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Please solve the following math problem. | Points $D$ and $E$ are selected on sides $AB$ and $AC$ of equilateral triangle $ABC$ , such that $AD = 4$ , $BD = 2$ , and the perimeter of $BCED$ is twice the perimeter of $ADE$ . Given that the area of $\vartriangle ADE$ can be expressed in the form $\frac{\sqrt{m}-\sqrt{n}}{2}$ , determine the value of $m + n$ . | Alright, I have this geometry problem about an equilateral triangle, and I need to find the sum of two numbers related to the area of a smaller triangle. Let's break it down step by step.
**Problem Statement:**
Points \( D \) and \( E \) are selected on sides \( AB \) and \( AC \) of equilateral triangle \( ABC \), s... |
Please solve the following math problem. | Let $P(x)=2x^{59}-x^{2}-x-6$ . If $Q(x)$ is a polynomial whose roots are the $59$ th powers of the roots of $P(x)$ , then find the sum of the roots of $Q(x)$ . | Alright, I have this problem in front of me:
*Let \( P(x) = 2x^{59} - x^2 - x - 6 \). If \( Q(x) \) is a polynomial whose roots are the 59th powers of the roots of \( P(x) \), then find the sum of the roots of \( Q(x) \).*
First, I need to understand what's being asked. I have a polynomial \( P(x) \) of degree 59, an... |
Please solve the following math problem. | A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are colored red. An arrangement of the cubes is $\mathfrak{Intriguing}$ if there is exactly $1$ red unit cube in every $1\times1\times4$ rectangular box composed of $4$ unit cubes. Determine the number of $\mathfrak{Intriguing}$ color... | Alright, I have this problem about a $4 \times 4 \times 4$ cube made up of 64 unit cubes. Some of these unit cubes have their faces colored red, specifically 16 of them. The goal is to find out how many ways we can color these cubes so that in every $1 \times 1 \times 4$ rectangular box (which is just a straight line o... |
Please solve the following math problem. | Zeus, Athena, and Posideon arrive at Mount Olympus at a random time between $12:00$ pm and $12:00$ am, and stay for $3$ hours. All three hours does not need to fall within 1 $2$ pm to $12$ am. If any of the $2$ gods see each other during $12$ pm to $12$ am, it will be a good day. The probability of it being a good day ... | Alright, I have this problem about Zeus, Athena, and Poseidon arriving at Mount Olympus at random times between 12:00 pm and 12:00 am. They each stay for 3 hours, and if any two of them see each other during that time, it's considered a good day. I need to find the probability that it's a good day, expressed as a fract... |
Please solve the following math problem. | Suppose that \(|x_i| < 1\) for \(i = 1, 2, \dots, n\). Suppose further that
\(
|x_1| + |x_2| + \dots + |x_n| = 19 + |x_1 + x_2 + \dots + x_n|.
\)
What is the smallest possible value of \(n\)? | Alright, I have this problem in front of me, and I need to find the smallest possible value of \( n \) given certain conditions. Let's break it down and see if I can understand it.
**Problem Statement:**
Suppose that \( |x_i| < 1 \) for \( i = 1, 2, \dots, n \). Suppose further that
\[
|x_1| + |x_2| + \dots + |x_n| =... |
Please solve the following math problem. | Given $ x, y $ are positive real numbers that satisfy $ 3x+4y+1=3\sqrt{x}+2\sqrt{y} $ , then the value $ xy $ can be expressed as $ \frac{m}{n} $ , where $ m $ and $ n $ are relatively prime positive integers. Find $ m+n $ . | Alright, I have this problem:
Given \( x, y \) are positive real numbers that satisfy \( 3x + 4y + 1 = 3\sqrt{x} + 2\sqrt{y} \), then the value \( xy \) can be expressed as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m + n \).
Alright, let's break this down.
**Underst... |
Please solve the following math problem. | Compute the largest integer $k$ such that $2004^k$ divides $2004!$ . | Alright, I have this problem in front of me:
**Compute the largest integer \( k \) such that \( 2004^k \) divides \( 2004! \).**
First, I need to understand what the problem is asking. It wants the highest power of 2004 that divides 2004 factorial. Okay, so \( 2004! \) is the product of all positive integers up to 20... |
Please solve the following math problem. | Let $ABCD$ be a parallelogram such that $AC \perp AD.$ Point $P$ exists inside $ABCD$ such that the distances from $P$ to lines $AB,$ $BC,$ $CD,$ and $DA$ are $5,$ $7,$ $7,$ and $13$ respectively. Find the area of parallelogram $ABCD.$ | Alright, I have this geometry problem about a parallelogram, and I need to find its area. Let's break it down and see if I can make sense of it.
