problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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Richard and Shreyas are arm wrestling against each other. They will play $10$ rounds, and in each round, there is exactly one winner. If the same person wins in consecutive rounds, these rounds are considered part of the same “streak”. How many possible outcomes are there in which there are strictly more than $3$ strea... | To solve this problem, we need to count the number of possible outcomes in which there are strictly more than 3 streaks in 10 rounds of arm wrestling. We will use the complement principle to simplify our calculations.
1. **Total Possible Outcomes**:
Each round can be won by either Richard (R) or Shreyas (S). There... | 932 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$ | We are given a positive integer \( n \) such that \( n = s(n)^2 + 2s(n) - 2 \), where \( s(n) \) denotes the sum of the digits of \( n \). We need to find the largest positive integer \( n \) that satisfies this equation.
1. **Define the equation and constraints:**
\[
n = S^2 + 2S - 2
\]
where \( S = s(n) ... | 397 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sohom constructs a square $BERK$ of side length $10$. Darlnim adds points $T$, $O$, $W$, and $N$, which are the midpoints of $\overline{BE}$, $\overline{ER}$, $\overline{RK}$, and $\overline{KB}$, respectively. Lastly, Sylvia constructs square $CALI$ whose edges contain the vertices of $BERK$, such that $\overline{CA}$... | 1. **Identify the problem and given data:**
- We have a square \(BERK\) with side length \(10\).
- Points \(T\), \(O\), \(W\), and \(N\) are the midpoints of \(\overline{BE}\), \(\overline{ER}\), \(\overline{RK}\), and \(\overline{KB}\), respectively.
- A new square \(CALI\) is constructed such that its edges ... | 180 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A $4$-dimensional hypercube of edge length $1$ is constructed in $4$-space with its edges parallel to
the coordinate axes and one vertex at the origin. The coordinates of its sixteen vertices are given
by $(a, b, c, d)$, where each of $a, b, c,$ and $d$ is either $0$ or $1$. The $3$-dimensional hyperplane given
by $x +... | 1. **Identify the vertices of the hypercube:**
The vertices of a 4-dimensional hypercube (or tesseract) with edge length 1 and one vertex at the origin are given by all possible combinations of coordinates \((a, b, c, d)\) where \(a, b, c,\) and \(d\) are either 0 or 1. This gives us 16 vertices.
2. **Determine the... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$ is a real number such that $3a + 6$ is the greatest integer less than or equal to $a$ and $4a + 9$ is the least integer greater than or equal to $a$. Compute $a$. | Given the problem, we need to find the real number \( a \) such that:
\[ 3a + 6 \leq a < 4a + 9 \]
We can rewrite the problem using the floor and ceiling functions:
\[ \begin{cases}
3a + 6 = \lfloor a \rfloor \\
4a + 9 = \lceil a \rceil
\end{cases} \]
### Case 1: \( a \in \mathbb{Z} \)
If \( a \) is an integer, the... | -3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
In triangles ABC and DEF, DE = 4AB, EF = 4BC, and F D = 4CA. The area of
DEF is 360 units more than the area of ABC. Compute the area of ABC. | 1. Given that in triangles \( \triangle ABC \) and \( \triangle DEF \), the sides of \( \triangle DEF \) are 4 times the corresponding sides of \( \triangle ABC \):
\[
DE = 4AB, \quad EF = 4BC, \quad FD = 4CA
\]
2. Since the sides of \( \triangle DEF \) are 4 times the sides of \( \triangle ABC \), the rat... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $[n] = \{1, 2, 3, ... ,n\}$ and for any set $S$, let$ P(S)$ be the set of non-empty subsets of $S$. What is the last digit of $|P(P([2013]))|$? | 1. Define the set \( [n] = \{1, 2, 3, \ldots, n\} \). For any set \( S \), let \( P(S) \) be the set of non-empty subsets of \( S \). We need to find the last digit of \( |P(P([2013]))| \).
2. To avoid confusion with the power set notation, we will use \( Q(S) \) to denote the set of non-empty subsets of \( S \). If \... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, let $\sigma (n)$ be the sum of the divisors of $n$ (for example $\sigma (10) = 1 + 2 + 5 + 10 = 18$). For how many $n \in \{1, 2,. .., 100\}$, do we have $\sigma (n) < n+ \sqrt{n}$? | 1. **Understanding the problem**: We need to find how many integers \( n \) in the set \(\{1, 2, \ldots, 100\}\) satisfy the condition \(\sigma(n) < n + \sqrt{n}\), where \(\sigma(n)\) is the sum of the divisors of \(n\).
2. **Analyzing the condition for composite numbers**: Assume \( n > 1 \) and \( n \) is not a pri... | 26 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The expression below has six empty boxes. Each box is to be fi lled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved?
$$\dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\squa... | To find the maximum possible final result of the given expression, we need to evaluate different combinations of the numbers \(1, 2, 3, 4, 5, 6\) in the expression:
\[ \dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\square}{\square}} \]
We need to maximize the expression by strategically placing the nu... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can you rearrange the letters of ‘Alejandro’ such that it contains one of the words ‘ned’ or ‘den’? | 1. Let \( X \) be the block that is either "NED" or "DEN". There are 2 ways to choose what \( X \) is.
2. The word "Alejandro" has 9 letters, but we are treating "NED" or "DEN" as a single block, so we have 7 other letters: \( A, A, L, J, R, O, X \).
3. We need to arrange these 7 letters and the block \( X \). Since th... | 40320 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Compute
$$\sqrt{(\sqrt{63} +\sqrt{112} +\sqrt{175})(-\sqrt{63} +\sqrt{112} +\sqrt{175})(\sqrt{63}-\sqrt{112} +\sqrt{175})(\sqrt{63} +\sqrt{112} -\sqrt{175})}$$
[b]p2.[/b] Consider the set $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many distinct $3$-element subsets are there such that the sum of the elemen... | To solve this problem, we need to count the number of distinct 3-element subsets of the set \( S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \) such that the sum of the elements in each subset is divisible by 3.
1. **Identify the elements modulo 3:**
- The elements of \( S \) modulo 3 are:
\[
\{0, 1, 2, 0, 1, 2, 0... | 42 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Suppose the transformation $T$ acts on points in the plane like this:
$$T(x, y) = \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2 + y^2}\right).$$
Determine the area enclosed by the set of points of the form $T(x, y)$, where $(x, y)$ is a point on the edge of a length-$2$ square centered at the origin with sides parallel to... | 1. **Understanding the Transformation:**
The given transformation \( T \) is defined as:
\[
T(x, y) = \left( \frac{x}{x^2 + y^2}, \frac{-y}{x^2 + y^2} \right).
\]
This transformation can be interpreted in terms of complex numbers. If we let \( z = x + iy \), then the transformation \( T \) can be written... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Two players play a game on a pile of $n$ beans. On each player's turn, they may take exactly $1$, $4$, or $7$ beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many $n$ between $2014$ and $2050$ (inclusive) ... | 1. **Understanding the Game Dynamics:**
- The game involves two players taking turns to remove 1, 4, or 7 beans from a pile.
- The player who takes the last bean wins.
- We need to determine for which values of \( n \) (the initial number of beans) the second player has a winning strategy.
