inclusionAI/Ring-lite-rl-data huggingface dataset:
One of the receipts for a math tournament showed that  $72$  identical trophies were purchased for  $\$  $-$ 99.9$-, where the first and last digits were illegible.  How much did each trophy cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.11}
The area of a right triangle is $2 r^{2} / 3$, where $r$ is the radius of the circle that touches one leg, the extension of the other leg, and the hypotenuse. Find the sides of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r, \frac{4}{3}r, \frac{5}{3}r}
Pentagon \(ABCDE\) is inscribed in a circle. The distances from point \(A\) to the lines \(BC\), \(CD\), and \(DE\) are \(a\), \(b\), and \(c\) respectively.

Find the distance from vertex \(A\) to the line \(BE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{ac}{b}}
A school buys 60% of its light bulbs from supplier A. The light bulbs from supplier A have exponentially distributed lifetimes with a mean of 600 hours, and the lifetimes of all bulbs are independent of each other. Using a normal approximation, find the probability that 100 light bulbs from supplier A last more than 50,000 hours in total.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9522}
Given the function $y = \log_a(x + 3) - 1$ where $a > 0$ and $a \neq 1$, its graph always passes through a fixed point $A$. If point $A$ lies on the line $mx + ny + 1 = 0$, where $mn > 0$, then the minimum value of $\frac{1}{m} + \frac{2}{n}$ is $\_\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{The minimum value is } 8.}
A positive integer $n$ is picante if $n$ ! ends in the same number of zeroes whether written in base 7 or in base 8 . How many of the numbers $1,2, \ldots, 2004$ are picante?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation $u \otimes v = ac + bdi$ Suppose $z$ is a complex number such that $z\otimes z = z^{2}+40$. What is $|z|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5\sqrt{2}$}
Evaluate the limit: \[ \lim_{n \to +\infty}\sum_{k=n}^{3n} \binom{k-1}{n-1} \left(\frac{1}{3}\right)^n \left(\frac{2}{3}\right)^{k-n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In triangle \(\triangle ABC\), vertices \(A(0,a)\) and \(B(0,b)\) are fixed, where \(a, b > 0\), and vertex \(C(x,0)\) is variable. Determine the value of \(x\) that maximizes the angle \(ACB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm \sqrt{ab}}
Determine the mass percentage of Oxygen in a mixture of Bromic acid (HBrO3) and Sulfurous acid (H2SO3), given that the mixture contains 1 mol of HBrO3 and 2 mol of H2SO3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49.13\%}
The school principal, the janitor, and the parent committee, unable to agree with each other, each bought a rug for the school assembly hall sized $10 \times 10$. After thinking about what to do, they decided to place all three rugs as shown in the picture: the first rug $6 \times 8$ in one corner, the second rug $6 \times 6$ in the opposite corner, and the third rug $5 \times 7$ in one of the remaining corners (all dimensions are given in meters).

Find the area of the part of the hall covered by rugs in three layers (give your answer in square meters).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{CF}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Aarti can do a piece of work in 5 days. In how many days will she complete a certain multiple of work of the same type, if it takes her 15 days to complete that amount of work?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For what integers  $ n\ge 3$  is it possible to accommodate, in some order, the numbers  $ 1,2,\cdots, n$  in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$, $b$, and $c$. If $P(1, 2, 3) = 1$, compute $P(2, 4, 8)$.

Note: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
In a $4 \times 4$ grid, there should be 30 squares. As shown in the figure, 4 points have already been removed. How many more points need to be removed so that there is exactly one square left in the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, if $a \geqslant \frac{b+c}{3}$, then the following inequality holds:
$$
ac + bc - c^2 \leqslant \lambda \left( a^2 + b^2 + 3c^2 + 2ab - 4bc \right).
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{2} + 1}{7}}
Given an ellipse $M$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a > b > 0\right)$ with foci $F_{1}$ and $F_{2}$, and intersecting points with the coordinate axes at $A$, $B$, $C$, and $D$, find the possible values for the eccentricity of the ellipse $M$ that allow three of the points $F_{1}$, $F_{2}$, $A$, $B$, $C$, and $D$ to form an equilateral triangle. Express your answer as a list of possible values, separated by commas, and enclosed in a boxed notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}, \frac{\sqrt{3}}{2}}
Determine the smallest positive integer $n \ge 3$ for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where $A$ denotes the result when the numbers $2^{10}$, $2^{20}$, $\dots$, $2^{10n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A = 10241048576$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
4. $1991^{2000}$ divided by $10^{6}$, the remainder is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{880001}
Adam is playing Minesweeper on a  $9\times9$  grid of squares, where exactly  $\frac13$  (or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent squares are mines.

