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ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
For what value of \( a \) does the system of inequalities \(\left\{\begin{array}{l}x^{2}+y^{2}+2x \leq 1, \\ x-y+a=0\end{array}\right.\) have a unique solution, and find that solution?
Answer: | Given the system of inequalities:
\[
\begin{cases}
x^{2} + y^{2} + 2x \leq 1 \\
x - y + a = 0 \\
\end{cases}
\]
we are asked to find the value of \(a\) such that the system has a unique solution.
1. From the second inequality \( x - y + a = 0 \), we can express \( y \) in terms of \( x \):
\[
y = x + a
\]
2... | null | oa_0 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A construction team of 18 men can dig the foundation of a building in 5 days. Assuming the team digs at a consistent rate, calculate how many days it would take for 30 men to dig a deeper foundation that requires 1.5 times th... | Let:
- \( m \) be the number of men,
- \( d \) be the number of days,
- \( k \) be the constant of proportionality.
From the problem, \( 18 \times 5 = 90 = k \).
For the deeper foundation requiring 1.5 times the effort:
- The total effort needed increases to \( 1.5 \times k = 1.5 \times 90 = 135 \).
To find how man... | null | oa_1 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Mojmír cut a regular hexagon into 12 identical pieces. From these pieces (not necessarily all of them), he created various right-angled triangles.
What could Mojmír's assembled triangles look like? Draw at least four possibi... | 1. **Initial Hexagon Division**:
- Start by noting that the regular hexagon can be divided into 6 congruent equilateral triangles. Each of these equilateral triangles is then further bisected to yield 12 smaller congruent triangles.
- These triangles created by bisecting the equilateral triangles have ang... | null | oa_2 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given that $x + \sin y = 2023$ and $x + 2023 \cos y = 2022$, where $\frac{\pi}{2} \leq y \leq \pi$, find the value of $x + y$.
Answer: | From the problem we have two equations:
1. $x + \sin y = 2023$
2. $x + 2023 \cos y = 2022$.
Subtracting equation (2) from equation (1), we get:
$$ (x + \sin y) - (x + 2023 \cos y) = 2023 - 2022 $$
$$ \sin y - 2023\cos y = 1.$$
Since $\frac{\pi}{2} \leq y \leq \pi$, $\sin y$ can reach its maximum value of 1 at $y = \... | null | oa_3 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let $[a]$ denote the greatest integer less than or equal to $a$. Find all real solutions to the equation $x^{2[x]}=2022$.
Answer: |
1. Define the function \( f(x) = x^{2[x]} \), which depends on the greatest integer function, denoted \([a]\), representing the largest integer ≤ \(a\). This function can be expressed piecewise based on the intervals of \( x \):
\[
f(x) = x^{2[x]} =
\begin{cases}
x^{-4}, & x \in [-2, -1) \\
... | null | oa_4 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth?
$\text{(A)}\ \frac{3}{8} \qquad \text{(B)}\ \frac{1}{2} \qqu... | 1. **Define the variables:**
Let $f$ represent the value of one fish, $l$ represent the value of a loaf of bread, and $r$ represent the value of a bag of rice.
2. **Set up the equations based on the given trades:**
- According to the problem, three fish can be traded for two loaves of bread. This can be written... | null | oa_5 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Last winter, 35% of the birds residing at Lake View were ducks, 30% were swans, 20% were herons, and 15% were geese. What percent of the birds that were not ducks were swans?
$\textbf{(A)}\ 40\qquad\textbf{(B)}\ 46.15\qquad\t... | #### Step 1: Determine the percentage of birds that are not ducks.
Given that 35% of the birds are ducks, the percentage of birds that are not ducks is:
\[ 100\% - 35\% = 65\% \]
#### Step 2: Calculate the percentage of swans among the non-duck birds.
Since 30% of the total birds are swans, we need to find what perce... | null | oa_6 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the quadratic curve $\frac{x^{2}}{4} + \frac{y^{2}}{m} = 1$, determine the range of values for the eccentricity $e$ when $m \in [-2, -1]$. The options are:
A: $\[\frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}\]$
B: $\[\frac{\s... | When $m \in [-2, -1]$, the quadratic curve is a hyperbola. The hyperbola $\frac{x^{2}}{4} + \frac{y^{2}}{m} = 1$ can be rewritten as $\frac{x^{2}}{4} - \frac{y^{2}}{-m} = 1$.
We have $a^{2} = 4$, $b^{2} = -m$, and thus $c^{2} = 4 - m$. The eccentricity $e$ is given by the formula $e = \frac{c}{a} = \frac{\sqrt{4-m}}{... | null | oa_7 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A right triangle with sides 3, 4, and 5 is cut by a line perpendicular to the hypotenuse into a trapezoid and another right triangle. Determine the lengths of the sides of the trapezoid.
Answer: | Consider a right triangle $ABC$ with sides 3, 4, and 5. We will divide the triangle using a perpendicular line to the hypotenuse into a tangential quadrilateral and a right triangle. We need to identify the lengths of the sides of the quadrilateral formed in each scenario.
First, let's determine the area of triangle ... | null | oa_8 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the inequality about x: $\frac{{(a+1)x-3}}{{x-1}}＜1$.
(1) When a=1, find the solution to the inequality.
(2) When a is any real number, find the solution to the inequality.
Answer: | (1) **When a=1**
Given $\frac{{(a+1)x-3}}{{x-1}}＜1$, substitute $a=1$ to get $\frac{{(1+1)x-3}}{{x-1}}＜1$
This simplifies to $\frac{{2x-3}}{{x-1}}＜1$.
We then subtract $1$ from both sides and simplify:
\begin{align*}
\frac{{2x-3}}{{x-1}} - 1 &＜0 \\
\frac{{2x-3-(x-1)}}{{x-1}} &＜0 \\
\frac{{x-2}}{{x-1}} &＜0
\end{ali... | null | oa_9 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given an ellipse $C$ with foci at $(-2, 0)$ and $(2, 0)$, and it passes through $P(2, \frac{\sqrt{6}}{3})$.
(1) Find the standard equation of ellipse $C$;
(2) A line $l$ passes through the right focus $F$ of ellipse $C$ and i... | (1) Since the foci of ellipse $C$ are on the x-axis, let's assume its standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$.
By the definition of an ellipse, we have $2a = \sqrt{(2+2)^2+(\frac{\sqrt{6}}{3})^2} + \sqrt{(2-2)^2+(\frac{\sqrt{6}}{3})^2} = 2\sqrt{6}$,
thus $a = \sqrt{6}$.
Since $c = 2$, ... | null | oa_10 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given quadrilateral \(ABCD\) where \(AD \parallel BC\), \(\angle A = 90^\circ\), \(CD \perp BD\), and \(DE \perp BC\) with \(E\) being the midpoint of \(BC\), determine which of the following statements is incorrect:
(A) \(CD... | 1. **Given information**:
\[
\text{In quadrilateral } ABCD, \quad AD || BC, \quad \angle A = 90^\circ, \quad CD \perp BD, \quad DE \perp BC,
\]
where \( E \) is the midpoint of \( BC \).
2. **Identify and analyze triangle \( \triangle BDC \)**:
Since \( CD \perp BD \), we conclude that \( \triangle BD... | null | oa_11 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the function $f(x)= \frac {\cos 2x(\sin x+\cos x)}{\cos x-\sin x}$.
