inclusionAI/Ring-lite-rl-data huggingface dataset:
Stephanie has several household bills to settle. Her electricity bill is $180 and she's paid 75% of that. Her gas bill is $120 out of which she's paid 60%. She has an outstanding water bill of $200, of which she has only paid 25%. Steph has also been paying down her internet bill of $100 in six payments of $5, leaving the remaining balance with her.  Thereafter, her ISP offered a discount of 15% on the remaining balance. Stephanie received a subsidy from her housing association that reduced her heating bill by 30%; the initial bill amount was $150. She paid 45% of the adjusted bill striving to pay off her debts. Furthermore, she has a garbage disposal bill of $60 with a standing payment of half of the bill so far. In addition, she has to pay a phone bill of $90, but she's off to a start by paying 20% of that. Finally, she has a cable bill of $80, but she only has to pay 60% of the cost due to a promotional offer. With all these financial commitments, how much does Stephanie still have to pay?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$510.25}
52. \(ABCD\) is a circumscribed quadrilateral. The lengths of the segments from \(A\) to the points of tangency are equal to \(a\), and the lengths of the segments from \(C\) to the points of tangency are equal to \(b\). In what ratio does the diagonal \(AC\) divide the diagonal \(BD\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{b}}
Evaluate the integral $$J=\int _0 ^\infty \frac{\log(x^2+1/x^2)}{1+x^2}dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \log 2}
At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $\frac{a}{3}$ thrice instead. We say that a positive integer $n$ is persistent if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. Find the greatest persistent number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given the line $l: \sqrt{3}x-y-4=0$, calculate the slope angle of line $l$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let \(ABC\) be a triangle with incenter \(I\), centroid \(G\), and \(|AC| > |AB|\). If \(IG \parallel BC\), \(|BC| = 2\), and \(\text{Area}(ABC) = \frac{3\sqrt{5}}{8}\), then find the value of \(|AB|\). The original answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Evaluate the following limit without using L'Hôpital's Rule:
$$\lim_{x \rightarrow 0}\frac{\cos(1-\frac{\sin x}{x})+\cos(2-\frac{\sin(2x)}{x})+\cdots+\cos(k-\frac{\sin(kx)}{x})-k}{\cos(1-\cos(x))+\cos(2-2\cos(2x))+\cdots+\cos(k-k\cos(kx))-k}$$
This question was proposed to the Romanian Math Magazine by Jalil Hajimir.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/9}
A boat crossed a lake from north to west at a certain speed, entered a river and covered twice as much distance going upstream at 4 km/h. It then turned around and stopped at the south shore of the lake. The boat averaged 3.6 km/h that day. What was its approximate speed while crossing the lake if its downstream speed was 6 km/h?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.25}
If $a$ is an element randomly selected from the set $\{1, 2, 3, 4, 5, 6, 7\}$, then the probability that the circle $C: x^2 + (y-2)^2 = 1$ is contained inside the circle $O: x^2 + y^2 = a^2$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{7}}
Given a continuous function $f(x)$ where $f(0)<0$ and $f(1)>0$, and the evaluation time for $f(x)<0$ is $1$ unit while for $f(x)>0$ it's $C$ units ($C>1$), what is the optimal point to choose within the interval $[0,1]$ to perform bisection in order to minimize the expected time to reach a desired accuracy, assuming only the bisection method can be used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{1+C}\)}
There is a table with $n$ rows and $18$ columns. Each cell contains either a $0$ or a $1$. The table satisfies the following properties:

1. Every two rows are different.
2. Each row contains exactly $6$ cells that contain $1$.
3. For every three rows, there exists a column such that the intersection of the column with the three rows (the three cells) all contain $0$.

What is the greatest possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12376}
\(ABCD\) is a convex quadrilateral and \(E, F\) are the mid-points of \(BC\) and \(CD\) respectively. The line segments \(AE, AF\) and \(EF\) divide \(ABCD\) into four triangles, whose areas are four consecutive integers. Find the greatest possible area of \(\triangle ABD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the upper bound of the function g(x), where g'(x) = 1/(x^2 + g^2(x) + 1) and g(1) = 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + π/4}
On the sides of the acute angle $ABC$, points $A$ and $C$ are taken. One circle touches the line $AB$ at point $B$ and passes through point $C$. The second circle touches the line $BC$ at point $B$ and passes through point $A$. Point $D$ is the second common point of the circles. It is known that $AB = a$, $CD = b$, $BC = c$. Find $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b \left( \frac{a}{c} \right)^2}
In a row, there are $n$ integers such that the sum of any three consecutive numbers is positive, and the sum of any five consecutive numbers is negative. What is the maximum possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
设$a$，$b$为正实数$\frac { 1 } { a } + \frac { 1 } { b } \leq 2 \sqrt { 2 } , ( a - b ) ^ { 2 } = 4 ( a b ) ^ { 3 }$，则

