she gets the largest regular dodecagon possible. If the original pie wa... | 1. **Determine the radius of the original circular pie:**
The diameter of the pie is given as \(8\) inches. Therefore, the radius \(r\) is:
\[
r = \frac{8}{2} = 4 \text{ inches}
\]
2. **Calculate the area of the original circular pie:**
The area \(A_{\text{circle}}\) of a circle is given by the formula:... | 64 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Terry the Tiger lives on a cube-shaped world with edge length $2$. Thus he walks on the outer surface. He is tied, with a leash of length $2$, to a post located at the center of one of the faces of the cube. The surface area of the region that Terry can roam on the cube can be represented as $\frac{p \pi}{q} + a\sqrt{b... | 1. **Understanding the Problem:**
Terry the Tiger is tied to a post at the center of one face of a cube with edge length 2. The leash length is also 2. We need to find the surface area of the region Terry can roam on the cube, expressed in the form \(\frac{p \pi}{q} + a\sqrt{b} + c\), and then determine \(p + q + a ... | 14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If $r$ can be expressed in the form $\frac{\sqrt{a}+b}{c}$ where $a, b, c$ are i... | 1. **Understanding the problem**: We need to find the largest radius \( r \) such that four balls of radius \( r \) can be packed into a regular tetrahedron with side length 1. The balls must be tangent to each other and to the faces of the tetrahedron.
2. **Circumradius of the tetrahedron**: The circumradius \( R \) ... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XY Z$ be the triangle formed by drawing the tangents to $\omega$ at $A, B, C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XY Z$ form an a... | 1. **Understanding the Problem:**
- We have a triangle \( \triangle ABC \) inscribed in a unit circle \( \omega \).
- \( H \) is the orthocenter of \( \triangle ABC \).
- \( D \) is the foot of the perpendicular from \( A \) to \( BC \).
- \( \triangle XYZ \) is formed by drawing tangents to \( \omega \) at... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has $\overline{AB} = \overline{AC} = 20$ and $\overline{BC} = 15$. Let $D$ be the point in $\triangle ABC$ such that $\triangle ADB \sim \triangle BDC$. Let $l$ be a line through $A$ and let $BD$ and $CD$ intersect $l$ at $P$ and $Q$, respectively. Let the circumcircles of $\triangle BDQ$ and $\triangle ... | 1. **Given Data and Initial Setup:**
- Triangle \(ABC\) with \(AB = AC = 20\) and \(BC = 15\).
- Point \(D\) in \(\triangle ABC\) such that \(\triangle ADB \sim \triangle BDC\).
- Line \(l\) through \(A\) intersects \(BD\) and \(CD\) at \(P\) and \(Q\), respectively.
- Circumcircles of \(\triangle BDQ\) and... | 1291 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EF GH$ be another regular tetrahedron such that the volume of $EF GH$ is $\tfrac{1}{8}\text{-th}$ the volume of $ABCD$. The height of $EF GH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\tfrac{a}{b}}$, where... | 1. **Volume of a Regular Tetrahedron:**
The volume \( V \) of a regular tetrahedron with side length \( s \) is given by:
\[
V = \frac{s^3}{6\sqrt{2}}
\]
For the tetrahedron \(ABCD\) with side length \( s = 1 \):
\[
V_{ABCD} = \frac{1^3}{6\sqrt{2}} = \frac{1}{6\sqrt{2}}
\]
2. **Volume of Tetrah... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the $22\text{nd}$ positive integer $n$ such that $22^n$ ends in a $2$? (when written in base $10$).
| 1. We need to find the $22\text{nd}$ positive integer $n$ such that $22^n$ ends in a $2$ when written in base $10$.
2. Note that $22^n \equiv 2^n \pmod{10}$. This is because $22 \equiv 2 \pmod{10}$, and thus raising both sides to the power $n$ gives $22^n \equiv 2^n \pmod{10}$.
3. We need $2^n \equiv 2 \pmod{10}$. This... | 85 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$?
| To solve the problem, we need to find all positive integers \( n \) such that \(\text{lcm}(2n, n^2) = 14n - 24\).
