inclusionAI/Ring-lite-rl-data huggingface dataset:
In how many ways can you transform the number 0 into the number 100 in exactly 4 seconds, if each second you can add or subtract any factorial number (e.g., 1!, 2!, 3!, etc.) to your current number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
If the mean of some numbers including x, 50, 78, and 104 is 62, and the mean of 48, 62, 98, 124 and x is 76.4, what is the missing number in the first set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Let \( f: D_{18} \to GL(2, \mathbb{R}) \) be a homomorphism where \( |r| = 18 \) and \( f(r) = \begin{pmatrix} 1 & 1 \\ -1 & 0 \end{pmatrix} \). Determine the order of the kernel of \( f \), denoted as \( |\ker(f)| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the flux of the vector field \( \mathbf{F}(x,y,z) = \mathbf{i} + \mathbf{j} + \mathbf{k} \) across the surface \( \sigma \), which is the portion of the cone \( z = \sqrt{x^2 + y^2} \) between the planes \( z = 3 \) and \( z = 6 \), oriented by downward unit normals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-27\pi}
Determine the number of epimorphisms from the free group with 2 generators, $F_2$, to the cyclic group $\mathbb{Z}_5$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
For which values of $n$ is the graph $G_n$ with vertex set $V_n = \{1, 2, ..., n\}$, where two distinct vertices $x$ and $y$ are adjacent if $x$ is a divisor of $y$ or $y$ is a divisor of $x$, a planar graph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \leq 15}
Find the area of the region in the first quadrant bounded by the parabolas $y=x^2$, $y=2x^2$, $x=y^2$, and $x=2y^2$ using a change of coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Two circles are given: $\Gamma_1$ with radius $4$ and center $(0,0)$, and $\Gamma_2$ with radius $14$ and center $(25,0)$. Let $A$ and $B$ be variable points on $\Gamma_1$ and $\Gamma_2$, respectively, and $M$ be the midpoint of $\overline{AB}$. Find the area of the region $S$ formed by all possible locations of $M$. \( \text{Express your answer in square units.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56\pi}
If in the sequence $\{a_n\}$, the sum of any three consecutive terms is positive, and the sum of any four consecutive terms is negative, then the maximum value of the term number $n$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the values of \( \alpha \in \mathbb{R} \) for which the function \( f(x) = \mathbb{1}_{(0,1)}(x)x^\alpha \) belongs to the space \( L^p(\mathbb{R}) \), where \( p \in [1, \infty) \) and \( \lambda \) is the Lebesgue measure on \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( -\frac{1}{p}, \infty \right)}
If the inequality \(4x^2 + y^2 + 1 \geq kx(y+1)\) holds for any integers \(x\) and \(y\), find the maximum constant \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the radius of convergence of the power series \( f(x) = \sum_{k=1}^{\infty} \frac{x^{2k}}{(4+(-1)^k)^{3k}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
Consider the polyhedron P defined by the following inequalities:

$$\begin{align} x_1 + x_2 &\leq 5 \\ x_1 - x_2 &\geq 1 \\ x_2 &\leq 3 \\ x_1 &\geq 0 \\ x_2 &\geq 0 \end{align}$$

Determine the face F induced by the equations $x_1 + x_2 = 5$ and $x_1 - x_2 = 1$. Find an inequality $c x \leq \gamma$ that is valid for P and such that $F = \{x \in P \mid c x = \gamma\}$. Provide the inequality $c x \leq \gamma$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_2 \leq 2}
Write down an irrational number that is greater than $\sqrt{3}$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Find a value of $\epsilon$ such that for any $N$, there exists an $n > N$ such that the sequence $$a_{n} = \dfrac{7n^{3} - 3n^{4} -1}{4n^{2} + 3}$$ satisfies $|a_{n} + \frac{3n}{4}| \geq \epsilon$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Divide 6 volunteers into 4 groups, with two groups having 2 people each and the other two groups having 1 person each, to serve at four different pavilions of the World Expo. How many different allocation schemes are there? (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1080}
In rectangle $JKLM$, $P$ is a point on $LM$ so that $\angle JPL=90^{\circ}$. $UV$ is perpendicular to $LM$ with $LU=UP$, as shown. $PL$ intersects $UV$ at $Q$. Point $R$ is on $LM$ such that $RJ$ passes through $Q$. In $\triangle PQL$, $PL=25$, $LQ=20$ and $QP=15$. Find $VD$. [asy]
size(7cm);defaultpen(fontsize(9));
real vd = 7/9 * 12;
path extend(pair a, pair b) {return a--(10 * (b - a));}

// Rectangle
pair j = (0, 0); pair l = (0, 16); pair m = (24 + vd, 0); pair k = (m.x, l.y);
draw(j--l--k--m--cycle);
label("$J$", j, SW);label("$L$", l, NW);label("$K$", k, NE);label("$M$", m, SE);

