inclusionAI/Ring-lite-rl-data huggingface dataset:
In triangle \( ABC \), where \( BC \) is the longest side, the bisector of angle \( C \) intersects the altitudes \( AA_1 \) and \( BB_1 \) at points \( P \) and \( Q \) respectively, and the circumcircle of \( ABC \) at point \( L \). Given that \( AP = LQ \), determine the measure of \( \angle ACB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Given that $n \equiv 1(\bmod 4)$ and $n > 1$, let $P=\{a_{1}, a_{2}, \cdots, a_{n}\}$ be an arbitrary permutation of $\{1,2,3, \cdots, n\}$. Define $k_{p}$ as the largest index $k$ for which the following inequality holds:
$$
a_{1}+a_{2}+\cdots+a_{k}<a_{k+1}+a_{k+2}+\cdots+a_{n}.
$$

Find the total sum of $k_{p}$ for all possible different permutations $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}(n-1)(n!)}
Find the formula to calculate the centroid of the solid of revolution formed by rotating the curve \( y = \sin(x) \) from \( x = 0 \) to \( x = \pi \) around the \( x \)-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{\pi}{2}, 0, 0 \right)}
What is the smallest number of factors that needs to be removed from 99! so that the product of the remaining factors ends in 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Compute the expected value of $\sin(\pi XY)$ given the joint density function of continuous random variables $X$ and $Y$:

$$f_{\small X,Y}(x,y) = \begin{cases} \frac{1}{3}(x+y), & \text{for } 0 < x < 1 \text{ and } 0 < y < 2 \\
0, & \text{otherwise} \end{cases}$$

Evaluate the integral:

$$\mathbb{E}[\sin(\pi XY)] = \iint_{\mathbb{R}^2} \sin(\pi XY) \cdot f_{\small X,Y}(x,y) \,dx\,dy$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{\pi}}
Determine the maximum number of cuts that can be made along the holes of a triple torus (a donut with three holes) without it falling apart.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the circulation of the vector field \( \vec{A} = \hat{r} r \cos(\phi) + \hat{z} z \sin(\phi) \) around a circular sector defined by \( 0 \leq r \leq 3 \) and \( 0^{\circ} \leq \phi \leq 90^{\circ} \). Assume the path of circulation is clockwise around the sector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{9}{2}}
Compute the limit: $$\lim_{a \to 0^+} \left(a \int_{1}^{\infty} e^{-ax}\cos \left(\frac{2\pi}{1+x^{2}} \right)\,\mathrm dx\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the minimum value of \[ P = \frac{(x+1)^2(y+1)^2}{z^2+1} + \frac{(y+1)^2(z+1)^2}{x^2+1} + \frac{(z+1)^2(x+1)^2}{y^2+1} \] given that \( x > 0, y > 0, z > 0 \) and \( x + y + z = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Compute the distance between a unilateral shift operator $S$ on a separable, infinite-dimensional Hilbert space $\mathcal{H}$ and the set of compact operators on $\mathcal{H}$, denoted by $\mathbb{K}(\mathcal{H})$. Find $d(S, \mathbb{K}(\mathcal{H}))=\inf\{\|S-K\|: K\in \mathbb{K}(\mathcal{H})\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the variance of $W_1$, where $W_1 = \int_{0}^{1} W^2(t) dt$ and $W(t)$ is a Wiener process, using its characteristic function $E[e^{izW_1}] = \left( \cos \sqrt{2iz} \right)^{-\frac{1}{2}}$. What is the variance of $W_1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
How many distinct arrangements are there for 61 red cards and 61 blue cards such that no two cards of the same color are adjacent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \times (61!)^2}
Evaluate the limit: \[ \lim_{n \to \infty} \sum_{k=1}^{nt} \frac{1}{n} \left( 1 - \frac{1}{n} \right)^{k-1}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - e^{-t}}
Determine the number of homomorphisms from the symmetric group $S_3$ to the dihedral group $D_4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( A \) be an \( n \times n \) diagonal matrix with real entries and characteristic polynomial \((x-a)^p(x-b)^q\), where \( a \) and \( b \) are distinct real numbers. Define \( V = \{ B \in M(n, \mathbb{R}) : AB = BA \} \). What is the dimension of the subspace \( V \) of matrices that commute with \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + q^2}
Given the solutions \( t^3 \) and \( t e^t \sin(t) \) to an nth-order homogeneous differential equation with constant coefficients, determine a restriction on \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \geq 8}
Determine whether the limit \( \lim_{n\to \infty}\int_{0}^{1}nx^ng(x)dx \) exists for any continuous function \( g \) on the interval \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(1)}
In a regular hexagon \( ABCDEF \), the side \( AB \) is \( \sqrt{3} \) and acts as a chord of a circle, while the other sides lie outside the circle. The tangent \( CM \) from vertex \( C \) to the circle measures 3. Determine the diameter of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
Find the limit of the integral \( \int_{1/\sqrt{n}}^{1} |f_n(x)| \, dx \) as \( n \rightarrow \infty \), where \( f_n(x) = \dfrac{\sin 2 \pi x \,\sin 2\pi n x}{x^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Compute the quotient group \((\mathbb{Z} \oplus \mathbb{Z}) / \langle(3,6),(5,5)\rangle\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_{15}}
Two closed disks of radius \( \sqrt{2} \) are drawn centered at the points(1,0)and(-1,0)
 Let \( P \) be the
 region belonging to both disks. Two congruent non-intersecting open disks of radius \( r \)
 have all of
 their points in \( P \) . Find the maximum possible value of \( r \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{2}}{4}$
}
Let \(a, b, c\) be the sides of a triangle. Given that \(a^2 + b^2 + c^2 + 2abc = 4\), determine the maximum possible value of \(a + b + c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{3}}
Find the positive integer $n$ for which there exists a permutation $x_{1}, x_{2}, \cdots, x_{n}$ of the set $\{1, 2, \cdots, n\}$ such that the absolute differences $\left|x_{k}-k\right|$ for $k=1, 2, \cdots, n$ are all distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Identify all closed sets in \(\mathbb{R}\) that are subgroups of \((\mathbb{R}, +)\) and have at least one limit point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Given the set $A={(x,y)|$\frac{m}{2}$\leqslant {{(x-2)}^{2}}+{{y}^{2}}\leqslant {{m}^{2}},x,y\in R}$ and $B={(x,y)|2m\leqslant x+y\leqslant 2m+1,x,y\in R}$, determine the range of the real number $m$ if $A\cap B\ne \varphi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{1}{2},2+ \sqrt{2}]}
Mr. Karan borrowed a certain amount at 6% per annum simple interest for 9 years. After 9 years, he returned Rs. 8410. Find out the amount that he borrowed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5461}
Determine the values of $p > 0$ for which the function $g(x)$, defined as \( g(x) = \begin{cases} \cot(x) & x \in (0,\frac{\pi}{2}) \\ 0 & \text{else} \end{cases} \), is in the space $\mathcal{L}^p(\lambda)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
Lindsey bought 2 exercise bands to intensify her workout. Each band adds an extra 5 pounds of resistance to her workout. If she doubles up both sets of bands and places them around her legs and picks up a 10-pound dumbbell, how much weight will she squat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $A_1,A_2,A_3,\cdots,A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\{A_1,A_2,A_3,\cdots,A_{12}\} ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{183}
Evaluate the integral \( \int_0^1\int_0^{z\pi}\int_\frac{y}{\pi}^z \sin\frac{y}{x}dxdydz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Let \( a(n) \) be a sequence defined as follows: \( a(0)=\frac{1}{2} \), and for any natural number \( n \), \( a(n+1)=a(n)+\frac{a(n)^2}{2013} \). Determine the smallest natural number \( k \) such that \( a(k)>1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
In how many ways can 6 people buy ice cream bowls from a store that sells 5 different flavors, such that each flavor is chosen at least once? Each bowl contains three scoops of ice cream, and the order of scoops does not matter. Use the inclusion-exclusion principle to find the solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1528224670}
Two equally strong tennis players play against each other until one player wins three consecutive games. Each game is independent, and each player has a probability of 1/2 of winning any given game. What is the expected number of games they will play until one player wins three games in a row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In an $n$-dimensional space, $n$ vectors form a regular simplex. What is the cosine of the angle between any two of these vectors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{n}}
Find the coefficient of $x^{50}$ in the expansion of $\left(\frac{1}{(1+x)(1+x^2)(1+x^4)}\right)^3$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Consider an abelian group $G$ isomorphic to $\mathbb{Z}_{36} \times \mathbb{Z}_{40} \times \mathbb{Z}_{42}$. Determine the number of elements in $G$ that have order equal to the least common multiple (LCM) of 36, 40, and 42.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13824}
What is the probability that the combined lifetime of the first two type-B bulbs exceeds the lifetime of the first type-A bulb, given that each bulb is chosen randomly with equal probability of being either type A or type B? The lifetime of a type-A bulb follows an exponential distribution with a rate of 1, while a type-B bulb has an exponential distribution with a rate of 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{16}}
Given $x^{3}=4$, solve for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{4}}
2. Two plane mirrors serve as the faces of a dihedral angle. A light ray, perpendicular to the edge of the angle and parallel to the first mirror, is reflected from the second mirror, then from the first, then again from the second, again from the first, and finally, reflecting for the fifth time from the second mirror, returns along the same straight line. What is the magnitude of the dihedral angle between the mirrors?

