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What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction. | To determine the coefficient of \(x^2y^6\) in the expansion of \(\left(\frac{3}{5}x - \frac{y}{2}\right)^8\), we can use the binomial theorem.
The binomial theorem states:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
In this case, \(a = \frac{3}{5}x\), \(b = -\frac{y}{2}\), and \(n = 8\).
We are interes... |
In how many ways can all ten digits be arranged in a row such that the digits $2, 5, 8$ are placed next to each other? | To determine the number of ways to arrange all ten digits such that the digits \(2, 5, 8\) are placed next to each other, we can use the concept of "treating the group as a single element." Here's the step-by-step reasoning:
1. **Treat \(2, 5, 8\) as a single unit:** Since \(2, 5, 8\) must be next to each other, we ca... |
Let $S = \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004 \}$. Given that $5^{2004} = 5443 \cdots 0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$? | To solve the problem of finding how many elements of the set \( S = \{5^k | k \in \mathbb{Z}, 0 \le k \le 2004 \} \) begin with the digit 1, we need to understand the distribution of leading digits in powers of 5. This problem can be approached by utilizing properties of logarithms and Benford's Law.
First, we observe... |
Solve in prime numbers the equation \( p^2 - 6pq + q^2 + 3q - 1 = 0 \). | To solve the problem of finding prime numbers \( p \) and \( q \) that satisfy the equation \( p^2 - 6pq + q^2 + 3q - 1 = 0 \), we can proceed with the following steps:
1. **Understand the equation**: We need to find all prime pairs \((p, q)\) that satisfy the given quadratic equation.
2. **Simplify and analyze the e... |
If \( 10 \leq x \leq 20 \) and \( 40 \leq y \leq 60 \), what is the largest value of \( \frac{x^{2}}{2 y} \)?
(A) 5
(B) \( \frac{5}{6} \)
(C) \( \frac{10}{3} \)
(D) \( \frac{5}{4} \)
(E) 10 | To find the largest value of \( \frac{x^2}{2y} \) given the constraints \( 10 \leq x \leq 20 \) and \( 40 \leq y \leq 60 \), it is useful to analyze the function mathematically. We need to maximize the expression \( \frac{x^2}{2y} \) by choosing appropriate values of \( x \) and \( y \) within their respective ranges.
... |
Given that $a$, $b$, and $c$ are positive real numbers satisfying $a+b+c=1$, find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac {1}{c}$$. | To solve this problem, we'll use the method of Lagrange multipliers to find the minimum value of the function \( f(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \) subject to the constraint \( g(a, b, c) = a + b + c - 1 = 0 \).
Here's how we can do it step-by-step:
1. **Define the function and constraint:**
-... |
The number of solutions in positive integers of $2x+3y=763$ is:
$\textbf{(A)}\ 255 \qquad \textbf{(B)}\ 254\qquad \textbf{(C)}\ 128 \qquad \textbf{(D)}\ 127 \qquad \textbf{(E)}\ 0$ | To solve the problem of finding the number of solutions in positive integers \((x, y)\) for the equation \(2x + 3y = 763\), we need to follow these steps:
1. **Rewrite the equation**: Express \(y\) in terms of \(x\):
\[ 3y = 763 - 2x \]
\[ y = \frac{763 - 2x}{3} \]
2. **Conditions for \(y\) to be an integer**: For \(... |
The number $101$ is the smallest three-digit palindromic prime. What is the second-smallest one? | To find the second-smallest three-digit palindromic prime number, let's first understand the definition:
1. A three-digit palindromic number reads the same forward and backward. Thus, it has the form \( \overline{aba} \), where \( a \) and \( b \) are digits and \( a \neq 0 \).
2. A prime number is a natural number gr... |
On rectangular coordinates, point $A = (1,2)$ , $B = (3,4)$ . $P = (a, 0)$ is on $x$ -axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$ . | To solve this problem, we need to minimize the sum of distances \(AP + PB\) where \(P = (a, 0)\), \(A = (1, 2)\), and \(B = (3, 4)\).
First, we can define the distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), which is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For our scenario,... |
If the sequence \(\{a_n\}\) is defined by \(a_1=1\) and \(4a_n a_{n+1} = (a_n + a_{n+1} - 1)^2\) with \(a_n > a_{n-1}\), then \(a_n\) equals
A. \(n^2\)
B. \(n\)
C. \(n(n-1)\)
D. \(n^3\) | To solve the problem, we start by understanding and simplifying the recurrence relation:
\[ a_1 = 1 \]
\[ 4a_n a_{n+1} = (a_n + a_{n+1} - 1)^2 \]
We need to find a general expression for \(a_n\) that satisfies this recurrence relation. Let's proceed step-by-step:
1. **Rewrite the recurrence relation:**
\[ 4a_n a_... |
Four people, Jia, Yi, Bing, and Ding, are each required to formulate a quadratic equation with solutions \( x_1, x_2 \) of the form \( x^2 + px + q = 0 \) under the following conditions:
- Jia: \( p, q, x_1, x_2 \) are all integers that leave a remainder of 1 when divided by 3.
- Yi: \( p, q, x_1, x_2 \) are all intege... | Let's break down the problem and check if there are valid solutions for each person's conditions. For each person, we need to ensure that the coefficients \( p \) and \( q \), and the roots \( x_1 \) and \( x_2 \) satisfy the given modular constraints.