First, he clicks the square directly above the center square, which shows the number  $4$ . Next, he clicks the square directly below the center square, which shows the number  $1$ . What is the probability that the center square is a mine?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{88}{379}}
Solve
\[\arcsin (\sin x) = \frac{x}{2}.\]Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{2 \pi}{3}, 0, \frac{2 \pi}{3}}
A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. The number of degrees of angle $ADB$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30$}
Let \( f(x) = x^3 - 3x + 1 \). Find the number of distinct real roots of the equation \( f(f(f(x))) = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given the following equations: $$\sqrt {2}=2\cos \frac {π}{4}$$, $$\sqrt {2+ \sqrt {2}}=2\cos \frac {π}{8}$$, $$\sqrt {2+ \sqrt {2+ \sqrt {2}}}=2\cos \frac {π}{16}$$, ... please deduce the nth equation: $$\overset{}{n\text{ of }2}\_\_\_\_\_\_$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\cos \frac {π}{2^{n+1}}}
5. In triangle $A B C, \angle A=45^{\circ}$ and $M$ is the midpoint of $\overline{B C}$. $\overline{A M}$ intersects the circumcircle of $A B C$ for the second time at $D$, and $A M=2 M D$. Find $\cos \angle A O D$, where $O$ is the circumcenter of $A B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{8}}
Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the big circle. Calculate the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given the following program, determine the condition for the UNTIL statement that yields an output of s = 132.
i = 12
s = 1
DO
s = s * i
i = i - 1
LOOP UNTIL "condition"
PRINT s
END.
Express your answer as a conditional statement (e.g., i < 11).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i < 11}
Given a cyclic quadrilateral \(ABCD\) where \(AB = BC = AD + CD\) and \(\angle BAD = \alpha\), and \(AC = d\). Find the area of \(\triangle ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} d^2 \sin \alpha}
Evaluate the constant \( \alpha \) such that \( \iiint_{g(S)}(2x+y-2z)\,dxdydz = \alpha \iiint_S z\,dxdydz \), where the transformation \( g: \mathbb{R}^3 \to \mathbb{R}^3 \) is defined by \( g(x,y,z) = (3x+4z, 2x-3z, x+3y) \) and \( S = \{(x,y,z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
The diagram shows a rectangle $AEFJ$ inside a regular decagon $ABCDEFGHIJ$. What is the ratio of the area of the rectangle to the area of the decagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2:5$}
A child lines up  $2020^2$  pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below  $250$ . How many bricks remain in the end?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Jurgen is traveling to Waterloo by bus. He packs for 25 minutes. He then walks to the bus station, which takes 35 minutes. He arrives 60 minutes before his bus leaves. His bus leaves at 6:45 p.m. At what time did he start packing? Express your answer in p.m. time format (e.g., 4:45 p.m.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:45 p.m.}
Find the value of the integral:

$$\int_0^1 \int_1^2 \frac{\log(x) - \log(y)}{x - y} \, dy \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
2. Find all eight-digit numbers with this property: if we strike out the first two and the last two digits of the number, we get a four-digit number that is 2019 times smaller than the original number.