1. **Case 1: \( n \) is even**
If \( n \) is even, let \( n = 2k \) for some integer \( k \). Then:
\[
\text{lcm}(2n, n^2) = \text{lcm}(4k, 4k^2) = 4k^2
\]
We need:
\[
4k^2 = 14(2k) -... | 17 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest positive integer $n$ less than $10, 000$ such that in base $4$, $n$ and $3n$ have the same number of digits; in base $8$, $n$ and $7n$ have the same number of digits; and in base $16$, $n$ and $15n$ have the same number of digits? Express your answer in base $10$. | 1. **Base 4 Condition:**
- If \( n \) and \( 3n \) have the same number of digits in base 4, then:
\[
4^a \leq n < 4^{a+1}
\]
and
\[
4^a \leq 3n < 4^{a+1}
\]
Dividing the second inequality by 3, we get:
\[
\frac{4^a}{3} \leq n < \frac{4^{a+1}}{3}
\]
Combini... | 4369 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that $p \mid (a^{p-2} - a)$? | 1. **Identify the primes between 3 and 100:**
We are given that there are 24 primes between 3 and 100, inclusive. Let's denote these primes as \( p_1, p_2, \ldots, p_{24} \).
2. **Condition for ordered pairs \((p, a)\):**
We need to find the number of ordered pairs \((p, a)\) such that \( p \) is a prime between... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on th... | 1. **Determine the probabilities of the coin being weighted towards heads or tails:**
- The coin can be weighted towards heads or tails with equal probability, so:
\[
P(\text{Heads weighted}) = P(\text{Tails weighted}) = \frac{1}{2}
\]
2. **Calculate the probability of flipping two heads given the co... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Imagine a regular a $2015$-gon with edge length $2$. At each vertex, draw a unit circle centered at that vertex and color the circle’s circumference orange. Now, another unit circle $S$ is placed inside the polygon such that it is externally tangent to two adjacent circles centered at the vertices. This circle $S$ is a... | 1. **Understanding the Problem:**
- We have a regular $2015$-gon with edge length $2$.
- At each vertex, there is a unit circle centered at that vertex.
- Another unit circle $S$ is placed inside the polygon such that it is externally tangent to two adjacent vertex circles.
- Circle $S$ rolls freely inside ... | 2014 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the smallest positive integer $n$ such that $2^n - 1$ is a multiple of $2015$? | 1. **Identify the prime factorization of 2015:**
\[
2015 = 5 \times 13 \times 31
\]
We need to find the smallest positive integer \( n \) such that \( 2^n - 1 \) is divisible by 2015. This means \( 2^n \equiv 1 \pmod{2015} \). By the Chinese Remainder Theorem (CRT), it suffices to find \( n \) such that:
... | 60 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number n such that $\sigma_1(n) = 1854$?
| 1. We are given the function $\sigma_1 : \mathbb{N} \to \mathbb{N}$, which returns the sum of the positive integer divisors of $n$. We need to find the largest number $n$ such that $\sigma_1(n) = 1854$.
2. First, we factorize $1854$:
\[
1854 = 2 \times 927
\]
\[
927 = 3 \times 309
\]
\[
309 = 3... | 1234 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. What is
\[ \left\lfloor \frac{5^{2017015}}{5^{2015}+7} \right\rfloor \mod 1000?\] | 1. We start by simplifying the expression \(\left\lfloor \frac{5^{2017015}}{5^{2015} + 7} \right\rfloor\). Notice that \(5^{2017015}\) is a very large number, and \(5^{2015} + 7\) is relatively small in comparison. We can use the method of successive approximations to simplify the division.
2. First, we approximate:
... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The repeating decimal $2.0151515\ldots$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 1. We start by separating the decimal into two parts: \(2\) and \(0.0\overline{15}\).
2. Let \( x = 0.\overline{15} \). To convert this repeating decimal into a fraction, we multiply \( x \) by 100:
\[
100x = 15.\overline{15}
\]
Subtracting the original \( x \) from this equation, we get:
\[
100x - x... | 199 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer whose digits add to a multiple of 27 yet the number itself is not a multiple
of 27. For example, 87999921 is one such number.
| To find the least positive integer whose digits add to a multiple of 27 yet the number itself is not a multiple of 27, we need to follow these steps:
1. **Understand the divisibility rule for 27**:
A number is divisible by 27 if the sum of its digits is divisible by 27. This is because 27 is a factor of 999 (i.e., ... | 9990 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number o... | To solve this problem, we need to determine the number of different positive weights of chemicals that Gerome can measure using the given weights on a two-pan balance scale. The weights available are three 1-gram weights, three 5-gram weights, and three 50-gram weights.