1. **Jia's Conditions:**
- \( p, q, x_1, x_2 \equiv 1 \pmod{3} ... |
In the larger of two concentric circles, a chord of 32 units is drawn that is tangent to the smaller circle. Find the radius of each of the circles if the width of the resulting ring is 8 units. | Sure, let's solve the problem step-by-step using Python and SymPy.
### Understanding the Problem
1. We have two concentric circles.
2. A chord of length 32 units in the larger circle is tangent to the smaller circle.
3. The width of the resulting ring (the difference in radii) is 8 units.
### Problem Breakdown
1. Let... |
If six cards labeled 1, 2, 3, 4, 5, and 6 are to be placed into three different envelopes with each envelope containing two cards, and the cards labeled 1 and 2 must be placed in the same envelope, how many different placement methods are there? | To solve this problem, we need to count the number of ways to distribute six cards, labeled 1 through 6, into three envelopes with each envelope containing exactly two cards. Additionally, the cards labeled 1 and 2 must be placed in the same envelope.
Let's break down the problem into manageable steps:
1. **Group Car... |
Given an ellipse \( a^{2} x^{2} + b^{2} y^{2} = 5 c^{2} \) and a line \( a x + b y = c \) intersect at points \( A \) and \( B \). If \( a, b, c \) form a geometric sequence, then find the value of \( \frac{b}{a} \left( k_{O A} + k_{O B} \right) \). | To solve this problem, let's break it down step-by-step:
1. **Understand the Intersections:**
The ellipse \(a^2x^2 + b^2y^2 = 5c^2\) and the line \(ax + by = c\) intersect at points \(A\) and \(B\).
2. **Geometric Sequence:**
Given that \(a\), \(b\), and \(c\) form a geometric sequence, we can express \(b\) ... |
Let \(\mathbb{Z}_{>0}\) be the set of positive integers. A positive integer constant \(C\) is given. Find all functions \(f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}\) such that, for all positive integers \(a\) and \(b\) satisfying \(a+b>C\),
$$
a+f(b) \mid a^{2}+b f(a).
$$ | To solve the problem, we will try finding all possible functions \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0} \) that satisfy the given divisibility condition. We'll use sympy to help us with algebraic manipulation and verification.
Steps to approach the problem:
1. Rewrite the given divisibility condition:
\[... |
Given a function $f(x)$ for all $x$, $y$ satisfies $f(ab) = bf(a) + af(b)$,
(1) Find $f(0)$;
(2) Prove that $f(x)$ is an odd function;
(3) If $F(x) = af(x) + bx^5 + cx^3 + 2x^2 + dx + 3$, given $F(-5) = 7$, find $F(5)$. | Let's solve the given problem step by step using Python and SymPy.
### Step 1: Find \( f(0) \)
Given the functional equation:
\[ f(ab) = bf(a) + af(b) \]
By substituting \( a = 0 \) and \( b = 0 \), we get:
\[ f(0 \cdot 0) = 0 \cdot f(0) + 0 \cdot f(0) \]
\[ f(0) = 0 \]
So, \( f(0) = 0 \).
### Step 2: Prove that \... |
If $3x^{m}y$ and $-5x^{2}y^{n}$ are like terms, then $m+n=\_\_\_\_\_\_$. | To solve the problem, we need to determine the values of \(m\) and \(n\) given that the terms \(3x^{m}y\) and \(-5x^{2}y^{n}\) are like terms. Like terms must have the same variables raised to the same powers.
Steps to complete the problem:
1. Equate the exponents of \(x\) from both terms.
2. Equate the exponents of \... |
There are 35 mooncakes to be packed, using two types of packaging: large packages containing 9 mooncakes each and small packages containing 4 mooncakes each. The goal is to pack all mooncakes without any remaining. How many packages are used in total? | To solve the problem of packing 35 mooncakes using large packages containing 9 mooncakes each and small packages containing 4 mooncakes each without any remaining, we need to find a combination of these packages that will sum up to 35 mooncakes.
We can systematically try different counts of large packages and see if t... |
Given a sequence $\{a_n\}$ that satisfies: $a_{4n-3} = 1$, $a_{4n-1} = 0$, $a_{2n} = a_n$, for $n \in \mathbb{N}^*$, find the values of $a_{2009}$ and $a_{2014}$. | To solve the given sequence problem using Python and sympy, we need to follow these steps:
1. Understand and analyze the sequence definitions.
2. Implement the sequence generation based on the given rules.
3. Compute the values of \(a_{2009}\) and \(a_{2014}\).
The sequence rules are:
- \(a_{4n-3} = 1\)
- \(a_{4n-1} ... |
Solve the system
\[
\begin{cases}
3^x \cdot 2^y = 972 \\
\log_{\sqrt{3}}(x - y) = 2
\end{cases}
\] | To solve the system of equations:
\[
\begin{cases}
3^x \cdot 2^y = 972 \\
\log_{\sqrt{3}}(x - y) = 2
\end{cases}
\]
we will follow these steps:
1. Express the logarithmic equation in simpler terms using the change of base formula.
2. Solve the logarithmic equation for one of the variables.
3. Substitute the expressio... |
Carmen had 28 cats and 18 dogs before she gave 3 of the cats up for adoption. How many more cats than dogs does Carmen have now? | To determine how many more cats than dogs Carmen has now, we need to follow these steps:
1. **Initial counts**:
- Cats: 28
- Dogs: 18
2. **After adoption**:
- Cats given up for adoption: 3
3. **Final counts**:
- Updated number of cats = Initial number of cats - Cats given up for adoption
4. **Differ... |
$(81)^{\frac12}=3^m$. Find $m$. | To solve the equation \((81)^{\frac{1}{2}} = 3^m\) and find the value of \(m\), we can use properties of exponents. Let's break down the problem and use Python to verify our steps.
