inclusionAI/Ring-lite-rl-data huggingface dataset:
Rahim went on a book shopping spree and visited 4 different shops. At the first shop, he bought 42 books for Rs. 520 and received a 10% discount on the total purchase. At the second shop, he bought 22 books for Rs. 248 and paid an additional 5% in taxes. Then, at the third shop, he bought 35 books for Rs. 740 and received a 15% discount on the total purchase. Finally, at the fourth shop, he bought 18 books for Rs. 360 and paid an additional 8% in taxes. What is the average price Rahim paid per book after considering all the discounts and taxes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.93}
If $y<0$, find the range of all possible values of $y$ such that $\lceil{y}\rceil\cdot\lfloor{y}\rfloor=110$.  Express your answer using interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-11, -10)}
Jason is trying to figure out whether he can afford a new car. The car he wants costs $32,000, and he can make a down payment of $8,000. He'll have to get a loan for the rest of the amount and make 48 equal monthly payments on it. Each month, he also has to pay interest equal to 5% of that month's payment. How much will Jason's monthly payments be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$502.08}
Given a geometric sequence $\{a_n\}$ with positive terms, the sum of the first $n$ terms is $S_n$. If $-3$, $S_5$, and $S_{10}$ form an arithmetic sequence, calculate the minimum value of $S_{15} - S_{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The Chinese Academy of Social Sciences needs to conduct a stratified sampling survey of the economic conditions in the counties within the Poyang Lake Eco-Economic Zone in Jiangxi Province. The survey will cover 20 counties within 9 prefecture-level cities (excluding urban districts and county-level cities). If exactly 2 samples were taken from the 8 counties in the Jiujiang area that are part of the economic zone, then the total number of samples taken is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( A \in M_{5\times 6}(\mathbb{R}) \) be a matrix with rank 4. Consider the block matrix \( D = \begin{pmatrix} I_5 & A \\ A^T & 0 \end{pmatrix} \). Determine the rank of \( D \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given the function y = sin(2x + π/4), translate its graph left by π/4 units and then up by 2 units. Calculate one of the symmetry centers of the resulting graph. Express your answer as an ordered pair (x, y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(π/8, 2)}
已知函数 $f: \mathbf{R} \rightarrow \mathbf{R}$, 若对任意实数 $x 、 y 、 z$ 都有  
$$\frac{1}{3} f(x y)+\frac{1}{3} f(x z)-f(x) f(y z) \geqslant \frac{1}{9},$$
求 $\sum\limits_{i=1}^{100}[i f(i)]$. 其中, $[x]$ 表示不超过实数 $x$ 的最大整数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1650}
设a,b,c,a+b-c,a+c-b,b+c-a,a+b+c是7个两两不同的素数，且a，b，c中有两数之和是800．设d是这7个素数中最大数与最小数之差，求d的最大可能值．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1594}
In a checkered square with a side length of 2018, some cells are painted white and the rest are black. It is known that from this square, one can cut out a 10x10 square where all the cells are white, and a 10x10 square where all the cells are black. What is the smallest value for which it is guaranteed that one can cut out a 10x10 square in which the number of black and white cells differ by no more than?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
If the domain of the function $f(x)=x^{2}$ is $D$, and its range is ${0,1,2,3,4,5}$, then there are \_\_\_\_\_\_ such functions $f(x)$ (answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
The manager of a discount clothing store received a shipment of Fall clothing. The shipment contained 10 identical sweaters and 20 identical jackets. The second shipment at the same prices, contained 5 of the same sweaters and 15 of the same jackets. The second shipment was 550 dollars. What was the cost of the first shipment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1100}
Given $p$: $|1-\frac{x-1}{3}|\leqslant 2$, $q$: $x^2-2x+1-m^2\leqslant 0 (m > 0)$, if $\lnot p$ is a sufficient but not necessary condition for $\lnot q$, find the range of values for the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < m\leqslant 3}
Compute the limit: $$L=\lim_{x\to 0}\frac{\sin(\ln(x+1))-\ln(\sin(x)+1)}{\sin^4\left(\frac{x}{2}\right)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/3}
Given that $\sqrt{51.11}\approx 7.149$ and $\sqrt{511.1}\approx 22.608$, determine the value of $\sqrt{511100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{714.9}
