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There are 16 different cards, 4 of each in the colors red, yellow, blue, and green. If 3 cards are drawn at random, the requirement is that the 3 cards must not all be of the same color. The number of different ways this can be done is __________. | First, let's calculate the total number of ways to draw 3 cards out of 16 without any restrictions. This can be done using the combination formula:
$$ C(16, 3) = \frac{16!}{3!(16-3)!} $$
Now, let's identify the restricted scenarios that we need to exclude from the total count. Specifically, we need to exclude the case... | null | null |
Find all integer values of $a$ so that the polynomial
\[x^3 + 3x^2 + ax + 7 = 0\]has at least one integer root. Enter all possible values of $a,$ separated by commas. | To find all integer values of $a$ such that the polynomial
\[x^3 + 3x^2 + ax + 7 = 0\]
has at least one integer root, we apply the Integer Root Theorem. This theorem tells us that any integer root of the polynomial must be a divisor of the constant term, which in this case is 7. Therefore, the possible integer roots ar... | null | null |
For given numbers \( n \geq 2 \) and \( a > 0 \), find the maximum value of the sum \( \sum_{i=1}^{n-1} x_i x_{i+1} \) subject to the conditions \( x_i \geq 0 \) for \( i = 1, \ldots, n \) and \( x_1 + \ldots + x_n = a \). |
Given the problem of finding the maximum value of the sum
\[
\sum_{i=1}^{n-1} x_i x_{i+1}
\]
under the constraints \(x_i \geq 0\) for \(i = 1, \ldots, n\) and
\[
x_1 + x_2 + \ldots + x_n = a,
\]
we proceed as follows:
1. **Identify the Maximum Term:**
Let \( \max \{x_1, x_2, \ldots, x_n\} = x_k \). Without l... | null | null |
Express $213_{8}-142_{7}$ as a base 10 integer. | To express $213_{8}-142_{7}$ as a base 10 integer, we first convert each number from its given base to base 10.
For $213_{8}$:
\begin{align*}
213_{8} &= 2(8^2)+ 1(8^1) +3(8^0) \\
&= 2(64)+ 1(8) +3(1) \\
&= 128 + 8 + 3 \\
&= 139
\end{align*}
For $142_{7}$:
\begin{align*}
142_{7} &= 1(7^2)+ 4(7^1) +2(7^0) \\
&= 1(49)+ ... | null | null |
A right square pyramid with base edges of length 12 units each and a total height of 18 units is cut by a plane that is parallel to its base and 6 units above its base. What is the volume, in cubic units, of the smaller pyramid that is cut off by this plane? | Let the vertices of the large pyramid be \( A \) (apex) and \( B, C, D, E \) (base vertices).
- The pyramid height \( h = 18 \) units.
- The base edge length \( s = 12 \) units.
- The cutoff plane is parallel to the base and located 6 units above the base, making the height of the smaller pyramid \( h' = 6 \) units.
... | null | null |
A bag of chips weighs 20 ounces, and a tin of cookies weighs 9 ounces. If Jasmine buys 6 bags of chips and 4 times as many tins of cookies, how many pounds does she have to carry? | To solve this problem, we follow the steps below:
1. Calculate the total weight of the bags of chips Jasmine buys:
- Each bag weighs 20 ounces, and she buys 6 bags.
- Therefore, the total weight of the bags of chips is $20 \times 6 = 120$ ounces.
2. Determine the number of tins of cookies Jasmine buys:
- She... | null | null |
The farm tax is levied on the 60% of the cultivated land. The tax department collected a total of $5000 through the farm tax from the village of Mr. William. Mr. William paid only $480 as farm tax. What is the percentage of total land of Mr. William over the total taxable land of the village? | Let's denote the total taxable land of the village as \( T \) (which is 60% of the total cultivated land in the village). The tax collected from the entire village is $5000, and Mr. William paid $480 of that.
The percentage of the total land of Mr. William over the total taxable land of the village can be calculated b... | null | null |
Northside's Drum and Bugle Corps raised money for a trip. The drummers and bugle players kept separate sales records. According to the double bar graph, in what month did one group's sales exceed the other's by the greatest percent?
$\text{(A)}\ \text{Jan}\qquad\text{(B)}\ \text{Feb}\qquad\text{(C)}\ \text{Mar}\qquad\... | To determine in which month one group's sales exceeded the other's by the greatest percent, we need to calculate the percentage difference between the sales of the drummers and the bugle players for each month. The month with the highest percentage difference will be our answer.
Let's denote the sales of the drummers ... | null | null |
Soft drinks are on sale at the grocery store for 2.99 dollars for a 12 pack. How much would each can of soft drink cost if purchased individually, in dollars? | To find the cost of each can of soft drink when purchased individually, you would divide the total cost of the 12 pack by the number of cans in the pack.
Total cost of 12 pack = $2.99
Number of cans in the pack = 12
Cost per can = Total cost of 12 pack / Number of cans in the pack
Cost per can = $2.99 / 12
Cost per c... | null | null |
Find all nonnegative integers $a, b, c$ such that $$ \sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}. $$ | 1. **Initial Equation and Squaring Both Sides:**
Given the equation:
\[
\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}
\]
Squaring both sides, we get:
\[
(\sqrt{a} + \sqrt{b} + \sqrt{c})^2 = (\sqrt{2014})^2
\]
Simplifying, we obtain:
\[
a + b + c + 2\sqrt{ab} + 2\sqrt{bc} + 2\sqrt{ac} = 2... | null | null |
Two tangents from a point $A$ touch a circle at points $B$ and $C$. The arcs between $B$ and $C$ are in the ratio $3:5$. What is the degree measure of $\angle BAC$?
**A) 60°**
**B) 62.5°**
**C) 67.5°**
**D) 70°**
**E) 75°** | 1. **Geometry and Tangency Property**:
From $A$, two tangents touch a circle at points $B$ and $C$. The radii to these points ($OB$ and $OC$) are perpendicular to the tangents at points $B$ and $C$ respectively, making $\angle OBA = \angle OCA = 90^\circ$.
2. **Cyclic Quadrilateral Insight**:
With $\angle OBA$ ... | null | null |
Find the shortest distance from a point on the curve y = ln(x) to the line x - y + 3 = 0. | According to the given problem, we have $y' = \frac{1}{x}$.
Let $\frac{1}{x}$ = -1, then x = 1.
So the point of tangency is (1, ln(1)) = (1, 0).
Now we'll find the distance between this point (1, 0) and the given line x - y + 3 = 0.
First, rewrite the line equation in standard form: $x - y + 3 = 0$ becomes $x - y = ... | null | null |
Suppose $S = i^n + i^{k(-n)}$, where $i = \sqrt{-1}$, $n$ is an integer, and $k = 2n \mod 4$. Find the possible distinct values for $S$.
A. 2
B. 3
C. 4
D. More than 4 | 1. **Restating the Problem**:
Given $S = i^n + i^{k(-n)}$, with $i^4 = 1$ and $k = 2n \mod 4$, the values of $i$ cycle every four powers. Hence, $i^5 = i$, $i^6 = -1$, etc.
2. **Derive Expressions**: Determine $k(-n)$:
- $k = 2n \mod 4$.
3. **Evaluate $S$ for Cyclic Powers**:
- $n=0$: $k = 0$ thus $k(-n)... | null | null |
The decreasing interval of the function $f(x)=x^{3}-3x$ is $(\quad\quad)$.
A: $(-\infty,-\frac{\sqrt{6}}{2})$ or $(\frac{\sqrt{6}}{2},+\infty)$
B: $(-1,1)$
C: $(-\infty,-1)$ or $(1,+\infty)$
D: $(-\frac{\sqrt{6}}{2},\frac{\sqrt{6}}{2})$ | The derivative of the function $f(x)=x^{3}-3x$ is $f'(x)=3x^{2}-3$. By solving $f'(x)<0$, we obtain $(-1<x<1)$.