(I) Find the domain of the function $f(x)$;
(II) Find the intervals on which $f(x)$ is increasing.
Answer: | Solution:
(I) According to the problem, we have $\cos x-\sin x\neq 0$,
which means $\tan x\neq 1$. Solving this gives $x\neq k\pi+ \frac {\pi}{4}$, where $k\in\mathbb{Z}$,
Therefore, the domain of the function $f(x)$ is $\{x|x\neq k\pi+ \frac {\pi}{4},k\in\mathbb{Z}\}$;
(II) Simplifying, we get $f(x)= \frac {\... | null | oa_12 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
The seventh term of an arithmetic sequence is $20$. If the difference between two consecutive terms is $2$, what is the product of the first two terms?
Answer: | To find the first term, `a`, and the second term, `a + d` (where `d` is the common difference):
1. The seventh term can be expressed as \( a + 6 \cdot d = 20 \).
2. Given the difference between the terms \( d = 2 \), substitute into the equation:
\[ a + 6 \cdot 2 = 20 \]
\[ a + 12 = 20 \]
\[ a = 20 - 12 = 8 \... | null | oa_13 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
In a dance class, the average weight of 8 boys is 155 pounds each, and the average weight of 5 girls is 125 pounds each. What is the average weight of all the students in the class?
A) 141 pounds
B) 142 pounds
C) 143 pounds
D... | 1. **Calculate the total weight of the boys**:
\[
S_B = 8 \times 155 = 1240 \text{ pounds}
\]
2. **Calculate the total weight of the girls**:
\[
S_G = 5 \times 125 = 625 \text{ pounds}
\]
3. **Calculate the total weight of all students**:
\[
S = S_B + S_G = 1240 + 625 = 1865 \text{ poun... | null | oa_14 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
How many even integers are there between $200$ and $900$ whose digits are all different and come from the set $\{1, 3, 4, 6, 7, 9\}$?
$\text{(A)}\,8 \qquad\text{(B)}\,12 \qquad\text{(C)}\,16 \qquad\text{(D)}\,20 \qquad\text{(... | To solve this, we need even integers between $200$ and $900$ using distinct digits from $\{1, 3, 4, 6, 7, 9\}$.
1. **The number must be even**: Hence, the units digit must be $4$ or $6$, the even digits in the set.
2. **The number must be between $200$ and $900$**: The hundreds digit can be $2$, $4$, or $6$.
3. **A... | null | oa_15 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A train of certain length traveling at 72 kmph takes 13.998880089592832 seconds to cross a bridge of 170 m in length. How long is the train?
Answer: | To find the length of the train, we need to calculate the total distance covered by the train while crossing the bridge. This distance is the sum of the length of the train and the length of the bridge.
First, we convert the speed of the train from km/h to m/s to match the units of the bridge length:
\[ 72 \text{ km... | null | oa_16 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given that the pole coincides with the origin of the rectangular coordinate system and the polar axis coincides with the positive semi-axis of the x-axis, the polar coordinate equation of the curve C is ρ - ρcos2θ - 12cosθ = ... | (I) Since ρ<sup>2</sup>\-ρ<sup>2</sup>(2cos<sup>2</sup>θ-1)-12ρcosθ=0,
∴ρ<sup>2</sup>\-ρ<sup>2</sup>cos<sup>2</sup>θ-6ρcosθ=0, x<sup>2</sup>+y<sup>2</sup>\-x<sup>2</sup>\-6x=0, hence y<sup>2</sup>\=6x
From $$\begin{cases} x-2=- \frac {4}{5}t \\ y= \frac {3}{5}t\end{cases}$$, by comparing the two equations, we get $$\f... | null | oa_17 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Mrs. Heine is buying Valentine's Day treats for some dogs. She wants to buy them 5 heart biscuits each and a set of puppy boots each. She will buy 12 items. How many dogs is she buying treats for?
Answer: | Mrs. Heine is buying 5 heart biscuits and 1 set of puppy boots for each dog, which makes a total of 6 items per dog. She is buying 12 items in total.
To find out how many dogs she is buying treats for, we divide the total number of items by the number of items per dog:
12 items ÷ 6 items/dog = 2 dogs
Mrs. Heine is... | null | oa_18 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given that $\{a_n\}$ is a geometric sequence, $a_2 - a_1 = 1$, $a_5 - a_4 = 8$, then the common ratio of $\{a_n\}$ is ( )
A: 1
B: 2
C: -2
D: 2 or -2
Answer: | Let the common ratio of the geometric sequence $\{a_n\}$ be $q$,
From $a_2 - a_1 = a_1(q - 1) = 1$ (1), $a_5 - a_4 = a_1q^4 - a_1q^3 = a_1q^3(q - 1) = 8$ (2),
Substituting (1) into (2) gives: $q^3 = 8$,
Solving this, we get: $q = 2$.
Therefore, the correct choice is $\boxed{\text{B}}$.
| null | oa_19 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Ship \( P \) spots ship \( Q \), which is moving in a direction perpendicular to \( PQ \), maintaining its course. Ship \( P \) chases \( Q \), always heading directly towards \( Q \); the speed of both ships at any moment is... | 1. Let $\alpha$ denote the instantaneous angle between the direction of $P Q$ and the course of the ship $Q$, and let $v$ be the speed of ships $P$ and $Q$ at the same moment.
2. The relative closing speed between ships $P$ and $Q$ is influenced by the speed $v$ of ship $P$ directed towards $Q$. Additionally, it is i... | null | oa_20 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Henry scored some points on his Geography test, some points on his Math test, and 66 on his English test. His History score is the average of these 3 scores. His total score across all 4 subjects was 248. What were his scores... | Let's denote Henry's Geography score as G and his Math score as M. We know that his English score is 66 and his History score is the average of the three tests (Geography, Math, and English).
The History score (H) can be calculated as:
H = (G + M + 66) / 3
We also know that the total score across all 4 subjects is ... | null | oa_21 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
The sum of the $x$-coordinates of the vertices of a triangle in the Cartesian plane equals $10$. Find the sum of the $x$-coordinates of the midpoints of the sides of the triangle.
Answer: | To solve this problem, let's break down the solution into detailed steps:
1. **Identify the $x$-coordinates of the vertices:** Let's denote the $x$-coordinates of the vertices of the triangle as $a$, $b$, and $c$. According to the problem, the sum of these coordinates is given as $a + b + c = 10$.
2. **Find the $x$-... | null | oa_22 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Consider the following scenario where three stores offer different discounts on a television priced at $149.99$:
\begin{tabular}{|l|l|}
\hline
\textbf{Store} & \textbf{Sale Price for Television $Y$} \\
\hline
Value Market & $... | First, calculate the sale prices for each store:
- **Value Market**: $149.99 - 10 = \$139.99$
- **Tech Bargains**: $149.99 \times (1 - 0.30) = \$104.99$
- **The Gadget Hub**: $149.99 \times (1 - 0.20) = \$119.99$
Next, identify the cheapest and most expensive prices:
- Cheapest: Tech Bargains at \$104.99
- Most Expen... | null | oa_23 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let \[g(x) =
\begin{cases}
3x + 4 &\text{if }x<-3, \\
6-3x&\text{if }x\ge -3.