$\log _ { a } b $值为多少？ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Jake splits some shots of vodka with his friend. Each shot of vodka is 1.5 ounces. The vodka is 50% pure alcohol. Jake drank 3 ounces of pure alcohol. How many shots of vodka did Jake split with his friend?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A dozen eggs and a certain amount of pounds of oranges are currently at the same price. If the price of a dozen eggs rises by 9 percent and the price of oranges rises by 6%, it will cost $15 more to buy a dozen eggs and the oranges. How many pounds of oranges are in the group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Two trains of length 120 m and a certain length are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. They will be clear of each other from the moment they meet in 20.99832013438925 seconds. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{299.97}
A football association stipulates that in the league, a team earns $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where $a$ and $b$ are real numbers such that $a > b > 0$. If a team has 2015 possible total scores after $n$ games, find the minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only six copies of today's handout. No freshman should get more than one handout, and any freshman who does not get one should be able to read a neighbor's. If the freshmen are distinguishable but the handouts are not, how many ways are there to distribute the six handouts subject to the above conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. Find the digit $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the following sum:

$$ \frac{1}{\log_2\left(\frac{1}{7}\right)} + \frac{1}{\log_3\left(\frac{1}{7}\right)} + \frac{1}{\log_4\left(\frac{1}{7}\right)} + \frac{1}{\log_5\left(\frac{1}{7}\right)} + \frac{1}{\log_6\left(\frac{1}{7}\right)} - \frac{1}{\log_7\left(\frac{1}{7}\right)} - \frac{1}{\log_8\left(\frac{1}{7}\right)} - \frac{1}{\log_9\left(\frac{1}{7}\right)} - \frac{1}{\log_{10}\left(\frac{1}{7}\right)} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Kevin collected various toys to use as prizes at the fair. He collected x stuffed animals, y frisbees, z yo-yos, and w puzzles. The total number of stuffed animals, frisbees, and puzzles was 80. 

Additionally, he discovered that 3/5 of the toys he collected were toy cars, while 1/5 of the total collected toys were toy robots. The number of toy cars and toy robots adds up to 15 more than the total number of other toys (stuffed animals, frisbees, yo-yos, and puzzles).

If Kevin collected a total of 300 prizes, determine how many yo-yos (z) did Kevin collect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{145}
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8x^3 - 4x^2 - 4x - 1 = 0}
Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB$, $\angle BFC$, $\angle CGD$, $\angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
A shop owner sells three different types of cloth with varying selling prices per meter, which are also subject to fluctuating market prices. The base prices are as follows: Cloth A is sold at $15 per meter, Cloth B at $20 per meter, and Cloth C at $25 per meter. On a given day, the market price for Cloth A increases by 8%, Cloth B decreases by 6%, and Cloth C remains unchanged. The owner sells 10 meters of Cloth A, 15 meters of Cloth B, and 5 meters of Cloth C. In addition, a 5% sales tax is applied to all types of cloth, and a varying discount rate is offered on the total purchase of Cloth B based on the quantity purchased: 5% for up to 10 meters, and 15% for more than 10 meters. Consider the cost prices of the cloths as $10 per meter for Cloth A, $15 per meter for Cloth B, and $20 per meter for Cloth C. Calculate the total profit percent earned by the shop owner on the given day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30.13\%}
The base of a right triangular prism is an isosceles triangle \( ABC \) with \( AB = BC = a \) and \(\angle BAC = \alpha\). A plane is drawn through the side \( AC \) at an angle \(\varphi(\varphi<\pi / 2)\) to the base. Find the area of the cross-section, given that the cross-section is a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2 \sin 2\alpha}{2 \cos \varphi}}
Given a sequence \( A = (a_1, a_2, \cdots, a_{10}) \) that satisfies the following four conditions:
1. \( a_1, a_2, \cdots, a_{10} \) is a permutation of \{1, 2, \cdots, 10\};
2. \( a_1 < a_2, a_3 < a_4, a_5 < a_6, a_7 < a_8, a_9 < a_{10} \);
3. \( a_2 > a_3, a_4 > a_5, a_6 > a_7, a_8 > a_9 \);
4. There does not exist \( 1 \leq i < j < k \leq 10 \) such that \( a_i < a_k < a_j \).

