inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $0 \leq \alpha \leq \pi$. Consider the sequence $u_n$ where $u_n = \cos(n\alpha)$. Denote by $V_n(\alpha)$ the count of sign changes in this sequence. Determine the limit of the ratio $\frac{V_n(\alpha)}{n}$ as $n$ approaches infinity, given that the sign changes in $u_n$ are irregular.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\alpha}{\pi}}
Tim buys a book of esoteric words. He decides to learn 10 words a day, but only on weekdays, as he takes weekends off for leisure. In 2 years, with occasional vacations (30 days in total) during which he doesn't learn new words, the book has increased the number of words he knows by 50%. Additionally, Tim forgets 2 words per week. How many words did he know originally?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9384}
For a $k$-element subset $T$ of the set $\{1,2,\cdots,242\}$, every pair of elements (which may be the same) in $T$ has a sum that is not an integer power of 3. Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{121}
An isosceles triangle with a base \( a \) and a base angle \( \alpha \) is inscribed in a circle. Additionally, a second circle is constructed, which is tangent to both of the triangle's legs and the first circle. Find the radius of the second circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{2 \sin \alpha (1 + \cos \alpha)}}
The least common multiple of a positive integer $n$ and $18$ is $180$, and the greatest common divisor of $n$ and $45$ is $15$. What is the sum of the digits of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the function $$f(x)=\sin(x)+\sqrt3 \cos(x)$$ Find the real number $m$ such that $$f'(x)+f''(x)=\ln(m^2-4m+5)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Which quadrant does the line \( l: ax + by + c = 0 \) not pass through when \( ab < 0 \) and \( bc < 0 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Fourth quadrant}
Determine the symmetry center of the function $y=\cos (2x-\frac{\pi }{4})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{\pi }{8},0)}
Find the remainder when the value of $m$ is divided by 1000 in the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_6 \le 1500$ such that $a_i-i$ is odd for $1\le i \le 6$. The total number of sequences can be expressed as ${m \choose n}$ for some integers $m>n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{752}
During a unique three-team soccer game, in the first half, Team A scored 8 points, Team B scored half as many as Team A, and Team C scored twice as many as Team B. Additionally, Team A missed one penalty kick, and Team C missed two penalty kicks. In the second half, Team A scored the same number of goals as Team C did in the first half, Team B scored as many goals as Team A in the first half, and Team C scored three more goals than Team B in the second half. Team A scored one successful penalty kick, while Team B scored two successful penalty kicks. How many goals did all three teams score during the whole match, including the penalty kicks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
实数$x _ { 1 } , x _ { 2 } , \cdots , x _ { 2 0 0 1 }$满足$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert x _ { k } - x _ { k + 1 } \vert = 2 0 0 1 ,$，令$y_ { k }= \frac { 1 } { k } ( x _ { 1 } + x _ { 2 } + \cdots + x _ { k } ) , k = 1 ,$,2,⋯,2001,，求$\sum _ { k = 1 } ^ { 2 0 0 0 } \vert y _ { k } - y _ { k + 1 } \vert$的最大可能值．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Given that the decomposition rate $v$ of a certain type of garbage approximately satisfies the relationship $v=a\cdot b^{t}$, where $a$ and $b$ are positive constants, and the decomposition rate is $5\%$ after $6$ months and $10\%$ after $12$ months, calculate the time it takes for this type of garbage to completely decompose.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
8. For the cubic function $y=x^{3}+a x^{2}+b x+c$, its graph intersects the $x$-axis at points $A, T$, and $B$ in sequence. The lines $A P$ and $B Q$ are tangent to the cubic curve at points $P$ and $Q$ (where $P$ does not coincide with $A$, and $Q$ does not coincide with $B$). Then the ratio of the projections of $\overrightarrow{A B}$ and $\overrightarrow{P Q}$ on the $x$-axis is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Dacid obtained 72, 45, 72, 77, and 75 marks (out of 100) in English, Mathematics, Physics, Chemistry, and Biology, respectively. Given that each subject has a different credit hour value - English: 3 credit hours, Mathematics: 4 credit hours, Physics: 4 credit hours, Chemistry: 3 credit hours, and Biology: 2 credit hours - calculate his weighted average marks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66.56}
Find the sum of the decimal digits of \[ \left\lfloor \frac{51525354555657\dots979899}{50} \right\rfloor. \] Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{457}
Let
$$
S=\sum_{r=0}^{n}\binom{3 n+r}{r}
$$

Evaluate \( S \div (23 \times 38 \times 41 \times 43 \times 47) \) when \( n = 12 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1274}
Given that the function $f(x)$ is an even function, and shifting its graph one unit to the right results in an odd function, and $f(2)=-1$, calculate the sum of $f(1)+f(2)+...+f(2017)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
