inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $a$ and $b$ be two non-perpendicular skew lines. Planes $\alpha$ and $\beta$ are constructed passing through lines $a$ and $b$ respectively. Consider the following four situations: $b \parallel \alpha$, $b \perpendicular \alpha$, $\alpha \parallel \beta$, $\alpha \perpendicular \beta$. How many of these situations are possible? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that \(a, b, x, y\) are all positive numbers, \(n\) is a positive integer, and \(x + y = 1\), find the minimum value of \(\frac{a}{x^n} + \frac{b}{y^n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt[n+1]{a} + \sqrt[n+1]{b})^{n+1}}
Compute the value of the expression:
\[ 2(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4))))))))) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{699050}
In triangle \(ABC\), \(AB = 32\), \(AC = 35\), and \(BC = x\). What is the smallest positive integer \(x\) such that \(1 + \cos^2 A\), \(\cos^2 B\), and \(\cos^2 C\) form the sides of a non-degenerate triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
设 A 是有限集，且 |A|\geq2. A_1,A_2,\cdots,A_n 是 A 的子集，满足：(1) |A_1|=|A_2|=\cdots=|A_n|=k, k>\df{|A|}{2}；(2) 对任意 a,b\in A，存在 1\leq r<s<t\leq n，使得 a,b\in A_r\cap A_s\cap A_t；(3) 对任意 1\leq i<j\leq n，|A_i\cap A_j|\leq3.求当 k 取最大值时 n 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given a solid $\Omega$ which is the larger part obtained by cutting a sphere $O$ with radius $4$ by a plane $\alpha$, and $\triangle ABC$ is an inscribed triangle of the circular section $O'$ with $\angle A=90^{\circ}$. Point $P$ is a moving point on the solid $\Omega$, and the projection of $P$ on the circle $O'$ lies on the circumference of $O'$. Given $OO'=1$, the maximum volume of the tetrahedron $P-ABC$ is \_\_\_\_\_\_
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
How many elements in $S_8$ are conjugate to $(12)(345)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1120}
Compute the smallest positive integer \( n \) for which
\[
\sqrt{100+\sqrt{n}} + \sqrt{100-\sqrt{n}}
\]
is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6156}
Let $a$ be a number with $n$ digits ($n > 1$). A number $b$ with $2n$ digits is obtained by writing two copies of $a$ one after the other. If $\frac{b}{a^2}$ is an integer $k$, find the possible values of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The sides of a triangle, as well as the inradius \( r \) and the radii of the excircles \( r_1, r_2, r_3 \), are integers, and the radii are even. Determine the sides if

\[ r r_{1} r_{2} + r r_{2} r_{3} + r r_{3} r_{1} + r_{1} r_{2} r_{3} = r r_{1} r_{2} r_{3} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(6, 8, 10)}
在平面直角坐标系内, 有四个定点 $A(-3,0), B(1,-1), C(0,3), D(-1,3)$及一个动点 $P$ ，则 $|P A|+|P B|+|P C|+|P D|$ 的最小值为a\sqrt{b}+c\sqrt{d}的形式。请给出该形式中所有系数的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find all values of \( a \) such that for each \( x \), the largest of the two numbers \( x^{3}+3x+a-9 \) and \( a+2^{5-x}-3^{x-1} \) is positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > -5}
From an external point \(A\), a tangent \(AB\) and a secant \(ACD\) are drawn to a circle. Find the area of triangle \(CBD\), given that the ratio \(AC : AB = 2 : 3\) and the area of triangle \(ABC\) is 20.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let \( W \) be the hypercube \(\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mid 0 \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 1\right\}\). The intersection of \( W \) and a hyperplane parallel to \( x_{1}+x_{2}+x_{3}+x_{4}=0 \) is a non-degenerate 3-dimensional polyhedron. What is the maximum number of faces of this polyhedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a cube with edge length 1, named ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, if point P is a point on an edge, then the number of points P that satisfy |PA|+|PC<sub>1</sub>|=2 is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let the set $A=\{x|y= \sqrt {x^{2}-4x+3}\}$, and $B=\{y|y=x+ \frac {m}{x}(m > 0),x\in\complement_{\mathbb{R}}A\}$, if $2 \sqrt {m}\in B$, then the range of $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,9)}
Evaluate the following double integral:
$$\int_F \arctan\left(\frac{x}{y}\right)\ dx\ dy,$$