2. **Analyzing Winn... | 14 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) -f(n - 1)$. For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even? | 1. **Define the function \( f(n) \):**
\[
f(n) = \sum_{d=1}^n \left\lfloor \frac{n}{d} \right\rfloor
\]
This function sums the floor of the division of \( n \) by each integer from 1 to \( n \).
2. **Express \( f(n) \) in terms of the number of divisors:**
We claim that:
\[
f(n) = \sum_{k=1}^n \ta... | 90 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] The following number is the product of the divisors of $n$.
$$2^63^3$$
What is $n$?
[b]p2.[/b] Let a right triangle have the sides $AB =\sqrt3$, $BC =\sqrt2$, and $CA = 1$. Let $D$ be a point such that $AD = BD = 1$. Let $E$ be the point on line $BD$ that is equidistant from $D$ and $A$. Find the angle $\a... | To find the number of positive integers \( n \) satisfying \( \lfloor n / 2014 \rfloor = \lfloor n / 2016 \rfloor \), we start by setting \( m = \lfloor n / 2014 \rfloor = \lfloor n / 2016 \rfloor \).
1. **Determine the range for \( n \) given \( m \):**
\[
m \leq \frac{n}{2014} < m+1 \quad \text{and} \quad m \l... | 1015056 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$? | 1. Since \( p+q+r+s \) is prime and \( p, q, r, s \) are distinct primes, one of \( p, q, r, s \) must be equal to \( 2 \). If this were not the case, then \( p+q+r+s \) would be even and greater than \( 2 \), hence not prime. Therefore, we can assume without loss of generality that \( p = 2 \).
2. Given \( p = 2 \), ... | 23 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x)$ be the polynomial of degree at most $6$ which satisfies $P(k)=k!$ for $k=0,1,2,3,4,5,6$. Compute the value of $P(7)$. | To solve the problem, we will use the method of finite differences to find the value of \( P(7) \). The polynomial \( P(x) \) is of degree at most 6 and satisfies \( P(k) = k! \) for \( k = 0, 1, 2, 3, 4, 5, 6 \).
1. **List the values of \( P(k) \) for \( k = 0, 1, 2, 3, 4, 5, 6 \):**
\[
\begin{array}{c|c}
k ... | 3186 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has side lengths $AB=18$, $BC=36$, and $CA=24$. The circle $\Gamma$ passes through point $C$ and is tangent to segment $AB$ at point $A$.
Let $X$, distinct from $C$, be the second intersection of $\Gamma$ with $BC$. Moreover, let $Y$ be the point on $\Gamma$ such that segment $AY$ is an angle bisector o... | 1. **Power of a Point**: By the power of a point theorem, we have:
\[
AB^2 = BX \cdot BC
\]
Substituting the given values:
\[
18^2 = BX \cdot 36 \implies 324 = BX \cdot 36 \implies BX = \frac{324}{36} = 9
\]
2. **Angle Relationships**: Since $\Gamma$ is tangent to $AB$ at $A$, $\angle BAX \cong \a... | 69 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\alpha$ be the unique real root of the polynomial $x^3-2x^2+x-1$. It is known that $1<\alpha<2$. We define the sequence of polynomials $\left\{{p_n(x)}\right\}_{n\ge0}$ by taking $p_0(x)=x$ and setting
\begin{align*}
p_{n+1}(x)=(p_n(x))^2-\alpha
\end{align*}
How many distinct real roots does $p_{10}(x)$ have? | 1. **Define the sequence of polynomials**:
We start with \( p_0(x) = x \) and the recursive relation:
\[
p_{n+1}(x) = (p_n(x))^2 - \alpha
\]
2. **Analyze the polynomial \( p_1(x) \)**:
\[
p_1(x) = (p_0(x))^2 - \alpha = x^2 - \alpha
\]
3. **Analyze the polynomial \( p_2(x) \)**:
\[
p_2(x) = ... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Anita plays the following single-player game: She is given a circle in the plane. The center of this circle and some point on the circle are designated “known points”. Now she makes a series of moves, each of which takes one of the following forms:
(i) She draws a line (infinite in both directions) between two “known p... | 1. **Initial Setup**: We start with a circle centered at point \( O \) and a known point \( P \) on the circle.
2. **First Move**: Draw a line through the known point \( P \) and the center \( O \). This line will intersect the circle at another point, which we will call \( X \). Now, \( X \) is a known point.
3. **S... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Two robots race on the plane from $(0, 0)$ to $(a, b)$, where $a$ and $b$ are positive real numbers with $a < b$. The robots move at the same constant speed. However, the first robot can only travel in directions parallel to the lines $x = 0$ or $y = 0$, while the second robot can only travel in directions p... | To solve this problem, we need to use the extended law of sines and some properties of triangles. Let's go through the steps in detail.
1. **Using the Extended Law of Sines:**
The extended law of sines states that for any triangle \( \triangle ABC \) with circumradius \( R \):
\[
a = 2R \sin A, \quad b = 2R \... | 72 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\vartriangle ABC$ be a triangle such that the area$ [ABC] = 10$ and $\tan (\angle ABC) = 5$. If the smallest possible value of $(\overline{AC})^2$ can be expressed as $-a + b\sqrt{c}$ for positive integers $a, b, c$, what is $a + b + c$?
| 1. Given that the area of triangle $\triangle ABC$ is $10$ and $\tan (\angle ABC) = 5$, we need to find the smallest possible value of $(\overline{AC})^2$ expressed in the form $-a + b\sqrt{c}$ and determine $a + b + c$.
2. Since $\tan (\angle ABC) = 5$, we can find $\sin (\angle ABC)$ and $\cos (\angle ABC)$ using th... | 42 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, ... | 1. **Identify the sides of the right triangle:**
Given a right triangle \(ABC\) with side lengths \(3\) and \(4\), we can determine the hypotenuse using the Pythagorean theorem:
\[
c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Therefore, the hypotenuse of the triangle is \(5\).
2. **Determine ... | 60 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Caltech's 900 students are evenly spaced along the circumference of a circle. How many equilateral triangles can be formed with at least two Caltech students as vertices? | 1. **Choosing 2 out of 900 students:**
We start by choosing 2 out of the 900 students to form the base of the equilateral triangle. The number of ways to choose 2 students from 900 is given by the binomial coefficient:
\[
\binom{900}{2} = \frac{900 \times 899}{2} = 404550
\]
2. **Considering the orientatio... | 808800 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]Beaver-number[/i] is a positive 5 digit integer whose digit sum is divisible by 17. Call a pair of [i]Beaver-numbers[/i] differing by exactly $1$ a [i]Beaver-pair[/i]. The smaller number in a [i]Beaver-pair[/i] is called an [i]MIT Beaver[/i], while the larger number is called a [i]CIT Beaver[/i]. Find the positive... | 1. **Understanding the Problem:**
- A *Beaver-number* is a 5-digit integer whose digit sum is divisible by 17.
- A *Beaver-pair* consists of two consecutive *Beaver-numbers*.
- The smaller number in a *Beaver-pair* is called an *MIT Beaver*, and the larger number is called a *CIT Beaver*.