if the function q is defined by the formula q = 5w^2 / (4d^2 (z^3 + x^2)), by what factor will q be multiplied if w is quadrupled, d is doubled, z is tripled, and x is halved?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16/27}
集合$\{ x \vert - 1 \leq \log _ { \frac { 1 } { x } } 1 0 < - \frac { 1 } { 2 } , x \in N _ { + } \}$的真子集的个数是$\underline{\hspace{1cm}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 ^ { 9 0 } - 1$}
There are 7 balls of each of the three colors: red, blue, and yellow. Each set of 7 balls of the same color is numbered from 1 to 7. When randomly selecting 3 balls with different numbers, find the number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
How many distinct functions $f: \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \to \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ are there such that $(f \circ f)(1) = 2$ and $f$ is onto (surjective)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35280}
In how many ways can two distinct squares be chosen from an $8 \times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
1021. Among all rectangular parallelepipeds having a given volume $\mathrm{V}$, find the one with the smallest total surface area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{V}}
Let $s(m)$ denote the sum of the digits of the positive integer $m$. Find the largest positive integer that has no digits equal to zero and satisfies the equation \[2^{s(n)} = s(n^2).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1111}
Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. Find the value of $AR+BR+CR+DR$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
There is a $6 \times 6$ grid of lights. There is a switch at the top of each column and on the left of each row. A light will only turn on if the switches corresponding to both its column and its row are in the "on" position. Compute the number of different configurations of lights.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3970}
Let f(x) be a non-decreasing function on the interval [0, 1], meaning that for any 0 ≤ x1 ≤ x2 ≤ 1, we have f(x1) ≤ f(x2). The function satisfies the following conditions:
1. f(0) = 0;
2. f(x/3) = (1/2)f(x);
3. f(1-x) = 1 - f(x).

Find the value of f(1/2016). Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/128}
若存在正整数 a,b，使得 \df{k}{2023}(a+b)=[a,b]，则整数 k 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
For each positive integer $n$, let $H_n = \frac{1}{1} + \frac{1}{2} + \cdots + \frac{1}{n}$. If \[ \sum_{n=4}^{\infty} \frac{1}{nH_nH_{n-1}} = \frac{M}{N} \] for relatively prime positive integers $M$ and $N$, compute $100M+N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{611}
In an isosceles triangle \( ABC \) with \(\angle B = \arctan \frac{8}{15}\), a circle with radius 1 is inscribed in angle \( C \) and touches side \( CB \) at point \( M \) and cuts off a segment \( KE \) from the base. It is known that \( MB = \frac{15}{8} \). Find the area of triangle \( KMB \), given that points \( A, K, E, B \) follow in the mentioned order along the base \( AB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{375}{272}}
liam is pulled over for speeding just as he is arriving at work . he explains to the police officer that he could not afford to be late today , and has arrived at work only four minutes before he is to start . the officer explains that if liam had driven 5 mph slower for his whole commute , he would have arrived at work exactly on time . if liam ' s commute is 40 miles long , how fast was he actually driving ? ( assume that liam drove at a constant speed for the duration of his commute . )
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\cdots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of $67$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
In a group of 500 students, each student is registered for at least one of 5 classes - History, Math, English, Science, and Geography. 120 students are registered for History, 105 students are registered for Math, 145 students are registered for English, 133 students are registered for Science, and 107 students are registered for Geography. If 15 students are registered for all 5 classes, 40 students are registered for both History and Math, 35 students are registered for both English and Science, and 25 students are registered for both Math and Geography, how many students are registered for exactly two classes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
What is the minimum degree of the Lagrange polynomial required to approximate $f(x) = \sin(x)$ in the interval $[0, \pi]$ such that the error, $| \sin(x) - L_n(x) |$, is less than or equal to $0.001$? In other words, given the Lagrange interpolation error formula:
$$| \sin(x) - L_n(x) | = \left| \frac{\sin^{(n+1)}(c)}{(n+1)!} \right| (x - x_0)(x - x_1) \cdots (x - x_n) \leq 0.001, \quad 0 \leq c \leq \pi$$
How do you determine $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
At a party, there are $2011$ people, each with a glass of fruit juice, sitting around a circular table. Every second, they clink glasses according to the following two rules:
1. They do not clink glasses crosswise.
2. At each point in time, everyone can clink glasses with at most one other person.

How many seconds are needed, at a minimum, for everyone to have clinked glasses with everyone else?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2011}
Determine the largest positive integer $n$ such that the following statement holds:

If $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > a$, and $ca > b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{107}
John Smith buys 3 cakes for $12 each and a pack of cookies for $5. He has a 10% discount coupon off the total cost and there is a 5% sales tax applied. After applying the discount and tax, he splits the cost with his brother. How much did each brother pay?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$19.38}
6. Ms. Olga Ivanovna, the class teacher of 5B, is staging a "Mathematical Ballet." She wants to arrange the boys and girls so that exactly 2 boys are 5 meters away from each girl. What is the maximum number of girls that can participate in the ballet if it is known that 5 boys are participating?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Calculate the limit, if it exists, of the expression $$\lim_{x\to 0}\left[\left[\log \left(1+\frac{1}{2x}\right)\right]^2-\left[\log \left(1+\frac{1}{x}\right)\right]^2\right].$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the inverse function of the given function $f: \mathbb{R} \to (0,2)$ defined by $f(x) = \frac{e^x-e^{-x}}{e^x+e^{-x}} + 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2}\ln\left(\frac{x}{2-x}\right)\)}
In a park, there are 2 maple trees and 5 poplar trees currently. Today, park workers will plant 3 times the current number of poplar trees as maple trees. If there are also 4 oak trees in the park, how many maple, poplar, and oak trees will the park have in total when the workers are finished planting today?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
For which natural numbers \( n \) can a board of size \( n \times n \) be divided along the grid lines into dominos of size \( 1 \times 2 \) so that the number of vertical and horizontal dominos is equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \text{ is divisible by 4}}
Given two sequences $\{a_n\}$ and $\{b_n\}$ satisfying $a_1= \frac{1}{2}$, $a_n+b_n=1$, $b_{n+1}= \frac{b_n}{1-a_n^2}$, for $n\in\mathbb{N}_+$, find the value of $b_{2016}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2016}{2017}}
72 vertices of a regular 3600-gon are painted red in such a way that the painted vertices form the vertices of a regular 72-gon. In how many ways can 40 vertices of this 3600-gon be chosen so that they form the vertices of a regular 40-gon and none of them are red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Given real numbers \(x\) and \(y\) satisfying \(x^{2}+(y-2)^{2} \leq 1\), determine the range of values for \(\frac{x+\sqrt{3} y}{\sqrt{x^{2}+y^{2}}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, 2]}
There are 20 points on a plane, and each pair of points is connected by a segment (edges) - each point with every other. Then, 35 edges are randomly selected and removed. Find the probability that the resulting graph is connected (i.e., it is possible to reach any point from any other point via edges).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{20 \binom{171}{16}}{\binom{190}{35}}}
A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the prime factors of $N$, including multiplicity. (For example, if $N = 12$, the answer would be $2 + 2 + 3 = 7$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Given the sets \( A = \{(x, y) \mid |x| + |y| = a, a > 0\} \) and \( B = \{(x, y) \mid |xy| + 1 = |x| + |y| \} \), if the intersection \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \sqrt{2}}
In $\triangle ABC$, $\angle A = 42^{\circ}$. The trisectors of $\angle B$ and $\angle C$ meet at points $D$ and $E$ respectively. Find the measure of $\angle BDC$. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88^\circ}
Example 7 $n$ people greet each other by phone during a holiday, it is known that each person has called at most 3 of their friends; any 2 people have had at most 1 phone call; and among any 3 people, at least 2 of them, one has called the other. Find the maximum value of $n$. (Original problem)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
把某公司的 $9$ 男 $3$ 女共 $12$ 名员工平均分配到甲、乙、丙三个部门。求甲、乙、丙三个部门各有 $1$ 名女员工的概率.用分数表示，写出分子加分母的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