First, recall that:
- \(81\) can be expressed as a power of 3, since \(81 = 3^4\).
- Taking the square root of \(81\) (\(81^{\frac{1}{2}}\... |
The distances from three points lying in a horizontal plane to the base of a television tower are 800 m, 700 m, and 500 m, respectively. From each of these three points, the tower is visible (from the base to the top) at certain angles, with the sum of these three angles being $90^{\circ}$.
A) Find the height of the t... | To solve this problem, we will use trigonometry. Let's denote the height of the television tower as \(h\).
Given:
- The distances from the three points to the base of the television tower are 800 m, 700 m, and 500 m.
- The angles of elevation from each of these points to the top of the tower sum up to \(90^\circ\).
L... |
Simplify
\[\frac{\tan^3 75^\circ + \cot^3 75^\circ}{\tan 75^\circ + \cot 75^\circ}.\] | Let's break down the given expression step-by-step to solve it accurately using Python code. The expression is:
\[
\frac{\tan^3 75^\circ + \cot^3 75^\circ}{\tan 75^\circ + \cot 75^\circ}.
\]
Here's the strategy:
1. Calculate \(\tan 75^\circ\).
2. Calculate \(\cot 75^\circ\).
3. Compute \(\tan^3 75^\circ\) and \(\cot... |
A square with a side length of \(1 \mathrm{~m}\) was cut into small squares with side lengths of \(1 \mathrm{~mm}\), with cuts parallel to its sides. Placing these small squares side by side, without overlapping, a rectangle with a width of \(1 \mathrm{~mm}\) was formed. What is the length of this rectangle? | To solve this problem, let's break it down step by step:
1. **Understand the Problem:**
- A square with a side length of \(1 \mathrm{~m}\) is cut into small squares with side lengths of \(1 \mathrm{~mm}\).
- These small squares are placed side by side to form a rectangle with a width of \(1 \mathrm{~mm}\).
- ... |
The bookshelf in the library has three tiers. The first tier contains 3 different mathematics books, the second tier has 5 different language books, and the third tier has 8 different English books. Now, taking out one book randomly, there are (\quad) different ways of selecting a book.
A: 120
B: 16
C: 64
D: 39 | To solve this problem, we need to determine the total number of ways to select one book from all the books on the three tiers of the bookshelf. Here's the step-by-step approach:
1. **Determine the number of books on each tier:**
- First tier: 3 mathematics books
- Second tier: 5 language books
- Third tier: 8... |
What is the base five product of the numbers $121_{5}$ and $11_{5}$? | To solve this problem, we will follow these steps:
1. Convert the base 5 numbers to base 10 (decimal).
2. Multiply the decimal numbers.
3. Convert the resulting product back to base 5.
Let's start by converting the base 5 numbers \(121_5\) and \(11_5\) to base 10.
### Converting base 5 to base 10
- \(121_5\):
\[ 12... |
In the tetrahedron \(KLMN\), the edge lengths are given as \(KL = MN = 9\), \(KM = LN = 15\), and \(KN = LM = 16\). The points \(P, Q, R,\) and \(S\) are the centers of the circles inscribed in the triangles \(KLM\), \(KLN\), \(KMN\), and \(LMN\), respectively. Find the volume of the tetrahedron \(PQRS\). Round the ans... | To solve this problem, we'll follow these steps:
1. **Define the vertices and calculate the coordinates**: We'll place the tetrahedron in a 3D coordinate system. For simplicity, we'll position \(K\) at the origin, \(L\) on the x-axis, and \(M\) in the xy-plane. Using the given edge lengths, we'll find the coordinates ... |
In the sequence $\{a_n\}$, $a_1 = 1$, $a_2 = 2$, and $a_{n+2} - a_n = 1 + (-1)^n$ for $n \in \mathbb{N}^*$. Find the sum of the first 100 terms, $S_{100}$. | To solve the problem, we need to analyze the recurrence relation given for the sequence \(\{a_n\}\):
\[ a_{n+2} - a_n = 1 + (-1)^n \]
Given:
- \( a_1 = 1 \)
- \( a_2 = 2 \)
We need to find the sum of the first 100 terms, \( S_{100} \).
Let's break down the problem into manageable parts step-by-step:
1. **Understan... |
The sky currently has 4 times as many cirrus clouds as cumulus clouds, and 12 times as many cumulus clouds as cumulonimbus clouds. If the sky currently has 3 cumulonimbus clouds, how many cirrus clouds are in the sky at this moment? | To solve this problem, we need to determine the number of cirrus clouds based on the relationships given:
1. There are 4 times as many cirrus clouds as cumulus clouds.
2. There are 12 times as many cumulus clouds as cumulonimbus clouds.
3. There are 3 cumulonimbus clouds.
Let's define:
- \( C_{cirrus} \) = Number of ... |
Given that $f(a+b)=f(a)⋅f(b)$, where $(a,b∈N)$, and $f(1)=2$, find the value of $\frac {f(2)}{f(1)}+ \frac {f(4)}{f(3)}+ \frac {f(6)}{f(5)}+ \frac {f(8)}{f(7)}+ \frac {f(10)}{f(9)}$. | To solve this problem, we need to determine the values of the function \( f \) for the given natural numbers and then compute the required sum. Given the function \( f \) satisfies the functional equation \( f(a + b) = f(a) \cdot f(b) \) and the initial condition \( f(1) = 2 \), we can deduce a general form for \( f(n)... |
The number of diagonals that can be drawn in a polygon of 100 sides is:
$\textbf{(A)}\ 4850 \qquad \textbf{(B)}\ 4950\qquad \textbf{(C)}\ 9900 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 8800$ | To solve the problem of finding the number of diagonals in a polygon with \( n \) sides, we can use the formula:
\[
\text{Number of Diagonals} = \frac{n(n-3)}{2}
\]
For a polygon with 100 sides (\( n = 100 \)), we substitute \( n = 100 \) into the formula.