- We need to find ... | 79200 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that
\[ zP(y) = P(y - n) + P(y + n) \]
for all reals $n$. Evaluate $P(y)$. | 1. Given the polynomial \( P(x) = x^3 - 6x^2 - 5x + 4 \), we need to evaluate \( P(y) \) under the condition:
\[
zP(y) = P(y - n) + P(y + n)
\]
for all real numbers \( n \).
2. First, let \( n = 0 \):
\[
zP(y) = P(y) + P(y)
\]
This simplifies to:
\[
zP(y) = 2P(y)
\]
Since \( P(y) \n... | -22 | Algebra | other | Yes | Yes | aops_forum | false |
Let $S$ be the sum of all positive integers $n$ such that $\frac{3}{5}$ of the positive divisors of $n$ are multiples of $6$ and $n$ has no prime divisors greater than $3$. Compute $\frac{S}{36}$. | 1. **Form of \( n \)**:
Since \( n \) has no prime divisors greater than 3, \( n \) must be of the form \( 2^x 3^y \) where \( x \) and \( y \) are non-negative integers.
2. **Divisors of \( n \)**:
The total number of divisors of \( n \) is given by \((x+1)(y+1)\). The divisors of \( n \) that are multiples of ... | 2345 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define
\[
S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1).
\]
Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 1. We start by considering the given expression for \( S \):
\[
S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1).
\]
2. We use the fact that the arctangent of a complex number can be expressed as the argument of that complex number. Specifically, for a complex number \( z = x + yi \), we have:... | 2023 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $n \ge 3$ be a positive integer. Suppose that $\Gamma$ is a unit circle passing through a point $A$. A regular $3$-gon, regular $4$-gon, \dots, regular $n$-gon are all inscribed inside $\Gamma$ such that $A$ is a common vertex of all these regular polygons. Let $Q$ be a point on $\Gamma$ such that $Q$ is a vertex ... | 1. **Convert to Complex Coordinates**: Place point \( A \) at \( 1 \) in the complex plane. The unit circle \(\Gamma\) is then represented by the set of complex numbers \( z \) such that \( |z| = 1 \).
2. **Primitive Roots of Unity**: The vertices of a regular \( n \)-gon inscribed in \(\Gamma\) are the \( n \)-th roo... | 63 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A unit circle is centered at $(0, 0)$ on the $(x, y)$ plane. A regular hexagon passing through $(1, 0)$ is inscribed in the circle. Two points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon into at most three pieces. The probability that each piec... | 1. **Understanding the Problem:**
- We have a unit circle centered at \((0, 0)\) on the \((x, y)\) plane.
- A regular hexagon is inscribed in this circle, passing through the point \((1, 0)\).
- We need to find the probability that two randomly selected points from the interior of the circle, when horizontal l... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$. | To find the smallest positive integer \( k \) such that there is exactly one prime number of the form \( kx + 60 \) for the integers \( 0 \le x \le 10 \), we need to analyze the expression \( kx + 60 \) for each \( x \) in the given range.
1. **Identify the form of the expression:**
\[
kx + 60
\]
We need t... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \le 10^{10}$ that satisfy the equation
\[
S(n) = (S(S(n)))^2.
\]
Find the remainder when $K$ is divided by $1000$. | 1. **Identify the possible values of \( S(n) \):**
- We need to find all nonnegative integers \( k \) such that \( S(n) = k \) and \( k = (S(S(n)))^2 \).
- Since \( S(n) \) is the sum of the digits of \( n \), the maximum value \( S(n) \) can take for \( n \leq 10^{10} \) is \( 9 \times 10 = 90 \).
2. **Solve th... | 632 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}} = \left(\frac{p}{q}\right)^{i \pi},
\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
[i]Note: for a complex number $z = re^{i \thet... | 1. We start with the given product:
\[
\prod_{n=1}^{\infty}\left(\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-i\cot\left(\frac{n\pi}{2n+1}\right)}\right)^{\frac{1}{n}}
\]
We need to simplify the term inside the product.
2. Consider the expression:
\[
\frac{1+i\cot\left(\frac{n\pi}{2n+1}\right)}{1-... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has circumcircle $\Omega$. Chord $XY$ of $\Omega$ intersects segment $AC$ at point $E$ and segment $AB$ at point $F$ such that $E$ lies between $X$ and $F$. Suppose that $A$ bisects arc $\widehat{XY}$. Given that $EC = 7, FB = 10, AF = 8$, and $YF - XE = 2$, find the perimeter of triangle $ABC$. | 1. **Define Variables and Given Information:**
- Let \( YF = x + 2 \), \( XE = x \), and \( EF = y \).
- Let \( AX = AY = z \).
- Given: \( EC = 7 \), \( FB = 10 \), \( AF = 8 \), and \( YF - XE = 2 \).
2. **Apply Stewart's Theorem:**
- Stewart's Theorem states that for a triangle \( \triangle ABC \) with ... | 51 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For two base-10 positive integers $a$ and $b$, we say $a \sim b$ if we can rearrange the digits of $a$ in some way to obtain $b$, where the leading digit of both $a$ and $b$ is nonzero. For instance, $463 \sim 463$ and $634 \sim 463$. Find the number of $11$-digit positive integers $K$ such that $K$ is divisible by $2$... | To solve this problem, we need to find the number of 11-digit positive integers \( K \) that satisfy the following conditions:
1. \( K \) is divisible by \( 2 \), \( 3 \), and \( 5 \).
2. There exists some positive integer \( K' \) such that \( K' \sim K \) and \( K' \) is divisible by \( 7 \), \( 11 \), \( 13 \), \( 1... | 3628800 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similar... | To solve this problem, we need to determine the probability that exactly 3 of the 5 trapezoidal pieces contain both dark and white chocolate. We will use combinatorial methods and probability calculations to achieve this.
1. **Understanding the Problem:**
- We have a rectangular chocolate bar divided into two halve... | 39 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$, point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and ... | 1. **Coordinate Setup**:
Let's place the right triangle \( \triangle ABC \) in the coordinate plane with \( B \) at the origin \((0,0)\), \( A \) on the \( y \)-axis, and \( C \) on the \( x \)-axis. Thus, \( A = (0, b) \) and \( C = (c, 0) \).
2. **Finding Coordinates of Points**:
- Since \( AE = 9 \) and \( E... | 18 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The residents of the local zoo are either rabbits or foxes. The ratio of foxes to rabbits in the zoo is $2:3$. After $10$ of the foxes move out of town and half the rabbits move to Rabbitretreat, the ratio of foxes to rabbits is $13:10$. How many animals are left in the zoo? | 1. Let the number of foxes be \(2x\) and the number of rabbits be \(3x\). This is based on the given ratio of foxes to rabbits, which is \(2:3\).
2. After 10 foxes move out, the number of foxes becomes \(2x - 10\).
3. After half of the rabbits move out, the number of rabbits becomes \(\frac{3x}{2}\).
4. The new rati... | 690 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $P(x)$ is a quadratic polynomial with integer coefficients satisfying the identity \[P(P(x)) - P(x)^2 = x^2+x+2016\] for all real $x$. What is $P(1)$? | 1. Let \( P(x) = ax^2 + bx + c \). Given the identity:
\[
P(P(x)) - P(x)^2 = x^2 + x + 2016
\]
we need to substitute \( P(x) \) into the equation and simplify.