// Extra points and lines
pair q = (24, 7); pair v = (q.x, 0); pair u = (q.x, l.y);
pair r = IP(k--m, extend(j, q));
pair p = (12, l.y);
draw(q--j--p--m--r--cycle);draw(u--v);
label("$R$", r, E); label("$P$", p, N);label("$Q$", q, 1.2 * NE + 0.2 * N);label("$V$", v, S); label("$U$", u, N);

// Right angles and tick marks
markscalefactor = 0.1;
draw(rightanglemark(j, l, p)); draw(rightanglemark(p, u, v)); draw(rightanglemark(q, v, m));draw(rightanglemark(j, p, q));
add(pathticks(l--p, 2, spacing=3.4, s=10));add(pathticks(p--u, 2, spacing=3.5, s=10));

// Number labels
label("$16$", midpoint(j--l), W); label("$25$", midpoint(j--p), NW); label("$15$", midpoint(p--q), NE);
label("$20$", midpoint(j--q), 0.8 * S + E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{28}{3}}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}+b^{2}+4 \sqrt {2}=c^{2}$ and $ab=4$, find the minimum value of $\frac {\sin C}{\tan ^{2}A\cdot \sin 2B}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {3 \sqrt {2}}{2}+2}
A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere. The base of the prism is \( A B C \), and the lateral edges are \( A A_{1}, B B_{1}, C C_{1} \). Segment \( C D \) is the diameter of the sphere. Points \( K \) and \( L \) are the midpoints of edges \( A A_{1} \) and \( A B \) respectively. If \( D L = \sqrt{2} \) and \( D K = \sqrt{3} \), find the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For any positive integer, if the number of $2$'s in its digits is greater than the number of $3$'s in its digits, we call that a **good** number. Conversely, if the number of $3$'s in its digits is greater than the number of $2$'s, we call that a **bad** number. For example, the number $2023$ has two $2$'s and one $3$, so $2023$ is a good number. However, in the number $123$, both the number of $2$'s and $3$'s are one, so $123$ is neither a good number nor a bad number.  

Find the difference between the number of good numbers and bad numbers among the positive integers not greater than $2023$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Evaluate the double integral: $$\int_{0}^{1}\int_{0}^{1}e^{\max(x^{2},y^{2})}dxdy$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 1}
Let $B_t$ be a standard Brownian motion, and let $L(x, t)$ denote the local time at $x$ for time $t$. Define $L_t = L(0, t)$. Determine whether the following holds with probability one: $$\limsup_{t \to 0} \frac{L_t}{\sqrt{t}} = \infty.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{902}
Given the function $f(x) = 3^{x} + \frac{1}{2}x - 2$, identify the interval where the zero point of the function lies. Express your answer in the form (a, b).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1)}
On the radius \( AO \) of a circle with center \( O \), a point \( M \) is chosen. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 16 and \( \sin \alpha = \frac{\sqrt{55}}{8} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The circle touches the extensions of two sides $AB$ and $AD$ of the square $ABCD$ with a side length of $\sqrt{2+\sqrt{2}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is $45^\circ$, and it is known that $\sin 22.5^\circ = \frac{\sqrt{2 - \sqrt{2}}}{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2} + \sqrt{2 - \sqrt{2}}}
Determine the asymptotic behavior of the ratio \( \frac{{2n \choose k}}{{n \choose k}} \) as \( n \to \infty \), given that \( k = O(f(n)) \) for some function \( f(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}
Given an ellipse and a hyperbola,\[\frac{x^2}{16} - \frac{y^2}{25} = 1\]and \[\frac{x^2}{K} + \frac{y^2}{25} = 1\], have the same asymptotes. Find the value of $K$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Calculate the rate of heat flow inward across the cylindrical surface defined by \(y^2 + z^2 = 324\) for \(-10 \le x \le 3\), given the temperature distribution \(u(x,y,z) = 4y^2 + 4z^2\) and the conductivity constant \(K = 5\). Use the formula \(\iint_S F \cdot dS = -K \iint_S \nabla u \cdot dS\) to find the rate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336960\pi}
Let $a, b$, and $c$ be real numbers such that $3^a = 125$,  $5^b = 49$,and $7^c = 8$1.
Find the product $abc$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
棱长为 $\sqrt{2}$ 的正四面体 $A B C D$ 中, 已知 $\overrightarrow{A P}=\frac{1}{2} \overrightarrow{A B}, \overrightarrow{A Q}=\frac{1}{3} \overrightarrow{A C}, \overrightarrow{A R}=\frac{1}{4} \overrightarrow{A D}$, 若点 $K$ 为 $\triangle B C D$ 的重心, 则四面体 $K P Q R$ 的体积为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
设函数 $f_{0}(x)=|x|, f_{1}(x)=\left|f_{0}(x)-1\right|, f_{2}(x)=\left|f_{1}(x)-2\right|$ ，则函数 $f_{2}(x)$ 的图像与 $x$ 轴所围成图形中的封闭部分的面积是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Harry is joining a marathon and diversifies his workouts with different activities, periods of rest, and varying terrain. On Monday, he ran 10 meters per hour on a flat terrain. On Tuesday, he spent half the day strength training, which made him run 30% slower on a slightly inclined terrain. On Wednesday, after a whole day rest on Tuesday, he ran 50% faster than he did on Monday on a hilly terrain, causing him to exert 10% more effort. On Thursday, he ran on a mixed terrain at the same speed as adjusted for Wednesday despite a downhill stretch reducing his effort by 5%. Then, after an intensive yoga session on Friday morning, which improved his stamina, Harry ran 60% faster than he ran on Thursday on a continuous uphill incline, decreasing his running efficiency by 15%. How fast did Harry run on Friday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
The world-renowned Marxist theorist *Joric* is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer  $n$ , he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the *defect* of the number  $n$ . Determine the average value of the defect (over all positive integers), that is, if we denote by  $\delta(n)$  the defect of  $n$ , compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\]
*Iurie Boreico*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Mr. Thompson, a traditionalist, enjoys spending his afternoons reading classic books in his quiet living room. He has a collection of 72 books on his wooden bookshelf. Every month, he decides to read 3 new books that he carefully selects from his local library, avoiding any digital versions. However, he also gives away 1 book each month to his neighbor, who shares his love for timeless stories. If Mr. Thompson continues this pattern for 6 months, how many books will he have on his shelf at the end of that period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
A barrel of water is filled to a maximum height of 90 cm. After it stops raining, the water leaks out at a rate proportional to the square root of the water depth. If the water level drops from 90 cm to 85 cm in one hour, how long will it take for the barrel to become empty?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18 + 3\sqrt{34}}
$64$ teams with distinct skill levels participate in a knockout tournament. In each of the $6$ rounds, teams are paired into match-ups and compete; the winning team moves on to the next round and the losing team is eliminated. After the second-to-last round, winners compete for first and second and losers compete for third and fourth. Assume that the team with higher skill level always wins. What is the probability that the first, second, and third place teams have the highest, second highest, and third highest skill levels, respectively?