(Remember, according to the laws of reflection, the incident and reflected rays lie in a plane perpendicular to the mirror plane and form equal angles with the mirror plane.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A list contains 20 numbers. For each positive integer \( n \), from 1 to 20, the \( n \)-th number in the list is either \( \frac{n}{2} \) or \( 2n \). If the mean of the numbers in the list is exactly 12 and the product of the numbers is \( P \), what is the greatest possible value of \( \frac{P}{20!} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The altitude of a right triangle, dropped to its hypotenuse, divides the bisector of the acute angle in the ratio $4:3$, counting from the vertex.

Find the measure of this angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos^{-1} \left( \frac{4}{7} \right)}
Given \( x \neq y \) and two arithmetic sequences \( x, a_1, a_2, a_3, y \) and \( b_1, x, b_2, b_3, y, b_4 \), find the value of \( \frac{b_4 - b_3}{a_2 - a_1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{3}}
Given that $f''(x)$ is the derivative of $f(x)$, $f(1)=\frac{1}{e}$, and $f(x)-f''(x) > 0$ for any real number, find the solution set for the inequality $f(x) < e^{x-2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, +\infty)}
Let $\mathbb{N}$ denote the set of positive integers. For how many positive integers $k\le 2018$ do there exist a function $f: \mathbb{N}\to \mathbb{N}$ such that $f(f(n))=2n$ for all $n\in \mathbb{N}$ and $f(k)=2018$?

[i]Proposed by James Lin
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1512}
Find the intersection point \((x^*, t^*)\) of the shock wave and the rarefaction wave for the equation \(\partial_t u + \partial_x(\frac{u^2}{2}) = 0\) with the initial conditions:
\[
u(x,0) = g(x) = \begin{cases} 2, & \text{if } x<1 \\ 1, & \text{if } 1<x<2 \\ 2, & \text{if } x>2 \end{cases}\]
The characteristic equation is given by:
\[x(t) = \xi + g(\xi)t = \begin{cases} \xi + 2t, & \text{if } \xi \leq 1 \\ \xi + t, & \text{if } 1<\xi<2 \\ \xi + 2t, & \text{if } \xi>2 \end{cases}\]
The shock wave from \((1,0)\) is described by:
\[x = s(t) = \frac{3}{2}t + 1\]
The rarefaction wave from \((2,0)\) is between \(x(t) = 1+t\) and \(x(t) = 2+t\) with:
\[u(x,t) = (q')^{-1}\left(\frac{x-2}{t}\right) = \frac{x-2}{t}\]
Determine the intersection \((x^*, t^*)\) of these waves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(4, 2)}
设 $m, n$ 是给定的整数， $m \geq n \geq 3$ 。求具有下述性质的最小正整数 $k$ ：若将 $1,2, \cdots, k$ 中的每个数任意染为红色或者蓝色，则或者存在 $m$ 个红色的数 $x_{1}, x_{2}, \cdots, x_{m}$ （允许相同），满足 $x_{1}+x_{2}+\cdots+x_{m-1}<x_{m}$ ，或者存在 $n$ 个蓝色的数 $y_{1}, y_{2}, \cdots, y_{n}$ （允许相同），满足 $y_{1}+y_{2}+\cdots+y_{n-1}<y_{n}$。请提供当 $m = 3, n = 3$ 时的答案。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
You have twelve boxes, each containing between 1 and 6 marbles. Each marble weighs 1 gram, and the marbles are fixed in place within the boxes. Using a simple balance scale and a reference weight of your choosing, determine the number of marbles in each box. What reference weight should you choose?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the expected claim size for an insurance policy with the claim size distribution given by:

$$F(x)=\begin{cases}
0.1+0.0001x&0\le x<5000\\
0.6&x=5000\\
1-0.4e^{(x-5000)/10000}&x>5000
\end{cases}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7250}
How may 50 cities be connected by the least possible number of airlines so that from any city one could get to any other by changing airplanes not more than once (i.e., using two planes)? Please provide the number of airlines needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
设 $S=\{1,2, \cdots, 100\}$. 求最小的自然数 $n$, 使得 $S$ 的每个 $n$ 元子集都含有 4 个两两互质的数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Determine the growth order of the function \( f(z) = \sum_{n=0}^\infty \frac{z^{2n}}{((2n)!)^a} \), where \( a \) is a positive real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{a}}
Maggie has an after-school job selling magazine subscriptions. She earns different commissions based on whether she sells the subscription to a family member or to a neighbor. She earns $7 for each subscription sold to a family member and $6 for each subscription sold to a neighbor. Additionally, her boss has promised her a bonus if she sells more than 10 subscriptions. This bonus is $10 plus $1 for each subscription sold over 10.

