inclusionAI/Ring-lite-rl-data huggingface dataset:
A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
An inscribed circle with radius \( R \) is placed in a certain angle, and the length of the chord connecting the tangent points is \( a \). Two tangents are drawn parallel to this chord, forming a trapezoid. Find the area of this trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8R^3}{a}}
Find the smallest integer value of $n$ such that when a fair coin is tossed $n$ times, the probability that the proportion of heads is between 0.49 and 0.51 is greater than 0.95.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9604}
Given the expression \( N=\frac{1987^{4(\sqrt{3}+2)^{1987}+1988}}{1987^{(\sqrt{3}+2)^{1988}}+(\sqrt{3}+2)^{1988}} \), calculate the last digit of \( N \). Express your answer as a single digit (0-9).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the expected value of the piecewise function \( u(x,y) \) given two independent continuous random variables \( x \) and \( y \) with uniform distribution in the range \([0,b]\). The joint PDF is \( f(x,y)=\frac{1}{b^2} \). The piecewise function is defined as:

\[
u(x,y) = \begin{cases} 
0 & \text{if } x,y < b/2 \\ 
b/2 & \text{if } (y<b/2 \text{ and } x > b/2) \text{ or } (x < b/2 \text{ and } y > b/2) \\ 
x & \text{if } y,x>b/2 \text{ and } y>x \\ 
y & \text{if } y,x>b/2 \text{ and } x>y
\end{cases}\]

Calculate \( E(u(x,y)) = \int_0^b\int_0^b \frac{u(x,y)}{b^2} \, dx \, dy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5b}{12}}
In a circle of 100 balls, 3 are red and 97 are blue. The balls are randomly placed in the circle. Determine the most probable minimum interval between the red balls.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the improper integral: $$\lim_{n \to \infty } \int_0^{\infty} x^{\frac{n}{n+1}} e^{-x^2} \, dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given that 2017 is a prime number, find the smallest positive integer \( n \) for which \( 2017^{{2017}^{2017}} \) divides \( n! \). Use the formula:

\[ v_{2017}(n!) = \frac{n - S_{2017}(n)}{2016} \]

where \( S_{p}(n) \) represents the sum of the standard base-\( p \) digits of \( n \). Determine this smallest value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016 × 2017^{2017} + 2017}
In a 10 by 10 grid of dots, what is the maximum number of lines that can be drawn connecting two dots on the grid so that no two lines are parallel?\n\n[i]Proposed by Aaron Lin[\/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Suppose $f(x,y)$ is differentiable at $(1,1)$, and $f(1,1)=\frac{\partial f}{\partial x}(1,1)=\frac{\partial f}{\partial y}(1,1)$. Find the value of $\frac{d}{dx}f(x,f(x,x))$ at $x=1$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a certain code, 'range' is coded as 12345 and 'random' is coded as a sequence of numbers. The code for the word 'rand' is 1236. What is the code for the word 'random'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123678,}
Given a rectangle \(ABCD\). A line through point \(A\) intersects segment \(CD\) at point \(X\) such that the areas of the resulting shapes satisfy \(S_{AXD}: S_{ABCX} = 1:2\). A line through point \(X\) intersects segment \(AB\) at point \(Y\) such that \(S_{AXY}: S_{YBCX} = 1:2\). Finally, a line through point \(Y\) intersects segment \(XC\) at point \(Z\) such that \(S_{XYZ}: S_{YBCZ} = 1:2\).