Let's perform this calculation using Python and the SymPy li... |
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$ , where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$ , satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does ... | To find how many sets of lattice points \(\{P_1, P_2, \ldots, P_M\}\) satisfy that the slope of the line \(P_iP_j\) is positive for each \(1 \le i < j \le M\), we observe the following:
1. The slope between two points \((x_i, y_i)\) and \((x_j, y_j)\) is \(\frac{y_j - y_i}{x_j - x_i}\).
2. For the slope to be positive... |
When \\(x > -1\\), the minimum value of the function \\(y=x+ \dfrac {1}{x+1}\\) is \_\_\_\_\_. | To solve this problem, we need to find the minimum value of the function \( y = x + \dfrac{1}{x+1} \) for \( x > -1 \). Here are the steps we will take:
1. **Find the First Derivative:** The first derivative of the function will help locate the critical points.
2. **Set the First Derivative to Zero:** Solving this equ... |
John weighs 220 pounds when he starts exercising. He manages to lose 10% of his body weight. He then gains back 2 pounds. How much does he weigh at the end? | To solve the problem, we'll break it down into the following steps:
1. Determine the weight John loses by calculating 10% of his initial weight.
2. Subtract the lost weight from his initial weight to get his weight after losing 10%.
3. Add the 2 pounds he gains back to get his final weight.
Let's begin by performing ... |
Suppose we roll a standard fair 6-sided die. What is the probability that a perfect square is rolled? | First, let's identify the possible outcomes when rolling a standard 6-sided die. The possible outcomes are:
\[ \{1, 2, 3, 4, 5, 6\} \]
Next, let's determine which of these outcomes are perfect squares. The numbers from 1 to 6 that are perfect squares are:
- 1 (since \(1^2 = 1\))
- 4 (since \(2^2 = 4\))
So there ar... |
In a factory that employed 852 people, 25% more workers have just been hired. How many employees are there now in the factory? | To determine the total number of employees after a 25% increase, we can follow these steps:
1. Calculate 25% of the initial number of employees.
2. Add this 25% to the initial number of employees to get the final count.
Let's break down the calculations:
- Initial number of employees = 852
- Percentage increase = 25... |
Consider a three-digit number where the unit digit is non-zero. When the digits of the number are reversed, the resulting number is called the reverse number of the original. For example, the reverse number of 325 is 523, and the reverse number of 1376 is 6731. There are two three-digit numbers whose sum is 1372. Find ... | Given the problem, we need to identify two three-digit numbers whose sum is 1372. Then, we need to find the maximum possible sum of their reverse numbers. Here’s the step-by-step approach:
1. Iterate over all possible pairs of three-digit numbers \( (a, b) \) such that \( a + b = 1372 \).
2. Check the conditions that ... |
Given vectors $\vec{a}=(1,2)$ and $\vec{b}=(x,-2)$. If $\vec{a}$ is perpendicular to $\vec{b}$, then find the value of $x$. | To solve the problem, we need to find the value of \(x\) such that the vectors \(\vec{a} = (1, 2)\) and \(\vec{b} = (x, -2)\) are perpendicular.
Two vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular if and only if their dot product is zero.
The dot product of \(\vec{a}\) and \(\vec{b}\) is given by:
\[ \vec{a} ... |
Find the smallest number in which all digits are different and the sum of all digits equals 32. | To solve this problem, we'll use a systematic approach to find the smallest number where all digits are different, and their sum equals 32. Here's the plan:
1. We'll start with the largest possible digits and work our way down, making sure each digit is unique.
2. We'll ensure that the sum equals 32.
3. Python can han... |
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 $10$ . | To solve the problem, we need to find a positive integer \( n \) less than 10,000 that satisfies the following conditions:
1. In base 4, \( n \) and \( 3n \) have the same number of digits.
2. In base 8, \( n \) and \( 7n \) have the same number of digits.
3. In base 16, \( n \) and \( 15n \) have the same number of d... |
Given the complex number $\frac{1+i}{1-i}$ represented in the form of $a+bi (a,b \in \mathbb{R})$, find the value of $a+b$. | To solve the problem, we need to simplify the complex number \(\frac{1+i}{1-i}\) and then express it in the form \(a + bi\). Finally, we will find the value of \(a + b\).
Here are the steps:
1. **Simplify the complex number:**
To simplify \(\frac{1+i}{1-i}\), we can multiply the numerator and the denominator by th... |
Use the Horner's method to calculate the polynomial $f(x) = x^7 - 2x^6 + 3x^3 - 4x^2 + 1$ when $x = 2$, find $v_4$. | To solve the problem using Horner's method, we need to evaluate the polynomial \( f(x) = x^7 - 2x^6 + 3x^3 - 4x^2 + 1 \) at \( x = 2 \).
Horner's method is a numerical algorithm to evaluate polynomials efficiently. The method rewrites the polynomial in a nested form.