\end{cases}
\]Find $g(-8).$
Answer: | Given the function $g(x)$, we first identify which part of the piecewise function applies. Since $-8 < -3$, we use the first case:
\[ g(x) = 3x + 4. \]
Substituting $x = -8$ into this expression, we get:
\[ g(-8) = 3(-8) + 4 = -24 + 4 = -20. \]
Thus, the value of $g(-8)$ is $\boxed{-20}$.
| null | oa_24 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A company has a total of 1000 employees across various departments. A stratified sampling method is employed to draw a sample of 80 employees from the entire workforce. It is known that 4 employees were sampled from the adver... | Let $x$ denote the number of employees in the advertising department.
Since stratified sampling is used, the proportion of employees from the advertising department in the sample should be the same as the proportion of employees from the advertising department in the entire company. Thus, we can set up the equation:
... | null | oa_25 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A can finish a work in 12 days and B can do the same work in half the time taken by A. Working together, what part of the same work can they finish in a day?
Answer: | Let's first determine how long it takes for B to finish the work. Since B can do the work in half the time taken by A, and A takes 12 days, B will take 12 / 2 = 6 days to finish the work.
Now, let's calculate the work done by A and B in one day.
A's work in one day = 1/12 of the work
B's work in one day = 1/6 of the... | null | oa_26 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let the acute $\triangle ABC$ be inscribed in circle $\odot O$, and let the perpendiculars $AA_1$, $BB_1$, and $CC_1$ from vertices $A$, $B$, and $C$ intersect the circle $\odot O$ at points $A_2$, $B_2$, and $C_2$ respective... | To demonstrate that:
\[
\frac{A A_{2}}{A A_{1}} + \frac{B B_{2}}{B B_{1}} + \frac{C C_{2}}{C C_{1}} = 4
\]
1. Let's denote \(BC = a\), \(CA = b\), \(AB = c\). The area of \(\triangle ABC\) will be denoted as \(S\).
2. By examining triangles \(\triangle AHB\) and \(\triangle AC_{2}B\) within \(\triangle ABC\):
... | null | oa_27 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Suppose the sum of the first $n$ terms of a geometric sequence $\{a\_n\}$ is $S\_n=a⋅3^n-2$, then $a\_2=$ \_\_\_\_\_\_.
Answer: | Given that the sum of the first $n$ terms of a geometric sequence $\{a\_n\}$ is $S\_n=a⋅3^n-2$,
Let's set $n=1$, $2$, $3$, we get: $a\_1=3a-2$, $a\_1+a\_2=9a-2$, $a\_1+a\_2+a\_3=27a-2$,
Solving these equations, we get: $a\_1=3a-2$, $a\_2=6a$, $a\_3=18a$,
Since $(6a)^{2}=(3a-2)(18a)$,
Solving for $a$, we get: $a=2$... | null | oa_28 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Find the locus of the centers of circles that pass through two fixed points $A$ and $B$ in the plane.
$\textbf{(A) }\text{a point}\qquad\textbf{(B) }\text{a straight line}\qquad\textbf{(C) }\text{two straight lines}\qquad\tex... | 1. **Understanding the New Problem**: We need to find the locus of centers of circles passing through two fixed points, $A$ and $B$.
2. **Properties of Circles Passing Through Two Points**: For a circle passing through two points $A$ and $B$, the line segment $AB$ connecting these points will be a chord of the circle... | null | oa_29 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
The average of a set of data $x_1, x_2, \ldots, x_n$ is 5, and the variance is 16, where $n$ is a positive integer. Then, the average and standard deviation of another set of data $3x_1+2, 3x_2+2, \ldots, 3x_n+2$ are respecti... | **Answer:** Since the average of $x_1, x_2, \ldots, x_n$ is 5, then $x_1+x_2+\ldots+x_n=5n$.
Therefore, the average of the new set of data is
$$
\frac{1}{n}[(3x_1+2)+\ldots+(3x_n+2)]=\frac{1}{n}[3(x_1+x_2+\ldots+x_n)+2n]=17,
$$
The variance of the new set of data is
$$
\frac{1}{n}[(3x_1+2-17)^2+(3x_2+2-17)^2+\ldo... | null | oa_30 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Mathematician Augustus De Morgan, who died in 1871, liked to say that he turned $x$ years old in the year $x^{2}$. Jasper Jenkins, wanting to outdo him, told me in 1925 that he was $\mathrm{a}^{2}+b^{2}$ years old in the year... | 1. Let's begin with De Morgan's case. He claimed he was \( x \) years old in the year \( x^2 \):
\[
x \text{ years old in year } x^2
\]
Given the information, he loved to say he was \( 43 \) years old in \( 43^2 \) year:
\[
43^2 = 1849
\]
Hence, De Morgan was born in:
\[
1849 - ... | null | oa_31 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A 2-mile long train exits a tunnel exactly 4 minutes after the front of the train entered the tunnel. If the train is moving 30 miles per hour, how many miles long is the tunnel?
Answer: | 1. **Train Speed and Distance Covered**: At 30 miles per hour, the train travels $\frac{30 \text{ miles}}{60 \text{ minutes}} = 0.5$ miles per minute.
2. **Total Distance in 4 Minutes**: In 4 minutes, the train covers $0.5 \text{ miles/minute} \times 4 \text{ minutes} = 2$ miles.
3. **Determining Tunnel Length**: Sinc... | null | oa_32 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given that point $A(-4,1)$ is the projection of point $B$ on the line $l: (2m+1)x-(m-1)y-m-5=0 (m\in R)$, then the maximum value of the distance from point $B$ to point $P(3,-1)$ is ( ).
A: $5-\sqrt{10}$
B: $5$
C: $5+\sqrt... | To solve this problem, let's follow the steps based on the given solution:
1. **Rearrange the given line equation**: The line $l: (2m+1)x-(m-1)y-m-5=0$ can be rearranged by grouping terms with $m$ and those without $m$ together, which gives us $(2x-y-1)m+(x+y-5)=0$.
2. **Solve the system of equations**: From the rea... | null | oa_33 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Consider the quadratic function $-3x^2 + 9x + 4$. Which of the following statements is true?
$\textbf{(A)}\ \text{The function always decreases as }x\text{ increases.}\\ \textbf{(B)}\ \text{The function always decreases as }x... | 1. **Identify the nature of the quadratic function:**
The function is $f(x) = -3x^2 + 9x + 4$, with $a = -3$, $b = 9$, and $c = 4$. Since $a < 0$, the parabola opens downward, indicating that the function has a maximum point.
2. **Finding the vertex:**
The vertex $x$-coordinate can be found using $x = -\frac{b}... | null | oa_34 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Spartak fans tell the truth when Spartak wins and lie when they lose. The same applies to the fans of Dynamo, Zenit, and Lokomotiv. After two matches involving these four teams, each of which ended in a victory for one of th... | 1. Let's denote:
- $x$ and $y$ as the number of fans for the winning teams,
- $z$ and $t$ as the number of fans for the losing teams.
2. Based on the problem, the responses for fans of each team (taking into account that they tell the truth when their team wins and lie when it loses) are:
- For the question ... | null | oa_35 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Solve in prime numbers the equation \( p^2 - 6pq + q^2 + 3q - 1 = 0 \).