Calculate the ratio of the areas \(S_{AXD}: S_{AXZY}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{10}}
If  $x > 10$ , what is the greatest possible value of the expression
\[
  {( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ? 
\]
All the logarithms are base 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is  $5:9$ . Compute the ratio of the secant line versus circle radius.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{10}}{5}}
Diagonals connecting the second and third neighboring vertices of a regular octagon determine two convex octagons. What is the ratio of their areas?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2+\sqrt{2} : 1}
Let $f(x)$ be a concave function from $[0,1]$ to the reals such that $f(0) = f(1) = 0$. Consider the approximation of the integral $\int_0^1 f(x) \, dx$ by dividing $[0,1]$ into $n$ sub-intervals of equal length and using the maximum value of $f(x)$ on each sub-interval to form rectangles. Determine the maximum relative error of this approximation as a function of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{n}}
Given that  $a,b,c$  are integers with  $abc = 60$ , and that complex number  $\omega \neq 1$  satisfies  $\omega^3=1$ , find the minimum possible value of  $|a + b\omega + c\omega^2|$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
现有2个红球，3个黑球，5个白球（同色球完全相同），从中任取6个球，有多少种不同的取法？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
Three spheres with radii $\sqrt{6}, 1$, and 1 touch each other pairwise. A plane $\gamma$ is passed through the line containing the centers $A$ and $B$ of the second and third spheres, such that the center $O$ of the first sphere is at a distance of 1 from this plane. Find the angle between the projections of the lines $OA$ and $OB$ on the plane $\gamma$ and compare it to $\arccos \frac{4}{5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos \frac{\sqrt{6}}{3}}
Suppose $n,k\to\infty$ such that $k/n^2\to a>0$. What is the limit of $\dfrac{{{k-1}\choose n}}{{{n+k-1}\choose n}}$ using the correct asymptotic analysis?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\exp(-1/a)}
The door of a rabbit hutch is made from a wooden frame and wire mesh with square openings. The wooden slats of the frame are $5 \mathrm{~cm}$ wide. Some nodal points of the mesh align with the inner edges of the frame as shown in the diagram. The inner (mesh) part of the door has an area of $432 \mathrm{~cm}^{2}$.

Determine the external dimensions (i.e., width and height) of the entire door.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28 \text{ cm} \times 34 \text{ cm}}
Given the identity $S(n, n - 3) = a \binom n 4 + b \binom n 5 + c \binom n 6$, find $a, b, c$. $S(n, k)$ denotes a Stirling number of the second kind, i.e., the number of ways to place $n$ labeled balls into $k$ non-labeled non-empty boxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 10, 15}
Starting from the number $123456789$, at each step, we are swaping two adjacent numbers which are different from zero, and then decreasing the two numbers by $1$. What is the sum of digits of the least number that can be get after finite steps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
Determine the sum of all values of the parameter \( a \) for which the equation \( f(x) = p(x) \), where \( f(x) = \left|\frac{2 x^{3}-x^{2}-18 x+9}{(1.5 x+1)^{2}-(0.5 x-2)^{2}}\right| \) and \( p(x) = |-2 x+2| + a \), has exactly one solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMT? Here are examples of possible sequences:
  [center]
        C,CA,CAM,CCAM,CCAMT.
    [/center]
   [center]
        A,AT,CAT,CAMT,CCAMT.
  [/center]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Given matrices $A$ and $B$, compute the gradient $\nabla_X g$ of the scalar field $g(X) = \operatorname{tr}(AXB)$. Express your answer in terms of $A$ and $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^\top B^\top}
In an acute triangle  $ABC$ , the segment  $CD$  is an altitude and  $H$  is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle  $DHB$ , determine all possible values of  $\angle CAB$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
A hemisphere with a radius of $10 \mathrm{~cm}$ protrudes from a horizontal plane (to the annoyance of drivers). A cylinder rolls on the plane and collides with the hemisphere with its side. Let $\alpha$ be the angle between the common tangent plane at the point of contact of the two bodies and the horizontal plane. What is the minimum radius of the cylinder if $\alpha \leq 30^{\circ}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10(3+2\sqrt{3})\approx64.64}
24. S2 (POL) $)^{1 \mathrm{MO}}$ The positive real numbers $x_{0}, x_{1}, \ldots, x_{1996}$ satisfy $x_{0}=$ $x_{1995}$ and
$$
x_{i-1}+\frac{2}{x_{i-1}}=2 x_{i}+\frac{1}{x_{i}}
$$
for $i=1,2, \ldots, 1995$. Find the maximum value that $x_{0}$ can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{997}}
Points \( M \) and \( N \) divide side \( AC \) of triangle \( ABC \) into three equal parts, each of which is 5, with \( AB \perp BM \) and \( BC \perp BN \). Find the area of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{75 \sqrt{3}}{4}}
How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the norm of the projection operator \( P: c \to c_0 \) defined by \( P(x_0,x_1,x_2,\ldots) = (x_0-\lim\limits_{n\to\infty}x_n,x_1-\lim\limits_{n\to\infty}x_n,x_2-\lim\limits_{n\to\infty}x_n,\ldots) \) in the space of convergent sequences with respect to the supremum norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a=\sqrt{0.5}$, $b=\sqrt{0.3}$, and $c=\log_{0.3}{0.2}$. Determine the order of $a$, $b$, and $c$, expressing your answer as an inequality using the variables $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b<a<c}
What is the molecular weight of 7 moles of KBrO3
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1169.00}
Given a sum of a sequence and the number of iterations \( T \), determine the base number \( x \) such that the sequence \( x^2 \) repeated \( T \) times and summed results in the given sum. For example, if the sum is 126 and \( T = 6 \), find the value of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{21}}
We call a positive integer  "special" if the number is divided by all of its digits (except the $0$s). At most how many consequtive special numbers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
12
}
The number of games won by six volleyball teams are displayed in a graph, but the names of the teams are missing. The following clues provide information about the teams:

1. The Falcons won more games than the Hawks.
2. The Warriors won more games than the Knights but fewer than the Royals.
3. The Knights won more than 25 games.