2. First, compute \( P(P(x)) \):
\[
P(P(x)) = P(ax^2 + bx + c) = a(ax^2 + bx + c)^2 + b(ax^2 + bx + c) + c
\]
3. Next, compute \( P(x)^2 ... | 1010 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $P$ is a quintic polynomial with real coefficients with $P(0)=2$ and $P(1)=3$ such that $|z|=1$ whenever $z$ is a complex number satisfying $P(z) = 0$. What is the smallest possible value of $P(2)$ over all such polynomials $P$? | 1. **Identify the roots and form of the polynomial:**
Given that \(P\) is a quintic polynomial with real coefficients and that all roots \(z\) of \(P(z) = 0\) satisfy \(|z| = 1\), we can write the roots as \(-1, a, \overline{a}, b, \overline{b}\), where \(a\) and \(b\) are complex numbers on the unit circle (i.e., \... | 54 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence $\{a_{n}\}_{n=1}^{\infty}$ via $a_{1} = 1$ and $a_{n+1} = a_{n} + \lfloor \sqrt{a_{n}} \rfloor$ for all $n \geq 1$. What is the smallest $N$ such that $a_{N} > 2017$? | To solve the problem, we need to find the smallest \( N \) such that \( a_N > 2017 \). We start by analyzing the sequence defined by \( a_1 = 1 \) and \( a_{n+1} = a_n + \lfloor \sqrt{a_n} \rfloor \).
1. **Initial Terms Calculation:**
- \( a_1 = 1 \)
- \( a_2 = a_1 + \lfloor \sqrt{a_1} \rfloor = 1 + \lfloor 1 \r... | 45 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Robert colors each square in an empty 3 by 3 grid either red or green. Find the number of colorings such that no row or column contains more than one green square. | 1. **Case 1: 0 green squares**
If there are no green squares, then all squares are red. There is only one way to color the grid in this case.
\[
\text{Number of ways} = 1
\]
2. **Case 2: 1 green square**
We need to place one green square in the grid. We can choose any of the 3 rows and any of the 3 co... | 34 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$. | 1. **Identify the problem constraints**: We need to find the maximum subset \( S \) of \(\{1, 2, \ldots, 2017\}\) such that for any two distinct elements in \( S \), both their sum and product are not divisible by 7.
2. **Analyze the product condition**: The product of two numbers is divisible by 7 if at least one of ... | 865 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that
\[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\]
Eval... | 1. **Understanding the problem**: We need to find the maximum and minimum number of planes required to divide $\mathbb{R}^3$ such that no region contains more than one point from a set $S$ of 200 points. We denote these values as $M(200)$ and $m(200)$ respectively, and we need to evaluate $M(200) \cdot m(200)$.
2. **M... | 2189 | Combinatorics | other | Yes | Yes | aops_forum | false |
What is the minimum number of times you have to take your pencil off the paper to draw the following figure (the dots are for decoration)? You must lift your pencil off the paper after you're done, and this is included in the number of times you take your pencil off the paper. You're not allowed to draw over an edge tw... | To solve this problem, we need to determine the minimum number of times we have to lift the pencil to draw the given figure. This involves understanding the properties of the graph formed by the figure.
1. **Identify the vertices and their degrees**:
- First, we need to identify all the vertices in the figure and c... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In acute triangle $ABC$, points $D$ and $E$ are the feet of the angle bisector and altitude from $A$ respectively. Suppose that $AC - AB = 36$ and $DC - DB = 24$. Compute $EC - EB$. | 1. Let \( AB = x \) and \( DB = y \). We know that \( AC = x + 36 \) and \( DC = y + 24 \).
2. From the Angle Bisector Theorem, we have:
\[
\frac{AC}{AB} = \frac{DC}{DB}
\]
Substituting the given values, we get:
\[
\frac{x + 36}{x} = \frac{y + 24}{y}
\]
Cross-multiplying, we obtain:
\[
y(... | 54 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The parabola $\mathcal P$ given by equation $y=x^2$ is rotated some acute angle $\theta$ clockwise about the origin such that it hits both the $x$ and $y$ axes at two distinct points. Suppose the length of the segment $\mathcal P$ cuts the $x$-axis is $1$. What is the length of the segment $\mathcal P$ cuts the $y$-a... | 1. **Rotation and Intersection Points:**
- The parabola \( y = x^2 \) is rotated by an acute angle \(\theta\) about the origin.
- After rotation, the parabola intersects the \(x\)-axis at two distinct points. Let these points be \((a, 0)\) and \((b, 0)\) with \(a < b\).
- Given that the length of the segment o... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Jesse has ten squares, which are labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors? | To solve this problem, we need to ensure that for all \(1 \le i < j \le 10\), if \(i\) divides \(j\), then the \(i\)-th and \(j\)-th squares have different colors. We will use a step-by-step approach to count the number of valid colorings.
1. **Color the 1st square:**
- There are 4 possible colors for the 1st squar... | 324 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n=2017$ and $x_1,\dots,x_n$ be boolean variables. An \emph{$7$-CNF clause} is an expression of the form $\phi_1(x_{i_1})+\dots+\phi_7(x_{i_7})$, where $\phi_1,\dots,\phi_7$ are each either the function $f(x)=x$ or $f(x)=1-x$, and $i_1,i_2,\dots,i_7\in\{1,2,\dots,n\}$. For example, $x_1+(1-x_1)+(1-x_3)+x_2+x_4+(1-... | 1. We are given \( n = 2017 \) and \( x_1, \dots, x_n \) as boolean variables. A \( 7 \)-CNF clause is an expression of the form \( \phi_1(x_{i_1}) + \dots + \phi_7(x_{i_7}) \), where each \( \phi_j \) is either the identity function \( f(x) = x \) or the negation function \( f(x) = 1 - x \), and \( i_1, i_2, \dots, i_... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be an isosceles trapezoid with $AD\parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP\perp CD$ and $BQ\perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX=3$ and $XQ=1$. Compute the largest possible area o... | 1. **Identify the given information and setup the problem:**
- $ABCD$ is an isosceles trapezoid with $AD \parallel BC$.
- Points $P$ and $Q$ are on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$.
- Point $X$ is the intersection of these two altitudes.
- G... | 32 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $... | 1. **Scaling Down the Triangle:**
We start by scaling down the triangle \( \triangle ABC \) with sides \( AB = 104 \), \( BC = 112 \), and \( CA = 120 \) to a \( 13-14-15 \) triangle. This is done by dividing each side by 8:
\[
AB = \frac{104}{8} = 13, \quad BC = \frac{112}{8} = 14, \quad CA = \frac{120}{8} = ... | 49 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Say an integer polynomial is $\textit{primitive}$ if the greatest common divisor of its coefficients is $1$. For example, $2x^2+3x+6$ is primitive because $\gcd(2,3,6)=1$. Let $f(x)=a_2x^2+a_1x+a_0$ and $g(x) = b_2x^2+b_1x+b_0$, with $a_i,b_i\in\{1,2,3,4,5\}$ for $i=0,1,2$. If $N$ is the number of pairs of polynomia... | 1. **Understanding the Problem:**
We need to find the number of pairs of polynomials \((f(x), g(x))\) such that their product \(h(x) = f(x)g(x)\) is primitive. A polynomial is primitive if the greatest common divisor (gcd) of its coefficients is 1.
