inclusionAI/Ring-lite-rl-data huggingface dataset:
## Task 5 - 230935

In a triangle $A B C$, a line parallel to $A B$, whose position is otherwise not specified, intersects the side $A C$ at a point $A_{1}$ between $A$ and $C$, and it intersects the side $B C$ at $B_{1}$. Furthermore, let $P$ be a point on $A B$ between $A$ and $B$, whose position is also not otherwise specified.

The area of triangle $A B C$ is $F_{0}$, and the area of triangle $A_{1} B_{1} C$ is $F_{1}$.

Determine the area $F$ of the quadrilateral $A_{1} P B_{1} C$ in terms of $F_{0}$ and $F_{1}$!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{F_{1}\cdotF_{0}}}
In Class 1 of Grade 12, there are a total of 56 students, numbered consecutively from 1 to 56. A systematic sampling method is used to draw a sample of 4 students. It is known that students with the numbers 6, 34, and 48 are included in the sample. Determine the number of the fourth student in the sample.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given that the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy the conditions $|\overrightarrow{a}| = 2|\overrightarrow{b}| = 2$ and $|\overrightarrow{a} - 4\overrightarrow{b}| = 2\sqrt{7}$, find the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}}
A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
At a tanning salon, customers are charged a certain amount for their first visit in a calendar month and $8 for each visit after that in the same calendar month. In the last calendar month, 100 customers visited the salon, of which 30 made a second visit, and 10 made a third visit. All other customers made only one visit. The revenue for the last calendar month at the salon was $1240. What is the charge for the first visit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$9.20}
Find the smallest real number \(\lambda\) such that the inequality
$$
5(ab + ac + ad + bc + bd + cd) \leq \lambda abcd + 12
$$
holds for all positive real numbers \(a, b, c, d\) that satisfy \(a + b + c + d = 4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A natural number A has three digits added to its end, forming a new number. This new number is equal to the sum of all natural numbers from 1 to A. Find the value of A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1999}
Given that the area of △ABC is S, and $| \overrightarrow {BC}|^{2}$ = $\overrightarrow {CA}$ • $\overrightarrow {CB}$ + 2S, determine the value of ∠B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Given a $202 \times 202$ matrix $A$ with $\|A\|_2 = 100$ and $\|A\|_f = 101$, determine the sharpest lower bound on the 2-norm condition number $k(A)$. Assume $k(A) = \|A\|_2 \cdot \|A^{-1}\|_2$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Consider a geometric progression with a non-zero natural number as the common ratio. The logarithm of the nth term, with the base equal to the common ratio, is 6. The logarithm of the product of the first n terms, with the same base, is 20. The product of the first and the nth term of the sequence is 243. Determine the sum of the first n terms of this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3280}{3}}
Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9720}
Let real numbers x and y satisfy the following equations:
x^3 = 3y^2x + 5 - sqrt(7)
y^3 = 3x^2y + 5 + sqrt(7)

Find the value of x^2 + y^2. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
1. In each cell of a $75 \times 75$ table, a natural number is written. In each row, there are at least 15 different numbers, and in any three consecutive rows, there are no more than 25 different numbers. What is the maximum number of different numbers that can be in the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{385}
A sequence of $28$ letters consists of $14$ of each of the letters $A$ and $B$ arranged in random order. The expected number of times that $ABBA$ appears as four consecutive letters in that sequence 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{145}
2. Solve the equation $\frac{x}{3}+\frac{x}{\sqrt{x^{2}-9}}=\frac{35}{12}$. Write the sum of all obtained solutions in the answer.

(5 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.75}
Robert has $4$ indistinguishable gold coins and $4$ indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the $8$ coins.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{630}
Let \( f:(0,1) \rightarrow (0,1) \) be a differentiable function with a continuous derivative such that for every positive integer \( n \) and odd positive integer \( a < 2^n \), there exists an odd positive integer \( b < 2^n \) such that \( f\left( \frac{a}{2^n} \right) = \frac{b}{2^n} \). Determine the set of possible values of \( f^{\prime} \left( \frac{1}{2} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-1, 1\}}
Two players, A and B, are competing in a table tennis match. The winner of each game scores 1 point, and the loser scores 0 points. The match ends when one player has 2 points more than the other or after 6 games, whichever comes first. The probability of A winning a game is $\frac{2}{3}$, and the probability of B winning a game is $\frac{1}{3}$. The outcomes of the games are independent. Find the expected number of games $\xi$ played when the match ends, $\mathrm{E} \xi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{266}{81}}
Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$248$}
In a campus singing competition, 15 students participated and each student had a unique score. The top 7 students based on their scores will advance to the final round. What statistical measure should Xiao Ming know, along with his own score, to determine whether he can advance to the final round?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{median}
Simplify the following expression:

$$
\frac{bx(a^{2}x^{2} + 2a^{2}y^{2} + b^{2}y^{2})}{bx + ay} + \frac{ay(a^{2}x^{2} + 2b^{2}x^{2} + b^{2}y^{2})}{bx + ay}
$$

Express your answer as a simplified algebraic expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(ax + by)^{2}}
The equation
$$
1 - 3 \sin \theta + 5 \cos 3\theta = 0 \quad (0 < \theta < 2\pi)
$$
has how many distinct solutions? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In the equilateral triangle \(PQR\), \(S\) is the midpoint of \(PR\), and \(T\) is on \(PQ\) such that \(PT=1\) and \(TQ=3\). Many circles can be drawn inside the quadrilateral \(QRST\) such that no part extends outside of \(QRST\). The radius of the largest such circle is closest to:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.10}
In $\triangle ABC$, $\angle A = 42^{\circ}$. If the trisectors of $\angle B$ and $\angle C$ meet at points $D$ and $E$ respectively, then what is the measure of $\angle BDC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88^{\circ}}
At a school trip, there are 8 students and a teacher. They want to take pictures in groups where each group consists of either 4 or 5 students. How many different group combinations can they make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
Chang's Garden has six kinds of apples: sweet, sour, crunchy, soft, juicy, and tangy apples. The sale prices for each apple variety depend on their weight. The prices and proportions for each kind of apple are as follows:

Sweet apples:
- Light (under 100 grams): $.4
- Heavy (100 grams or more): $.6
- Proportion of total apples: 40%

Sour apples:
- Light: $.1
- Heavy: $.15
- Proportion of total apples: 20%

Crunchy apples:
- Light: $.25
- Heavy: $.35
- Proportion of total apples: 15%

Soft apples:
- Light: $.15
- Heavy: $.25
- Proportion of total apples: 10%

Juicy apples:
- Light: $.2
- Heavy: $.3
- Proportion of total apples: 10%

Tangy apples:
- Light: $.05
- Heavy: $.1
- Proportion of total apples: 5%

For each type of apple, 60% of them are light, and 40% are heavy. If Chang earns $120, how many apples did his trees give him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{392}
For a positive integer $n$, an $n$-branch $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to appear in $B$ if it is an element of the last set $S_m$. Define an $n$-plant to be an (unordered) set of $n$-branches $\{ B_1, B_2, \dots, B_k\}$, and call it perfect if each of $1$, $2$, \dots, $n$ appears in exactly one of its branches.

Let $T_n$ be the number of distinct perfect $n$-plants (where $T_0=1$), and suppose that for some positive real number $x$ we have the convergence \[ \ln \left( \sum_{n \ge 0} T_n \cdot \frac{\left( \ln x \right)^n}{n!} \right) = \frac{6}{29}. \] If $x = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Find a six-digit number $\overline{xy243z}$ that is divisible by 396.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{432432}
There are several chairs inside the church. If 9 students sit per chair, one student cannot sit, and if 10 students sit per chair, 1 chair becomes vacant. How many students went to church?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2 - 4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6900}
Given the series $\sum_{n=1}^{\infty} \left(\sqrt[4]{n^4+2n^2}-\sqrt[3]{p(n)}\right)$, find a polynomial $p(n)$ such that this series converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^3 + \frac{3n}{2}}
Let $x, y, z$ be positive real numbers satisfying $x^3 + y^3 + z^3 = 3$. Find the minimum value of the expression \[ P = \frac{1}{2-x} + \frac{1}{2-y} + \frac{1}{2-z}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9 + 2\sqrt[3]{3} + \sqrt[3]{9}}{5}}
A large circular table has 60 chairs around it. What is the largest number of people who can sit around the table so that each person is only sitting next to exactly one other person?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A cashier, upon checking the account before leaving work, finds that the cash is 153 yuan less than the account book. She knows the actual amount collected cannot be wrong, so it must be due to a decimal point error during bookkeeping. What is the actual amount of the cash that was recorded incorrectly?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
a group of people participate in some curriculum , 25 of them practice yoga , 18 study cooking , 10 study weaving , 4 of them study cooking only , 5 of them study both the cooking and yoga , 4 of them participate all curriculums . how many people study both cooking and weaving ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The altitudes $A D$ and $B E$ of an acute-angled triangle $A B C$ intersect at point $H$. The circumcircle of triangle $A B H$ intersects sides $A C$ and $B C$ at points $F$ and $G$ respectively. Find $F G$, if $D E=5$ cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The tax on a commodity is diminished by 20%, and its consumption increased by some percentage. The effect on revenue is 8%. What is the percentage by which the consumption increased?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15\%}
Let's determine the area enclosed by the curves $y=\sin x$ and $y=(4 / \pi)^{2} \sin (\pi / 4) x^{2}$ (the latter being a quadratic function)!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\frac{\sqrt{2}}{2}(1+\frac{\pi}{12})}
Solve the linear recurrence relation with binomial coefficients:

\[ a_n = \binom{9}{1}a_{n-1}-\binom{9}{2}a_{n-2}+\binom{9}{3}a_{n-3}-\dots+\binom{9}{9}a_{n-9} \]

Given the initial conditions:
\[ a_0=a_1=a_2=\dots=a_7=0, \quad a_8=1, \quad \text{for} \quad n\ge9 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{n}{8}}
Find (in terms of $n \geq 1$) the number of terms with odd coefficients after expanding the product: $\prod_{1 \leq i<j \leq n}\left(x_{i}+x_{j}\right)$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
When a wire was divided by intervals of 120 centimeters (cm), it was split into two parts, with 2.4 centimeters (cm) being left. Find how many centimeters (cm) of wire you'll get if you divide this wire into three parts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80.8}
The function $f(x) = 2^{x-1} - \ln x - a$ has exactly two distinct zeros. Find the range of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, +\infty)}
Let $G_{1} G_{2} G_{3}$ be a triangle with $G_{1} G_{2}=7, G_{2} G_{3}=13$, and $G_{3} G_{1}=15$. Let $G_{4}$ be a point outside triangle $G_{1} G_{2} G_{3}$ so that ray $\overrightarrow{G_{1} G_{4}}$ cuts through the interior of the triangle, $G_{3} G_{4}=G_{4} G_{2}$, and $\angle G_{3} G_{1} G_{4}=30^{\circ}$. Let $G_{3} G_{4}$ and $G_{1} G_{2}$ meet at $G_{5}$. Determine the length of segment $G_{2} G_{5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{169}{23}}
设数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=a_{2}=1$, 且 $a_{n+2}=\frac{1}{a_{n+1}}+a_{n}, n=1,2, \cdots$. 求 $a_{2004}$. 原始答案是 \(\frac{m!!}{n!!}\) 格式，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4005}
给定整数 $m 、 n(2 \leqslant m<n,(m, n)=1)$ 。求最小的整数 $k$ ，满足对集合 $\{1,2, \cdots, n\}$ 的任意 $m$ 元子集 $I$, 若 $\sum_{i \in I} i>k$, 则存在 $n$ 个实数 $a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$, 使得
$$  \frac{1}{m} \sum_{i \in I} a_{i}>\frac{1}{n} \sum_{i=1}^{n} a_{i}  $$  ，设m=2，n=3，求出k的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{99}\)}
Two trains of length 100 m and 160 m are running towards each other on parallel lines at 42 kmph and another speed respectively. They will be clear of each other in 12.998960083193344 seconds from the moment they meet. What is the speed of the second train in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Evaluate \( \frac{1}{\int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\, dx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2007}
设 $x_{1}=3, x_{n+1}=\sqrt{x_{n}+14}-\sqrt{x_{n}+2}\left(n \in N^{*}\right)$, 求当$n=100$时$\left[x_{1}+x_{2}+\cdots+x_{n}\right]$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{151}
记$F = \underset { 1 \leq x \leq 3 }\max\vert x ^ { 3 } - a x ^ { 2 } - b x - c \vert$.当$a$、$b$、$c$取遍所有实数时，求$F$的最小值．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { 1 } { 4 }$}
Points $M$ and $N$ are taken on the sides $AB$ and $AD$ of parallelogram $ABCD$ such that the lines $MC$ and $NC$ divide the parallelogram into three equal areas. Find $MN$, if $BD = d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{d}{3}}
A trapezoid with a top base of 5 and a side of 6 can have an inscribed circle and a circumscribed circle. Calculate the area of the pentagon formed by the radii of the inscribed circle, perpendicular to the sides of the trapezoid, its bottom base, and the corresponding segments of the side lengths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{2} \sqrt{35}}
Find the smallest integer $N \in \mathbb{N}$ such that the Taylor polynomial $T_Nf(x,0)$ of $f(x) = \cos(x)e^x$ centered at $x = 0$ satisfies the inequality $$|f(x) - T_Nf(x,0)| \leq 10^{-6}$$ for all $x \in [-\frac{1}{10}, \frac{1}{10}]$ without using the Lagrange remainder theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given \( x, y, z \in \mathbf{Z}_{+} \) and \([x, y] = 72\), \([x, z] = 600\), \([y, z] = 900\), where \([m, n]\) represents the least common multiple of integers \( m \) and \( n \), find the number of ordered triples \((x, y, z)\) satisfying these conditions. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Consider a hyperbola given by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$. Let $F_1$ be its left focus, and let $MN$ be a chord perpendicular to the real axis with $A$ as the right vertex. If $\overrightarrow{AM} \cdot \overrightarrow{AN} = 0$, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Tracy initially had a bag of candies. She first ate $\frac{1}{4}$ of them, then gave $\frac{1}{3}$ of the remaining candies to her friend Rachel. After that, Tracy and her mom each ate 12 candies from what was left. Lastly, her brother took between two to six candies, leaving Tracy with five candies. How many candies did Tracy have at the beginning?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
A train 110 m long running at the speed of 90 km/hr takes 9.679225661947045 seconds to cross a bridge of a certain length. What is the length of the bridge in meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131.98}
What is the largest number of digits that can be erased from the 1000-digit number 201820182018....2018 so that the sum of the remaining digits is 2018?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{741}
A bug is on a corner of a cube. A healthy path for the bug is a path along the edges of the cube that starts and ends where the bug is located, uses no edge multiple times, and uses at most two of the edges adjacent to any particular face. Find the number of healthy paths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the domain of the function $y= \sqrt {\sin x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2kπ,2kπ+π]}
When \(0 < x < \frac{\pi}{2}\), the value of the function \(y = \tan 3x \cdot \cot^3 x\) cannot take numbers within the open interval \((a, b)\). Find the value of \(a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Given that $x > 0$, $y > 0$, if ${2}^{x}\cdot {8}^{y}=16$, then ${2}^{-1+{\mathrm{log}}_{}\left(2x\right)}+{\mathrm{log}}_{}^{27y}=\_\_\_\_\_\mathrm{.}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the mass of the solid $E$ where $E$ is outside the sphere $x^2 + y^2 + z^2 = 2z$, but inside the hemisphere $x^2 + y^2 + z^2 = 4$, $z\geq 0$ and its density is $\rho(x,y,z)=z\sqrt{x^2 +y^2 +z^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32π/7}
What is the minimum possible value of the probability $P[Y=0]$, given that $Y$ is a non-negative integer-valued random variable with $E[Y]=1$, $E[Y^2]=2$, and $E[Y^3]=5$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
5. Given the number $500 \ldots 005$ (80 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 165. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17280}
Find the value of \(x\) to three decimal places for the equation \(4^x = \frac{5}{x}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.095}
Find the unique 3-digit number $N=\underline{A}\ \underline{B}\ \underline{C}$, whose digits $(A, B, C)$ are all nonzero, with the property that the product $P=\underline{A}\ \underline{B}\ \underline{C} \times \underline{A}\ \underline{B} \times \underline{A}$ is divisible by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{875}
Given that point P (a, b) is outside the circle C: x^2 + y^2 = 1, describe the positional relationship between the line ax + by + 1 = 0 and circle C. Express your answer as a single geometric term (e.g. Tangent, Separate, Intersect).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Intersect}
When the child was born, neither of his parents was yet 40 years old, but they were already adults. When the child turned 2 years old, the age of exactly one of the parents was divisible by 2; when the child turned 3 years old, the age of exactly one of the parents was divisible by 3; and so on. How long could this pattern have continued?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that \( x - \frac{1}{x} = 5 \), find the value of \( x^4 - \frac{1}{x^4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{727}