Answer: | To solve the equation \( p^{2} - 6pq + q^{2} + 3q - 1 = 0 \) in prime numbers, we can follow these steps:
1. **Analyzing Modulo Conditions:**
Consider the equation modulo 3:
\[
p^{2} - 6pq + q^{2} + 3q - 1 \equiv p^{2} + q^{2} - 1 \mod 3
\]
Noting that \(6 \equiv 0 \mod 3\) and \(3q \equiv 0 \mod 3\):
... | null | oa_36 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$ . If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the v... | 1. **Identify the given information and draw the pentagon:**
- $\angle A = 90^\circ$
- $\angle B = 105^\circ$
- $\angle C = 90^\circ$
- $AB = 2$
- $BC = CD = DE = \sqrt{2}$
2. **Analyze the triangle $\triangle ABD$:**
- Since $\angle A = 90^\circ$ and $\angle B = 105^\circ$, the remaining angle $\an... | null | oa_37 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
If $m > 0$, $n > 0$, $m+n=1$, and the minimum value of $\frac{t}{m} + \frac{1}{n} (t > 0)$ is $9$, then $t =$ \_\_\_\_\_\_.
Answer: | Given that $m > 0$, $n > 0$, and $m+n=1$, we have:
$$
\begin{align}
\frac{t}{m} + \frac{1}{n} &= \frac{t}{m} + \frac{1}{n}(m+n) \\
&= t + 1 + \frac{tn}{m} + \frac{m}{n} \\
&\geq t + 1 + 2\sqrt{t} \quad (\text{by AM-GM inequality})
\end{align}
$$
Since the minimum value of this expression is $9$, we have:
$$
t + 1 + 2\... | null | oa_38 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
For how many values of $a$ is it true that the line $y=2x+a$ passes through the vertex of the parabola $y=x^2+a^2$?
Answer: | The vertex of the parabola $y = x^2 + a^2$ is at $(0, a^2)$. The line $y=2x+a$ passes through this point if and only if substituting $x = 0$ into the line equation yields the y-coordinate of the vertex. Thus, substituting $x = 0$ gives $y = a$.
Setting this equal to the y-coordinate of the vertex, we have:
\[ a = a^2... | null | oa_39 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
The distance $d$ from the point $P(1,0)$ to the line $x-2y+1=0$ is given by $d=$ .
A: $\frac{\sqrt{5}}{5}$
B: $\frac{2\sqrt{5}}{5}$
C: $\frac{3\sqrt{5}}{5}$
D: $\frac{4\sqrt{5}}{5}$
Answer: | The distance $d$ from a point $(x_0,y_0)$ to a line $Ax+By+C=0$ is given by the formula:
$$d = \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}.$$
Applying this formula to find the distance from point $P(1,0)$ to the line $x-2y+1=0$, we substitute $x_0=1$, $y_0=0$, $A=1$, $B=-2$, and $C=1$.
Thus, we get:
$$d = \frac{|(1)(1)+(-2... | null | oa_40 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
If the hundreds digit of a three-digit natural number is 5, the tens digit is 1, and the units digit is 3, what is this number?
Answer: | The three-digit number with a hundreds digit of 5, a tens digit of 1, and a units digit of 3 is $\boxed{513}$ .
| null | oa_41 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
The line $y=kx+b$ passes through the first, third, and fourth quadrants. Then, $k$ __ .
Answer: | Given that the graph of the linear function $y=kx+b$ passes through the first, third, and fourth quadrants,
and knowing that when $k>0$, the line must pass through the first and third quadrants,
we can conclude that $k>0$.
Therefore, the answer is $k>0$, which can be written as $\boxed{k>0}$.
| null | oa_42 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let \( k, m, n \) be integers satisfying \( 1 < n \leq m - 1 \leq k \). Given \( S \subset \{1, 2, \ldots, k\} \) such that the sum of any \( n \) different elements in \( S \) is not equal to \( m \), find the maximum size o... | ### Problem:
Given the problem involves integer parameters $k$, $m$, and $n$, where $1 < n \leq m - 1 \leq k$, and a subset $S$ of $\{1,2, \ldots, k\}$ such that the sum of any $n$ distinct elements from $S$ is not equal to $m$, find the maximum size of the subset $S$.
#### Step 1: Initial Considerations
Let \( T = \... | null | oa_43 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given $z=2-i$, then $z(\overline{z}+i)=\left(\ \ \right)$
A: $6-2i$
B: $4-2i$
C: $6+2i$
D: $4+2i$
Answer: | Given $z=2-i$, we need to find the value of $z(\overline{z}+i)$.
First, let's find the conjugate of $z$, denoted as $\overline{z}$. For $z=2-i$, the conjugate $\overline{z}=2+i$ because the imaginary part changes its sign.
Now, let's calculate $z(\overline{z}+i)$ step by step:
\[
\begin{align*}
z(\overline{z}+i) & ... | null | oa_44 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\circ}$ and the largest angle is $160^{\circ}$, then $n$ equals:
$\text{(A) } 9\quad \text{(B... | 1. **Identify the formula for the sum of interior angles of a polygon**: The sum of the interior angles of a polygon with $n$ sides is given by the formula:
\[
S = 180^\circ (n-2)
\]
2. **Set up the arithmetic sequence**: Given that the angles are in arithmetic progression with a common difference of $5^\cir... | null | oa_45 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
With the rise in popularity of science fiction films and TV shows, more and more high school students are enjoying reading science fiction books. A certain school library purchased two types of science fiction books, labeled ... | ### Step-by-Step Solution
#### Part 1: Finding the Cost Price per Book for Type A and Type B
Let's denote the cost price per book for type B as $m$ yuan. Given that the cost price of each type A book is $1.5$ times the cost price of each type B book, the cost price per book for type A can be expressed as $1.5m$ yuan... | null | oa_46 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A rope of which a calf is tied is increased from some length to 23 m. The calf can graze an additional 1348.2857142857142 square meters of grassy ground. What was the initial length of the rope?
Answer: | The area that a calf can graze is given by the formula for the area of a circle, A = πr^2, where r is the length of the rope. When the rope is increased to 23 meters, the new area that the calf can graze is the area of a circle with a radius of 23 meters.
Let's denote the initial length of the rope as r meters. The i... | null | oa_47 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
you enter a weight loss challenge game and manage to lose 12 % of your body weight . for the final weigh in you are forced to wear clothes that add 2 % to your weight . what percentage of weight loss is measured at the final ... | Let's assume your initial body weight is \( W \) pounds.
You lose 12% of your body weight, so the weight you lose is \( 0.12W \).
Your new weight after losing 12% is \( W - 0.12W = 0.88W \).
At the final weigh-in, you have to wear clothes that add 2% to your weight. The weight of the clothes is \( 0.02 \times 0.88W... | null | oa_48 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given $n>4$ points in the plane, no three collinear. Prove that there are at least $\frac{(n-3)(n-4)}{2}$ convex quadrilaterals with vertices amongst the $n$ points.
Answer: | 1. **Base Case:**
- First, we need to show that given any 5 points in the plane, no three of which are collinear, there is always a convex quadrilateral. This is a well-known result in combinatorial geometry. Given 5 points, at least one subset of 4 points will form a convex quadrilateral. This is because if all 5 ... | null | oa_49 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given a circle with center at the origin O and radius of 5, denoted as ⊙O, determine the positional relationship between the point P(-3, 4) and ⊙O.