1. **Understanding the Balance Scale:**
- On ... | 63 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Jamie, Linda, and Don bought bundles of roses at a flower shop, each paying the same price for each
bundle. Then Jamie, Linda, and Don took their bundles of roses to a fair where they tried selling their
bundles for a fixed price which was higher than the price that the flower shop charged. At the end of the
fair, Jamie, ... | 1. Let \( b \) be the price of one bundle when bought at the flower shop, and let \( s \) be the price of one bundle when sold at the fair.
2. Jamie bought 20 bundles and sold 15 of them. Her profit is given by:
\[
15s - 20b = 60
\]
3. Linda bought 34 bundles and sold 24 of them. Her profit is given by:
\[
... | 252 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in cen... | 1. **Identify the midpoints of \(A'C'\) and \(B'D'\):**
- The midpoint of \(A'C'\) is the average of the heights of \(A'\) and \(C'\).
- The midpoint of \(B'D'\) is the average of the heights of \(B'\) and \(D'\).
2. **Calculate the height of the midpoint of \(A'C'\):**
\[
\text{Height of midpoint of } A'C... | 14 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The diagram below shows an $8$x$7$ rectangle with a 3-4-5 right triangle drawn in each corner. The lower two triangles have their sides of length 4 along the bottom edge of the rectangle, while the upper two
triangles have their sides of length 3 along the top edge of the rectangle. A circle is tangent to the hypotenus... | 1. **Identify the problem and given information:**
- We have an $8 \times 7$ rectangle with a 3-4-5 right triangle drawn in each corner.
- The lower two triangles have their sides of length 4 along the bottom edge of the rectangle.
- The upper two triangles have their sides of length 3 along the top edge of th... | 47 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The five numbers $17$, $98$, $39$, $54$, and $n$ have a mean equal to $n$. Find $n$. | 1. We are given five numbers: $17$, $98$, $39$, $54$, and $n$. The mean of these numbers is equal to $n$.
2. The mean of a set of numbers is calculated by dividing the sum of the numbers by the number of elements in the set. Therefore, we have:
\[
\frac{17 + 98 + 39 + 54 + n}{5} = n
\]
3. Simplify the sum in t... | 52 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The two diagonals of a quadrilateral have lengths $12$ and $9$, and the two diagonals are perpendicular to each other. Find the area of the quadrilateral. | 1. Given that the diagonals of the quadrilateral are perpendicular to each other, we can use the property that the area of such a quadrilateral is given by half the product of the lengths of the diagonals. This is because the diagonals divide the quadrilateral into four right triangles.
2. Let the lengths of the diago... | 54 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all the real values of x satisfying $(x+\frac{1}{x}-17)^2$ $= x + \frac{1}{x} + 17.$ | 1. Let \( x + \frac{1}{x} = y \). The given equation becomes:
\[
(y - 17)^2 = y + 17
\]
2. Expand and simplify the equation:
\[
(y - 17)^2 = y + 17 \implies y^2 - 34y + 289 = y + 17
\]
\[
y^2 - 34y + 289 - y - 17 = 0
\]
\[
y^2 - 35y + 272 = 0
\]
3. Solve the quadratic equation \( y... | 35 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The product $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$ can be expressed in the form $m$ $\cdot$ $n^3$, where m and n are positive integers, and m is not divisible by the cube of any prime. Find m. | 1. We start with the product \( 20! \cdot 21! \cdot 22! \cdot 23! \cdot 24! \cdot 25! \cdot 26! \cdot 27! \cdot 28! \).
2. We aim to express this product in the form \( m \cdot n^3 \), where \( m \) is not divisible by the cube of any prime.
3. First, we regroup the factorials to identify cubes:
\[
(20! \cdot 21!... | 825 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The diagram below shows a parallelogram ABCD with $AB = 36$ and $AD = 60$. Diagonal BD is
perpendicular to side AB. Points E and F bisect sides AD and BC, respectively. Points G and H are the
intersections of BD with AF and CE, respectively. Find the area of quadrilateral EGFH The diagram below shows a parallelogram AB... | 1. **Identify the given information and draw the diagram:**
- Parallelogram \(ABCD\) with \(AB = 36\) and \(AD = 60\).
- Diagonal \(BD\) is perpendicular to side \(AB\).
- Points \(E\) and \(F\) bisect sides \(AD\) and \(BC\), respectively.
- Points \(G\) and \(H\) are the intersections of \(BD\) with \(AF\... | 288 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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