Binhao has a fair coin. He writes the number $+1$ on a blackboard. Then he flips the coin. If it comes up heads (H), he writes $+\frac12$ , and otherwise, if he flips tails (T), he writes $-\frac12$ . Then he flips the coin again. If it comes up heads, he writes $+\frac14$ , and otherwise he writes $-\frac14$ . Binhao continues to flip the coin, and on the nth flip, if he flips heads, he writes $+ \frac{1}{2n}$ , and otherwise he writes $- \frac{1}{2n}$ . For example, if Binhao flips HHTHTHT, he writes $1 + \frac12 + \frac14 - \frac18 + \frac{1}{16} -\frac{1}{32} + \frac{1}{64} -\frac{1}{128}$ . The probability that Binhao will generate a series whose sum is greater than $\frac17$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + 10q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13, 19, 20, 25$, and $31$, although this is not necessarily their order around the pentagon. What is the area of the pentagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{745}
Two circles with radii 2 and 3 touch each other externally at point $A$. Their common tangent passing through point $A$ intersects their other two common tangents at points $B$ and $C$. Find $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{6}}
Find the Galois group of $x^6+1$ over $F=\mathbb{F}_2=\{0,1\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\mathbb{Z}/2\mathbb{Z}$}
How many of the smallest 2401 positive integers written in base 7 include the digits 4, 5, or 6?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2146}
Given a tetrahedron $ABCD$, each edge is colored red or blue with equal probability. What is the probability that point $A$ can reach point $B$ through red edges within the tetrahedron $ABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
In the right-angled triangle $ABC$, the foot of the altitude from point $A$ to the hypotenuse $AB$ is point $D$. The angle bisectors of $\angle ACD$ and $\angle BCD$ intersect the hypotenuse $AB$ at points $E$ and $F$ respectively. Determine the ratio of the inradius of triangle $ABC$ to the circumradius of triangle $CEF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
从一个正方体的八个顶点中取出三个. 则以这三个点为顶点构成直角三角形的概率是 $\qquad$。
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
In a triangle $XYZ$, angle $\angle ZXY$ is $72^\circ$. It is given that side $XY$ is twice as long as side $YZ$. What is the measure of angle $\angle XYZ$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\angle XYZ = 36^\circ}
Given the equation:

$$\int_{0}^ {\pi/2}(\sin x+a\cos x)^3dx-\frac{4a}{\pi-2}\int_{0}^{\pi/2}x\cos x dx=2$$

find the coefficient of the linear term in the cubic polynomial expression inside the first integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Usually, I go up the escalator in the subway. I have calculated that when I walk up the moving escalator, I ascend 20 steps, and it takes me exactly 60 seconds. My wife walks up the stairs more slowly and only ascends 16 steps; therefore, her total time to ascend the escalator is longer - it is 72 seconds.

How many steps would I have to climb if the escalator suddenly breaks down?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
57 Given a regular $(2n+1)$-gon. Consider all triangles whose vertices are the vertices of this $(2 n+1)$-gon. Find the number of triangles that contain the center of the regular $(2 n+1)$-gon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n+1)(2n+1)}{6}}
Given that the lengths of the sides of an acute triangle are $3$, $4$, and $a$, the range of values for $a$ is \_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{7},5)}
A school club designed an unconventional rectangular checkerboard that measures 10 by 6. Calculate the probability that a randomly chosen unit square does not touch any edge of the board or is not adjacent to any edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
Each cell of a $2 \times 5$ grid of unit squares is to be colored white or black. Compute the number of such colorings for which no $2 \times 2$ square is a single color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{634}
Distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=BC=CD=1$. Points $E$ and $F$ lie on a second line, parallel to the first, with $EF=1$. A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3$}
In a cube with 7 visible corner cubies, determine the number of different permutations and orientations possible for the hidden edges, given that the visible corners and edges are correctly positioned and oriented.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find all integers $n$ such that $\frac{5^n - 1}{3}$ is a prime or a perfect square of an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a parallelepiped \( ABCD A_1 B_1 C_1 D_1 \), point \( X \) is chosen on edge \( A_1 D_1 \) and point \( Y \) is chosen on edge \( BC \). It is known that \( A_1 X = 5 \), \( B Y = 3 \), and \( B_1 C_1 = 14 \). The plane \( C_1 X Y \) intersects the ray \( DA \) at point \( Z \). Find \( DZ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Consider a unit cube in a coordinate system with vertices $A(0,0,0)$, $A'(1,1,1)$, and other vertices placed accordingly. A regular octahedron has vertices placed at fractions $\frac{1}{3}$ and $\frac{2}{3}$ along the segments connecting $A$ with $A'$'s adjacent vertices and vice versa. Determine the side length of this octahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{3}}