[i]2019 CCA Math Bonanza Lightning Round #3.3[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{512}{1953}}
How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
In a certain brick wall, each row of bricks above the bottom row contains one less brick than the row just below it. There are some rows in all and a total of 100 bricks in the wall. The bottom row contains 18 bricks. How many rows are there in the wall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
设 $w=\cos \frac{2 \pi}{5}+\mathrm{i} \sin \frac{2 \pi}{5}, P(x)=x^{2}+x+2$, 求 $P(w) P\left(w^{2}\right) P\left(w^{3}\right) P\left(w^{4}\right)$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the degree of the field extension \(\mathbb{Q}[a]\) over \(\mathbb{Q}[\sqrt[3]{2}]\), where \(a=\sqrt{ \sqrt[3]{2}-1}\) and \(a\) is a root of the polynomial \(X^6 + 3X^4+3X^2-1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $\{a_n\}$ be a positive sequence such that $a_{n+1} = a_n + \frac{n^4}{a_n}$. Determine the rate of growth of $a_n$. Provide your answer in terms of asymptotic notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n^{5/2})}
Determine the value of \(a\) such that the equation \(f(x) = a\) has a solution for \(x \in (0,1]\), where \(f: (0,\infty) \to \mathbb{R}\) is defined by \(f(x) = x^n - n\ln x + 1\) and \(n \neq 0\), \(n \in \mathbb{N}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the sum of all fractions with a denominator of 2014, where each fraction $\frac{1}{2014}, \frac{2}{2014}, \cdots, \frac{2012}{2014}, \frac{2013}{2014}$ is reduced to its simplest form?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{468}
In a 3x3x3 tic-tac-toe game with three players A, B, and C taking turns in order, how many unique board states are possible after 6 moves? Assume that each player places their mark in an empty cell during their turn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26640900}
What time is it 2017 minutes after 20:17?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{05:54}
Determine the number of matrices similar to \( A = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\ \end{pmatrix} \) with entries from \( \mathbb{Z}/3\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{117}
Given real numbers $\alpha$ and $\beta$ where $1 < \alpha < \beta$, find the largest positive integer $r$ such that any coloring of the positive integers with $r$ colors ensures there exist two positive integers $x$ and $y$ of the same color satisfying $\alpha \leqslant \frac{x}{y} \leqslant \beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lceil \log_\alpha \beta \right\rceil}
Given that for a function \( g(t) \), \( \int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2} \) for any real value \( \omega \), and \( y(t)=\int_{-\infty}^{t}g(\tau)d\tau \), calculate the value of \( \int_{-\infty}^{\infty} y(t)dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-j}
Given the presence of hereditary traits determined by paired genes $A$ and $a$ in peas, where $A$ is dominant and $a$ is recessive, and the offspring $Aa$ is yellow, determine the probability that the offspring will produce yellow peas after self-crossing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{3}{4}"]}
Given that \( \int \frac{f(x)}{x^2(x+1)^3}\,dx \) is a rational function and \( f(x) \) is a quadratic function with \( f(0) = 1 \), find \( f'(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is chosen, and on one side of \(AO\) on the circle, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\) if the radius of the circle is 8, and \(\cos \alpha = \frac{3}{4}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the value of: ${9}^{\frac{1}{2}\log_{3}10}+(\frac{1}{81})^{\frac{1}{4}}-0.027^{-\frac{1}{3}}+\log_{6}2+\frac{1}{1+\log_{3}2}=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the number of solutions to the equation \(x_1 + x_2 + x_3 + \ldots + x_9 = 42\) where each variable satisfies \(0 \le x_n \le 9\) for \(1 \le n \le 9\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44865975}
Task B-4.8. A circle is described around the focus of the parabola $y^{2}=16 x$ which touches the directrix of the parabola. At what angle do these two curves intersect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
The natural numbers \(a, x\) and \(y\), each greater than 100, satisfy the equation \(y^{2} - 1 = a^{2}(x^{2} - 1)\). What is the smallest value that the fraction \(\frac{\alpha}{x}\) can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the functions \( f(x) \) and \( g(x) \) as defined in Theorem 1, with \( T_1 = 1 \) and \( T_2 = \frac{1}{m} \) where \( m \in \mathbb{N} \) and \( m > 1 \), determine the smallest positive period of the function \( h(x) = f(x) + g(x) \). The period is \( \frac{1}{k} \), where \( k = 1 \) or \( k \in \mathbb{N} \), \( k \) is not a multiple of \( m \), and \( m \) is not a multiple of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimum number of coin flips, $n$, required to ensure that the estimated probability $\hat{p} = X/n$ is within 5% of the true probability $p$, given that $0.4 \leq p \leq 0.6$, with a confidence level of at least 0.95. That is, find $n$ such that \( \mathbb{P}[0.95p \leq \hat{p} \leq 1.05p] \geq 0.95 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2305}
In the Intense Goblet competition, 10 teams compete in a "round robin" format: that is, each team plays every other team exactly once. A team gets 4 points for a win, 2 points for a tie (a draw), and 1 point for a loss. What is the difference between the maximum total points and the minimum total points that can be gained by all teams (added together) in the Intense Goblet competition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Consider distinct natural numbers $a, b, c \neq 0$. Express them in the form $3^a + 3^b + 3^c$. When these numbers are ordered in increasing order, what is the sum $a + b + c$ for the 101st number in this sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
[ Combinatorics (miscellaneous) $]$  $[$ Estimation + example ]