A: Inside ⊙O
B: On ⊙O
C: Outside ⊙O
D: Cannot be determined
Answer: | First, we need to determine the distance of point P from the origin O. The distance is calculated using the Euclidean distance formula in a Cartesian coordinate system. We have point P at coordinates (-3, 4), so the distance from the origin, denoted as d, is given by:
$$ d = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqr... | null | oa_50 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the curve $C: y = x + \frac{1}{x}$,
(1) Prove that the slope of the tangent line at any point on curve $C$ is less than 1;
(2) Find the equation of the tangent line on curve $C$ that has a slope of 0.
Answer: | (1) To prove that the slope of the tangent line is less than 1, let's start by finding the derivative of the function $y = x + \frac{1}{x}$.
The derivative $y'$ is obtained as follows:
$$
y' = \frac{d}{dx} \left( x + \frac{1}{x} \right) = \frac{d}{dx} x + \frac{d}{dx} \left( \frac{1}{x} \right) = 1 - \frac{1}{x^2}.
$$... | null | oa_51 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the set $M=\{x|x^{2}-5x+4\leq 0\}$ and the set $N=\{0,1,2,3\}$, find the number of elements in the intersection $M \cap N$.
A: $4$
B: $3$
C: $2$
D: $1$
Answer: | Firstly, let's factorize the quadratic inequality in the definition of set $M$.
The factored form of $x^{2}-5x+4$ is $(x-1)(x-4)$. The inequality $x^{2}-5x+4 \leq 0$ then becomes $(x-1)(x-4) \leq 0$. By analyzing the critical points $x=1$ and $x=4$, we deduce that $M$ consists of the real numbers between 1 and 4, inc... | null | oa_52 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A bag contains fifteen balls, some of which are red, some are yellow, and the rest are white. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{2}{35}$. How many balls in t... | Let $r$ represent the number of red balls in the bag. The probability that the first ball is red is $\frac{r}{15}$, and the probability that the second ball is red, given the first is red, is $\frac{r-1}{14}$. Thus, the probability that both balls are red is given by:
\[
\frac{r}{15} \cdot \frac{r-1}{14} = \frac{2}{35... | null | oa_53 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the function $f(x)=2\sin(2x-\frac{\pi}{3})-1$, a real number $a$ is randomly selected from the interval $\left[0,\frac{\pi}{2}\right]$. The probability that $f(a) > 0$ is ( ).
A: $\frac{5}{6}$
B: $\frac{2}{3}$
C: $\f... | **Analysis**
This question examines the application of geometric probability. Given the length of the interval is $\frac{\pi}{2}$, we need to find the range of $x$ that satisfies $f(a) > 0$, calculate the length of this interval, and then use the formula for geometric probability to find the result.
**Solution**
To... | null | oa_54 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Find a positive integer that is divisible by 14 and has a square root between 25 and 25.3.
Answer: | To find a positive integer that is divisible by 14 and has a square root between 25 and 25.3, we follow these steps:
1. Calculate the lower bound of the square root, which is $25^2 = 625$.
2. Calculate the upper bound of the square root, which is approximately $(25.3)^2 \approx 640$.
3. Identify the range in which th... | null | oa_55 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
The workforce of company X is 60% female. The company hired 28 additional male workers, and as a result, the percent of female workers dropped to 55%. How many employees did the company have after hiring the additional male w... | Let's denote the total number of employees before hiring the additional male workers as \( T \).
Before hiring the additional male workers, the number of female workers was \( 0.60T \) because the workforce was 60% female.
The number of male workers before hiring the additional ones was \( T - 0.60T = 0.40T \).
Aft... | null | oa_56 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
To make fried green tomatoes, Thelma cuts each green tomato into 8 slices before cooking them up. If 20 slices of fried green tomato make a meal for a single person, Thelma needs to make enough fried green tomatoes to feed a ... | If Thelma cuts each green tomato into 8 slices, and she has 20 tomatoes, then the total number of slices she will have is:
20 tomatoes * 8 slices per tomato = 160 slices
If 20 slices make a meal for a single person, then the number of people Thelma can feed with 160 slices is:
160 slices / 20 slices per person = 8 ... | null | oa_57 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Prove that the ratio of the area of any triangle to the area of its circumscribed circle is less than $2 / 3$.
Answer: | 1. Let the sides and angles of triangle $ \triangle ABC $ be $ a, b, c $ and $ \alpha, \beta, \gamma $ respectively. Let $ R $ be the radius of the circumscribed circle around the triangle $ \triangle ABC $.
2. The area \( S_{ABC} \) of the triangle can be calculated using the formula involving sides $ b $ and $ c $ ... | null | oa_58 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A bowl was filled with 10 ounces of water, and 0.0008 ounce of the water evaporated each day during a certain period. During this period, 0.4% of the original amount of water evaporated. How many days did this period last?
An... | To find out how many days the period lasted, we first need to determine the total amount of water that evaporated during this period.
We know that 0.4% of the original amount of water evaporated. The original amount of water was 10 ounces. To find out how much 0.4% of 10 ounces is, we calculate:
0.4% of 10 ounces = ... | null | oa_59 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Find the equation of the normal line to the given curve at the point with the x-coordinate \( x_{0} = 1 \).
\[ y = 2x^{2} - 3x + 1 \, (x_{0} = 1) \]
Answer: | 1. Given the function:
\[
y = 2x^2 - 3x + 1
\]
and the point where the normal line is to be found is \(x_0 = 1\).
2. First, we need to find the derivative of \(y\) with respect to \(x\):
\[
y' = \frac{d}{dx}(2x^2 - 3x + 1) = 4x - 3
\]
3. Evaluate the derivative at \(x_0 = 1\):
\[
y'(x_0) =... | null | oa_60 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
if a , b , c , d , e and f are integers and ( ab + cdef ) < 0 , then what is the maximum number a of integers that can be negative ?
Answer: | To maximize the number of negative integers, we need to consider the expression (ab + cdef) and the condition that it must be less than 0.
Let's analyze the two parts of the expression separately:
1. The product ab: For this product to be negative, one of the integers a or b must be negative, and the other must be p... | null | oa_61 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Find the area of the triangle with vertices $(1,8,11),$ $(0,7,9),$ and $(-3,10,9).$
Answer: | Let $A = (1,8,11),$ $B = (0,7,9),$ and $C = (-3,10,9)$. To find the side lengths, we apply the distance formula:
- $AB = \sqrt{(1-0)^2 + (8-7)^2 + (11-9)^2} = \sqrt{1 + 1 + 4} = \sqrt{6}$,
- $BC = \sqrt{(0+3)^2 + (7-10)^2 + (9-9)^2} = \sqrt{9 + 9 + 0} = \sqrt{18} = 3\sqrt{2}$,
- $AC = \sqrt{(1+3)^2 + (8-10)^2 + (11-9... | null | oa_62 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let \( x \) and \( y \) be two different positive numbers, and denote the following:
\[ R = \sqrt{\frac{x^2 + y^2}{2}}, \]
\[ A = \frac{x + y}{2}, \]
\[ G = \sqrt{xy}, \]
\[ H = \frac{2xy}{x + y}. \]
Determine the order of t... | ### Problem:
设 $x, y$ 是两个不同的正数, 记
$$
R=\sqrt{\frac{x^{2}+y^{2}}{2}}, A=\frac{x+y}{2}, G=\sqrt{x y}, H=\frac{2 x y}{x+y} .
$$
试确定 $R-A, A-G, G-H$ 三个数的大小关系.
:
1. **Show that \( G < A \).**
- Given that \( A = \frac{x + y}{2} \) and \( G = \sqrt{xy} \).