where \(F\) is the region in the \(xy\)-plane bounded by the curves \(x^2+y^2=1\), \(x^2+y^2=4\), \(y=-|x|\), and \(y=|x|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There is a product $a_{1} \cdot a_{2} \cdot \cdots \cdot a_{100}$ written on the blackboard, where $a_{1}, a_{2}, \cdots, a_{100}$ are positive integers. If one of the multiplication symbols is changed to an addition symbol (while keeping the rest multiplication symbols), it is found that 32 out of the 99 resultant sums are even. How many even numbers can there be at most among $a_{1}, a_{2}, \cdots, a_{100}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Find the number of ordered tuples  $\left(C,A,M,B\right)$  of non-negative integers such that \[C!+C!+A!+M!=B!\]

*2019 CCA Math Bonanza Team Round #4*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Karen is considering her winter coat options. One coat costs $300 and will last for 15 years. The other coat costs $120 and will last for a certain number of years. Karen would save $120 over 30 years by buying the more expensive coat. How many years will the less expensive coat last?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that the graph of a power function passes through the points $(2,16)$ and $(\frac{1}{2},m)$, find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
On the radius \(A O\) of a circle with center \(O\), a point \(M\) is chosen. Points \(B\) and \(C\) are chosen on the circle on one side of \(A O\) such that \(\angle A M B = \angle O M C = \alpha\). Find the length of \(B C\) if the radius of the circle is 12 and \(\cos \alpha = \frac{3}{4}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
If \( M = \left\{(x, y) \mid \tan \pi y + \sin^2 \pi x = 0\right\} \) and \( N = \left\{(x, y) \mid x^2 + y^2 \leqslant 2\right\} \), what is the number of elements in \( M \cap N \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$. What is the smallest positive integer $n$ such that
\[\sum_{t \mid n} d(t)^3\]is divisible by $35$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864}
Larry and Diane start $100$ miles apart along a straight road. Starting at the same time, Larry and Diane
drive their cars toward each other. Diane drives at a constant rate of 30 miles per hour. To make it
interesting, at the beginning of each 10 mile stretch, if the two drivers have not met, Larry ﬂips a fair coin.
If the coin comes up heads, Larry drives the next 10 miles at 20 miles per hour. If the coin comes up tails,
Larry drives the next 10 miles at 60 miles per hour. Larry and Diane stop driving when they meet. The expected number of times that Larry ﬂips the coin 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{413}
In a regular hexagonal pyramid, the center of the circumscribed sphere lies on the surface of the inscribed sphere. Find the ratio of the radius of the circumscribed sphere to the radius of the inscribed sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+\sqrt{\frac{7}{3}}}
Let $D$ be a regular ten-sided polygon with edges of length 1. A triangle $T$ is defined by choosing three vertices of $D$ and connecting them with edges. How many different (non-congruent) triangles $T$ can be formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the number of pairs $(a, b)$, where $1 \le a \le b \le 100$ are positive integers, so that $\frac{a^3+b^3}{a^2+b^2}$ is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{122}
Given a regular square pyramid where all edges are of length 2, calculate the angle formed between a lateral edge and the base. Express your answer in degrees, including the degree symbol (°).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45°}
A pad of paper comes with 60 sheets. Evelyn uses a pad of paper writing notes at work for a certain period. She takes Monday and Friday off from work. She uses 12 sheets of paper per day at work. How often does she use a pad of paper?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $p$ be a prime with $p \equiv 3\pmod{4}$. Find the remainder of $$\sum_{i=1}^{p-1}\Big(\frac{i^8+1}{p}\Big)$$ divided by $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p-2}
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$, which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$, which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ such that $T_n = T_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a$, $b$, and $c$ form an arithmetic sequence, and $\sin A$, $\sin B$, $\sin C$ form a geometric sequence, then the angle $B=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
A total of $2010$ coins are distributed in $5$ boxes. Initially, the quantities of coins in the boxes are consecutive natural numbers. Martha should choose and take one of the boxes, but before that, she can perform the following transformation any number of times: from a box with at least $4$ coins, she can transfer one coin to each of the other boxes.