**Problem Statement:**
Let \( ABCD \) be a parallelogram such that \( AC \perp AD \). Point \( P \) exists inside \( ABCD \) such that the distances from \( P \) to lines \(... |
Please solve the following math problem. | $x_1$ , $x_2$ , and $x_3$ are complex numbers such that
$$$x_1 + x_2 + x_3 = 0$$$
$$$x^2_1 + x^2_2 + x^2_3 = 16$$$
$$$x^3_1 + x^3_3 + x^3_3 =-24$$$
Let $\gamma = \min (|x_1|, |x_2|, |x_3|)$ , where $|a + bi|=\sqrt{a^2 + b^2}$ and $i = \sqrt{-1}$ .
Determine the value of $\gamma^6- 15\gamma^4 + \gamma^3 + 56\gamma^2$ . | Alright, I have this problem about three complex numbers \( x_1, x_2, x_3 \) with some given conditions, and I need to find the value of a specific expression involving the minimum of their magnitudes. Let's break it down step by step.
### Understanding the Problem
First, let's write down what's given:
1. \( x_1 + x... |
Please solve the following math problem. | Consider the set $S=\{1, 2, 3, 4, 5, 6, 7, 8\}$ . We say a subset $s \in S$ , with $|s| \geq 2$ , is prime-like if every two-element subset of $s$ will always contain at least one prime number. Find the number of prime-like subsets. | Alright, I have this problem in front of me, and I need to figure out how many prime-like subsets there are in the set \( S = \{1, 2, 3, 4, 5, 6, 7, 8\} \). Let's break it down step by step.
### Understanding the Problem
First, I need to understand what a **prime-like subset** is. According to the problem:
- A subse... |
Please solve the following math problem. | Compute the number of positive integer triplets $(a,b,c)$ with $1\le a,b,c\le 500$ that satisfy the following properties:
(a) $abc$ is a perfect square,
(b) $(a+7b)c$ is a power of $2$ ,
(c) $a$ is a multiple of $b$ . | Alright, I have this problem in front of me, and it's a bit challenging, but I'll try to break it down step by step. The problem is about finding the number of positive integer triplets \((a, b, c)\) where \(1 \leq a, b, c \leq 500\) that satisfy three specific conditions:
1. **Condition (a):** \(abc\) is a perfect sq... |
Please solve the following math problem. | Circles of radii \(5, 5, 8,\) and \(\frac mn\) are mutually externally tangent, where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n.\) | Alright, I have this problem about four circles that are all mutually externally tangent. The radii of three of them are 5, 5, and 8, and the fourth one has a radius of \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. I need to find \(m + n\).
First, I need to understand what "mutually ex... |
Please solve the following math problem. | Find the remainder when \(9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\) is divided by \(1000\). | Alright, I have this problem:
**Find the remainder when \(9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\) is divided by \(1000\).**
First, let's understand what's being asked. We have a product of numbers, each of which is a sequence of 9's, starting from a single 9 up to a numbe... |
Please solve the following math problem. | A circle $\omega_1$ of radius $6\sqrt2$ is internally tangent to a larger circle $\omega_2$ of radius $12\sqrt2$ such that the center of $\omega_2$ lies on $\omega_1$ . A diameter $AB$ of $\omega_2$ is drawn tangent to $\omega_1$ . A second line $\ell$ is drawn from $B$ tangent to $\omega_1$ . Let the line tangent to $... | Alright, I have this geometry problem in front of me, and I need to find the area of triangle \( ABC \). Let's break it down step by step.
### Understanding the Problem
First, let's make sure I understand all the parts of the problem:
1. **Circles \( \omega_1 \) and \( \omega_2 \):**
- \( \omega_1 \) has a radius... |
Please solve the following math problem. | Find the positive solution to
<center>\(\frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0\)</center> | Alright, I have this equation to solve:
\[
\frac{1}{x^2 - 10x - 29} + \frac{1}{x^2 - 10x - 45} - \frac{2}{x^2 - 10x - 69} = 0
\]
First, I notice that all the denominators have the same quadratic expression \( x^2 - 10x \) but with different constants subtracted. Maybe I can simplify this by letting \( y = x^2 - 10x \... |
Please solve the following math problem. | Let \(S\) be a set with six elements. Let \(\mathcal{P}\) be the set of all subsets of \(S.\) Subsets \(A\) and \(B\) of \(S\), not necessarily distinct, are chosen independently and at random from \(\mathcal{P}\). The probability that \(B\) is contained in one of \(A\) or \(S-A\) is \(\frac{m}{n^{r}},\) where \(m\), \... | Alright, I have this probability problem to solve. Let's break it down step by step.
**Problem Statement:**
Let \( S \) be a set with six elements. Let \( \mathcal{P} \) be the set of all subsets of \( S \). Subsets \( A \) and \( B \) of \( S \), not necessarily distinct, are chosen independently and at random from ... |
Please solve the following math problem. | Define a regular \( n \)-pointed star to be the union of \( n \) line segments \( P_1P_2, P_2P_3,\ldots, P_nP_1 \) such that
* the points \( P_1, P_2,\ldots, P_n \) are coplanar and no three of them are collinear,
* each of the \( n \) line segments intersects at least one of the other line segments at a point other t... | Alright, I have this problem about regular \( n \)-pointed stars, and I need to figure out how many non-similar regular 1000-pointed stars there are. Let's break it down step by step.