Find the number of such sequences \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Find the smallest positive integer solution to the equation $2^{2^k} \equiv k \pmod{29}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Inside a right triangle \(ABC\) with hypotenuse \(AC\), a point \(M\) is chosen such that the areas of triangles \(ABM\) and \(BCM\) are one-third and one-quarter of the area of triangle \(ABC\) respectively. Find \(BM\) if \(AM = 60\) and \(CM = 70\). If the answer is not an integer, round it to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Shelly writes down a vector \( v = (a, b, c, d) \), where \( 0 < a < b < c < d \) are integers. Let \( \sigma(v) \) denote the set of 24 vectors whose coordinates are \( a, b, c \), and \( d \) in some order. For instance, \( \sigma(v) \) contains \( (b, c, d, a) \). Shelly notes that there are 3 vectors in \( \sigma(v) \) whose sum is of the form \( (s, s, s, s) \) for some \( s \). What is the smallest possible value of \( d \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In a certain high school, student A's math scores in 10 mock exams in the third year of high school are arranged in ascending order as $94$, $96$, $98$, $98$, $100$, $101$, $101$, $102$, $102$, $103$. The 40th percentile of student A's math scores in these 10 mock exams is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
What is the maximum number of digits that a natural number can have, where all the digits are different, and the number is divisible by each of its digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let  $\mathbb{Z}_{\geq 0}$  be the set of nonnegative integers. Let  $f: \mathbb{Z}_{\geq0} \to \mathbb{Z}_{\geq0}$  be a function such that, for all  $a,b \in \mathbb{Z}_{\geq0}$ : \[f(a)^2+f(b)^2+f(a+b)^2=1+2f(a)f(b)f(a+b).\] 
Furthermore, suppose there exists  $n \in \mathbb{Z}_{\geq0}$  such that  $f(n)=577$ . Let  $S$  be the sum of all possible values of  $f(2017)$ . Find the remainder when  $S$  is divided by  $2017$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{597}
Calculate the following expression:

$$
\operatorname{tg} \frac{\pi}{43} \cdot \operatorname{tg} \frac{2 \pi}{43}+\operatorname{tg} \frac{2 \pi}{43} \cdot \operatorname{tg} \frac{3 \pi}{43}+\ldots+\operatorname{tg} \frac{k \pi}{43} \cdot \operatorname{tg} \frac{(k+1) \pi}{43}+\ldots+\operatorname{tg} \frac{2021 \pi}{43} \cdot \operatorname{tg} \frac{2022 \pi}{43}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2021}
A list of five positive integers has a mean of $12$ and a range of $18$. The mode and median are both $8$. Find how many different values are possible for the second largest element of the list.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB = 1$. If the radius of the circle is $r$, find the value of $100r$ to the nearest integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{237}
Dany owns a farm, in his farm he has 4 cows and 3 sheep that eat 2 bushels a day. He also has 7 chickens that eat a certain amount of bushels a day. He should have 35 bushels to suffice the animals for a day. How many bushels do the chickens eat per day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A hurricane system consisting of three storms, A, B, and C, passed over a location. Storm A dropped 5 inches of rain in the first 30 minutes and then dropped half that amount of rain in the next 30 minutes. Storm B dropped 3 inches of rain in the first 45 minutes, and in the next hour, the rainfall increased by 50%. Storm C consisted of light rainfall of 1/2 inch per hour for 3 hours. Calculate the overall average rainfall amount for the entire hurricane system.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.87}
In triangle \( ABC \), angle \( A \) is equal to \( 60^{\circ} \). The distances from vertices \( B \) and \( C \) to the center of the inscribed circle of triangle \( ABC \) are 3 and 4, respectively. Find the radius of the circumscribed circle around triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{37}{3}}}
A circle inscribed in triangle \(ABC\) touches sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. Point \(I_A\) is the center of the excircle opposite side \(BC\) of triangle \(ABC\). Points \(K\) and \(L\) are the midpoints of segments \(DI_A\) and \(EI_A\) respectively. Lines \(BK\) and \(CL\) intersect at point \(F\) located inside angle \(BAC\). Find \(\angle BFC\), given that \(\angle BAC = 50^\circ\). (The excircle touches side \(BC\) and the extensions of sides \(AB\) and \(AC\) beyond points \(B\) and \(C\) respectively.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130^
}
In triangle $XYZ$, $XY=15$, $YZ=18$, and $ZX=21$. Point $G$ is on $\overline{XY}$, $H$ is on $\overline{YZ}$, and $I$ is on $\overline{ZX}$. Let $XG = p \cdot XY$, $YH = q \cdot YZ$, and $ZI = r \cdot ZX$, where $p$, $q$, and $r$ are positive and satisfy $p+q+r=3/4$ and $p^2+q^2+r^2=1/2$. The ratio of the area of triangle $GHI$ to the area of triangle $XYZ$ can be written in the form $m/n$, 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{41}
A \(20 \times 19\) rectangle is plotted on the Cartesian plane with one corner at the origin and with sides parallel to the coordinate axes. How many unit squares do the two diagonals of this rectangle pass through?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number of successful teams that could have participated in the tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
10.4. In triangle $A B C$, points $D$ and $F$ are marked on sides $A B$ and $A C$ respectively such that lines $D C$ and $B F$ are perpendicular to each other and intersect at point $E$ inside triangle $A B C$. It turns out that $A D=D C$ and $D E \cdot E C=F E \cdot E B$. What degree measure can angle $B A C$ have? (7 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given the function f(x) = ax^3 + b*sin(x) + 4, where a, b are real numbers, and f(lg(log_2 10)) = 5, determine the value of f(lg(lg2)). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A positive integer, which when added to a certain number, gives a sum which is greater than when it is multiplied by that number. This positive integer is 1. What is the certain number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The diagonals of parallelogram $ABCD$ intersect at $E$ . The bisectors of $\angle DAE$ and $\angle EBC$ intersect at $F$. Assume $ECFD$ is a parellelogram . Determine the ratio $AB:AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
${\sqrt{3}}$}
In how many ways can \(a, b, c\), and \(d\) be chosen from the set \(\{0,1,2, \ldots, 9\}\) so that \(a<b<c<d\) and \(a+b+c+d\) is a multiple of three?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
What is the minimum number of L-shaped figures made of 3 cells that need to be painted in a $5 \times 5$ grid so that no more L-shaped figures can be painted? (The painted L-shaped figures must not overlap.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Point \( A \) lies on the line \( y = \frac{5}{12} x - 11 \), and point \( B \) lies on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6311}{624}}
Quadrilateral $A B C D$ has $\angle B C D=\angle D A B=90^{\circ}$. The perimeter of $A B C D$ is 224 and its area is 2205. One side of $A B C D$ has length 7. The remaining three sides have integer lengths. The sum of the squares of the side lengths of $A B C D$ is $S$. What is the integer formed by the rightmost two digits of $S$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Given a rectangular coordinate system with origin point $O$, vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ satisfy $|\overrightarrow {a}|=|\overrightarrow {b}|=1$ and $\overrightarrow {a}\cdot \overrightarrow {b}=\frac {1}{2}$. Let $\overrightarrow {c} = (m, 1-m)$ and $\overrightarrow {d} = (n, 1-n)$. For any real numbers $m$ and $n$, the inequality $|\overrightarrow {a} - \overrightarrow {c}| + |\overrightarrow {b} - \overrightarrow {d}| \geq T$ holds. Determine the range of possible values for the real number $T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6} - \sqrt{2}}{4}}
Find the maximum value of the function
$$
f(x) = \sqrt{3} \sin 2x + 2 \sin x + 4 \sqrt{3} \cos x.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{2}}
In triangle $ABC$, angle $A$ satisfies $\sin A + \cos A > 0$ and $\tan A < \sin A$. Determine the range of values for angle $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{\pi}{2}, \frac{3\pi}{4})}
$S$ is the sum of the first 6 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$, consisting of integers. It is known that $a_{10} a_{16} > S + 39$ and $a_{11} a_{15} < S + 55$. Specify all possible values of $a_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10}
A rectangle-shaped puzzle is assembled with $2000$ pieces that are all equal rectangles and similar to the large rectangle. The sides of the small rectangles are parallel to those of the large one. The shortest side of each piece measures $1$. Determine the minimum possible value of the area of the large rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
The polynomial $f(x)=x^{2007}+17x^{2006}+1$ has distinct zeroes $r_1,\ldots,r_{2007}$.  A polynomial $P$ of degree $2007$ has the property that $P\left(r_j+\dfrac{1}{r_j}\right)=0$ for $j=1,\ldots,2007$.  Determine the value of $P(1)/P(-1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{289}{259}}
Find the $2016$th smallest positive integer that satisfies the congruence $x^x \equiv x \pmod{5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3360}