On Monday, she sells 4 subscriptions to her parents and 1 to her grandfather. On Tuesday, she sells 2 to the next-door neighbor and twice that amount to another neighbor. On Wednesday, she sells 3 to her teacher and 1 to her school principal.

How much money did Maggie earn by the end of Wednesday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$114}
29 boys and 15 girls came to the ball. Some boys danced with some girls (no more than once per pair). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have reported?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Calculate the covariance between $X$ and $Y^3$ for a bivariate normal distribution $(X,Y)$ with mean vector $\begin{pmatrix} 0\\ 0 \end{pmatrix}$ and covariance matrix $\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\rho}
On the board, 2017 digits are written. From these digits, several numbers are formed, the sum of the digits of these numbers is calculated, and then the sum of the sums of their digits is subtracted from the sum of all numbers. The resulting number is broken down into digits, and the above operation is repeated again. After performing this operation several times, only one digit remained on the board for the first time. What is this digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The midpoints of the sides \(PQ\), \(QR\), and \(RP\) of triangle \(PQR\) are \(A\), \(B\), and \(C\) respectively. Triangle \(ABC\) is enlarged from its centroid \(S\) by a factor of \(k\), where \(1 < k < 4\). The sides of the resulting enlarged triangle intersect the sides of triangle \(PQR\) at points \(D_{1}D_{2}\), \(E_{1}E_{2}\), and \(F_{1}F_{2}\) respectively. Determine the value of \(k\) for which the area of the hexagon \(D_{1}D_{2}E_{1}E_{2}F_{1}F_{2}\) is twice the area of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 - \sqrt{6}}
If five points are given on a plane, then by considering all possible triples of these points, 30 angles can be formed. Denote the smallest of these angles by $\alpha$. Find the maximum value of $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36^\circ}
Mr. Noeth's BC Calculus class has 17 students who live in consecutive houses on the same street. He needs to select at least three out of every four consecutive students to take a makeup test. How many ways can Mr. Noeth select the students to take the makeup tests?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{345}
A bird is building a nest from twigs. The bird has put twelve twigs together already to make a circle. For each twig it placed into the circle, it wants to weave in six more twigs to fill out the nest. It knows a tree that has dropped a third of the twigs it needs. How many twigs will the bird still need to find to finish its nest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Find the largest positive integer  $n$  such that  $n\varphi(n)$  is a perfect square.  ( $\varphi(n)$  is the number of integers  $k$ ,  $1 \leq k \leq n$  that are relatively prime to  $n$ )
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Suppose that \( X \) is a random variable with the probability mass function \( \mathbb{P}(X=k)=e^{-3}\frac{3^{k-1}}{k!}(1+\beta k) \) for \( k = 0, 1, 2, \ldots \). Determine the value of \( \beta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the period of a function that satisfies the condition \( f(a-x) = f(a+x) \), where \( a \) is any positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a chess match against a skilled opponent, there is a 75% chance that any given game will end in a draw. For games that do not end in a draw, there is a 2/3 chance that you will win and a 1/3 chance that you will lose. The first player to win 2 games in a row is declared the winner of the match. What is the probability that you will win the match?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{26}{33}}
Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$.

[i]Proposed by A. Antropov[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Ben rolls 6 fair 12-sided dice, with each die numbered from 1 to 12. What is the probability that exactly three of the dice show a prime number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{857500}{2985984}}
Given the system of equations: \(x^2 + xy + y^2 = 25\), \(x^2 + xz + z^2 = 49\), and \(y^2 + yz + z^2 = 64\), find the value of \(x + y + z\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{129}}
Given two dependent Poisson processes, $X_t$ with a rate of 15 events per hour and $Y_t$ with a rate of 10 events per hour, where the probability of an event occurring in $X_t$ given that it occurs in $Y_t$ is 3/5, calculate the probability of observing exactly 10 events in $X_t$ and 5 events in $Y_t$ within a 30-minute period.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0154}
Let $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\overline{BC}$, and denote the circle with diameter $\overline{AD}$ by $\Omega$. If the region inside $\Omega$ and outside $\triangle ABC$ has area $800\pi-600\sqrt{3}$, find the length of $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
If the square root of (20% of 45% of 60% of the cube of a number) is 35.7, then what is the number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28.7}
A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After a total of $n$ jumps, the position of the grasshopper is 162 cm to the west and 158 cm to the south of its original position. What is the sum of the squares of the digits of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
A tangent line PA is drawn from a point P outside a circle, touching the circle at point A. Another secant line PBC intersects the circle at points B and C, respectively. Given that PA = 6, AC = 8, and BC = 9, find the length of AB.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
在一个圆周上给定 8 个点 A_1,A_2,\cdots,A_8. 求最小的正整数 n，使得在以这 8 个点为顶点的任意 n 个三角形中，必存在两个有公共边.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}