### Understanding the Problem
First, I need to understand what a regular \( n \)-pointed star is. According to the problem:
1. **Poin... |
Please solve the following math problem. | A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the wa... | Alright, I have this problem about a moving walkway and three people: Al, Bob, and Cy. I need to figure out at what point one of them is exactly halfway between the other two and then find the distance from the start of the walkway to that middle person. Let's break it down step by step.
### Understanding the Problem
... |
Please solve the following math problem. | When a right triangle is rotated about one leg, the volume of the cone produced is \(800\pi \;\textrm{ cm}^3\). When the triangle is rotated about the other leg, the volume of the cone produced is \(1920\pi \;\textrm{ cm}^3\). What is the length (in cm) of the hypotenuse of the triangle? | Alright, I have this problem about a right triangle that forms cones when rotated around its legs. The volumes of these cones are given, and I need to find the length of the hypotenuse of the triangle. Let's break it down step by step.
**Understanding the Problem:**
First, let's make sure I understand what's being as... |
Please solve the following math problem. | Kelvin the Frog's favorite quadratic function is $f(x) = x^2- 42x + 1337$ . His favorite quartic function is $f(f(x))$ , and in general his favorite function of degree $2^n$ is $f^n(x)$ . Compute the sum of the roots of Kelvin's favorite $16$ th degree function. | Alright, I have this problem about Kelvin the Frog and his favorite functions. Let's try to understand what's being asked and how to approach it.
**Problem Statement:**
Kelvin the Frog's favorite quadratic function is \( f(x) = x^2 - 42x + 1337 \). His favorite quartic function is \( f(f(x)) \), and in general, his f... |
Please solve the following math problem. | In the Cartesian plane let \(A = (1,0)\) and \(B = \left( 2, 2\sqrt{3} \right)\). Equilateral triangle \(ABC\) is constructed so that \(C\) lies in the first quadrant. Let \(P=(x,y)\) be the center of \(\triangle ABC\). Then \(x \cdot y\) can be written as \(\tfrac{p\sqrt{q}}{r}\), where \(p\) and \(r\) are relatively ... | Alright, I have this problem about an equilateral triangle in the Cartesian plane, and I need to find the product of the coordinates of its centroid. Let's break it down step by step.
**Given:**
- Point \( A = (1, 0) \)
- Point \( B = \left( 2, 2\sqrt{3} \right) \)
- Equilateral triangle \( ABC \) with \( C \) in the ... |
Please solve the following math problem. | A sequence $\{R_n\}_{n \ge 0}$ obeys the recurrence $7R_n = 64-2R_{n-1} +9R_{n-2}$ for any integers $n \ge 2$ . Additionally, $R_0 = 10 $ and $R_1 = -2$ . Let
$$$S = \sum^{\infty}_{i=0} \frac{R_i}{2^i}.$$$
$S$ can be expressed as $\frac{m}{n}$ for two relatively prime positive integers $m$ and $n$ . Determine the valu... | Alright, I have this problem about a sequence defined by a recurrence relation, and I need to find the sum of an infinite series involving this sequence. Let's break it down step by step.
### Understanding the Problem
First, let's make sure I understand what's given:
- A sequence \(\{R_n\}_{n \ge 0}\) satisfies the ... |
Please solve the following math problem. | Two positive integers differ by \(60\). The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? | Alright, I have this problem in front of me:
*Two positive integers differ by \(60\). The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?*
First, I need to understand what's being asked. There are two positive integers, let... |
Please solve the following math problem. | $\{A_n\}_{n\ge1}$ is a sequence of positive integers such that
$$$a_n = 2a_{n-1} + n^2$$$
for all integers $n > 1$ . Compute the remainder obtained when $a_{2004}$ is divided by $1000$ if $a_1 = 1$ . | Alright, I have this problem about a sequence of positive integers defined by the recurrence relation:
\[ a_n = 2a_{n-1} + n^2 \]
for all integers \( n > 1 \), and \( a_1 = 1 \). I need to find the remainder when \( a_{2004} \) is divided by 1000.
First, I need to understand the sequence and find a way to express \(... |
Please solve the following math problem. | The points \(A\), \(B\) and \(C\) lie on the surface of a sphere with center \(O\) and radius \(20\). It is given that \(AB=13\), \(BC=14\), \(CA=15\), and that the distance from \(O\) to \(\triangle ABC\) is \(\frac{m\sqrt{n}}k\), where \(m\), \(n\), and \(k\) are positive integers, \(m\) and \(k\) are relatively prim... | Alright, I have this geometry problem about a sphere and a triangle on its surface. Let's try to understand what's given and what's being asked.
**Given:**
- A sphere with center \( O \) and radius \( 20 \).
- Points \( A \), \( B \), and \( C \) lie on the surface of this sphere.
- The lengths of the sides of triangl... |
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