For example, the polynomial \( f(x) = x^7 - 2x^6... |
Calculate: $\sqrt{9}-\sqrt{4}=\_\_\_\_\_\_$. | To solve the given problem, we will follow these steps:
1. Define the square root expressions.
2. Calculate the square roots.
3. Subtract the results to get the final answer.
Let's implement this in Python using the `sympy` library:
```python
import sympy as sp
# Define the square root expressions
sqrt_9 = sp.sqrt(... |
All possible sequences of numbers $-1$ and $+1$ of length $100$ are considered. For each of them, the square of the sum of the terms is calculated. Find the arithmetic average of the resulting values. | To solve this problem, we will need to consider all possible sequences of \(-1\) and \(+1\) of length 100. Since each term in the sequence can either be \(-1\) or \(+1\), there are \(2^{100}\) possible sequences. Calculating the sum and the square of the sum for each of these sequences would be extremely computationall... |
Greta received 10 more letters in the mail than her brother. Greta's mother received twice the total number of letters Greta and her brother received. Calculate the total number of letters the three received if Greta's brother received 40 letters. | Let's solve the problem step by step.
1. Let the number of letters Greta's brother received be \( B \), which is given as 40.
2. Greta received 10 more letters than her brother, so the number of letters Greta received (\( G \)) is \( B + 10 \).
3. Greta's mother received twice the total number of letters Greta and her... |
Elizabeth has 10 reusable water bottles. She loses 2 water bottles at school. Then someone steals 1 of her water bottles at dance practice. To avoid any future theft of her water bottles, Elizabeth places 3 stickers on each of her remaining bottles. How many stickers does Elizabeth use in total on her water bottles? | We can solve the problem step-by-step:
1. Start with the initial number of water bottles.
2. Subtract the number of bottles lost at school.
3. Subtract the number of bottles stolen at dance practice.
4. Calculate the remaining number of bottles.
5. Multiply the number of remaining bottles by the number of stickers per... |
Given point $P$ on the parabola $x^{2}=2y$, draw a line $PQ$ perpendicular to the $x$-axis with foot $Q$. Let $M$ be the midpoint of $PQ$.
(I) Find the equation of the locus $E$ of point $M$.
(II) Let point $S(-4,4)$, and a line $l$ through $N(4,5)$ intersects the locus $E$ at points $A$ and $B$. If the slopes of lines... | Let's break down the problem into smaller, manageable parts and solve each part step-by-step using sympy.
### Part (I): Find the equation of the locus \( E \) of point \( M \).
Given:
1. A point \( P \) on the parabola \( x^2 = 2y \).
2. A line \( PQ \) perpendicular to the \( x \)-axis with foot \( Q \).
3. \( M \) ... |
Given that the first term of the geometric sequence {a<sub>n</sub>} is 1, and a<sub>6</sub> + a<sub>4</sub> = 2(a<sub>3</sub> + a<sub>1</sub>), find the value of a<sub>1</sub>a<sub>2</sub>a<sub>3</sub>…a<sub>7</sub>. | To solve this problem, we need to utilize the properties of geometric sequences and the given conditions to find the value of the product \(a_1 \cdot a_2 \cdot a_3 \cdot \ldots \cdot a_7\).
In a geometric sequence, the terms can be represented as:
\[a_n = a_1 \cdot r^{n-1}\]
where \(a_1\) is the first term and \(r\) i... |
Given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms being $S_n$, if $S_4 = 8$ and $S_8 = 20$, find the value of $a_{11} + a_{12} + a_{13} + a_{14}$. | To solve this problem, we need to follow these steps:
1. Express the sum of the first \( n \) terms of an arithmetic sequence \( \{a_n\} \) in terms of the first term \( a \) and the common difference \( d \).
2. Use the given conditions \( S_4 = 8 \) and \( S_8 = 20 \) to form a system of equations.
3. Solve the syst... |
In an opaque box, there are 20 blue lucky stars, some red lucky stars, and 15 yellow lucky stars. All these lucky stars are identical except for their colors. Xiao Ming randomly picks out a lucky star from the box, notes its color, and puts it back. Through a large number of repeated experiments, he finds that the freq... | To solve this problem, let's use the concept of probability.
Given:
- 20 blue lucky stars
- Some red lucky stars (let's denote this number as \( R \))
- 15 yellow lucky stars
- The probability of picking a red lucky star is 0.5
First, let's calculate the total number of lucky stars. The total number of lucky stars \... |
Define "geometric sequence": In a sequence, if the product of each term and its next term is the same constant, then this sequence is called a geometric sequence, and this constant is called the common product of the sequence. It is known that the sequence $\{a_n\}$ is a geometric sequence, and $a_1 = 3$, the common pr... | To solve the problem and verify the given geometric sequence using Python and SymPy, we need to follow these steps:
1. **Understand the Problem:**
- We are given that \(a_1 = 3\) and the common product is 15.
- A geometric sequence is defined such that the product of each term and its next term is the same const... |
Gordon owns 3 restaurants, his first restaurant serves 20 meals, his second restaurant serves 40 meals, and his third restaurant serves 50 meals per day. How many meals do his 3 restaurants serve per week? | Let's begin by calculating the total number of meals served by all three restaurants per day. Then, we can multiply that number by 7 to find the total number of meals served per week.