- By the Arithmetic-Geometric Mean Inequality (AM-GM Ine... | null | oa_63 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given \( f(x)=\frac{a x^{2}+1}{b x+c} \) where \( a, b, c \in \mathbf{Z} \) is an odd function, and \( f(1)=2 \), \( f(2)<3 \).
(1) Find the values of \( a, b, \) and \( c \); (2) Determine the monotonicity of the function \(... | #### Part (1)
We are given the function \( f(x) = \frac{ax^2 + 1}{bx + c} \), and we know \(f(x)\) is an odd function. This gives us the condition \( f(-x) = -f(x) \).
1. Express the condition mathematically:
\[
f(-x) = \frac{ax^2 + 1}{-bx + c}, \quad \text{and} \quad -f(x) = -\frac{ax^2 + 1}{bx + c}
\]
S... | null | oa_64 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A rectangular storage unit is 15 yd long, 12 yd wide, and 8 yd high. It is to be painted on all its walls both inside and outside, and on the ceiling, but not on the floor or the two windows, each measuring 3 yd by 4 yd. Calc... | #### Step 1: Wall Area Calculations
There are four walls, with two sets of opposite walls.
1. **First pair of walls (Length x Height):**
- Area of each wall = \(15 \text{ yd} \times 8 \text{ yd} = 120 \text{ sq yd}\).
- Two walls total \(120 \text{ sq yd} \times 2 = 240 \text{ sq yd}\).
2. **Second pair of wal... | null | oa_65 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
The simple interest on a certain sum of money for some years at 10% per annum is half the compound interest on Rs. 4000 for 2 years at 10% per annum. The sum placed on simple interest is Rs. 1400.0000000000014. For how many y... | First, let's calculate the compound interest on Rs. 4000 for 2 years at 10% per annum.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, including interest.
P = the principal amount (the initial amount of money)
r = the annual interest rate (decimal)
... | null | oa_66 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let \( n \in \mathbf{N}^{*} \), \( x_0 = 0 \), \( x_i > 0 \) for \( i=1,2, \cdots, n \), and \( \sum_{i=1}^{n} x_i = 1 \). Prove that
\[ 1 \leqslant \sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+\cdots+x_{i-1}} \sqrt{x_i+x_{i+1}+\... | Let \( n \in \mathbb{N}^{*} \), \( x_{0}=0 \), \( x_{i}>0 \) for \( i=1,2, \cdots, n \), and \( \sum_{i=1}^{n} x_{i}=1 \).
We want to prove that
\[ 1 \leq \sum_{i=1}^{n} \frac{x_{i}}{\sqrt{1+x_{0}+\cdots+x_{i-1}} \sqrt{x_{i}+x_{i+1}+\cdots+x_{n}}} < \frac{\pi}{2} . \]
1. **Right Inequality:**
Using the inequalit... | null | oa_67 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Prove that if the angles at the base of a trapezoid are equal, then the trapezoid is isosceles.
Answer: | 1. **Given Information:**
We are given a trapezoid where the angles at the bases are equal, and we need to prove that it is isosceles.
2. **Label the Trapezoid:**
Let the trapezoid be \(ABCD\) with bases \(AB\) and \(CD\), where the angles at the bases are \(\angle A = \angle D\) and \(\angle B = \angle C\).
3... | null | oa_68 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A rectangle has a diagonal of length $x$ and its length is three times its width. What is the area of the rectangle?
A) $\frac{1}{4}x^2$
B) $\frac{3}{10}x^2$
C) $\frac{1}{2}x^2$
D) $x^2$
E) $\frac{3}{2}x^2$
Answer: | 1. **Define the dimensions of the rectangle**: Let the width of the rectangle be $w$. Since the rectangle's length is three times its width, the length is $3w$.
2. **Apply the Pythagorean Theorem**: The diagonal of the rectangle forms a right triangle with the rectangle's length and width. Therefore, using the Pythag... | null | oa_69 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
For how many integers \( a \) with \( 1 \leq a \leq 10 \) is \( a^{2014} + a^{2015} \) divisible by 5?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
Answer: | Consider the expression \( a^{2014} + a^{2015} \). We start by factoring this expression:
\[ a^{2014} + a^{2015} = a^{2014}(1 + a) \]
To determine when \( a^{2014} + a^{2015} \) is divisible by 5, we need to consider when at least one of the factors \( a^{2014} \) or \( 1 + a \) is divisible by 5. We will check each... | null | oa_70 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given that $\tan \alpha = 2 \tan \frac{\pi}{5}$, find the value of $\frac{\cos \left( \alpha - \frac{3}{10} \pi \right)}{\sin \left( \alpha - \frac{\pi}{5} \right)} = \boxed{?}$
A: $1$
B: $2$
C: $3$
D: $4$
Answer: | This problem requires simplifying trigonometric functions and applying trigonometric identities.
First, given $\tan \alpha = 2 \tan \frac{\pi}{5}$, we want to find the value of $\frac{\cos \left( \alpha - \frac{3}{10} \pi \right)}{\sin \left( \alpha - \frac{\pi}{5} \right)}$.
We can start by using the trigonometric ... | null | oa_71 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
In the first year of Shishi High School, there are 570 male students. If a stratified random sampling method is used to draw a sample of size 110 from the first-year students, including 53 female students, then the total numb... | To solve the problem, let's denote the total number of first-year students at Shishi High School as $N$. The problem states that a stratified random sampling method is used to draw a sample of size 110, which includes 53 female students. Given that there are 570 male students, the number of male students in the sample... | null | oa_72 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Quincy just bought a car using a 5-year loan with an annual interest rate of 4%. He put $5,000.00 down as a down payment, making his monthly payment $250.00. What is the price of the car?
Answer: | To find the price of the car, we need to calculate the total amount Quincy will pay over the life of the loan and then add the down payment.
First, let's calculate the total amount paid over the life of the loan:
Monthly payment = $250.00
Loan term = 5 years
Total number of payments = 5 years * 12 months/year = 60 ... | null | oa_73 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the function $y = 4\cos^2 x + 4\sqrt{3}\sin x \cos x - 2$, with $x \in \mathbb{R}$.
1. Find the smallest positive period of the function.
2. Find the maximum value of the function and the corresponding $x$ values.
3. Wr... | 1. The function can be rewritten using the double-angle formulas $\cos^2 x = \frac{1 + \cos 2x}{2}$ and $2\sin x\cos x = \sin 2x$:
$$
\begin{aligned}
y &= 4\left(\frac{1 + \cos 2x}{2}\right) + 4\sqrt{3}\left(\frac{\sin 2x}{2}\right) - 2 \\
&= 2 + 2\cos 2x + 2\sqrt{3}\sin 2x - 2 \\
&= 2\cos 2x + 2\sqrt{3... | null | oa_74 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let $m, n \in \mathbf{N}^{+}$, with $\gcd(m, n)=1$, and let $p, q$ be two distinct odd prime numbers. Find the greatest common divisor of $\left(p^{m}+q^{m}\right)$ and $\left(p^{n}+q^{n}\right)$, denoted as $\gcd\left(p^{m}+... | Given a problem and solution, let's outline a detailed step-by-step approach.
#### Problem:
Let $m, n \in \mathbf{N}^{+}$, such that $\gcd(m, n)=1$, and let $p, q$ be two odd prime numbers which are not equal. Find the greatest common divisor of $p^m + q^m$ and $p^n + q^n$, i.e., $\gcd(p^m + q^m, p^n + q^n)$.