Determine the minimum distance $d(C)$ of a linear code $C$ with a parity-check matrix $H$ over $\Bbb F_2$. The first seven rows of $H$ are the identity matrix $I_7$, and the remaining rows consist of all weight-$3$ vectors in $\{0,1\}^7$. The rank of $H$ is 7. What is the minimum distance $d(C)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $f(x)=\cos(\omega x+\varphi)$, where $\omega\gt 0$ and $0\lt \varphi\lt \pi$. The smallest positive period of the function is $T$. If $f(T)=\frac{\sqrt{3}}{2}$ and $x=\frac{π}{9}$ is a zero of $f(x)$, then the minimum value of $\omega$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Compute the flux of water through the parabolic cylinder $S: y = x^2$, for $0 \le x \le 2$ and $0 \le z \le 3$, given the velocity vector field $F(x,y,z) = 3z^2\mathbf{i} + 6\mathbf{j} + 6xz\mathbf{k}$. What is the flow rate of the fluid through this surface?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Based on a $city^{'}s$ rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
45
}
Given the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \; (a, b>0)$ with left and right foci at $F_{1}(-c, 0)$ and $F_{2}(c, 0)$, respectively, and an eccentricity of 2. If the line $y=2(x-c)$ intersects the hyperbola at points $A$ and $B$, and the incenters of $\triangle A F_{1} F_{2}$ and $\triangle B F_{1} F_{2}$ are $I_{1}$ and $I_{2}$ respectively, and $\left|I_{1} I_{2}\right|=2 \sqrt{5}$, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Suppose that $f'(z)$ exists on $\mathbb{C}$ and $\Im(f'(z))=6x(2y-1)$, with $f(0)=3-2i$ and $f(1)=5-5i$. Find $f(1+i)$. \(z = x + yi\) is a complex number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 + 2i}
Let $D$ and $E$ be points on respectively $[AB]$ and $[AC]$ of $\triangle ABC$ where $|AB|=|AC|$, $m(\widehat{BAC})=40^{\circ}$. Let $F$ be a point on $BC$ such that $C$ is between $B$ and $F$. If $|BE|=|CF|$, $|AD|=|AE|$ and $m(\widehat{BEC})=60^{\circ}$, then what is $m(\widehat{DFB})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
25°
}
Calculate the limit of the following integral as \( n \to \infty \):

\[ \lim_{n \to \infty } \frac{1}{n^2} \int_0^n \frac{ \sqrt{n^2-x^2} }{2+x^{-x}} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Find an upper bound for the smallest positive integer $b$ such that the inequality \(2^{3^{4^{\cdots^a}}} \leq b^{(b-1)^{\cdots^{3^{2}}}}\) holds for a given positive integer \(a \geq 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + 1}
Evaluate the integral \( \int_{0}^{\infty} \frac{d x}{\left(x^{3}+\frac{1}{x^{3}}\right)^{2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{18}}
On circle in clockwise order are written positive integers from $1$ to $2010$. Let us cross out number $1$, then number $10$, then number $19$, and so on every $9$th number in that direction.  How many numbers at that moment are not crossed out?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1340}
Anya, Vanya, Danya, Sanya, and Tanya were collecting apples. It turned out that each of them collected a whole percentage of the total number of apples, and all these numbers were different and greater than zero. Then Tanya, who collected the most apples, ate her apples. After that, it turned out that each of the remaining children still had a whole percentage, but now from the number of apples left. What is the minimum number of apples that could have been collected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given a relation $R$, determine the smallest set of pairs of elements that must be added to $R$ to make the resulting relation $R^*$ reflexive, symmetric, and transitive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R^* \setminus R}
What is the product of an empty family of similar algebras, \( \prod\langle \mathbf{A}_i \mid i \in I \rangle \), where \( I = \emptyset \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{1}}
Find the area enclosed by the inverse function of \( f(x) = x^3 + 3x + 1 \), between \( x = -3 \) and \( x = 5 \), and the x-axis. Use techniques of finding area by definite integrals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Suppose $A$ and $B$ are $N \times N$ matrices with $\text{rank}(A) = N$ and $\text{rank}(B) = 1$. Determine the lower bound of $\text{rank}(A + B)$. Provide your answer as a single number or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N - 1}
Compute the limit: \[ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the limit: \[ \lim_{n\rightarrow \infty } \int_0^1 \sqrt{ \frac 1 x + n^2 x^{2n} } \ \mathrm dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the domain of \( z \) such that the Möbius transformation \( w = T(z) = \frac{1+z}{1-z} \) maps to the region \( \{ \Re(w) > 0 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z| < 1}