How many games did the Warriors win? The wins recorded are 20, 26, 30, 35, 40, and 45.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
我从1,2，···，144中任取一个数，你每次取一个｛1,2，···，144｝的子集，问我取的数是否在该子集中，最终确定我取的数.如果每次回答，答案是Yes，则你需付出2元，答案是No，则你需付出1元．为保证能找出我取的数，你需要付出的钱的最小值为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine the value of the parameter $a$ such that the function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = \begin{cases} \sin\frac{1}{x} & \text{for } x \in \mathbb{R} \setminus \{0\} \\ a & \text{for } x = 0 \end{cases} \) has an antiderivative on \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The numbers from 1 to 10 were written in some order, resulting in the numbers \( a_{1}, a_{2}, a_{3}, \ldots, a_{10} \). Then the sums \( S_{1} = a_{1} \), \( S_{2} = a_{1} + a_{2} \), \( S_{3} = a_{1} + a_{2} + a_{3} \), \(\ldots\), \( S_{10} = a_{1} + a_{2} + a_{3} + \ldots + a_{10} \) were calculated. What is the maximum number of prime numbers that could appear among the numbers \( S_{1}, S_{2}, S_{3}, \ldots, S_{10} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the interval of uniform convergence for the series \(\sum_{n=1}^{\infty} \left(1+\frac{1}{n}\right)^{np} e^{-nx}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
For any integer $n$, define $\lfloor n\rfloor$ as the greatest integer less than or equal to $n$. For any positive integer $n$, let $$f(n)=\lfloor n\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n}{3}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor.$$ For how many values of $n, 1 \leq n \leq 100$, is $f(n)$ odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Fix an odd integer  $n > 1$ . For a permutation  $p$  of the set  $\{1,2,...,n\}$ , let S be the number of pairs of indices  $(i, j)$ ,  $1 \le i \le  j \le  n$ , for which  $p_i +p_{i+1} +...+p_j$  is divisible by  $n$ . Determine the maximum possible value of  $S$ .

Croatia
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(n+1)(n+3)}{8}}
Mom gave Vasya money for 30 pencils. It turned out that the pencil factory is running a promotional campaign in the store: in exchange for a receipt for the purchase of a set of 20 pencils, they return 25% of the cost of the set, and in exchange for a receipt for the purchase of a set of 5 pencils, they return 10%. What is the maximum number of pencils Vasya can buy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
The product 
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positive integers. Find $s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \iint_D \sin(x^2+y^2)~dA \) by transforming it to polar coordinates, where \( D \) is the region in the first quadrant bounded by the lines \( x=0 \) and \( y=\sqrt{3}\cdot x \) and the circles \( x^2+y^2=2\pi \) and \( x^2+y^2=3\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
In a directed graph with  $2013$  vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach  $k$  vertices using at most two edges. Find the maximum value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012}
At a recent math contest, Evan was asked to find  $2^{2016} \pmod{p}$  for a given prime number  $p$  with  $100 < p < 500$ . Evan has forgotten what the prime  $p$  was, but still remembers how he solved it:


- Evan first tried taking  $2016$  modulo  $p - 1$ , but got a value  $e$  larger than  $100$ .
- However, Evan noted that  $e - \frac{1}{2}(p - 1) = 21$ , and then realized the answer was  $-2^{21} \pmod{p}$ .