(Pavel Calábek)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19990119}
For some complex number  $\omega$  with  $|\omega| = 2016$ , there is some real  $\lambda>1$  such that  $\omega, \omega^{2},$  and  $\lambda \omega$  form an equilateral triangle in the complex plane. Then,  $\lambda$  can be written in the form  $\tfrac{a + \sqrt{b}}{c}$ , where  $a,b,$  and  $c$  are positive integers and  $b$  is squarefree. Compute  $\sqrt{a+b+c}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4032}
Determine the number of ideals of norm 100 in the ring of integers of the field $K=\mathbb{Q}(\sqrt{-7})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The cells of a $5 \times 5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through 9 cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly 3 red cells, exactly 3 white cells, and exactly 3 blue cells no matter which route he takes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
6. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system

$$
\left\{\begin{array}{l}
y=-b-x^{2}, \\
x^{2}+y^{2}+8 a^{2}=4+4 a(x+y)
\end{array}\right.
$$

has at least one solution $(x ; y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b\leq2\sqrt{2}+\frac{1}{4}}
The function $f(x) = x^3 - 3ax + 2$ is a decreasing function on the interval $(-2, 2)$. Determine the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \geq 4}
What are the numbers whose triples, when added to 1, result in a prime number between 70 and 110?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24, \, 26, \, 32, \, 34, \, 36}
Let $n\ge 1$  be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$  , where $p$  runs over polynomials of degree less than $n$  with real coefficients and $f$  runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$   defined on the closed interval $[0,1]$ , where $c_k \ge 0$  and $\sum_{k=n}^{\infty} c_k=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{-2n+1} }
Consider the sum $$S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$
The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3031}
For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Given the ellipse $\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ with foci $F_1$ and $F_2$ at $(-\frac{\sqrt{2}}{2}, 0)$ and $(\frac{\sqrt{2}}{2}, 0)$ respectively, a tangent line $l$ is drawn at any point $P$ on the ellipse. Let $d_1$ and $d_2$ denote the distances from $F_1$ and $F_2$ to the line $l$, respectively. Calculate the product $d_1 \cdot d_2$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Suppose $W(t)$ is a standard Brownian motion. Calculate the probability $P(-1 < W(3) - W(1) < 1)$. Provide your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.05\%}
The polynomial \( x^3 - 3x^2 + 1 \) has three real roots \( r_1, r_2, \) and \( r_3 \). Compute:
\[ 
\sqrt[3]{3r_1 - 2} + \sqrt[3]{3r_2 - 2} + \sqrt[3]{3r_3 - 2}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2010}
A soccer store typically sells replica jerseys at various discounts ranging from 20 percent to 80 percent off the list price, depending on the popularity of the team and the age of the jersey design. During the annual summer sale, everything in the store has an additional discount policy. Items that were already discounted 40 percent or less receive an additional 25 percent off, while items that were discounted between 41 percent and 80 percent receive an additional 10 percent off the original list price. If a replica jersey's list price is $120 and comes with varying discounts and additional summer sale offers, approximately what percent of the list price is the lowest possible total sale price for the jersey?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\%}
Rs. 116000 is divided among 3 persons A, B, and C such that the ratio of the shares of A and B is 3 : 4 and that of B : C is 5 : 6. What is the share of A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29491.53}
Find \( \mathop{\lim }\limits_{{n \rightarrow  \infty }}\frac{\pi }{n}\left\{  {\frac{1}{\sin \frac{\pi \left( {n + 1}\right) }{4n}} + \frac{1}{\sin \frac{\pi \left( {n + 2}\right) }{4n}} + \cdots  + \frac{1}{\sin \frac{\pi \left( {n + n}\right) }{4n}}}\right\} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$4\ln(\sqrt{2} + 1)$
}
In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than  $12$  employees have a more difficult work;
(ii) At least  $30$  employees take a higher salary.
Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Consider  $7$ -gons inscribed in a circle such that all sides of the  $7$ -gon are of different length. Determine the maximal number of  $120^\circ$  angles in this kind of a  $7$ -gon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For any positive real numbers \(a_1, a_2, \cdots, a_5\), if \(\sum_{i=1}^{5} \frac{a_i}{\sqrt{a_i^2 + 2^{i-1} a_{i+1} a_{i+2}}} \geq \lambda\), find the maximum value of \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A merry mouse moves in a maze. If it is at time $n$ in a room with $k$ horizontal or vertical adjacent rooms, it will be at time $n+1$ in one of the $k$ adjacent rooms, choosing one at random, each with probability $1 / k$. A fat lazy cat remains all the time in room $3,$ and a piece of cheese waits for the mouse in room $5$. The mouse starts in room $1$. The cat is not completely lazy: If the mouse enters the room inhabited by the cat, the cat will eat it. Also, if the mouse eats the cheese, it rests forever. Let $X_{n}$ be the position of the mouse at time $n$. What is the probability that the mouse ever gets to eat the cheese?