2. **Primitive Polynomial Property:**
The product of two primiti... | 689 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For how many triples of positive integers $(a,b,c)$ with $1\leq a,b,c\leq 5$ is the quantity \[(a+b)(a+c)(b+c)\] not divisible by $4$? | To determine the number of triples \((a, b, c)\) such that \((a+b)(a+c)(b+c)\) is not divisible by 4, we need to analyze the conditions under which this product is not divisible by 4.
1. **Understanding the divisibility by 4**:
- For a product of three numbers to be divisible by 4, at least one of the numbers must ... | 117 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
One can define the greatest common divisor of two positive rational numbers as follows: for $a$, $b$, $c$, and $d$ positive integers with $\gcd(a,b)=\gcd(c,d)=1$, write \[\gcd\left(\dfrac ab,\dfrac cd\right) = \dfrac{\gcd(ad,bc)}{bd}.\] For all positive integers $K$, let $f(K)$ denote the number of ordered pairs of pos... | 1. We start by understanding the given definition of the greatest common divisor (gcd) for two positive rational numbers. For positive integers \(a, b, c, d\) with \(\gcd(a, b) = \gcd(c, d) = 1\), the gcd of two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\) is given by:
\[
\gcd\left(\frac{a}{b}, \frac{c}{d... | 2880 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
[list]
[*] $D(1) = 0$;
[*] $D(p)=1$ for all primes $p$;
[*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$.
[/list]
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$. | 1. **Understanding the arithmetic derivative:**
- The arithmetic derivative \( D(n) \) is defined by:
- \( D(1) = 0 \)
- \( D(p) = 1 \) for all primes \( p \)
- \( D(ab) = D(a)b + aD(b) \) for all positive integers \( a \) and \( b \)
2. **Applying the product rule:**
- For \( n = p_1^{a_1} p_2^{a... | 31 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of ordered triples $(a,b,c) \in \{1, \ldots, 2016\}^{3}$ such that $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$. What are the last three digits of $N$? | 1. **Understanding the Problem:**
We need to find the number of ordered triples \((a, b, c)\) such that \(a^2 + b^2 + c^2 \equiv 0 \pmod{2017}\). Here, \(a, b, c\) are elements of the set \(\{1, 2, \ldots, 2016\}\).
2. **Simplifying the Problem:**
Since \(2017\) is a prime number, we can use properties of quadra... | 000 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest prime $p$ for which there exist positive integers $a,b$ such that
\[
a^{2} + p^{3} = b^{4}.
\] | 1. We start with the equation \(a^2 + p^3 = b^4\). We can rewrite this as:
\[
p^3 = b^4 - a^2 = (b^2 + a)(b^2 - a)
\]
Since \(p\) is a prime number, the factors \(b^2 + a\) and \(b^2 - a\) must be powers of \(p\).
2. We consider two cases:
- Case 1: \(b^2 + a = p^2\) and \(b^2 - a = p\)
- Case 2: \(b... | 23 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For each positive integer $n$, define \[g(n) = \gcd\left\{0! n!, 1! (n-1)!, 2 (n-2)!, \ldots, k!(n-k)!, \ldots, n! 0!\right\}.\] Find the sum of all $n \leq 25$ for which $g(n) = g(n+1)$. | 1. **Understanding the function \( g(n) \)**:
The function \( g(n) \) is defined as the greatest common divisor (gcd) of the set:
\[
\{0! n!, 1! (n-1)!, 2! (n-2)!, \ldots, k! (n-k)!, \ldots, n! 0!\}
\]
This set contains \( n+1 \) terms, each of the form \( k! (n-k)! \) for \( k = 0, 1, 2, \ldots, n \).
... | 82 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the integer $n$ such that
\[n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017.\] Here, as usual, $\lfloor\cdot\rfloor$ denotes the floor function. | 1. We start with the equation:
\[
n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017
\]
2. Let \( k = \left\lfloor \sqrt{n} \right\rfloor \). Then, \( k \leq \sqrt{n} < k+1 \), which implies:
\[
k^2 \leq n < (k+1)^2
\]
3. Similarly, let \( m = \left\lfloor \sqrt{... | 1967 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Say an odd positive integer $n > 1$ is $\textit{twinning}$ if $p - 2 \mid n$ for every prime $p \mid n$. Find the number of twinning integers less than 250. | 1. **Understanding the Problem:**
We need to find the number of odd positive integers \( n \) less than 250 such that for every prime \( p \) dividing \( n \), \( p - 2 \) also divides \( n \). These integers are called twinning integers.
2. **Analyzing the Condition:**
For \( n \) to be twinning, if \( p \) is ... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \$3, buy or sell bundles of three wheat for \$12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollars mu... | 1. **Determine the resources needed to build a city:**
- Misha needs 3 ore and 2 wheat to build a city.
2. **Calculate the ore Misha will have after three days:**
- Misha produces 1 ore each day.
- After 3 days, Misha will have \(3 \text{ ore}\).
3. **Determine the cost of obtaining wheat:**
- Misha can b... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x>1$ is a real number such that $x+\tfrac 1x = \sqrt{22}$. What is $x^2-\tfrac1{x^2}$? | 1. Given the equation \( x + \frac{1}{x} = \sqrt{22} \), we need to find \( x^2 - \frac{1}{x^2} \).
2. First, square both sides of the given equation:
\[
\left( x + \frac{1}{x} \right)^2 = (\sqrt{22})^2
\]
\[
x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 22
\]
\[
x^2 + 2 + \frac{1}{x^2} =... | 18 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
2018 little ducklings numbered 1 through 2018 are standing in a line, with each holding a slip of paper with a nonnegative number on it; it is given that ducklings 1 and 2018 have the number zero. At some point, ducklings 2 through 2017 change their number to equal the average of the numbers of the ducklings to their ... | 1. Let \( d_i \) be the number that duckling \( i \) originally has, and \( d_i' \) be the number that duckling \( i \) has after changing their numbers. We are given that \( d_1 = 0 \) and \( d_{2018} = 0 \).
2. According to the problem, ducklings 2 through 2017 change their number to the average of the numbers of th... | 2000 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if \[k = \sum_{i=2}^n a_i F_i,\] where $a_i\in\{0,1\}$ for all $i$, $a_n=1$, and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ($F_0=0$, $F_1=1$, and $F_i=F_{i-1}+F_{i-2}$ for all $i\ge2$). This representation is said to be $... | 1. **Understanding the Fibonacci Representation**:
- The Fibonacci representation of a positive integer \( k \) is given by:
\[
k = \sum_{i=2}^n a_i F_i,
\]
where \( a_i \in \{0, 1\} \) for all \( i \), \( a_n = 1 \), and \( F_i \) denotes the \( i^{\text{th}} \) Fibonacci number. The Fibonacci s... | 1596 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a_0,a_1,\ldots, a_{2018}$ are integers such that \[(x^2-3x+1)^{1009} = \sum_{k=0}^{2018}a_kx^k\] for all real numbers $x$. Compute the remainder when $a_0^2 + a_1^2 + \cdots + a_{2018}^2$ is divided by $2017$. | 1. Given the polynomial \((x^2 - 3x + 1)^{1009} = \sum_{k=0}^{2018} a_k x^k\), we need to compute the remainder when \(a_0^2 + a_1^2 + \cdots + a_{2018}^2\) is divided by 2017.