What is the maximum number of coins that Martha can take away?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2004}
From a circular log with radius \( R \), it is required to cut out a rectangular beam of maximum strength. It is known that the strength of the beam is directly proportional to the product of its width and the square of its height. What should the dimensions of the beam be to maximize its strength?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2R}{\sqrt{3}}, \frac{2R\sqrt{2}}{\sqrt{3}}}
Let $C\subseteq\mathbb{P}^{3}$ be the conic of equations $$ C=V(X_{3}, X_{0}X_{2}-X_{1}^{2})=\{(t_{0}^{2}:t_{0}t_{1}:t_{1}^{2}:0)\in\mathbb{P}^{3}:(t_{0}:t_{1})\in\mathbb{P}^{1}\}. $$ Determine the dimension of the set of surfaces $S\subseteq\mathbb{P}^{3}$ of degree $d$ (that we identify with $\mathbb{P}^{\binom{d+3}{3}-1}$) such that $C\subseteq S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{d+3}{3}-1-(2d+1)}
Let \( a, b, c \) be prime numbers such that \( a^5 \) divides \( b^2 - c \), and \( b + c \) is a perfect square. Find the minimum value of \( abc \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1958}
On graph paper, two right triangles are drawn. Find the sum of the angles BCA and \(\mathrm{B}_{1} \mathrm{C}_{1} \mathrm{~A}_{1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
# Task 8.2

For a natural number $N$, all its divisors were listed, and then the sum of digits for each of these divisors was calculated. It turned out that among these sums, all numbers from 1 to 9 were found. Find the smallest value of $\mathrm{N}$.

## Number of points 7
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
Each of the twelve letters in ``MATHEMATICS'' is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word ``CALM''? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{12}}
Masha came up with the number \( A \), and Pasha came up with the number \( B \). It turned out that \( A + B = 2020 \), and the fraction \( \frac{A}{B} \) is less than \( \frac{1}{4} \). What is the maximum value that the fraction \( \frac{A}{B} \) can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{403/1617}
A light flashes in one of three different colors: red, green, and blue. Every  $3$  seconds, it flashes green. Every  $5$  seconds, it flashes red. Every  $7$  seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after  $671$  seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{154}
设正整数$n \geq 3 , ( a _ { 1 } , a _ { 2 } , \cdots , a _ { n } )$是任意$n$元不同的实数，且其和是正数.若它的一个排列$( b _ { 1 } , b _ { 2 } , \cdots , b _ { n } )$满足对于任意的$k = 1 ,2 , \cdots , n ,$均有$b _ { 1 } + b _ { 2 } + \cdots + b _ { k } > 0$，则这个排列称为好的.求好的排列至少有多少个？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$( n - 1 ) !$}
For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \le 20$ and $r_1 + r_2 = \sqrt{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Tegwen has the same number of brothers as she has sisters. Each one of her brothers has 50% more sisters than brothers. Using algebraic relationships, calculate the total number of children in Tegwen's family. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $x_1$ and $x_2$ be positive real numbers and define, for $n>2$:$$\displaystyle x_{n+1}=\sum_{k=1}^{n}\sqrt[n]{x_k}$$ Evaluate $\displaystyle \lim_{n\to\infty}\frac{x_n-n}{\ln n}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Patrick buys season passes for each of his 4 children to visit the local amusement park as often as they want. The passes have different prices based on the children's ages: $100 for the oldest, $90 for the second oldest, $80 for the third oldest, and $70 for the youngest. The oldest child went to the park 35 times, the second oldest went 25 times, the third oldest went 20 times, and the youngest went 15 times. Calculate the overall average cost per trip to the park for all children combined.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$3.58}
S 是集合 \{1,2,\cdots,2023\} 的子集，满足任意两个元素的平方和不是 9 的倍数，则 |S| 的最大值是__________.（这里 |S| 表示 S 的元素个数）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1350}
Find the number of $7$-tuples $(n_1,\ldots,n_7)$ of integers such that \[\sum_{i=1}^7 n_i^6=96957.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2688}
Let $c$ and $d$ be real numbers such that
\[\frac{c}{d} + \frac{c}{d^3} + \frac{c}{d^6} + \dots = 9.\]
Find
\[\frac{c}{c + 2d} + \frac{c}{(c + 2d)^2} + \frac{c}{(c + 2d)^3} + \dotsb.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{11}}
10. There is a cup filled with a $16 \%$ saltwater solution. There are large, medium, and small iron balls, with their volume ratios being $10: 4: 3$. First, the small ball is submerged in the saltwater cup, causing $10 \%$ of the saltwater to overflow, and then the small ball is removed; next, the medium ball is submerged in the saltwater cup and then removed; then the large ball is submerged in the saltwater cup and removed; finally, pure water is added to the cup until it is full. What is the concentration of the saltwater in the cup at this point? (Round to one decimal place)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.7}
Trains A and B start simultaneously from stations A and B, respectively, and travel towards each other. The speed ratio of Train A to Train B is 3:2. Station C is located between stations A and B. Train A reaches station C at 9 AM and Train B reaches station C at 7 PM. Determine the time at which trains A and B meet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13:00}
Determine the largest natural number such that all its decimal digits are different and it is divisible by each of its digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9867312}
Xiaohua has a number of apples that he wants to distribute into several baskets. If he puts 4 apples in each basket, he will have 20 apples left undistributed. If he puts 8 apples in each basket, there will be one basket that does not have enough apples. Calculate the original number of apples Xiaohua had. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Solve the following quadratic equation and find the positive difference between its solutions:  
$$x^2 - 5x + m = 13 + (x+5)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{27 - m}}
9 people decided to split the restaurant bill evenly . if the bill was $ 514.16 dollars , how much money did they 1 cent is the smallest unit ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$57.13}
Two numbers \(a\) and \(b\) with \(0 \leq a \leq 1\) and \(0 \leq b \leq 1\) are chosen at random. The number \(c\) is defined by \(c=2a + 2b\). The numbers \(a\), \(b\), and \(c\) are each rounded to the nearest integer to give \(A\), \(B\), and \(C\), respectively. Find the probability that \(2A + 2B = C\). Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
B2. An integer $n$ is called a combi-number if every pair of different digits from all possible digits $0 \mathrm{t} / \mathrm{m} 9$ appear next to each other at least once in the number. Thus, in a combi-number, the digits 3 and 5 appear next to each other somewhere. It does not matter whether they appear in the order 35 or 53. We agree that a combi-number does not start with the digit 0.