The transition matrix is as follows:
$$P=\begin{pmatrix}0 & 1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 1/2 & 0 & 0 \\
0 & 1/2 & 0 & 1/2 & 0 \\
1/3 & 0 & 1/3 & 0 & 1/3 \\
0 & 0 & 0 & 1 & 0 \\
\end{pmatrix}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$.  How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Over all integers $1 \leq n \leq 100$, find the maximum value of $\phi(n^2+2n) - \phi(n^2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Evaluate the contour integral \( \int_C \frac{z+2}{(z^2-2z+1)^2}dz \) using the residue theorem, where \( C \) is the semi-circle in the right half-plane with radius \( R > 1 \) and its diameter lies on the imaginary axis. The contour is traced counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A sequence of positive integers is given by $a_{1}=1$ and $a_{n}=\operatorname{gcd}\left(a_{n-1}, n\right)+1$ for $n>1$. Calculate $a_{2002}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Compute the integral \( \int_0^{\infty} xe^{-x^2/2}y \, dx \), where \( y = \sum\limits_{n \ge 0} \frac{(x/2)^{2n}}{(n!)^2} \) is the unique solution to the initial value problem \( xy''+y'-xy=0, \ y(0)=1, \ y'(0) = 0. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Given a circle with center \( O \) and radius 1. From point \( A \), tangents \( AB \) and \( AC \) are drawn. Point \( M \) lies on the circle such that the quadrilaterals \( OBMC \) and \( ABMC \) have equal areas. Find \( MA \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Over all real numbers $x$ and $y$, find the minimum possible value of $$ (x y)^{2}+(x+7)^{2}+(2 y+7)^{2} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given $T' = \{(0,0),(5,0),(0,4)\}$ be a triangle in the coordinate plane, determine the number of sequences of three transformations—comprising rotations by $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, and reflections across the lines $y = x$ and $y = -x$—that will map $T'$ onto itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let \(\left\{a_{n}\right\}\) be a sequence of positive integers such that \(a_{1}=1\), \(a_{2}=2009\) and for \(n \geq 1\), \(a_{n+2} a_{n} - a_{n+1}^{2} - a_{n+1} a_{n} = 0\). Determine the value of \(\frac{a_{993}}{100 a_{991}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89970}
Determine the mass percentage of Al in a mixture containing 2 moles of Aluminium sulfide (Al2S3), 3 moles of Calcium carbonate (CaCO3), and 5 moles of Potassium chloride (KCl).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11.09\%}
In triangle \( PQR \), points \( N \) and \( L \) are located on sides \( PQ \) and \( PR \) respectively, such that \( NQ = LR \). The intersection point \( A \) of segments \( QL \) and \( NR \) divides \( QL \) in the ratio \( m:n \) from point \( Q \). Determine the ratio \( \frac{PN}{PR} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{m}}
Given a function $y=f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+2)=f(x)$, and when $x \in (-1,1]$, $f(x)=|x|$, find the number of real roots of $\log_{3}|x|-f(x)=0$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the line integral \(\int_{C} \left(\frac{\sin(3x)}{x^2+1}-6x^2y\right) dx + \left(6xy^2+\arctan\left(\frac{y}{7}\right)\right) dy\) using Green's Theorem, where \(C\) is the circle \(x^2+y^2=8\) oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192\pi}
设三位数 $n=abc$，若以 $a, b, c$ 为三条边长可以构成一个等腰（含等边）三角形，则这样的三位数 $n$ 有多少个？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}