Hence, the decreasing interval of the function is $\boxed{(-1,1)}$.
To find the decreasing interval of the function $f(x)=x^{3}-3x$, we differentiate the function and then find the range of $x$ for which th... | null | null |
The perimeter of triangle \( ABC \) is 2. On side \( AC \), point \( P \) is marked, and on segment \( CP \), point \( Q \) is marked such that \( 2AP = AB \) and \( 2QC = BC \). Prove that the perimeter of triangle \( BPQ \) is greater than 1. |
1. **Introduce the Variables:**
Let $AB = c$, $AC = b$, and $BC = a$. The perimeter of triangle $ABC$ is given to be $2$.
\[
a + b + c = 2
\]
2. **Conditions and Pivotal Points:**
Given conditions:
- Point $P$ on $AC$ such that $2AP = AB \Rightarrow AP = \frac{c}{2}$.
- Point $Q$ on $CP$... | null | null |
Given a sequence $\{a_n\}$ with the sum of the first $n$ terms denoted as $S_n$, where $S_n=\frac{a_n-1}{3}$ $(n\in \mathbb{N}^*)$.
(Ⅰ) Find $a_1$ and $a_2$;
(Ⅱ) Prove that the sequence $\{a_n\}$ is a geometric sequence. | (Ⅰ) Solution: Since $S_n= \frac{1}{3}(a_n−1)$ $(n\in \mathbb{N}^*)$,
When $n=1$, we have $a_1= \frac{1}{3}(a_1−1)$,
Solving this gives $a_1= -\frac{1}{2}$,
When $n=2$, we have $a_1+a_2= \frac{1}{3}(a_2−1)$,
Solving this gives $a_2= \frac{1}{4}$.
(Ⅱ) Proof: When $n\geqslant 2$, $a_n=S_n-S_{n-1}= \frac{(a_n-1)}{3}- ... | null | null |
Triangle $XYZ$ has $XZ=5$, $YZ=12$, and $XY=13$. Point $W$ is on $\overline{XY}$, and $\overline{ZW}$ bisects $\angle YZX$. The inscribed circles of $\triangle ZWX$ and $\triangle ZWY$ have radii $r_x$ and $r_y$, respectively. Calculate the ratio $\frac{r_x}{r_y}$.
A) $\frac{1}{12}$
B) $\frac{1}{10}$
C) $\frac{1}{6}$
D... | 1. **Type of Triangle and Pythagorean Theorem**: Triangle $XYZ$ with sides $XZ=5$, $YZ=12$, and $XY=13$ is a right triangle because $5^2 + 12^2 = 13^2$. Thus, $\angle ZYX = 90^\circ$.
2. **Applying the Angle Bisector Theorem**: Since $\overline{ZW}$ bisects the angle $\angle YZX$, by the Angle Bisector Theorem:
\[
... | null | null |
Among the following sets of data, which pair has equal numerical values? ( )
A: $ (25)_{10} $ and $ (10110)_2 $
B: $ (13)_{10} $ and $ (1101)_2 $
C: $ (11)_{10} $ and $ (1100)_2 $
D: $ (10)_{10} $ and $ (10)_2 $ | For option A, let's convert $ (25)_{10} $ into binary:
\[
\begin{align*}
25 \div 2 &= 12 \text{ remainder } 1 \\
12 \div 2 &= 6 \text{ remainder } 0 \\
6 \div 2 &= 3 \text{ remainder } 0 \\
3 \div 2 &= 1 \text{ remainder } 1 \\
1 \div 2 &= 0 \text{ remainder } 1 \\
\end{align*}
\]
Reversed, the remainders give us $ (11... | null | null |
The cubic polynomial $q(x)$ satisfies $q(1) = 3,$ $q(6) = 23,$ $q(12) = 17,$ and $q(17) = 31.$ Find
\[
q(0) + q(1) + q(2) + \dots + q(18).
\] | The cubic polynomial $q(x)$ passes through the points $(1,3),$ $(6,23),$ $(12,17),$ and $(17,31).$ From these points, we hypothesize symmetric properties, as the center point here is $(9, 24)$ based on the average of the x-values and y-values.
Let $f(x) = q(x + 9) - 24.$ Then
\begin{align*}
f(-8) &= q(1) - 24 = -21,... | null | null |
Given a set $I=\{(x_1,x_2,x_3,x_4)|x_i\in\{1,2,\cdots,11\}\}$ . $A\subseteq I$ , satisfying that for any $(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\in A$ , there exists $i,j(1\leq i<j\leq4)$ , $(x_i-x_j)(y_i-y_j)<0$ . Find the maximum value of $|A|$ . | 1. **Define the set \( I \) and subset \( A \):**
- The set \( I \) is defined as \( I = \{(x_1, x_2, x_3, x_4) \mid x_i \in \{1, 2, \cdots, 11\}\} \).
- The subset \( A \subseteq I \) must satisfy the condition that for any two elements \((x_1, x_2, x_3, x_4)\) and \((y_1, y_2, y_3, y_4)\) in \( A \), there exis... | null | null |
Given: The solution to the equation $\dfrac{x+m}{3}-\dfrac{2x-1}{2}=m$ is a non-positive number. Find the range of values for $m$. | To solve the given equation $\dfrac{x+m}{3}-\dfrac{2x-1}{2}=m$ for $x$ and find the range of values for $m$ such that $x$ is a non-positive number, we proceed as follows:
1. Start with the given equation:
\[
\dfrac{x+m}{3}-\dfrac{2x-1}{2}=m
\]
2. To eliminate the fractions, find a common denominator, which i... | null | null |
At his cafe, Milton sells apple pie and peach pie slices. He cuts the apple pie into 8 slices. He cuts the peach pie into 6 slices. On the weekend, 56 customers ordered apple pie slices and 48 customers ordered peach pie slices. How many pies did Milton sell during the weekend? | To determine how many pies Milton sold during the weekend, we calculate the number of apple pies and peach pies separately, and then sum these quantities.
First, for the apple pies:
- Each apple pie is cut into 8 slices.
- With 56 customers ordering apple pie slices, the number of apple pies sold is calculated as foll... | null | null |
A school bought pencils and pens. A pencil costs $2.50, while a pen costs $3.50. The school receives a 10% discount on the purchase of more than 30 pencils and a 15% discount on the purchase of more than 50 pens. If the cost of their purchase exceeds $250, the school will get an additional 5% discount on the entire ord... | First, let's calculate the cost of the pencils and pens before any discounts.
Cost of 38 pencils = 38 * $2.50 = $95.00
Cost of 56 pens = 56 * $3.50 = $196.00
Total cost before discounts = $95.00 + $196.00 = $291.00
Now, let's apply the discounts.
Since the school bought more than 30 pencils, they get a 10% discount... | null | null |
Given a regular tetrahedron $OABC$ with side length 1. Let $D,\ E$ be the midpoints of the sides $AB,\ OC$ respectively.
Find the scalar product of two vectors $\overrightarrow{DE}$ and $\overrightarrow{AC}$ .
2012 Tokyo Institute of Technology entrance exam, problem 1-A | 1. **Define the vectors:**
Let $\overrightarrow{O} = \mathbf{0}$, $\overrightarrow{A} = \mathbf{a}$, $\overrightarrow{B} = \mathbf{b}$, and $\overrightarrow{C} = \mathbf{c}$. Since $OABC$ is a regular tetrahedron with side length 1, we have:
\[
|\overrightarrow{OA}| = |\overrightarrow{OB}| = |\overrightarrow{O... | null | null |
Let $\theta$ be an acute angle, and let
\[\sin \frac{\theta}{2} = \sqrt{\frac{x - 1}{2x}}.\]Express $\tan \theta$ in terms of $x.$ | To express $\tan \theta$ in terms of $x$, we start by using the given information and the double-angle formula. The problem gives us:
\[\sin \frac{\theta}{2} = \sqrt{\frac{x - 1}{2x}}.\]
From the double-angle formula for cosine, we have:
\[\cos \theta = 1 - 2 \sin^2 \frac{\theta}{2}.\]
Substituting the given value of... | null | null |
Distinct planes $p_1, p_2, ..., p_k$ intersect the interior of a regular tetrahedron $T$. Let $S$ be the union of the faces of $T$ and let $P = \bigcup_{j=1}^k p_j$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $T$. What... | #### Step 1: Identify the types of segments
- **Segment types**: Each face of the tetrahedron has edges, the segments connect midpoints of any two edges on the same face.