What is the smallest number of digits a combi-number can consist of?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last $2$ marbles. What is the probability that Cheryl gets $2$ marbles of the same color? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In a right-angled triangle, the angle bisector of the right angle and the median corresponding to the hypotenuse form a $16^{\circ}$ angle with each other. What are the angles of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61^\circ, 29^\circ, 90^\circ}
What is the mass percentage of N in Dinitrogen pentoxide?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25.94\%}
Given that $a$, $b$, $c$ are the sides opposite to the angles $A$, $B$, $C$ respectively, with $a=2$, and $\frac{\sin A-\sin B}{\sin C}=\frac{c-b}{2+b}$, the maximum area of $\Delta ABC$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
In a class of 20 students, all but 4 of the students put their names on a typed assignment. If the teacher randomly guesses, what is the probability that she correctly guesses which paper belongs to each of the four remaining students? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{24}}
以正方体的顶点为顶点的锐角三角形有 个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Integer  $n>2$  is given. Find the biggest integer  $d$ , for which holds, that from any set  $S$  consisting of  $n$  integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by  $d$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n-1 }
Compare: $\tan \left( - \frac {13\pi}{7} \right)$ ______ $\tan \left( - \frac {15\pi}{8} \right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
The product of the two $99$-digit numbers
$303,030,303,...,030,303$ and $505,050,505,...,050,505$
has thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Square $ABCD$ and rectangle $BEFG$ are placed as shown in the figure. $AG = CE = 2$ cm. What is the difference in the area between square $ABCD$ and rectangle $BEFG$ in square centimeters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let  $\mathcal{P}$  be a parallelepiped with side lengths  $x$ ,  $y$ , and  $z$ .  Suppose that the four space diagonals of  $\mathcal{P}$  have lengths  $15$ ,  $17$ ,  $21$ , and  $23$ .  Compute  $x^2+y^2+z^2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{371}
A capricious mathematician writes a book with pages numbered from  $2$  to  $400$ . The pages are to be read in the following order. Take the last unread page ( $400$ ), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be  $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$ . What is the last page to be read?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{397}
Find the number of primes $p$ between $100$ and $200$ for which $x^{11}+y^{16}\equiv 2013\pmod p$ has a solution in integers $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
It is half past eight on the clock. What is the angle between the hour and minute hands?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}