2. Notice that \(x^2 - 3x + 1\) is a palindromic polynomial, meaning it reads the same forwards and backwards. Therefore, \((x^2 - 3x + 1)^{10... | 9 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence of polynomials $F_n(x)$ by $F_0(x)=0, F_1(x)=x-1$, and for $n\geq 1$, $$F_{n+1}(x)=2xF_n(x)-F_{n-1}(x)+2F_1(x).$$ For each $n$, $F_n(x)$ can be written in the form $$F_n(x)=c_nP_1(x)P_2(x)\cdots P_{g(n)}(x)$$ where $c_n$ is a constant and $P_1(x),P_2(x)\cdots, P_{g(n)}(x)$ are non-constant polynomials... | 1. **Define the sequence and transform the recursion:**
We start with the given sequence of polynomials \( F_n(x) \) defined by:
\[
F_0(x) = 0, \quad F_1(x) = x - 1,
\]
and for \( n \geq 1 \),
\[
F_{n+1}(x) = 2x F_n(x) - F_{n-1}(x) + 2F_1(x).
\]
To simplify the recursion, let us define \( G_n... | 24 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Ninety-eight apples who always lie and one banana who always tells the truth are randomly arranged along a line. The first fruit says "One of the first forty fruit is the banana!'' The last fruit responds "No, one of the $\emph{last}$ forty fruit is the banana!'' The fruit in the middle yells "I'm the banana!'' In how ... | 1. Let's denote the fruits as \( F_1, F_2, \ldots, F_{99} \), where \( F_i \) represents the fruit in the \( i \)-th position.
2. The first fruit \( F_1 \) claims that "One of the first forty fruits is the banana." Since the banana always tells the truth and apples always lie, if \( F_1 \) is the banana, this statement... | 21 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3\times 3\times 3$ cube such that each $1\times1\times3$ row or column contains exactly one white cube. Note that rotations are considered $\textit{distinct}$. | 1. **Split the cube into layers**: We start by splitting the $3 \times 3 \times 3$ cube into three $3 \times 3 \times 1$ layers. Each layer must contain exactly 3 white cubes because each $1 \times 1 \times 3$ column must contain exactly one white cube.
2. **Determine the placement of white cubes in each layer**:
... | 60480 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Nine distinct light bulbs are placed in a circle, each of which is off. Determine the number of ways to turn on some of the light bulbs in the circle such that no four consecutive bulbs are all off. | 1. **Define the problem in terms of a linear arrangement:**
Let \( f(n) \) denote the number of ways to configure \( n \) bulbs in a line such that no four consecutive bulbs are all off.
2. **Establish the recurrence relation:**
We can derive the recurrence relation for \( f(n) \) by considering the possible st... | 367 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider the following two vertex-weighted graphs, and denote them as having vertex sets $V=\{v_1,v_2,\ldots,v_6\}$ and $W=\{w_1,w_2,\ldots,w_6\}$, respectively (numbered in the same direction and way). The weights in the second graph are such that for all $1\le i\le 6$, the weight of $w_i$ is the sum of the weights of... | 1. Let's denote the weight of vertex \( v_i \) in the original graph as \( a_i \) for \( i = 1, 2, \ldots, 6 \).
2. According to the problem, the weight of vertex \( w_i \) in the second graph is the sum of the weights of the neighbors of \( v_i \) in the original graph. Let's denote the weight of vertex \( w_i \) as \... | 32 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider the natural implementation of computing Fibonacci numbers:
\begin{tabular}{l}
1: \textbf{FUNCTION} $\text{FIB}(n)$: \\
2:$\qquad$ \textbf{IF} $n = 0$ \textbf{OR} $n = 1$ \textbf{RETURN} 1 \\
3:$\qquad$ \textbf{RETURN} $\text{FIB}(n-1) + \text{FIB}(n-2)$
\end{tabular}
When $\text{FIB}(10)$ is evaluated, how m... | 1. Define \( c(n) \) as the number of recursive calls made by the function \(\text{FIB}(n)\).
2. Observe that for the base cases:
\[
c(0) = 0 \quad \text{and} \quad c(1) = 0
\]
because no further recursive calls are made when \( n = 0 \) or \( n = 1 \).
3. For \( n \geq 2 \), the function \(\text{FIB}(n)\) ... | 176 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We consider a simple model for balanced parenthesis checking. Let $\mathcal R=\{\texttt{(())}\rightarrow \texttt{A},\texttt{(A)}\rightarrow\texttt{A},\texttt{AA}\rightarrow\texttt{A}\}$ be a set of rules for phrase reduction. Ideally, any given phrase is balanced if and only if the model is able to reduce the phrase to... | To determine the number of balanced phrases of length 14 that are not correctly detected by the given set of rules $\mathcal{R}$, we need to follow a systematic approach. Let's break down the solution step-by-step.
1. **Define the Problem:**
We need to find the number of balanced parenthetical phrases of length 14 ... | 392 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Select points $T_1,T_2$ and $T_3$ in $\mathbb{R}^3$ such that $T_1=(0,1,0)$, $T_2$ is at the origin, and $T_3=(1,0,0)$. Let $T_0$ be a point on the line $x=y=0$ with $T_0\neq T_2$. Suppose there exists a point $X$ in the plane of $\triangle T_1T_2T_3$ such that the quantity $(XT_i)[T_{i+1}T_{i+2}T_{i+3}]$ is constant f... | 1. **Define the coordinates of the points:**
- \( T_1 = (0, 1, 0) \)
- \( T_2 = (0, 0, 0) \) (origin)
- \( T_3 = (1, 0, 0) \)
- \( T_0 = (0, 0, z) \) (since \( T_0 \) is on the line \( x = y = 0 \) and \( T_0 \neq T_2 \))
- Let \( X = (x, y, 0) \) be a point in the plane of \(\triangle T_1T_2T_3\).
2. *... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with circumradius $17$, inradius $4$, circumcircle $\Gamma$ and $A$-excircle $\Omega$. Suppose the reflection of $\Omega$ over line $BC$ is internally tangent to $\Gamma$. Compute the area of $\triangle ABC$. | 1. **Define Variables and Given Information:**
- Let $BC = a$, $CA = b$, and $AB = c$.
- Let $K$ be the area of $\triangle ABC$.
- Let $R = 17$ be the circumradius.
- Let $r = 4$ be the inradius.
- Let $s = \frac{a+b+c}{2}$ be the semiperimeter.
- Let $r_a$ be the $A$-exradius.
2. **Reflection and Ta... | 128 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be a complex number, and set $\alpha$, $\beta$, and $\gamma$ to be the roots of the polynomial $x^3 - x^2 + ax - 1$. Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of $a$. | 1. **Using Vieta's Relations:**
Given the polynomial \(x^3 - x^2 + ax - 1\), by Vieta's formulas, the roots \(\alpha, \beta, \gamma\) satisfy:
\[
\alpha + \beta + \gamma = 1,
\]
\[
\alpha\beta + \beta\gamma + \gamma\alpha = a,
\]
\[
\alpha\beta\gamma = 1.