#### Step 2: Determine the planes containing these segments
- **Case 1: Plane containing all segments on one face**
- Each face of the tetrahedron... | null | null |
If the polar coordinate equation of line $l$ is $\rho \cos (\theta - \frac{\pi}{4}) = 3 \sqrt{2}$, and the distance between a point on curve $C: \rho = 1$ and line $l$ is $d$, find the maximum value of $d$. | This problem involves the conversion between polar and rectangular coordinate equations, the relationship between a line and a circle, the formula for the distance from a point to a line, and computational skills. We will find the rectangular coordinate equation of the line, the equation of the curve (circle), and then... | null | null |
(64 + 5 * 12) / (180 / 3) + sqrt(49) - 2^3 * (4!) = ? | Let's solve the equation step by step:
1. Calculate the value inside the parentheses and the factorial:
- 5 * 12 = 60
- 4! = 4 * 3 * 2 * 1 = 24
2. Add the result to 64:
- 64 + 60 = 124
3. Divide 180 by 3:
- 180 / 3 = 60
4. Divide the sum from step 2 by the result from step 3:
- 124 / 60 = 2.06666667 ... | null | null |
The sum of the coefficients of the expansion of the binomial $(x+ \frac{a}{x})^{5}$ is $-1$. Find the maximum value of the coefficients in this expansion. | **Analysis**
This question examines the application of the binomial theorem. According to the sum of all coefficients being $(1+a)^{5}=-1$, we can find $a=-2$, and then determine the coefficient of $x$ in the expansion to find the result.
**Solution**
Let $x=1$ in the expansion of $\left(x+ \frac{a}{x}\right)^{5}$, ... | null | null |
Let \(R\) be the set of real numbers. Consider a function \(f : R \to R\) that is twice-differentiable. There exists a function \(g : R \to R\) such that \(g(x) \geq 0\) and the equation \(f(x) + f''(x) = -xg(x)f'(x)\) holds for all \(x\). Prove that \(f(x)\) is bounded. |
Given that the function \( f : \mathbb{R} \to \mathbb{R} \) is twice-differentiable, and there exists a function \( g : \mathbb{R} \to \mathbb{R} \) such that \( g(x) \geq 0 \) and the equation
\[
f(x) + f''(x) = -xg(x)f'(x)
\]
holds for all \( x \in \mathbb{R} \), we are required to prove that \( f(x) \) is bounde... | null | null |
Let \( a \) and \( b \) be positive real numbers. Find the maximum value of
\[ 2(a - x)(x - \sqrt{x^2 + b^2}) \] in terms of \( a \) and \( b \). | Let \( t = x - \sqrt{x^2 + b^2} \). Then, \( t + x = -\sqrt{x^2 + b^2} \), and squaring both sides gives
\[ (t + x)^2 = x^2 + b^2. \]
Expanding and simplifying, we have
\[ t^2 + 2tx + x^2 = x^2 + b^2, \]
\[ x = \frac{b^2 - t^2}{2t}. \]
Substituting back into the expression:
\[
2(a - x)(x - \sqrt{x^2 + b^2}) = 2 \left(... | null | null |
Teresa is collecting pencils. She has 14 colored pencils and 35 black pencils. Her three younger siblings need pencils for class and their dad asks her to share all her pencils, giving each an equal number of pencils, regardless of color. He tells her she can keep 10 of them for herself. How many pencils does each sibl... | To solve the problem, we first determine the total number of pencils Teresa has. She has colored pencils and black pencils. The total can be calculated as follows:
- Total pencils = Number of colored pencils + Number of black pencils
- Total pencils = 14 + 35
- Total pencils = $49$
Teresa decides to keep 10 pencils f... | null | null |
Yella's computer usage last week was 91 hours. If she plans to use the computer 8 hours a day for this week, how much less is her computer usage for this week? | To solve this problem, we first need to calculate the total hours Yella plans to use the computer this week. Since she plans to use it for 8 hours a day for 7 days, we calculate the total hours as follows:
\[8 \text{ hours/day} \times 7 \text{ days} = 56 \text{ hours}\]
Next, we need to find out how much less her com... | null | null |
Julia rode 45 miles at 15 miles per hour and 15 miles at 45 miles per hour. What was her average speed, in miles per hour, for the entire trip? | Julia rode a total of $45 + 15 = 60$ miles. The 45-mile segment took $\frac{45}{15} = 3$ hours, and the 15-mile segment took $\frac{15}{45} = \frac{1}{3}$ hours. Therefore, the total time for the trip was $3 + \frac{1}{3} = \frac{10}{3}$ hours.
The average speed for the entire trip is calculated by dividing the total ... | null | null |
Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, with each edge measuring 1 meter. A bug starts from vertex $A$ and follows a new rule: at each vertex, it chooses one of the three edges to proceed, with the edge leading back to the previous vertex being half as likely to be chosen as each of the oth... | Let $Q(n)$ denote the probability that the bug is at vertex $A$ after crawling $n$ meters. At each node, the bug has a $\frac{1}{4}$ probability of returning to the previous vertex and $\frac{3}{8}$ probability for each of the other two options. This gives the recursion
\[ Q(n+1) = \frac{1}{4}Q(n) + \frac{3}{4} \left(... | null | null |
A rectangular array of books is an arrangement of the books in rows and columns such that each row contains the same number of books as every other row and each column contains the same number of books as every other column. If there must be at least three books in every row and column and all of the books in the room ... | To solve this problem, we first factorize 48, which is $48 = 2^4 \cdot 3$. The possible divisors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. We eliminate 1 and 2, as each row and column must contain at least three books. Thus, the valid divisors are 3, 4, 6, 8, 12, 16, 24. Each divisor corresponds to a unique arrangeme... | null | null |
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
$\textbf{(A)}\ 2\ \text{by}\ 4\qquad\textbf{(B)}\ \ 2\ \text{by}\ 6\qquad\textbf{(C)}\ \ 2\ \text{by}\ 8\qquad\textbf{(D)}\ 4\ \text{by}\ 4\qquad\textbf{(E)}\ 4\ \text{by}\ 8$ | 1. **Identify the original dimensions of the square**: The square has a side length of 8 units. Therefore, each side of the square is 8 units long.
2. **Understand the effect of cutting the square in half**: When the square is cut in half, one dimension is halved while the other remains the same. This is because the c... | null | null |
Given that $x$ and $y$ are nonzero real numbers such that $x + \frac{1}{y} = 7$ and $y + \frac{1}{x} = \frac{1}{3}$, find all possible values for $x$. | First, multiply the equation $x + \frac{1}{y} = 7$ by $y$ and the equation $y + \frac{1}{x} = \frac{1}{3}$ by $x$ to eliminate the denominators:
\[
xy + 1 = 7y,
\]
\[
xy + 1 = \frac{1}{3}x.
\]
Since both expressions equal $1 + xy$, equate them:
\[
7y = \frac{1}{3}x.
\]
This simplifies to:
\[
y = \frac{1}{21}x.