#First... | null | oa_75 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Prove that \(\sin x + \frac{1}{2} \sin 2x + \frac{1}{3} \sin 3x > 0\) for \(0 < x < 180^\circ\).
Answer: | **Step 1: Express \(\sin 2x\) and \(\sin 3x\) in terms of \(\sin x\) and \(\cos x\)**
We start by noting the known trigonometric identities:
\[
\sin 2x = 2 \sin x \cos x
\]
and
\[
\sin 3x = 3 \sin x - 4 \sin^3 x
\]
**Step 2: Substitute the identities into the given expression**
We are asked to prove:
\[
\sin x + \fra... | null | oa_76 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Mary sees a flock of various types of birds crossing the street. There are 2 ducks with 5 ducklings each, 6 ducks with 3 ducklings each, and 9 ducks with 6 ducklings each. Additionally, she sees 4 geese with 7 goslings each a... | First, let's calculate the total number of birds (adults and offspring) for each type:
Ducks:
- 2 ducks with 5 ducklings each: 2 * 5 = 10 ducklings + 2 adults = 12
- 6 ducks with 3 ducklings each: 6 * 3 = 18 ducklings + 6 adults = 24
- 9 ducks with 6 ducklings each: 9 * 6 = 54 ducklings + 9 adults = 63
Geese:
- 4 ge... | null | oa_77 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given sets $A=\{1,2,3\}$ and $B=\{0,2,3,4\}$, find $A\cap B$.
Answer: | Given the sets $A=\{1,2,3\}$ and $B=\{0,2,3,4\}$, to find the intersection $A\cap B$, we compare elements from both sets to determine which elements they have in common.
- First, we examine if $1$ from set $A$ is in set $B$: it is not.
- Then, we check if $2$ from set $A$ is in set $B$: it is.
- Next, we ascertain if... | null | oa_78 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
In which set are both elements metalloids? ${ \textbf{(A)}\ \text{Cr and Mo}\qquad\textbf{(B)}\ \text{Ge and As}\qquad\textbf{(C)}\ \text{Sn and Pb}\qquad\textbf{(D)}}\ \text{Se and Br}\qquad $
Answer: | 1. **Identify Metalloids on the Periodic Table:**
Metalloids are elements that have properties intermediate between metals and nonmetals. They are typically found in a diagonal region of the periodic table, often referred to as the "staircase" or "L" shape, which includes the elements: Boron (B), Silicon (Si), Germ... | null | oa_79 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Ed and Jacob already had 2 shells in their shell collection. Then they went to the beach to collect even more. Ed found 7 limpet shells, 2 oyster shells, 4 conch shells, and 3 scallop shells. Jacob found 2 more shells than Ed... | First, let's find out how many shells Ed found at the beach. He found 7 limpet shells, 2 oyster shells, 4 conch shells, and 3 scallop shells. So in total, Ed found:
7 + 2 + 4 + 3 = 16 shells
Jacob found 2 more shells than Ed, so he found:
16 + 2 = 18 shells
Marissa found 5 limpet shells, 6 oyster shells, 3 conch she... | null | oa_80 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the function $f(x)$ defined on the set of positive integers that satisfies $f(1)=2$, $f(2)=-3$, and $f(n+2)=f(n+1)-f(n)$, find the value of $f(2014)$.
A. $2$
B. $-2$
C. $3$
D. $-3$
Answer: | This problem involves the application of periodic sequences. The key is to find the period of the sequence using its recursive formula.
Analysis:
The function $f(x)$ satisfies $f(1)=2$, $f(2)=-3$, and $f(n+2)=f(n+1)-f(n)$.
We can find subsequent terms of the sequence as follows:
- $f(3)=f(2)-f(1)=-3-2=-5$
- $f(4)=-5... | null | oa_81 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given an ellipse $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ with $a>b>0$, where its foci are $F_1$ and $F_2$, $b=4$, and the eccentricity is $$\frac {3}{5}$$. A line passing through $F_1$ intersects the ellipse at point... | Since the given ellipse is $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ with foci $F_1$ and $F_2$, $b=4$, and the eccentricity is $$\frac {3}{5}$$, we have:
$$
\begin{cases}
b=4 \\
\frac {c}{a}= \frac {3}{5} \\
a^{2}=b^{2}+c^{2}
\end{cases}
$$
Solving these equations, we find $a=5$, $b=4$, and $c=3$.
For a lin... | null | oa_82 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. How many diagonals does a regular decagon have?
Answer: | Using the formula \( \frac{n(n-3)}{2} \) to find the number of diagonals in a regular n-sided polygon, where \( n \) is the number of sides:
1. Substitute \( n = 10 \) into the formula:
\[
\frac{n(n-3)}{2} = \frac{10(10-3)}{2}
\]
2. Simplify the expression:
\[
\frac{10 \times 7}{2} = \frac{70}{2} = 35
... | null | oa_83 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given the sequence $\{a\_n\}$, its first $n$ terms sum is $S\_n$. The points $(n, S\_n) (n \in \mathbb{N}^*)$ lie on the graph of the function $y = 2x^2 + x$. Determine the general term formula of the sequence $\{a\_n\}$.
Ans... | Since the first $n$ terms sum of the sequence $\{a\_n\}$ is $S\_n$ and the points $(n, S\_n)$ lie on the graph of the function $y = 2x^2 + x$, we have:
$$S\_n = 2n^2 + n$$
For $n \geq 2$, we also have:
$$S\_{n-1} = 2(n-1)^2 + (n-1)$$
The $n$-th term of the sequence is given by:
$$a\_n = S\_n - S\_{n-1} = (2n^2 + ... | null | oa_84 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A man is 46 years older than his son. In a certain number of years, his age will be twice the age of his son. The present age of his son is 44 years. How many years from now will the man's age be twice the age of his son?
Ans... | Let's denote the present age of the son as S and the present age of the man (father) as M.
According to the information given:
S = 44 years (present age of the son)
M = S + 46 (since the man is 46 years older than his son)
Now, we can calculate the present age of the man:
M = 44 + 46
M = 90 years (present age of the... | null | oa_85 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
In the $xy$-plane, the segment with endpoints $(-6,0)$ and $(18,0)$ is the diameter of a circle. If the point $(x,10)$ is on the circle, then $x=$
A. $6 + 2\sqrt{11}$
B. $6 - 2\sqrt{11}$
C. $8\sqrt{11}$
D. $12$
E. $6$
Answer: | 1. **Identify the center and radius of the circle**:
The endpoints of the diameter of the circle are $(-6,0)$ and $(18,0)$. The center of the circle, $C$, is the midpoint of the diameter:
\[
C = \left(\frac{-6 + 18}{2}, \frac{0 + 0}{2}\right) = (6, 0)
\]
The radius, $r$, is half the length of the diame... | null | oa_86 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let the set $M=\{x | x < 2011\}$, and the set $N=\{x | 0 < x < 1\}$. Which of the following relations is correct?
A. $M \cup N = \mathbb{R}$
B. $M \cap N = \emptyset$
C. $N \in N$
D. $M \cap N = \{x | 0 < x < 1\}$
Answer: | Given that $M=\{x | x < 2011\}$ and $N=\{x | 0 < x < 1\}$:
- The set $N$ is a subset of $M$ because all elements in $N$ are less than 2011.