Given that the even function $f(x)$ satisfies $f(x-2) = -f(x)$ for all $x \in \mathbb{R}$, and when $x \in [-1, 0]$, $f(x) = 2^x$. Find the value of $f(2015)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Dima needed to arrive at the station at 18:00, where his father was supposed to pick him up by car. However, Dima caught an earlier train and arrived at the station at 17:05. Instead of waiting for his father, he started walking towards him. They met on the way, Dima got into the car, and they arrived home 10 minutes earlier than the calculated time. What was Dima's walking speed before he met his father, if the car's speed was $60 \ \text{km/h}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 \text{ km/h}}
Given that the function f(x) = 4x^2 - kx - 8 has monotonicity on the interval (∞, 5], determine the range of values for the real number k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[40, +\infty)}
A number x is 12 times another number y, and the third number z is 20% greater than x. Find the percentage that y is less than z.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93.06\%}
There is a basket of apples. After dividing them into three equal parts, 2 apples remain. Taking out two of these parts, and dividing them into three equal parts again, 2 apples remain. After taking out two of these parts again and dividing them into three equal parts, 2 apples remain. How many apples are in the basket, at minimum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Evaluate the integral $$\int_0^\infty e^{-ay^{2}-\frac{b}{y^2}}dy $$ where a and b are real and positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sqrt{\pi}}{2\sqrt{a}} e^{-2\sqrt{ab}}\)}
已知曲线 $C_{1}: y=\sqrt{-x^{2}+10 x-9}$ 及点 $A(1,0)$. 若曲线 $C_{1}$ 上存在相异的两点 $B 、 C$,其到直线 $l: 3 x+1=0$ 的距离分别为 $|A B|$ 、 $|A C|$, 则 $|A B|+|A C|=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A total of  $n$  people compete in a mathematical match which contains  $15$  problems where  $n>12$ . For each problem,  $1$  point is given for a right answer and  $0$  is given for a wrong answer. Analysing each possible situation, we find that if the sum of points every group of  $12$  people get is no less than  $36$ , then there are at least  $3$  people that got the right answer of a certain problem, among the  $n$  people. Find the least possible  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
若 T=\ds{k=1}{100}2^{99+k}\cdot3^{101-k}，则 T 的末尾数字 0 的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the maximum number of distinct positive integer solutions for the equation $x(x+1)(x+2)=ay^b$, where $a,b$ are positive integers and $b\ge3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are 20 students in the class. Each boy has made a list of all the girls in the class they find attractive (the list may be empty). It is known that there are no three boys whose lists contain the same number of girls. What is the minimum number of girls that could be in the class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the vertical distance from the end of the pole touching the wall to the ground in meters, or $\qquad$ meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16 - 4\sqrt{2}}
The origin coincides with the pole, and the positive $x$-axis coincides with the polar axis. What are the polar coordinates of the point $(-5-5\sqrt{3})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(10, \frac{4\pi}{3})}
A canister contains $2\frac{1}{2}$ cups of flour. Greg and Sally have a brownie recipe that calls for $1\frac{1}{3}$ cups of flour. They want to make $1\frac{1}{2}$ recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
If the following limit $$\lim_{x\to 0}\frac{\cos^2x-\cos x-e^x\cos x+e^x-\frac{x^3}{2}}{x^n}$$ is a non zero finite number, find $n$ where $n\in\mathbb{N}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$, respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt{3} \cdot PS$ and the area of $PQRS$ is $28\sqrt{3}$, what is the length of $PC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In the rectangular parallelepiped \(ABCD A_1 B_1 C_1 D_1\), two diagonals of the side faces \(A_1 C_1\) and \(C_1 D\) are drawn. Find the angle between them, given that the diagonal \(A_1 C_1\) is equal in length to one of the edges of the parallelepiped, and the diagonal \(C_1 D\) forms a 30-degree angle with the same edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos(\frac{\sqrt{3}}{6})}