\]
Subst... | null | null |
Let \( x(t) \) be a solution to the differential equation
\[
\left(x + x' \right)^{2} + x \cdot x'' = \cos t
\]
with initial conditions \( x(0) = x'(0) = \sqrt{\frac{2}{5}} \). Compute \( x\left(\frac{\pi}{4}\right) \). |
1. **Rewrite the given problem in terms of \( y(t) \):**
Given the differential equation:
\[
(x + x')^2 + x x'' = \cos t
\]
we make the substitution \( y = x^2 \), hence \( x = \sqrt{y} \). Differentiating, we get:
\[
x' = \frac{1}{2} \frac{dy}{dx} \cdot 2x = \frac{y'}{2\sqrt{y}}
\]
and for ... | null | null |
Using the method of stratified sampling, if we draw 10 balls from 15 identical red balls and 10 identical black balls and arrange them in a row, the number of different arrangements is ( )
A: $C_{10}^{4}$
B: $A_{10}^{4}$
C: $A_{10}^{6}$
D: $A_{10}^{10}$ | According to the problem, using the method of stratified sampling, we need to draw 6 red balls and 4 black balls. Since the red and black balls are identical, there is only one way to draw them. Then, arranging the drawn balls as required, there are $C_{10}^{4}$ different ways to do so. Therefore, the correct answer is... | null | null |
A supermarket receives a delivery of 15 cases of tins of beans. Each case contains 24 tins. If 5% of the tins are damaged and thrown away, how many tins of beans are left? | To solve this problem, we proceed as follows:
1. Calculate the total number of tins delivered:
\[15 \text{ cases} \times 24 \text{ tins/case} = 360 \text{ tins}\]
2. Determine the number of tins that are damaged and thrown away:
\[360 \text{ tins} \times 0.05 = 18 \text{ tins}\]
3. Subtract the damaged tins from the... | null | null |
Football is a popular sport. The opening match of the 2022 Qatar World Cup will kick off on November 21, 2022, and the final will be held on the evening of December 18, lasting a total of 28 days.
$(1)$ In order to investigate whether the preference for football is related to gender, a random sample of 100 male and 1... | ### Solution:
#### Part 1:
To test if the preference for football is related to gender, we perform an independence test. We assume the null hypothesis $H_{0}$: The preference for football is independent of gender.
The test statistic $K^{2}$ is calculated as follows:
$$K^{2} = \frac{N(O_{11}O_{22} - O_{12}O_{21})^{2}}... | null | null |
Suppose [$a$ $b$] denotes the average of $a$ and $b$, and {$a$ $b$ $c$} denotes the average of $a$, $b$, and $c$. Evaluate $\{\{\text{2 3 -1}\} \text{ [1 2] } 1\}$.
$\textbf{(A)}\ \frac{5}{9} \qquad\textbf{(B)}\ \frac{23}{18} \qquad\textbf{(C)}\ \frac{11}{18} \qquad\textbf{(D)}\ \frac{12}{9} \qquad\textbf{(E)}\ \frac{7... | 1. **Calculate $\{2, 3, -1\}$**:
The average of $2$, $3$, and $-1$ is calculated as follows:
\[
\{2, 3, -1\} = \frac{2 + 3 - 1}{3} = \frac{4}{3}
\]
2. **Calculate $[1, 2]$**:
The average of $1$ and $2$ is calculated as follows:
\[
[1, 2] = \frac{1 + 2}{2} = \frac{3}{2}
\]
3. **Calculate $\{\f... | null | null |
If an even function $f(x)$ and an odd function $g(x)$ defined on $\mathbb{R}$ satisfy $f(x) + g(x) = e^x$, then $g(x) = \text{( )}$.
A: $e^x - e^{-x}$
B: $(e^x + e^{-x})$
C: $(e^{-x} - e^x)$
D: $(e^x - e^{-x})$ | Since $f(x)$ is an even function, it holds that $f(-x) = f(x)$. And because $g(x)$ is an odd function, it holds that $g(-x) = -g(x)$. Given $f(x) + g(x) = e^x$, we can use the properties of even and odd functions to also describe the function for negative $x$:
For $x < 0$:
$$f(-x) + g(-x) = f(x) - g(x) = e^{-x}.$$
No... | null | null |
Let $A$ , $B$ , $C$ , and $D$ be points in the plane with $AB=AC=BC=BD=CD=36$ and such that $A \neq D$ . Point $K$ lies on segment $AC$ such that $AK=2KC$ . Point $M$ lies on segment $AB$ , and point $N$ lies on line $AC$ , such that $D$ , $M$ , and $N$ are collinear. Let lines $CM$ and $BN$ in... | 1. **Understanding the Geometry:**
We are given points \(A\), \(B\), \(C\), and \(D\) such that \(AB = AC = BC = BD = CD = 36\). This implies that \(A\), \(B\), and \(C\) form an equilateral triangle with side length 36. Additionally, \(D\) is equidistant from \(B\) and \(C\), suggesting that \(D\) lies on the perpe... | null | null |
A soccer ball is kicked into the air, and its height \( h \) in meters after \( t \) seconds is given by the equation \( h = -20t^2 + 40t + 20 \). What is the maximum height reached by the soccer ball? | To find the maximum height, we need to find the vertex of the parabola represented by the equation \( h = -20t^2 + 40t + 20 \).
1. Start by identifying the \( t \)-coordinate of the vertex using the formula \( t = -\frac{b}{2a} \) for a quadratic equation \( ax^2 + bx + c \). Here, \( a = -20 \), \( b = 40 \).
\[
... | null | null |
How many three-digit whole numbers have no 5's, no 6's, no 7's, and no 9's as digits? | 1. **Hundreds Digit**: The first digit of a three-digit number cannot be 0, 5, 6, 7, or 9. Therefore, the possible choices for the hundreds digit are 1, 2, 3, 4, and 8, which gives us 5 choices.
2. **Tens and Units Digits**: These digits cannot be 5, 6, 7, or 9 either. The possible choices are 0, 1, 2, 3, 4, and 8, wh... | null | null |
The diagonals of a convex quadrilateral ABCD intersect at point E. It is known that \(AB = BC = CD = DE = 1\). Prove that \(AD < 2\). |
1. Consider triangle \(DEC\). Since \( B C = C D = D E = 1 \), triangle \(CDE\) is an isosceles triangle.
2. Note that \(\angle CED < 90^\circ\) because it is an angle at the base of isosceles triangle \(CDE\). This implies that \(\angle BEC > 90^\circ\) because \(BCE\) is supplementary to \(\angle CED\) (i.e., \(... | null | null |
If $x$ satisfies $x^2 + 3x + \frac{3}x + \frac{1}{x^2} = 26$ and $x$ can be written as $a + \sqrt{b}$ where $a$ and $b$ are positive integers, then find $a + b$. | To solve the problem, let's follow the steps closely related to the given solution:
1. **Introduce a New Variable**: Let $k = x + \frac{1}{x}$. This step is crucial for simplifying the given equation.
2. **Express $k^2$ in Terms of $x$**: We know that $k^2 = \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}$. ... | null | null |
The line $y = \frac{7}{5}x - \frac{23}{5}$ is to be parameterized using vectors. Which of the following options are valid parameterizations?
(A) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 2 \end{pmatrix} + t \begin{pmatrix} -5 \\ -7 \end{pmatrix}$
(B) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{p... | 1. The direction vector for the line $y = \frac{7}{5}x - \frac{23}{5}$ can be derived from the slope, which is $\frac{7}{5}$. Thus, a direction vector is $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$.