Here are the individual daily meal counts:
- First restaurant: 20 meals
- Second restaurant: 40 meals
- Third restaurant: 50 meals
W... |
Given a triangle with vertices \( A(1,-1), B(-2,1), C(3,-5) \). Write the equation of the perpendicular from vertex \( A \) dropped onto the median drawn from vertex \( B \). | To solve this problem, we need to follow these steps:
1. Calculate the midpoint \(M\) of the segment \(AC\).
2. Determine the equation of the median \(BM\) from vertex \(B\) to the midpoint \(M\).
3. Find the slope of the line \(BM\) and then determine the slope of the line perpendicular to \(BM\).
4. Write the equati... |
Use the Euclidean algorithm to find the greatest common divisor (GCD) of 228 and 1995. | Let's solve this problem using the Euclidean algorithm to find the greatest common divisor (GCD) of 228 and 1995.
The Euclidean algorithm is a method for finding the GCD of two numbers based on the principle that the GCD of two numbers also divides their difference. The algorithm can be summarized as follows:
1. Give... |
(The full score for this question is 12 points) Zhengding Middle School organizes students from the east and west schools to use their Sunday time to participate in a charity activity at Hope Primary School, with at least one student from each school participating. It is known that the round-trip bus fare for each stud... | To solve this problem, we can break it down into the following steps:
1. Define the variables:
- \( x \): Number of students from the east school.
- \( y \): Number of students from the west school.
2. Define the constraints:
- Each student from the east school has a round-trip fare of 3 yuan: \( 3x \).
-... |
Let $\mathbf{a}$ and $\mathbf{b}$ be nonzero vectors such that
\[\|\mathbf{a}\| = \|\mathbf{b}\| = \|\mathbf{a} + \mathbf{b}\|.\]Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees. | To find the angle between vectors \(\mathbf{a}\) and \(\mathbf{b}\) given that \(\|\mathbf{a}\| = \|\mathbf{b}\| = \|\mathbf{a} + \mathbf{b}\|\), we need to use the properties of vector norms and dot products. Let's break down the problem into manageable steps:
1. **Understand the Given Conditions:**
We have three ... |
A florist received between 300 and 400 roses for decorating a celebration. When he placed them in vases with 21 roses in each, there were 13 roses left over. When arranging them with 15 roses in each vase, eight roses were missing. How many roses were there in total? | To solve the problem, we'll start by setting up some equations based on the conditions given:
1. Let \( x \) be the total number of roses.
2. From the first condition, placing the roses in vases with 21 roses each leaves a remainder of 13:
\[
x \equiv 13 \pmod{21}
\]
3. From the second condition, arranging t... |
Given a triangle \(ABC\). Points \(K, L,\) and \(M\) are placed on the plane such that triangles \(KAM, CLM,\) and \(KLB\) are all congruent to triangle \(KLM\). What inequality sign should be placed between the perimeter of triangle \(KLM\) and the semiperimeter of triangle \(ABC\)?
The vertices of the triangles are ... | To solve the problem, we need to compare the perimeter of triangle \(KLM\) with the semiperimeter of triangle \(ABC\) given that \(KAM \cong CLM \cong KLB \cong KLM\).
Here is the step-by-step breakdown:
1. **Understanding the congruence**:
- Each of the triangles \(KAM\), \(CLM\), \(KLB\), and \(KLM\) are congrue... |
On the board, there are three natural numbers $a, b, c$, for which the following holds:
- the greatest common divisor of the numbers $a$ and $b$ is 15,
- the greatest common divisor of the numbers $b$ and $c$ is 6,
- the product of the numbers $b$ and $c$ is 1800,
- the least common multiple of the numbers $a$ and $b$ ... | To solve this problem, we need to find the natural numbers \( a \), \( b \), and \( c \) that satisfy the given conditions. Let's break down the problem step-by-step:
1. **Understand the Given Conditions:**
- \(\gcd(a, b) = 15\)
- \(\gcd(b, c) = 6\)
- \(b \cdot c = 1800\)
- \(\text{lcm}(a, b) = 3150\)
2. ... |
Given the geometric sequence $\{a_{n}\}$, where $a_{2}=2$, $a_{6}=8$, find the value of $a_{3}a_{4}a_{5}$. | To solve the problem, we need to find the value of the product \(a_3 a_4 a_5\) for the geometric sequence \(\{a_n\}\) where the second term \(a_2 = 2\) and the sixth term \(a_6 = 8\).
Let's break it down into manageable steps:
1. **Understand the Terms of the Geometric Sequence:**
For a geometric sequence, the nth... |
Zuzka had five cubic blocks with side lengths of 1 to 5 units. She glued these blocks into a tower, placing smaller blocks on larger ones with their faces completely aligned. Afterward, Zuzka painted the entire tower except for the base face. Each paint container could paint an area of exactly 5 square units.
How many... | To determine the number of paint containers Zuzka needed, we need to calculate the total painted surface area of the tower.
Given that the cubes are stacked with smaller blocks on top of larger ones, the side lengths of the blocks are 1 unit, 2 units, 3 units, 4 units, and 5 units respectively from top to bottom.
###... |
Given a finite set of cards. Each card has either the number 1 or the number -1 written on it (each card has exactly one number). There are 100 more cards with -1 than cards with 1. If you find the product of the numbers on each pair of different cards and sum all these products, the result is 1000. How many cards have... | Let's denote the number of cards with \(1\) as \( x \) and the number of cards with \(-1\) as \( y \).
According to the problem:
1. \( y = x + 100 \) (there are 100 more cards with -1 than cards with 1).
2. The sum of the products of the numbers on each pair of different cards is \(1000\).
Firstly, let's consider the... |
The midpoint of one side of a triangle and the bases of the altitudes dropped on the two other sides form an equilateral triangle.