In a pond, 30 pikes were released, which gradually eat each other. A pike is considered full if it has eaten at least three pikes (full or hungry). What is the maximum number of pikes that can become full?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given sequences \( a_n \) and \( b_n \) of positive numbers such that \( \lim_{n\to\infty}a_n^n = a \) and \( \lim_{n\to\infty}b_n^n = b \), where \( a, b \in (0, \infty) \), and constants \( p \) and \( q \) such that \( p + q = 1 \), find the limit:
\[ \lim_{n\to\infty}(pa_n + qb_n)^n. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^p b^q}
Given the average highest monthly temperatures (in degrees Celsius) in our city in 2022 were $12$, $11$, $10$, $20$, $23$, $28$, $36$, $36$, $31$, $24$, $23$, $19$, find the 60th percentile of this data set. Express your answer as a single number in degrees Celsius.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A right rectangular prism $P_{}$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P_{}$ cuts $P_{}$ into two prisms, one of which is similar to $P_{},$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{040}
Assume that the failure rate of each airplane engine during flight is $1 - P$, and the events of each engine failing are independent. The airplane can successfully fly if at least $50\%$ of the engines are running normally. What is the range for the value of $P$ to ensure that a four-engine airplane is safer than a two-engine airplane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left(\frac{2}{3}, 1\right)$}
Given a function $y=f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+2)=f(x)$, and when $x \in [-1, 1]$, $f(x) = |x|$. The function $y=g(x)$ is defined on $\mathbb{R}$ as an odd function, and when $x \in (0, +\infty)$, $g(x) = \log_{3}x$. Find the number of intersection points between the graphs of $y=f(x)$ and $y=g(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}