2. Checking each option:
- (A): $\begin{pmatrix} -5 \\ -7 \end{pmatrix}$ is a scalar multiple of $\begin{pmatrix} 5 \\ 7 \... | null | null |
Hunter needs to make a square patch of land whose perimeter is twice as large as a rectangular patch of land. If the rectangular patch of land has a length of 400 feet and a width of 300 feet, calculate the length of one side of the square patch of land. | To solve for the length of one side of the square patch of land, given the dimensions of the rectangular patch of land and the relationship between their perimeters, we proceed as follows:
1. **Calculate the perimeter of the rectangular patch of land:**
The formula for the perimeter ($P$) of a rectangle is $P = 2 \ti... | null | null |
Let $T = TNFTPP$ , and let $S$ be the sum of the digits of $T$ . In triangle $ABC$ , points $D$ , $E$ , and $F$ are the feet of the angle bisectors of $\angle A$ , $\angle B$ , $\angle C$ respectively. Let point $P$ be the intersection of segments $AD$ and $B... | 1. **Determine the value of \( S \):**
Given \( T = 1801 \), we need to find the sum of the digits of \( T \):
\[
S = 1 + 8 + 0 + 1 = 10
\]
Therefore, \( S - 1 = 9 \).
2. **Set up the problem in triangle \( ABC \):**
- \( AP = 3PD \)
- \( BE = S - 1 = 9 \)
- \( CF = 9 \)
3. **Use the Angle Bis... | null | null |
There were 90 jellybeans in a jar. Samantha snuck 24 jellybeans out of the jar, without being seen. Shelby ate 12 jellybeans from the jar. Their mom, Shannon, refilled the jar with half as much as Samantha and Shelby took out. How many jellybeans are in the jar now? | Initially, the jar had 90 jellybeans. Let's break down the events to see how the number of jellybeans changes over time:
1. **Samantha's Action**: Samantha took 24 jellybeans out of the jar. This action reduces the number of jellybeans in the jar from 90 to:
\[
90 - 24 = 66
\]
So, after Samantha's action, ... | null | null |
If the roots of the quadratic equation $x^{2}-8x+m=0$ with respect to $x$ are $x_{1}$ and $x_{2}$, and $x_{1}=3x_{2}$, then the value of $m$ is ( )
A: $4$
B: $8$
C: $12$
D: $16$ | Given the quadratic equation $x^{2}-8x+m=0$, where the roots are $x_{1}$ and $x_{2}$, and it's given that $x_{1}=3x_{2}$, we aim to find the value of $m$.
From the properties of the roots of a quadratic equation, we know that the sum of the roots ($x_{1}+x_{2}$) is equal to the coefficient of $x$ taken with the opposi... | null | null |
Carol was playing a trivia game. In the first round she scored 17 points, in the second round she scored some points, and in the last round she lost 16 points. She had 7 points at the end of the game. How many points did she score in the second round? | Let's call the points Carol scored in the second round "x".
From the first round, Carol had 17 points.
After the second round, she had 17 + x points.
After the last round, she lost 16 points, so she had (17 + x) - 16 points.
At the end of the game, Carol had 7 points. Therefore, we can set up the following equation:
... | null | null |
If $5x - 8 = 15x + 4$, what is $3(x + 9)$? | 1. Start with the equation $5x - 8 = 15x + 4$.
2. Subtract $5x$ and add 8 to both sides to isolate the x terms:
\[
-8 + 8 = 15x - 5x + 4
\]
\[
0 = 10x + 4
\]
\[
-4 = 10x
\]
3. Divide both sides by 10 to solve for x:
\[
x = -\frac{4}{10} = -\frac{2}{5}
\]
4. Substitute $x = -\frac{2}... | null | null |
Given a circle and a point $P$ inside it. For each point $Q$ on the circle, draw a tangent.
The perpendicular dropped from the center of the circle to the line $PQ$ and the tangent intersect at point $M$.
Find the locus of point $M$. | ### Given Problem
A circle and a point \(P\) inside it are given. For any point \(Q\) on the circle, a tangent line is drawn from \(Q\). The perpendicular drawn from the center of the circle \(O\) to the line \(PQ\) and the tangent at \(Q\) intersect at a point \(M\). Find the locus of \(M\).
1. **Identify Key Points... | null | null |
If x is an integer such that 3 < x < 10, 5 < x < some number, 9 > x > -2, 8 > x > 0, and x + 1 < 9, then x is 7. What is the upper bound of the second inequality? | The second inequality is given as 5 < x < some number. Since we know that x is 7, we can determine the upper bound of this inequality by substituting x with 7:
5 < 7 < some number
This means that the "some number" must be greater than 7. However, we also have other inequalities that provide additional constraints on ... | null | null |
The sum of four positive integers that form an arithmetic sequence is 46. Of all such possible sequences, what is the greatest possible third term? | To solve the problem, let's start by defining the first term of the arithmetic sequence as $a$ and the common difference as $d$. This means the four terms of the sequence can be represented as $a$, $a + d$, $a + 2d$, and $a + 3d$. Given that the sum of these four terms is 46, we can write the equation:
\[a + (a + d) +... | null | null |
Side $AE$ of regular pentagon $ABCDE$ is extended past $E$ to point $Y$ such that $AY = 4AE$. Given that each side of the pentagon is $2$ units long, find the length of segment $DY$. Express your answer in simplest radical form. | Let $P$ be the foot of the perpendicular from $D$ to the line containing $AE$.
Since $\angle DAE = 108^{\circ}$ in a regular pentagon, $\angle PAD = 180^\circ - 108^\circ = 72^{\circ}$. Hence, $\triangle PAD$ is isosceles with $AD = DP$ and base angles of $72^\circ$ each.
Given $AD = 2$, $AE = 2$, then $AY = 4AE = 8$... | null | null |
Tony invested Rs. 3200 at a certain rate to obtain an income of Rs. 250. The dividend from Tony's stock is 6.640625. What is the rate at which Tony invested? | To find the rate at which Tony invested, we need to use the formula for calculating the dividend yield, which is given by:
Dividend Yield = (Annual Dividend / Investment) * 100
We are given the annual dividend as Rs. 250 and the investment as Rs. 3200. We are also given the dividend from Tony's stock as 6.640625, but... | null | null |
Let $A$ , $B$ , and $C$ be distinct points on a line with $AB=AC=1$ . Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$ . What is the degree measure of the acute angle formed by lines $EC$ and $BF$ ?
*Ray Li* | 1. **Identify the coordinates of points \(A\), \(B\), and \(C\)**:
- Let \(A\) be at the origin, i.e., \(A = (0, 0)\).
- Since \(AB = 1\) and \(AC = 1\), and \(B\) and \(C\) are on the same line, we can place \(B\) at \((1, 0)\) and \(C\) at \((-1, 0)\).
2. **Determine the coordinates of points \(D\) and \(E\) i... | null | null |
An ellipse is defined parametrically by
\[(x,y) = \left( \frac{2 (\sin t - 1)}{2 - \cos t}, \frac{3 (\cos t - 5)}{2 - \cos t} \right).\]Then the equation of the ellipse can be written in the form
\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integers, and $\gcd(|A|,|B|,|C|,|D|,|E|,|... | To find the equation of the ellipse defined parametrically by
\[(x,y) = \left( \frac{2 (\sin t - 1)}{2 - \cos t}, \frac{3 (\cos t - 5)}{2 - \cos t} \right),\]
we start by isolating $\cos t$ and $\sin t$ from the given parametric equations.
First, for the equation $y = \frac{3 (\cos t - 5)}{2 - \cos t},$ we solve for $... | null | null |
What is
\[\dfrac{3+5+7}{2+4+6} - \dfrac{2+4+6}{3+5+7} ?\]
- **A)** $\frac{7}{12}$
- **B)** $\frac{1}{20}$
- **C)** $\frac{9}{20}$
- **D)** $\frac{11}{24}$ | 1. **Calculate the sums in the numerators and denominators:**
- The sum of the numerators in the first fraction: $3 + 5 + 7 = 15$.
- The sum of the denominators in the first fraction: $2 + 4 + 6 = 12$.