\]
2. **Given Condition:**
We ... | 2009 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $ABCD$ is a trapezoid with $AB\parallel CD$ and $AB\perp BC$. Let $X$ be a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally, and denote $Y$ as the intersection of $AC$ and $BD$. If $AB=10$ and $CD=15$, compute the maximum possible value of $XY$. | 1. **Identify the given information and setup the problem:**
- $ABCD$ is a trapezoid with $AB \parallel CD$ and $AB \perp BC$.
- $X$ is a point on segment $\overline{AD}$ such that $AD$ bisects $\angle BXC$ externally.
- $Y$ is the intersection of $AC$ and $BD$.
- $AB = 10$ and $CD = 15$.
2. **Introduce po... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of integers $a$ with $1\leq a\leq 1007$ and the property that both $a$ and $a+1$ are quadratic residues mod $1009$. | To determine the number of integers \( a \) with \( 1 \leq a \leq 1007 \) such that both \( a \) and \( a+1 \) are quadratic residues modulo \( 1009 \), we need to analyze the properties of quadratic residues modulo a prime number.
1. **Quadratic Residue Definition**:
An integer \( x \) is a quadratic residue modul... | 251 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$, $b$, and $c$ are relatively prime integers such that \[\frac{a}{b+c} = 2\qquad\text{and}\qquad \frac{b}{a+c} = 3.\] What is $|c|$? | 1. Given the equations:
\[
\frac{a}{b+c} = 2 \quad \text{and} \quad \frac{b}{a+c} = 3
\]
we can rewrite these equations as:
\[
a = 2(b+c) \quad \text{and} \quad b = 3(a+c)
\]
2. Substitute \( a = 2(b+c) \) into \( b = 3(a+c) \):
\[
b = 3(2(b+c) + c)
\]
Simplify the equation:
\[
b... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$. | 1. According to Legendre's three-square theorem, the set \( S \) consists of natural numbers that can be written in the form \( 4^a(8b+7) \) where \( a \) and \( b \) are nonnegative integers.
2. We need to find the smallest \( n \in \mathbb{N} \) such that both \( n \) and \( n+1 \) are in \( S \).
3. Since \( n \) an... | 111 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a>1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct. | 1. We start with the sequence $\{a_n\}_{n \geq 1}$ defined such that $a_1 = a$ and for all $n \geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$.
2. We need to determine the smallest possible value of $a$ such that the numbers $a_1, a_2, \ldots, a_7$ are all distinct.
Let's analyze the sequence step by step... | 46 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\phi(n)\mid n$. | 1. **Understanding Euler's Totient Function**:
Euler's Totient Function, $\phi(n)$, counts the number of integers up to $n$ that are coprime with $n$. For a number $n$ with prime factorization $n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$, the function is given by:
\[
\phi(n) = n \left(1 - \frac{1}{p_1}\right) \l... | 396 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive di... | 1. **Identify the sequence of integers relatively prime to 91:**
- Since \(91 = 7 \times 13\), the integers between 1 and 91 that are relatively prime to 91 are those not divisible by 7 or 13.
- Using the principle of inclusion-exclusion, the count of such integers is:
\[
\phi(91) = 91 \left(1 - \frac{1... | 1054 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $BC=30$, $AC=50$, and $AB=60$. Circle $\omega_B$ is the circle passing through $A$ and $B$ tangent to $BC$ at $B$; $\omega_C$ is defined similarly. Suppose the tangent to $\odot(ABC)$ at $A$ intersects $\omega_B$ and $\omega_C$ for the second time at $X$ and $Y$ respectively. Compute $XY... | ### Part 1: Compute \( XY \)
1. **Identify the given elements and properties:**
- Triangle \( ABC \) with sides \( BC = 30 \), \( AC = 50 \), and \( AB = 60 \).
- Circle \( \omega_B \) passes through \( A \) and \( B \) and is tangent to \( BC \) at \( B \).
- Circle \( \omega_C \) passes through \( A \) and ... | 44 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered triples $(a,b,c)$ of integers satisfy the inequality \[a^2+b^2+c^2 \leq a+b+c+2?\]
Let $T = TNYWR$. David rolls a standard $T$-sided die repeatedly until he first rolls $T$, writing his rolls in order on a chalkboard. What is the probability that he is able to erase some of the numbers he's written su... | 1. We start with the inequality \(a^2 + b^2 + c^2 \leq a + b + c + 2\).
2. To simplify this inequality, we complete the square for each variable. Rewrite the inequality as:
\[
a^2 - a + b^2 - b + c^2 - c \leq 2
\]
3. Completing the square for each term:
\[
a^2 - a = \left(a - \frac{1}{2}\right)^2 - \frac... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$?
Let $T = TNYWR$, and let ... | 1. Given the triangle $\triangle ABC$ with $AB = 3$ and $AC = 5$, we need to find $BC^2$ under the given conditions.
- Points $D, E,$ and $F$ are on $\overline{BC}$ such that $\overline{AD} \perp \overline{BC}$, $\angle BAE = \angle CAE$, and $\overline{BF} = \overline{CF}$.
- $E$ is the midpoint of segment $\o... | 9602 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = 2^x + 3^x$. For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$? | 1. **Define the function and the problem:**
We are given the function \( f(x) = 2^x + 3^x \). We need to determine for how many integers \( 1 \leq n \leq 2020 \) the value \( f(n) \) is relatively prime to all of \( f(0), f(1), \dots, f(n-1) \).
2. **Identify the key property:**
The key property to consider is t... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The intramural squash league has 5 players, namely Albert, Bassim, Clara, Daniel, and Eugene. Albert has played one game, Bassim has played two games, Clara has played 3 games, and Daniel has played 4 games. Assuming no two players in the league play each other more than one time, how many games has Eugene played? | 1. **Understanding the Problem:**
We are given a squash league with 5 players: Albert (A), Bassim (B), Clara (C), Daniel (D), and Eugene (E). The number of games each player has played is given as follows:
- Albert: 1 game
- Bassim: 2 games
- Clara: 3 games
- Daniel: 4 games
- Eugene: ?
We need to... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider a $1$-indexed array that initially contains the integers $1$ to $10$ in increasing order.
The following action is performed repeatedly (any number of times):
[code]
def action():
Choose an integer n between 1 and 10 inclusive
Reverse the array between indices 1 and n inclusive
Reverse the array be... | 1. **Initial Setup**: We start with an array containing integers from 1 to 10 in increasing order:
\[
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
\]
2. **Action Description**: The action described involves two steps:
- Choose an integer \( n \) between 1 and 10 inclusive.
- Reverse the subarray from index 1 to \( ... | 20 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Point $A$, $B$, $C$, and $D$ form a rectangle in that order. Point $X$ lies on $CD$, and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$. If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle? | 1. **Identify the given areas and relationships:**
- The area of triangle $BCP$ is given as 3.
- The area of triangle $PXC$ is given as 2.