What was the prime  $p$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{211}
Given that $P$: $a < x < a+1$, and $q$: $x^{2}-7x+10\leqslant 0$, if $P$ is a necessary but not sufficient condition for $q$, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\leqslant a\leqslant 4}
Given the equation \(x^4 + 3\cos(ax^2 + bx +c) = 2(x^2-2)\) with parameters \(a, b, c \in (2,5)\), it has two solutions. Determine the maximum value of \(\frac{ac}{b^2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find all integer values of $n$ such that $n-2$ divides $n^{2} + 3n + 27$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 3, 39, -35}
设 $x, y, z$ 是三个两两不等且都大于 1 的正整数, 若 $x y z$ 整除 $(x y-1) \cdot (x z-1)(y z-1)$, 求 $x$ 所有可能值之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
New this year at HMNT: the exciting game of $R N G$ baseball! In RNG baseball, a team of infinitely many people play on a square field, with a base at each vertex; in particular, one of the bases is called the home base. Every turn, a new player stands at home base and chooses a number $n$ uniformly at random from \{0,1,2,3,4\}. Then, the following occurs: - If $n>0$, then the player and everyone else currently on the field moves (counterclockwise) around the square by $n$ bases. However, if in doing so a player returns to or moves past the home base, he/she leaves the field immediately and the team scores one point. - If $n=0$ (a strikeout), then the game ends immediately; the team does not score any more points. What is the expected number of points that a given team will score in this game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{409}{125}}
A roulette wheel can land on any number from 0 to 2007 with equal probability. The roulette wheel is spun repeatedly. Let \( P_{k} \) denote the probability that at some point the sum of the numbers that have come up in all the spins made equals \( k \). Determine which is greater: \( P_{2007} \) or \( P_{2008} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P_{2007}}
Perimeter of triangle  $ABC$  is  $1$ . Circle  $\omega$  touches side  $BC$ , continuation of side  $AB$  at  $P$  and continuation of side  $AC$  in  $Q$ . Line through midpoints  $AB$  and  $AC$  intersects circumcircle of  $APQ$  at  $X$  and  $Y$ .
Find length of  $XY$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
One side of an equilateral triangle of height $24$ lies on line $\ell.$ A circle of radius $12$ is tangent to $\ell$ and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line $\ell$ can be written as $a\sqrt{b} - c\pi,$ where $a,$ $b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a+b+c\,?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
75
}
How many ordered triples of nonzero integers  $(a, b, c)$  satisfy  $2abc = a + b + c + 4$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A hotel has three types of rooms: a triple room, a double room, and a single room, each type having one room available. Three adults and two children come to stay, and the children cannot stay in a room by themselves (they must be accompanied by an adult). Find the total number of different ways to arrange their accommodation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
As shown in the figure, a geometric shape is formed by stacking 26 identical small cubes. Its bottom layer is composed of $5 \times 4$ small cubes. If its entire outer surface (including the bottom) is painted red, then after this geometric shape is disassembled, the number of small cubes with 3 red faces is $\qquad$ cubes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $R$ be a non-commutative ring and consider a function $f: R \rightarrow R$ such that $f(ab) = f(a)f(b) + f(a)b + af(b)$ for all $a, b \in R$. Determine the expression for $f(x^n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(f(x) + x)^n - x^n}
Convex quadrilateral $A B C D$ has right angles $\angle A$ and $\angle C$ and is such that $A B=B C$ and $A D=C D$. The diagonals $A C$ and $B D$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $A M B$ and segment $C D$, respectively, such that points $P, M$, and $Q$ are collinear. Suppose that $m \angle A B C=160^{\circ}$ and $m \angle Q M C=40^{\circ}$. Find $M P \cdot M Q$, given that $M C=6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Segment $AB$ has midpoint $C$, and segment $BC$ has midpoint $D$. Semi-circles are constructed with diameters $\overline{AB}$ and $\overline{BC}$ to form the entire region shown. Segment $CP$ splits the region into two sections of equal area. What is the degree measure of angle $ACP$? Express your answer as a decimal to the nearest tenth.

[asy]
draw((0,0)--10dir(180),linewidth(2));
draw((0,0)--10dir(67.5),linewidth(2));
draw((0,0)--10dir(0),dashed);

draw(10dir(180)..10dir(90)..10dir(0),linewidth(2));

draw((5,0)+5dir(180)..(5,0)+5dir(-90)..(5,0)+5dir(0),linewidth(2));

dot((0,0));
dot((5,0));

label("A",10dir(180),W);
label("B",10dir(0),E);
label("C",(0,0),SW);
label("D",5dir(0),NE);
label("P",10dir(67.5),NE);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112.5}
Let  $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$  and  $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and  }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in  $S$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
A secret facility is a rectangle measuring $200 \times 300$ meters. There is one guard at each of the four corners of the facility. An intruder approaches the perimeter of the facility from the outside, and all the guards run towards the intruder by the shortest routes along the outer perimeter (while the intruder remains stationary). Three guards cover a total distance of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
How can we find all the numbers $m$ such that $\sigma(m)=24$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14, 15, 23}