Is it true that the original triangle is also equilateral? | To solve the problem, we need to determine whether the original triangle is also equilateral given that the midpoint of one side and the bases of the altitudes dropped on the two other sides form an equilateral triangle.
We'll break this down into smaller steps:
1. **Understand the Problem:**
- Let's denote the ve... |
The two real roots of the equation $x^{2}+x+c=0$ are $x_{1}$ and $x_{2}$. If $x_{1}^{2}x_{2}+x_{2}^{2}x_{1}=3$, then the real number $c$ is $\_\_\_\_\_\_$. | To solve this problem, we need to determine the real number \( c \) for which the quadratic equation \( x^2 + x + c = 0 \) has roots \( x_1 \) and \( x_2 \) such that the equation \( x_1^2 x_2 + x_2^2 x_1 = 3 \) holds true.
Here is the step-by-step plan:
1. **Use Vieta's formulas:**
For a quadratic equation \( ax^... |
For normal operation, the bus depot must have at least eight buses on the line, and currently, it has ten. The probability that each bus does not go on the line is 0.1. Find the probability of normal operation of the bus depot for the coming day. | To solve this problem, we need to calculate the probability that at least 8 out of 10 buses will be operational.
This can be modeled using the binomial distribution. The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials. In this case:
- The number of trials \(n... |
Find all polynomials \( P \in \mathbb{R}[X] \) such that \( P(0)=0 \) and
\[
P(x^2 + 1) = P(x)^2 + 1
\]
for all real numbers \( x \). | To find all polynomials \( P \in \mathbb{R}[X] \) that satisfy \( P(0) = 0 \) and the functional equation
\[
P(x^2 + 1) = P(x)^2 + 1
\]
for all real numbers \( x \), we will proceed as follows:
1. **Understand the given conditions:**
- The polynomial \( P \) should satisfy \( P(0) = 0 \).
- The polynomial sh... |
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
$\text... | To solve this problem, we need to simulate the game described and determine who is the last person remaining in the circle. The counting will start from Arn and proceed in order around the circle: Arn, Bob, Cyd, Dan, Eve, and Fon. Any person whose count either contains the digit 7 or is a multiple of 7 will leave the c... |
Compute the sum of the number $10 - \sqrt{2018}$ and its radical conjugate. | To solve this problem, we need to compute the sum of the number \(10 - \sqrt{2018}\) and its radical conjugate. The radical conjugate of a number of the form \(a - \sqrt{b}\) is \(a + \sqrt{b}\).
Given \( x = 10 - \sqrt{2018} \), its radical conjugate \( y \) would be \( 10 + \sqrt{2018} \).
We need to find the sum \... |
How many integers between $100$ and $150$ have three different digits in increasing order? One such integer is $129$. | To solve the problem, we need to count all integers between 100 and 150 that have three distinct digits arranged in increasing order.
We'll iterate through the integers from 100 to 150, check the conditions, and count how many numbers satisfy the criteria.
Here are the steps we'll follow:
1. Iterate through each num... |
If the maximum value and the minimum value of the function \( f(x) = \frac{a + \sin x}{2 + \cos x} + b \tan x \) sum up to 4, then find \( a + b \). | To solve for \( a + b \) given the function \( f(x) = \frac{a + \sin x}{2 + \cos x} + b \tan x \), and the information that the sum of its maximum and minimum values is 4, we need to:
1. Analyze the function \( f(x) \) to find its maximum and minimum values.
2. Use the given sum of these extrema to set up equations in... |
When a piece of paper with digits written on it is turned upside down, the digits $0, 1, 8$ remain the same, while $6$ and $9$ switch places and all other digits do not make sense when viewed upside down. Find the number of 9-digit numbers that remain the same when the piece of paper is turned upside down. | Let's start by identifying the conditions under which a 9-digit number remains the same when turned upside down:
- The digits involved must all be one of the valid digits: 0, 1, 8, 6, 9.
- The number must be symmetrical, meaning the digits must match their upside-down counterparts.
For a number to meet this condition... |
The formula for calculating the Richter magnitude $M$ is: $M = \lg A - \lg A_0$, where $A$ is the maximum amplitude recorded by the seismograph, and $A_0$ is the amplitude of the corresponding standard earthquake. Suppose in an earthquake, the seismograph records a maximum amplitude of 1000, and the amplitude of the st... | Let's solve the problem step-by-step using Python and SymPy.
### Step 1: Calculate the Earthquake Magnitude
Given:
- \(A = 1000\) (maximum amplitude recorded by the seismograph)
- \(A_0 = 0.001\) (amplitude of the corresponding standard earthquake)
The formula for Richter magnitude \(M\) is:
\[
M = \lg A - \lg A_0
\]... |
Given that $|\overrightarrow{OA}|=2$, $|\overrightarrow{OB}|=3$, $\angle AOB= \frac{\pi}{3}$, and $\overrightarrow{OC}= \frac{1}{3}\overrightarrow{OA}+ \frac{1}{9}\overrightarrow{OB}$, find the value of $\overrightarrow{OB} \cdot \overrightarrow{OC}$. | To solve the problem, we need to find the dot product \(\overrightarrow{OB} \cdot \overrightarrow{OC}\) given the magnitudes and angle between vectors \(\overrightarrow{OA}\), \(\overrightarrow{OB}\), and the expression for \(\overrightarrow{OC}\).