- We need to find the area of the entire rectangle $ABCD$.
2. **Use the given areas to find the ratio of segments:**
- Since the area of $\triangle BCP$ is 3 and the ar... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For every positive integer $k$, let $\mathbf{T}_k = (k(k+1), 0)$, and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\tfrac{1}{2}$ if $k$ is odd and $\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that
$$(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ... | 1. Start with the point \( P = (x, y) \).
2. Apply the homothety \(\mathcal{H}_1\) with ratio \(\frac{1}{2}\) centered at \(\mathbf{T}_1 = (2, 0)\):
\[
\mathcal{H}_1(P) = \left( \frac{x + 2}{2}, \frac{y}{2} \right)
\]
3. Apply the homothety \(\mathcal{H}_2\) with ratio \(\frac{2}{3}\) centered at \(\mathbf{T... | 256 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We say that a binary string $s$ [i]contains[/i] another binary string $t$ if there exist indices $i_1,i_2,\ldots,i_{|t|}$ with $i_1 < i_2 < \ldots < i_{|t|}$ such that $$s_{i_1}s_{i_2}\ldots s_{i_{|t|}} = t.$$ (In other words, $t$ is found as a not necessarily contiguous substring of $s$.) For example, $110010$ contain... | 1. **Understanding the Problem:**
We need to find the length of the shortest binary string \( s \) that contains the binary representations of all positive integers less than or equal to \( 2048 \).
2. **Binary Representation:**
The binary representation of \( 2048 \) is \( 100000000000 \), which has \( 12 \) b... | 22 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the unique 3 digit number $N=\underline{A}$ $\underline{B}$ $\underline{C},$ whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}$ $\underline{B}$ $\underline{C}$ $\times$ $\underline{A}$ $\underline{B}$ $\times$ $\underline{A}$ is divisible by $1000$.
[i]Proposed by Kyle ... | 1. We need to find a 3-digit number \( N = \underline{A}\underline{B}\underline{C} \) such that the product \( P = \underline{A}\underline{B}\underline{C} \times \underline{A}\underline{B} \times \underline{A} \) is divisible by 1000.
2. Since 1000 = \( 2^3 \times 5^3 \), \( P \) must be divisible by both \( 2^3 \) and... | 875 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a,b$ are positive real numbers such that $a+a^2 = 1$ and $b^2+b^4=1$. Compute $a^2+b^2$.
[i]Proposed by Thomas Lam[/i] | 1. First, solve for \(a\) from the equation \(a + a^2 = 1\):
\[
a^2 + a - 1 = 0
\]
This is a quadratic equation in the standard form \(ax^2 + bx + c = 0\). We can solve it using the quadratic formula \(a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 1\), and \(c = -1\):
\[
a = \frac{-... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose there are $160$ pigeons and $n$ holes. The $1$st pigeon flies to the $1$st hole, the $2$nd pigeon flies to the $4$th hole, and so on, such that the $i$th pigeon flies to the $(i^2\text{ mod }n)$th hole, where $k\text{ mod }n$ is the remainder when $k$ is divided by $n$. What is minimum $n$ such that there is at... | 1. **Understanding the Problem:**
We need to find the minimum \( n \) such that each pigeon flies to a unique hole. The \( i \)-th pigeon flies to the hole given by \( i^2 \mod n \). We need to ensure that no two pigeons fly to the same hole, i.e., \( i^2 \not\equiv j^2 \pmod{n} \) for \( i \neq j \).
2. **Analyzin... | 326 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be complex numbers such that $(a+1)(b+1)=2$ and $(a^2+1)(b^2+1)=32.$ Compute the sum of all possible values of $(a^4+1)(b^4+1).$
[i]Proposed by Kyle Lee[/i] | 1. Let \( (a^4+1)(b^4+1) = x \). We are given two equations:
\[
(a+1)(b+1) = 2
\]
\[
(a^2+1)(b^2+1) = 32
\]
2. From the second equation, we can expand and rearrange:
\[
(a^2+1)(b^2+1) = a^2b^2 + a^2 + b^2 + 1 = 32
\]
\[
a^2b^2 + a^2 + b^2 + 1 = 32
\]
\[
a^2b^2 + a^2 + b^2 = 31... | 1924 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the remainder when $$\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor$$ is divided by $10^4$.
[i]Proposed by Vijay Srinivasan[/i] | 1. **Identify the problem and simplify the expression:**
We need to find the remainder when
\[
\left \lfloor \frac{149^{151} + 151^{149}}{22499}\right \rfloor
\]
is divided by \(10^4\). Notice that \(22499 = 149 \times 151\). Let \(p = 149\) and \(q = 151\).
2. **Simplify using modular arithmetic:**
... | 7800 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$P(Q(x))=(x-1)(x-3)(x-5)(x-7)$$$$Q(R(x))=(x-2)(x-4)(x-6)(x-8)$$
for all real numbers $x.$ What is $P(0)+Q(0)+R(0)?$
[i]Proposed by Kyle Lee[/i] | 1. Let \( P(x) = (x-a)(x-b) \), \( Q(x) = (x-c)(x-d) \), and \( R(x) = (x-e)(x-f) \). Since these are monic quadratic polynomials, the leading coefficient of each polynomial is 1.
2. Given that \( P(Q(x)) = (x-1)(x-3)(x-5)(x-7) \), we can express \( P(Q(x)) \) as:
\[
P(Q(x)) = ((x-c)(x-d) - a)((x-c)(x-d) - b)
... | 129 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a$, $b$, $c$, and $d$ are non-negative integers such that
\[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\]
Find $a^3+b^3+c^3+d^3$.
[i]Proposed by Connor Gordon[/i] | 1. Given the equation:
\[
(a+b+c+d)(a^2+b^2+c^2+d^2)^2 = 2023
\]
we start by noting the prime factorization of \(2023\):
\[
2023 = 7 \cdot 17^2
\]
This implies that the only perfect square factors of \(2023\) are \(1\) and \(17^2\).
2. We consider the possible values for \((a^2 + b^2 + c^2 + d^... | 43 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider digits $\underline{A}, \underline{B}, \underline{C}, \underline{D}$, with $\underline{A} \neq 0,$ such that $\underline{A} \underline{B} \underline{C} \underline{D} = (\underline{C} \underline{D} ) ^2 - (\underline{A} \underline{B})^2.$ Compute the sum of all distinct possible values of $\underline{A} + \under... | 1. Let $\underline{AB} = x$ and $\underline{CD} = y$. The given equation is:
\[
100x + y = y^2 - x^2
\]
This can be rewritten using the difference of squares:
\[
100x + y = (y - x)(y + x)
\]
2. Note that $y - x$ must divide $100x + y$. Since $y - x$ also divides $y^2 - x^2$, it must divide $101x$:... | 21 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$?
[i]Proposed by Giacomo Rizzo[/i] | 1. We start with the equation given in the problem:
\[
2n^2 + 3n = p^2
\]
where \( p \) is an integer.
2. To transform this into a Pell-like equation, we multiply both sides by 8:
\[
8(2n^2 + 3n) = 8p^2 \implies 16n^2 + 24n = 8p^2
\]
3. We can rewrite the left-hand side as a complete square minus... | 444 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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