- The intersection $M \cap N$ consists of all elements that are in both $M$ and $N$. Since $N$ itself contains numbers between 0 and 1, and all such numbers are included in $M$ (s... | null | oa_87 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
A discrepancy of $0.01''$ is made in the measurement of a line $20''$ long, and an error of $0.3''$ is made in measuring a line that is $150''$ long. How do the relative errors for these two measurements compare?
$\textbf{(A)... | 1. **Calculate the Relative Error for the First Measurement:**
\[
\text{Relative Error}_{\text{first}} = \frac{0.01''}{20''} \times 100\% = 0.05\%
\]
2. **Calculate the Relative Error for the Second Measurement:**
\[
\text{Relative Error}_{\text{second}} = \frac{0.3''}{150''} \times 100\% = 0.2\%
\]... | null | oa_88 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
What number must be added to \(4 - 8i\) to obtain \(1 + 10i\)? Here, \(i^2 = -1\).
Answer: | We need to find the complex number \(z\) such that:
\[ 4 - 8i + z = 1 + 10i \]
Re-arranging the equation to solve for \(z\), we get:
\[ z = 1 + 10i - 4 + 8i \]
\[ z = (1 - 4) + (10i + 8i) \]
\[ z = -3 + 18i \]
Thus, the required number is:
\[ \boxed{-3 + 18i} \]
| null | oa_89 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Calculate $\log_{10} 50 + \log_{10} 20$.
Answer: | Using the logarithmic addition property $\log_b x + \log_b y = \log_b (xy)$ we can combine $\log_{10} 50$ and $\log_{10} 20$ into a single expression:
\[ \log_{10} 50 + \log_{10} 20 = \log_{10} (50 \cdot 20) \]
\[ = \log_{10} (1000) \]
We know that $1000=10^3$, thus:
\[ \log_{10} 1000 = 3 \]
Therefore, the solution ... | null | oa_90 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Andrew works in a company that provides a generous vacation allotment: for every 10 days worked, you get 1 vacation day. If last year Andrew worked 300 days and took 5 days off in March and twice as many in September, how man... | First, let's calculate how many vacation days Andrew earned last year. For every 10 days worked, he gets 1 vacation day. So if he worked 300 days:
300 days / 10 = 30 vacation days earned
Now let's calculate how many vacation days Andrew has already taken. He took 5 days off in March and twice as many in September:
... | null | oa_91 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
What is the positive number $x$ that satisfies the equation $(\log_2 x)(\log_x 7) = \log_2 7$?
**A)** All positive $x$ \\
**B)** Only $x = 7$ and $x = 2$ \\
**C)** All positive $x \neq 1$ \\
**D)** Only $x = 7$
Answer: | 1. **Rewrite the given equation using the change-of-base formula:**
\[
(\log_2 x)(\log_x 7) = \left(\frac{\log x}{\log 2}\right)\left(\frac{\log 7}{\log x}\right)
\]
2. **Simplify the expression:**
\[
\left(\frac{\log x}{\log 2}\right)\left(\frac{\log 7}{\log x}\right) = \frac{\log 7}{\log 2}
\]
... | null | oa_92 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
An alien species, called Triamoeba, is placed in an experimental container. Initially, there is one Triamoeba. Each day, every Triamoeba in the container triples in number (each Triamoeba splits into three Triamoebas). After ... | Initially, there is 1 Triamoeba. At the end of the first day, there are $3^1 = 3$ Triamoebas. At the end of the second day, each Triamoeba has tripled, so there are $3 \cdot 3 = 3^2 = 9$ Triamoebas. This pattern continues such that at the end of $n$ days, there are $3^n$ Triamoebas. Thus, after seven days, the number ... | null | oa_93 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
In a bag, there are 3 red balls and 2 white balls. If two balls are drawn successively without replacement, let A be the event that the first ball drawn is red, and B be the event that the second ball drawn is red. Find $p(B|... | When drawing two balls successively without replacement from a bag containing 3 red balls and 2 white balls, and defining the events A = {drawing a red ball first} and B = {drawing a red ball second}, we want to find the conditional probability $p(B|A)$.
According to the definition of conditional probability,
$$ p(B|... | null | oa_94 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ( $AB = BC$ ) so that $AK +LC = KL$ . A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$ , inter... | 1. **Identify the given information and setup the problem:**
- We have an isosceles triangle $\triangle ABC$ with $AB = BC$.
- Points $K$ and $L$ are chosen on sides $AB$ and $BC$ respectively such that $AK + LC = KL$.
- A line parallel to $BC$ is drawn through the midpoint $M$ of segment $KL$, intersecting s... | null | oa_95 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given in triangle $\triangle ABC$, point $A(1, 2)$, and the equations of the medians on sides $AB$ and $AC$ are $5x-3y-3=0$ and $7x-3y-5=0$ respectively, find the general form of the equation of the line on which side $BC$ li... | Let the coordinates of point $C$ be $(a, b)$.
Since point $C$ lies on the median of side $AB$,
we have $5a-3b-3=0$.
Also, since the midpoint of $AC$ has coordinates $\left( \frac {1+a}{2}, \frac {2+b}{2}\right)$,
and this midpoint lies on the median of side $AC$,
we have $7\times \frac {1+a}{2}-3\times \frac... | null | oa_96 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given an isosceles triangle \(ABC\) with \(AB = BC\). Point \(E\) is marked on the ray \(BA\) beyond point \(A\), and point \(D\) is marked on the side \(BC\). It is known that
\[
\angle ADC = \angle AEC = 60^\circ, \quad A... | 1. **Identifying Key Properties**: Given an isosceles triangle \( \triangle ABC \) with \( AB = BC \), point \( E \) on ray \( BA \) beyond \( A \), and point \( D \) on side \( BC \). We know:
\[
\angle ADC = \angle AEC = 60^\circ, \quad AD = CE = 13, \quad \text{and} \quad DC = 9.
\]
2. **Setting Up Aux... | null | oa_97 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Let $a$ and $b$ be real numbers, and the function
$$
f(x) = ax^{2} + b(x + 1) - 2.
$$
If the equation $f(x) = x$ has two distinct real roots for any real number $b$, determine the range of values for the real number $a$.
Ans... | 1. **Given equation**: We start with the function \( f(x) = ax^2 + b(x+1) - 2 \) and need to find the range of \( a \) such that \( f(x) = x \) has two distinct real roots for any real number \( b \).
2. **Rewriting the equation \( f(x) = x \)**:
\[
ax^2 + b(x+1) - 2 = x
\]
3. **Simplify** the equation:
... | null | oa_98 |
ServiceNow-AI/R1-Distill-SFT | The following is a question about knowledge and skills in advanced master-level STEM courses.
Given a circle with a point $A(2,3)$ on it, the point symmetric to $A$ with respect to the line $x+2y=0$ is also on the circle. Additionally, the chord formed by the intersection of the circle and the line $x-y+1=0$ has a len... | Let the center of the sought circle be $(a,b)$, and its radius be $r$.
Since the point $A(2,3)$ is on the circle, and its symmetric point $A'$ (with respect to the line $x+2y=0$) is also on the circle, the center of the circle $(a,b)$ must lie on the line $x+2y=0$. Therefore, we have equation $(1)$: $a+2b=0$.
Additi... | null | oa_99 |
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