2. **Calculate for the second fraction too:**
- The sum of the numerators in the second fraction: $2 + 4 + 6... | null | null |
Kevin collected toys to use as prizes at the fair. He collected 14 stuffed animals, 18 frisbees, several yo-yos, and 12 puzzles. Additionally, he discovered that 0.4 of the toys he collected were toy cars, while 1/10 of the total collected toys were toy robots. Kevin has 120 prizes in all. How many yo-yos did Kevin col... | Let's denote the number of yo-yos Kevin collected as Y.
We know that Kevin collected 14 stuffed animals, 18 frisbees, and 12 puzzles. So, without the yo-yos, the number of toys he collected is:
14 (stuffed animals) + 18 (frisbees) + 12 (puzzles) = 44 toys
We also know that 0.4 of the toys are toy cars and 1/10 of the... | null | null |
In the Cartesian coordinate system $xOy$, the graph of which of the following functions passes through the point $\left(0,0\right)$?
A: $y=x+1$
B: $y=x^{2}$
C: $y=\left(x-4\right)^{2}$
D: $y=\frac{1}{x}$ | To determine which graph of the given functions passes through the point $\left(0,0\right)$, we substitute $x=0$ and $y=0$ into each option and check for validity.
- **Option A:** For the line $y = x + 1$, substituting $x = 0$ gives $y = 0 + 1 = 1$. Since $y \neq 0$, this line does not pass through $\left(0,0\right)$.... | null | null |
"Determine whether \\(m > 10\\) is a condition for the equation \\( \\frac{x^2}{m-10} - \\frac{y^2}{m-8} = 1\\) to represent a hyperbola."
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Sufficient and necessary condition
D: Neither sufficient nor necessary condition | To determine the conditions under which the given equation represents a hyperbola, we need to ensure that the denominators in the equation are positive since the standard form of a hyperbola is \\( \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\) with \\( a^2 > 0 \\) and \\( b^2 > 0 \\).
If \\( m > 10 \\), then \\( m - 10 ... | null | null |
Let the function $g(x)$ satisfy
\[g(x) + g \left( \frac{x + 2}{2 - 4x} \right) = 2x\] for all $x \neq \frac{1}{2}.$ Find $g(3)$. | 1. Setting $x = 3$ in the given equation:
\[g(3) + g \left( \frac{3 + 2}{2 - 4 \cdot 3} \right) = 6.\]
\[g(3) + g \left( -\frac{5}{10} \right) = 6.\]
\[g(3) + g \left( -\frac{1}{2} \right) = 6.\]
2. Setting $x = -\frac{1}{2}$:
\[g \left( -\frac{1}{2} \right) + g \left( \frac{-\frac{1}{2} + 2}{2 - 4 \cdot (... | null | null |
A paper triangle with sides of lengths $5, 12,$ and $13$ inches is folded so that point $A$ falls on point $C$. What is the length in inches of the crease?
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7.25 \qquad \textbf{(C)}\ 7.5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 8.5$ | 1. **Identifying the Triangle Type**: Given side lengths $5, 12, 13$, we recognize $\triangle ABC$ as a right triangle because $5^2 + 12^2 = 13^2$.
2. **Folding Point A to Point C**: The crease formed by folding such that point $A$ falls on point $C$ will be a line segment that is equidistant from $A$ and $C$. This is... | null | null |
Victor works at Clucks Delux, a restaurant specializing in chicken. An order of Chicken Pasta uses 2 pieces of chicken, an order of Barbecue Chicken uses 3 pieces of chicken, and a family-size Fried Chicken Dinner uses 8 pieces of chicken. Tonight, Victor has 2 Fried Chicken Dinner orders, 6 Chicken Pasta orders, and 3... | To calculate the total number of pieces of chicken Victor needs for all the orders at Clucks Delux tonight, we can break down the calculation based on the type of order and then sum them up.
1. For the Fried Chicken Dinner orders:
- Each order uses 8 pieces of chicken.
- There are 2 orders.
- Therefore, the t... | null | null |
At the bake sale, Tamara makes $32 from the brownies. She made 2 pans of brownies which were all sold. The brownies were cut into 8 big square pieces. How much did each brownie cost? | To find out how much each brownie cost, we first need to determine the total number of brownie pieces Tamara sold. Since she made 2 pans of brownies and each was cut into 8 big square pieces, we calculate the total number of pieces as follows:
\[ \text{Total number of brownie pieces} = 2 \times 8 = 16 \]
Next, to fin... | null | null |
Given $\frac{1}{C_{5}^{m}}-\frac{1}{C_{6}^{m}}=\frac{7}{10 C_{7}^{m}}$, find $C_{21}^{m}$ . | Since $\frac{1}{C_{5}^{m}}-\frac{1}{C_{6}^{m}}=\frac{7}{10 C_{7}^{m}}$,
We have $\frac{m!(5-m)!}{5!}-\frac{m!(6-m)!}{6!}=\frac{7 \cdot m!(7-m)!}{10 \cdot 7!}$.
Simplifying, we get $6 \times(5-m)!-(6-m)!= \frac{7-m!}{10}$,
$6-(6-m)=\frac{(7-m)(6-m)}{10}$,
Thus, $m^{2}-23 m+42=0$,
Solving for $m$, we get $m=2$ or $m... | null | null |
Given that point $P(\tan \alpha, \cos \alpha)$ is located in the third quadrant, the quadrant in which angle $\alpha$ lies is $\underline{\qquad}$.
A: Quadrant I
B: Quadrant II
C: Quadrant III
D: Quadrant IV | Since $P(\tan \alpha, \cos \alpha)$ is in the third quadrant,
We have the following system of inequalities:
$\begin{cases} \tan \alpha < 0 \ \cos \alpha < 0 \end{cases}$.
From $\tan \alpha < 0$, we know that $\alpha$ is in either Quadrant II or Quadrant IV.
From $\cos \alpha < 0$, we know that $\alpha$ is in either Q... | null | null |
Given an arithmetic sequence $\{a_n\}$ where $a_1=2$ and $a_3+a_5=10$,
(1) Find the general formula for the sequence $\{a_n\}$;
(2) Let $b_n=a_n\cdot2^n$, find the sum of the first $n$ terms of the sequence $\left\{\frac{1}{b_n}\right\}$, denoted as $S_n$. | (1) Since in the arithmetic sequence $\{a_n\}$, $a_1=2$ and $a_3+a_5=10$,
we have $2a_4 = a_3 + a_5 = 10$.
Solving this equation, we find $a_4 = 5$.
Therefore, the common difference $d = \frac{a_4 - a_1}{4 - 1} = \frac{5 - 2}{3} = 1$.
Hence, the general term of the sequence is $a_n = a_1 + (n - 1)d = 2 + (n - 1)\cdot1 ... | null | null |
Let $p$ be the proposition that the equation $x^{2}-2ax+2a^{2}-a-6=0$ has real roots for $x$, and let $q$ be the proposition that $m-1\leq a\leq m+3$.
$(1)$ If the proposition $\neg p$ is true, find the range of real numbers for $a$.
$(2)$ If $p$ is a necessary but not sufficient condition for $q$, find the range o... | ### Solution
#### Part 1:
To determine the range of real numbers for $a$ when proposition $\neg p$ (not $p$) is true, we analyze the discriminant $\Delta$ of the given quadratic equation $x^{2}-2ax+2a^{2}-a-6=0$ since the real roots existence is directly related to the discriminant being non-negative.
For the equat... | null | null |
Given the function $f(x)=x\ln x-\frac{1}{2}mx^{2}-x$ $(m\in\mathbb{R})$.