Determine the smallest value of $a > 1$ such that the inequality \( \frac{a + \sin x}{a + \sin y} \leq e^{(y-x)} \) holds for all $x \leq y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given 2006 points on a plane, determine the maximum number of points among them that can form an acute-angled triangle with any other two points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Calculate the integral \( \int_{S^2} \frac{1}{\sqrt{(x-a)^2+(y-b)^2+(z-c)^2}} \, dS \) over the unit sphere \( S^2 \) in \( \mathbb{R}^3 \), given that \( a^2+b^2+c^2 < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
p13. Isosceles trapezoid \( {ABCD} \) has \( {AB} = {CD} \) and \( {AD} = {6BC} \) . Point \( X \) is the intersection of the diagonals \( {AC} \) and
 \( {BD} \) . There exist a positive real number \( k \) and a point \( P \) inside \( {ABCD} \) which satisfy
 \( \left\lbrack  {PBC}\right\rbrack   : \left\lbrack  {PCD}\right\rbrack   : \left\lbrack  {PDA}\right\rbrack   = 1 : k : 3, \)
 where \( \left\lbrack  {XYZ}\right\rbrack \) denotes the area of triangle \( {XYZ} \) . If \( {PX} \parallel  {AB} \) , find the value of \( k \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( k = \frac{219}{42} \)}
On an island, there are 100 knights and 100 liars. Every resident has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident said either the phrase "All my friends are knights" or the phrase "All my friends are liars," with exactly 100 people saying each phrase. Find the minimum possible number of pairs of friends where one is a knight and the other is a liar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Let \(OA\) be a fixed radius of a circle with center \(O\); \(B\) be the midpoint of the radius \(OA\); and \(M\) be an arbitrary point on the circumference of the circle. Find the maximum value of the angle \(\angle OMB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
Given that the average score of six students is 92.5, the highest score is 99, and the lowest score is 76, and that all scores are unique integers, calculate the minimum possible score of the student who ranks 3rd from the highest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
给定 $A(-2,2)$, 已知 $B$ 是椭圆 $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ 上的动点, $F$ 是左焦点, 当 $|A B|+\frac{5}{3}|B F|$ 取最小值时, $B$ 的坐标为$\left(-\frac{a \sqrt{b}}{c}, b\right)$.求$a+b+c+d$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the value of $r$ in the congruence $$\sum_{0\leq{k}<50}\binom{100}{2k+1}5^k \equiv r\pmod{ 2^{99}}$$ where $0 \leq{r} < 2^{99}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given $p: -3 \leqslant x \leqslant 3$, $q: x < a$ ($a$ is a real number). If $q$ is a sufficient and necessary condition for $p$, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(3, +\infty \right)}
若直线 \(x=\frac{\pi}{4}\) 被曲线 \(C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0\) 所截得的弦长为 \(d\)，当 \(a\) 变化时，求 \(d\) 的最小值。答案的形式为 \(\frac{k\pi}{m}\)，请给出 \(k + m\) 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1, 2, 3, 4, 5, 6, 7, 8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{421}
Consider the sequence of Fibonacci numbers $F_0, F_1, F_2, ... $, given by $F_0 = F_1= 1$ and $F_{n+1} =F_n + F_{n-1}$ for $n\ge 1$. Define the sequence $x_0, x_1, x_2, ....$ by $x_0 = 1$ and $x_{k+1} = x^2_k + F^2_{2^k}$ for $k \ge 0$. Define the sequence $y_0, y_1, y_2, ...$ by $y_0 = 1$ and $y_{k+1} = 2x_ky_k - y^2_k$ for $k \ge 0$. If $$\sum^{\infty}_{k=0} \frac{1}{y_k}= \frac{a -\sqrt{b}}{c}$$
for positive integers a$, b, c$ with $gcd (a, c) = 1$, find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
The eight vertices of a cube are randomly labelled with the integers from 1 to 8 inclusive. Judith looks at the labels of the four vertices of one of the faces of the cube. She lists these four labels in increasing order. After doing this for all six faces, she gets the following six lists: $(1,2,5,8),(3,4,6,7),(2,4,5,7),(1,3,6,8),(2,3,7,8)$, and $(1,4,5,6)$. What is the label of the vertex of the cube that is farthest away from the vertex labelled 2? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the interval of real numbers $a$ for which the function $f(x) = \lg(1-ax)$ is monotonically decreasing on the interval $(0,1)$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1]}
If $\frac{\pi}{4}$ is a zero point of the function $f(x) = \sin 2x + a \cos^2 x$ ($a \in \mathbb{R}$, and it's a constant), then the maximum value of $f(x)$ is $\_\_\_\_\_\_\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} - 1}
A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Given the lines ${l}_{1}:x-y+5=0$ and ${l}_{2}:x+4=0$, and the parabola $C:{y}^{2}=16x$, let $P$ be a moving point on $C$. The minimum value of the sum of the distances from $P$ to ${l}_{1}$ and ${l}_{2}$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9\sqrt{2}}{2}}
Triangle $PQR$ has $\angle Q = 45^{\circ}$ and $QR = 6$. Point $S$ is the midpoint of $QR$. If the altitude from point $P$ to $QR$ is 3 units, what is the largest possible value of $\tan{\angle PSQ}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}