Here's the step-by-step plan:
1. **Express \(\overrightarrow{OA}\) an... |
Let \( x, y, z, \) and \( t \) be non-negative numbers such that \( x + y + z + t = 5 \). Prove the inequality
\[
\sqrt{x^{2} + y^{2}} + \sqrt{x^{2} + 1} + \sqrt{z^{2} + y^{2}} + \sqrt{z^{2} + t^{2}} + \sqrt{t^{2} + 9} \geq 10
\] | To solve this problem, we need to establish whether the given inequality holds for \(\sqrt{x^2 + y^2} + \sqrt{x^2 + 1} + \sqrt{z^2 + y^2} + \sqrt{z^2 + t^2} + \sqrt{t^2 + 9} \geq 10\) given the constraint \(x + y + z + t = 5\).
We will use Python to generate potential values for \(x, y, z,\) and \(t\) that satisfy thi... |
Leah has $13$ coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth?
$\textbf {(A) } 33 \qquad \textbf {(B) } 35 \qquad \textbf {(C) } 37 \qquad \textbf {(D) } 39 \qquad \textbf {(E)... | To solve the problem, let's denote the number of pennies Leah has as \( p \) and the number of nickels as \( n \). According to the problem, we can set up the following equations based on the given information:
1. Leah has a total of 13 coins:
\[
p + n = 13
\]
2. If she had one more nickel, she would have... |
Greta worked 40 hours and was paid $12 per hour. Her friend Lisa earned $15 per hour at her job. How many hours would Lisa have to work to equal Greta's earnings for 40 hours? | To determine how many hours Lisa needs to work to match Greta's earnings for 40 hours, we can follow these steps:
1. Calculate Greta's total earnings by multiplying her hourly wage by the number of hours she worked.
2. Divide Greta's total earnings by Lisa's hourly wage to find the number of hours Lisa needs to work.
... |
Given the complex number $z= \frac{5}{3+4i}$, find the absolute value of its conjugate, $|\overline{z}|$. | To find the absolute value of the conjugate of the given complex number \( z = \frac{5}{3+4i} \), we can follow these steps:
1. **Find the conjugate of the complex number \( z \).**
2. **Calculate the absolute value of the conjugate.**
Let's break it down and solve it step-by-step.
### Step 1: Find the Conjugate of ... |
12 balls numbered 1 through 12 are placed in a bin. In how many ways can 3 balls be drawn, in order, from the bin, if each ball remains outside the bin after it is drawn? | To solve this problem, we need to determine the number of ways to draw 3 balls in a specific order from a set of 12 balls, where each ball is removed from the bin once it is drawn.
Given that the order in which the balls are drawn matters, this is a permutation problem without replacement.
For permutation problems ... |
Given the areas of the three squares in the figure, what is the area of the interior triangle? [asy]
/* AMC8 2003 #6 Problem */
draw((0,0)--(12,0)--(12,5)--cycle);
draw((12,0)--(17,0)--(17,5)--(12,5));
draw((0,0)--(0,-12)--(12,-12)--(12,0));
draw((0,0)--(-5,12)--(7,17)--(12,5));
draw((11,0)--(11,1)--(12,1));
label("169... | To determine the area of the interior triangle, let's start by examining the given data and forming equations around the figure:
The problem gives us the following square areas:
- Area of the first square: \(169\)
- Area of the second square: \(25\)
- Area of the third square: \(144\)
From these areas, we can determi... |
If the graph of the function \(y=f(x)\) is shifted 2 units to the left and 2 units downward, resulting in the graph of the function \( y=2^{x} \), then \( f(x) = \quad .\) | To solve the problem, we need to determine the original function \(f(x)\) given the transformed function \( y=2^x \), which is the result of shifting the graph of \( y=f(x) \) 2 units to the left and 2 units downward.
Let's break it down into manageable steps:
1. **Understand the Transformation:**
- Shifting a fun... |
If a positive four-digit number's thousand digit \\(a\\), hundred digit \\(b\\), ten digit \\(c\\), and unit digit \\(d\\) satisfy the relation \\((a-b)(c-d) < 0\\), then it is called a "Rainbow Four-Digit Number", for example, \\(2012\\) is a "Rainbow Four-Digit Number". How many "Rainbow Four-Digit Numbers" are there... | To solve this problem, we need to count the number of four-digit numbers \( abcd \) that satisfy the condition \( (a - b)(c - d) < 0 \). Let's break this down into steps:
1. **Understand the Condition \( (a - b)(c - d) < 0 \):**
- \( (a - b)(c - d) < 0 \) means either \( (a - b) \) is positive and \( (c - d) \) is ... |
The last digit of $2^{m+2006} + 2^m$ (where $m$ is a positive integer) is. | To solve the problem of finding the last digit of \(2^{m+2006} + 2^m\), we need to understand the pattern of the last digits of powers of 2.
Let's break down the steps required to solve the problem:
1. **Identify the Pattern of Last Digits:**
The last digits of powers of 2 repeat in a cycle: 2, 4, 8, 6. This cycle... |
Hannah is at a restaurant. She spends $23 in total on an entree and a dessert. If the entree costs $5 more than the dessert, how much does the entree cost, in dollars? | To solve the problem, we can use the following steps:
1. Let \( x \) be the cost of the dessert in dollars.
2. The cost of the entree, therefore, is \( x + 5 \) dollars (since it costs $5 more than the dessert).
3. The total cost Hannah spends is given as $23. So, the equation becomes:
\[
x + (x + 5) = 23
\]
4. Simpli... |
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