$(1)$ If the function $f(x)$ is decreasing on $(0,+\infty)$, find the range of the real number $m$;
$(2)$ If the function $f(x)$ has two extreme points $x_{1}$ and $x_{2}$ on $(0,+\infty)$, and $x_{1} < x_{2}$, prove that: $\ln x_{1}+\ln x_{2} >... | $(1)$ Solution: Since $f(x)=x\ln x- \frac{1}{2}mx^{2}-x$ $(m\in\mathbb{R})$ is decreasing on $(0,+\infty)$,
then $f′(x)=\ln x-mx\leqslant 0$ holds true in the domain $(0,+\infty)$,
thus $m\geqslant \left( \frac {\ln x}{x}\right)_{\min}$,
let $h(x)= \frac {\ln x}{x}$, then $h′(x)= \frac {1-\ln x}{x^{2}}$,
since ... | null | null |
Given a sequence $\{a_n\}$ with the first term $a_1=2$, and $a_{n+1} = \frac{2a_{n}}{a_{n+2}}$ (for $n=1,2,3,\ldots$), find the value of $a_{2012}$. | Since $a_{n+1} = \frac{2a_{n}}{a_{n+2}}$,
we have $\frac{1}{a_{n+1}} = \frac{a_{n}+2}{2a_{n}} = \frac{1}{2} + \frac{1}{a_{n}}$,
which implies $\frac{1}{a_{n+1}} - \frac{1}{a_{n}} = \frac{1}{2}$,
Therefore, the sequence $\left\{\frac{1}{a_{n}}\right\}$ is an arithmetic sequence with common difference $d = \frac{1}... | null | null |
Given the function $f(x)=a\ln x-(a+1)x-\frac{1}{x}(x > 0)$.
(1) When $a=-\frac{3}{2}$, discuss the monotonicity of $f(x)$;
(2) When $a=1$, if $g(x)=-x-\frac{1}{x}-1$, prove that the graph of $g(x)$ is always above the graph of $f(x)$ when $x > 1$;
(3) Prove that $\frac{\ln 2}{{{2}^{2}}}+\frac{\ln 3}{{{3}^{2}}}+…+\fr... | (1) When $a=-\frac{3}{2}$, $f(x)=-\frac{3}{2} \ln x+\frac{1}{2} x-\frac{1}{x} (x > 0)$,
then $f'(x)=-\frac{3}{2x} +\frac{1}{2} +\frac{1}{{x}^{2}} =\frac{(x−1)(x−2)}{2{x}^{2}}$.
Let $f'(x) > 0 \Rightarrow 0 < x < 1$ or $x > 2$, and $f'(x) < 0 \Rightarrow 1 < x < 2$.
Thus, the intervals where $f(x)$ is increasing are ... | null | null |
Given that $x=3$ is an extremum point of the function $f\left(x\right)=x^{3}-ax^{2}-9x+1$.
$(1)$ Find the value of the real number $a$.
$(2)$ Find the maximum and minimum values of the function $f\left(x\right)$ on the interval $\left[-2,0\right]$. | ### Step-by-Step Solution:
#### Part 1: Finding the value of $a$
Given that $x=3$ is an extremum point of the function $f\left(x\right)=x^{3}-ax^{2}-9x+1$, we first find the derivative of $f(x)$ to determine the conditions for an extremum point.
1. The derivative of $f(x)$ is given by:
\[
f'(x) = \frac{d}{dx}(... | null | null |
Given a sequence $\{a_{n}\}$ where $a_{1}=33$ and $a_{n+1}-a_{n}=2n$, find the minimum value of $\frac{a_n}{n}$. | To solve the given problem, let's break it down step by step:
1. **Expression for $a_n$:**
Given $a_1 = 33$ and $a_{n+1} - a_n = 2n$, we can express $a_n$ as a sum of the differences plus the first term:
\[
\begin{align*}
a_n &= (a_n - a_{n-1}) + (a_{n-1} - a_{n-2}) + \ldots + (a_2 - a_1) + a_1 \\
&= 2[1 + 2 + \ldot... | null | null |
An alloy of zinc and copper contains the metals in a certain ratio. The quantity of zinc to be added to 6 kg of the alloy so that the ratio of the metal may be 3 : 1 is 8 kg. What is the initial ratio of zinc to copper in the alloy? | Let's denote the initial quantity of zinc in the 6 kg alloy as \( Z \) kg and the quantity of copper as \( C \) kg. The initial ratio of zinc to copper is therefore \( Z : C \).
According to the problem, when 8 kg of zinc is added to the alloy, the new ratio of zinc to copper becomes 3 : 1. This means that after addin... | null | null |
Show that for any $\lambda>0$ the Boole transformation
$$
x \rightarrow x-\frac{\lambda}{x} \quad (x \neq 0)
$$
preserves the Lebesgue measure on $\mathbb{R}$, i.e., for any integrable function $f$ on $\mathbb{R}$, the following equality holds:
$$
\int_{\mathbb{R}} f(x) \, d x = \int_{\mathbb{R}} f\left(x-\frac{\la... |
We start by verifying the given transformation for \( \lambda = 1 \):
1. Consider the transformation:
\[
x \rightarrow x - \frac{1}{x}, \quad x \neq 0
\]
2. Split the integral over the entire real line into two parts, from \(0\) to \(\infty\) and from \(-\infty\) to \(0\):
\[
\int_{\mathbb{R}} f\left( x - \f... | null | null |
the average ( arithmetic mean ) of 5 numbers is 5 . if 2 is subtracted from each of 4 of the numbers , what is the new average ? | If the average of 5 numbers is 5, then the sum of those 5 numbers is 5 * 5 = 25.
If 2 is subtracted from each of 4 of the numbers, the total amount subtracted is 2 * 4 = 8.
The new sum of the 5 numbers is 25 - 8 = 17.
The new average is the new sum divided by the number of numbers, which is still 5.
So the new aver... | null | null |
In a school there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$ . | 1. **Initial Setup and Lemma**:
- We are given that there are 1200 students, each of whom must join exactly \( k \) clubs.
- Each club is joined by exactly 23 students, and no club is joined by all 1200 students.
- We need to find the smallest possible value of \( k \).
2. **Inductive Hypothesis**:
- We wi... | null | null |
A 40 meters rope was cut into 2 parts in the ratio of 2:3. How long is the shorter part? | To find the length of the shorter part, we first need to determine the total number of parts into which the rope was divided. The ratio is 2:3, which means there are 2 parts + 3 parts = 5 parts in total.
Now, we can find the length of one part by dividing the total length of the rope by the total number of parts:
40 ... | null | null |
Find the number of four-element subsets of $\{1,2,3,4,\dots, 20\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\{3,5,13,19\}$ and $\{6,10,20,18\}$ are two such subsets. | To solve this problem, we need to find the number of four-element subsets $\{a, b, c, d\}$ of $\{1, 2, 3, \dots, 20\}$ such that two distinct elements sum to $16$ and two distinct elements sum to $24$. We will consider two cases based on the given conditions.
#### Case 1: $a+b = 16$ and $c+d = 24$
1. **Finding pairs $... | null | null |
Evaluate the expression: $\left(\frac{1}{8}\right)^{-\frac{1}{3}}$
A) 1
B) 2
C) 3
D) 4 | 1. **Use the properties of exponents:**
The property of exponents $(\frac{1}{a})^b = a^{-b}$ is applied here:
\[
\left(\frac{1}{8}\right)^{-\frac{1}{3}} = 8^{\frac{1}{3}}
\]
2. **Compute the cube root of 8:**
The expression $8^{\frac{1}{3}}$ refers to the cube root of 8:
\[
8^{\frac{1}{3}} = \sqrt... | null | null |
There are 3 trains from location A to location B, 2 ferry lines from location B to location C, and additionally, there are 2 flights from location A to location C. The total number of travel options from location A to location C is ( ).
A: 7
B: 8
C: 10
D: 12 | In this problem, we apply the principle of counting.
Since there are 3 trains from location A to location B, and 2 ferry lines from location B to location C, the number of travel combinations using a train-then-ferry sequence is $3 \times 2 = 6$ (multiplying the number of options for each leg of the journey).
Also, c... | null | null |
Given point F(1, 0), and the moving point M's distance to line l: x = 4 is d. The ratio of $|MF|$ to $d$ is $\frac{1}{2}$. Let the trajectory of the moving point M be curve E.
1. Find the equation of curve E.
2. Draw two perpendicular lines through point F that intersect curve E at points A, B and C, D, respectively. ... | (This question is worth 12 points)
1. Let M(x, y). Since $|MF| = \frac{1}{2}d$, we have $\sqrt{(x - 1)^2 + y^2} = \frac{1}{2}|x - 4|$. Simplifying and squaring both sides, we get the equation for curve E as $\frac{x^2}{4} + \frac{y^2}{3} = 1$.
2. Solution 1: When the slope of line AB is 0, $|AB| = 2a = 4$ and $|CD| =... | null | null |
Given that the universal set $U=\mathbb{R}$, and let $A=\{x \mid 1 \leq x < b\}$, the complement of set $A$ in $U$, $\complement_U A$, is $\{x \mid x < 1 \text{ or } x \geq 2\}$. Find the value of the real number $b$. | We start with the given set $A = \{x \mid 1 \leq x < b\}$.
The complement of set $A$ in $U$ is therefore $\complement_U A = \{x \mid x < 1 \text{ or } x \geq b\}$, considering the universal set is $\mathbb{R}$.
Since we know that $\complement_U A = \{x \mid x < 1 \text{ or } x \geq 2\}$, we can compare this with the ... | null | null |
A man buys an article for some amount and sells it for $25. The gain percent is 150%. What was the original price of the article? | Let's denote the original price of the article as \( P \).
Given that the gain percent is 150%, this means that the man sold the article for 150% more than the original price. To find the gain in terms of the original price, we can express 150% as a decimal, which is 1.5.
The gain can be calculated as:
\[ \text{Gain}... | null | null |
For which values of \( n \) is the following expression divisible by:
a) \( n \)?
b) \( n+1 \)?
$$
\sum_{k=1}^{n} \sum_{i=1}^{k} i^{2}
$$ |
Before addressing the divisibility questions, let's transform the double sum notation:
\[
\begin{aligned}
\sum_{k=1}^{n} \sum_{i=1}^{k} i^{2} &= 1^{2}+\left(1^{2}+2^{2}\right)+\ldots+\left(1^{2}+2^{2}+\ldots+n^{2}\right) \\
&= n \cdot 1^{2}+(n-1) \cdot 2^{2} + \ldots + 1 \cdot n^{2} \\
&= n \left(1^{2} + 2^{2} + \ldo... | null | null |
Zachary paid for a $\$1$ burger with 32 coins and received no change. Each coin was either a penny or a nickel. What was the number of nickels Zachary used? | To solve for the number of nickels Zachary used, let's denote the number of pennies as $p$ and the number of nickels as $n$. From the problem, we have two equations based on the given conditions:
1. The total number of coins used is 32, which gives us the equation:
\[p + n = 32\]
2. The total value of the coins equal... | null | null |
Calculate as required.
(1) The lateral area of a cylinder with a base circumference of 1.8 meters and a height of 1.5 meters.
(2) The volume of a cylinder with a diameter of 3dm and a height of 8cm.
(3) The surface area of a cylinder with a base radius of 6 decimeters and a height of 5 decimeters.
(4) A cone ha... | **Solution**:
(1) $1.8 \times 1.5 = 2.7$ (square meters)
Answer: The lateral area of the cylinder is $\boxed{2.7 \text{ square meters}}$.
(2) $3$ decimeters $= 30$ centimeters
$3.14 \times (30 \div 2)^2 \times 8$
$= 3.14 \times 225 \times 8$
$= 5652$ (cubic centimeters)
Answer: The volume of the cylind... | null | null |
If the sum of $1! + 2! + 3! + \cdots + 49! + 50!$ is divided by $15$, what is the remainder? | To solve this problem, we need to understand two key points:
1. **Factorial Definition**: For any positive integer $n$, the factorial $n!$ is the product of all positive integers less than or equal to $n$. That is, $n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$.
2. **Divisibility by 15**: A number... | null | null |
There is a cricket team with some members. The captain is 26 years old and the wicket keeper is 5 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team, which is 24. How many members are in the cricket team? | Let's denote the number of members in the cricket team as \( n \).
The total age of the team is \( n \times 24 \) because the average age of the whole team is 24.
The captain is 26 years old, and the wicket keeper is 5 years older than the captain, so the wicket keeper is \( 26 + 5 = 31 \) years old.
The total age o... | null | null |
How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$? | To solve this problem, we start by identifying the nature of the triangle with sides $8,$ $15,$ and $17$. Recognizing the sides, we can see they form a Pythagorean triple, which means the triangle is a right triangle with the sides $8$ and $15$ being the legs, and $17$ being the hypotenuse.
In a right triangle, the al... | null | null |
In a city council election, there are five candidates running for office. The city uses ranked-choice voting to determine the winner. In the first round, the candidate with the least number of first-choice votes is eliminated. The votes for the eliminated candidate are then redistributed to the remaining candidates bas... | After the third elimination and redistribution, Candidate B has 305 votes, which is more than Candidate A's 295 votes. Since there are only two candidates left at this point, one of them must have a majority of the votes. Candidate B, with 305 votes, is the winner.
Therefore, the number of votes cast for the winning c... | null | null |
Johnny wrote an essay with 150 words. Madeline wrote an essay with a certain length, and Timothy wrote an essay that had 30 words more than Madeline's. Their essays fill 3 pages, with one page containing 260 words. What is the ratio of the number of words in Madeline's essay to the number of words in Johnny's essay? | Let's denote the number of words in Madeline's essay as M. According to the information given, Timothy's essay has M + 30 words.
The total number of words in the three essays is equal to the number of words that can fit on one page, which is 260 words. Therefore, the total number of words in all three essays is 3 page... | null | null |
Let $f(n)$ represent the sum of all the positive divisors of $n$ excluding $n$ itself. For instance, $f(10) = 1 + 2 + 5 = 8$. Compute $f(f(f(12)))$.
A) 12
B) 15
C) 9
D) 10 | First, calculate $f(12)$:
The divisors of $12$ are $1, 2, 3, 4, 6, 12$. The sum of all divisors except $12$ itself is:
\[
f(12) = 1 + 2 + 3 + 4 + 6 = 16.
\]
Next, calculate $f(16)$:
The divisors of $16$ are $1, 2, 4, 8, 16$. The sum of all divisors except $16$ itself is:
\[
f(16) = 1 + 2 + 4 + 8 = 15.
\]
Finally, cal... | null | null |
End of preview. Expand in Data Studio
NuminaMath Worked Solutions
Competition and school maths problems with step-by-step solutions.
| Rows | 3,000 |
| Domain | mathematics |
| Format | data.parquet, one row per example |
| Licence | apache-2.0 |
| Built for | supervised fine-tuning (SFT) experiments on Adaption AutoScientist |
Columns
| Column | Description |
|---|---|
original_prompt |
The prompt (user turn) as uploaded. |
original_completion |
The target response as uploaded. |
enhanced_prompt |
Empty in this dataset. |
enhanced_completion |
Empty in this dataset. |
How it was built
Sampled from NuminaMath-CoT, balanced across its sources and decontaminated against MATH-500, GSM8K test and the NuminaMath test split.
Sources and licence
- AI-MO/NuminaMath-CoT (Apache-2.0)
Notes
- Columns
enhanced_prompt,enhanced_completionare empty in this dataset (the Adaption export reserves them for rewritten text).
Loading
from datasets import load_dataset
ds = load_dataset("rodriguescarson/adaption-math-worked-solutions-raw", split="train")
import pandas as pd
df = pd.read_parquet("hf://datasets/rodriguescarson/adaption-math-worked-solutions-raw/data.parquet")
Published by Carson Rodrigues (Hugging Face, Kaggle).
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