inclusionAI/Ring-lite-rl-data huggingface dataset:
设等比数列 $\left\{a_{n}\right\}$ 的前 $n$ 项和为 $S_{n}, S_{n}=2^{n}+r$ ( $r$ 为常数),记 $b_{n}=2\left(1+\log _{2} a_{n}\right)\left(n \in \mathbf{N}^{*}\right)$.若对于任意的正整数 $n$, 都有 $\frac{1+b_{1}}{b_{1}} \cdot \frac{1+b_{2}}{b_{2}} \cdots \frac{1+b_{n}}{b_{n}} \geqslant k \sqrt{n+1}$ 成立, 求实数 $k$ 的最大值的$2\sqrt{2}$倍是多少.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $( \overrightarrow {a}-2 \overrightarrow {b})⊥(3 \overrightarrow {a}+ \overrightarrow {b})$, and $| \overrightarrow {a}|= \frac {1}{2}| \overrightarrow {b}|$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {2\pi}{3}}
Four people were guessing the number, $N$, of jellybeans in a jar. No two guesses were equally close to $N$. The closest guess was 80 jellybeans, the next closest guess was 60 jellybeans, followed by 49 jellybeans, and the furthest guess was 125 jellybeans. Find the sum of all possible values for $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1256}
In a speech contest held in a class, there are a total of $5$ contestants, including $3$ females and $2$ males. If the two males cannot appear consecutively, and female A cannot be the first, what is the number of possible sequences for their appearance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
In triangle $DEF$, $DE = 8$, $EF = 6$, and $FD = 10$.

[asy]
defaultpen(1);

pair D=(0,0), E=(0,6), F=(8,0);
draw(D--E--F--cycle);

label("\(D\)",D,SW);
label("\(E\)",E,N);
label("\(F\)",F,SE);

[/asy]

Point $Q$ is arbitrarily placed inside triangle $DEF$. What is the probability that $Q$ lies closer to $D$ than to either $E$ or $F$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
A polynomial $P$ of degree $2015$ satisfies the equation $P(n) = \frac{1}{n^2}$ for $n = 1, 2, \dots, 2016$. Find $\lfloor 2017P(2017) \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, for all $n\in N^{*}$, $S_{n}\geqslant S_{3}$. Find the range of $\frac{{{a_6}}}{{{a_5}}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[\frac{3}{2}, 2\right]}
How to compute the sum of \begin{align*} \sum\limits_{k=0}^n \frac{(-1)^k}{k! (2k+1)} \frac{1}{(n-k)!} \end{align*}?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2^{2n} n!}{(2n+1)!}}
A train of length 380 m moving at a speed of 72 kmph takes 91.9926405887529 seconds to cross another train of some length, moving at 36 kmph in the same direction. What is the length of the second train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{539.93}
If the oblique section diagram of a horizontally placed figure is an isosceles trapezoid with a base angle of $45^{\circ}$, legs and upper base all measuring $1$, then the area of the original plane figure is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \sqrt{2}}
For $a \in \mathbb R,a≠-1$,
$$\lim_{n\to\infty}\frac{1^a+2^a+\cdots +n^a}{(n+1)^{a-1}[(na+1)+(na+2)+(na+3)+\cdots+(na+n)]}=\frac{1}{60}$$
Find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the geometric sequence {a_n}, it is known that a_1 = 1/5 and a_3 = 5. Find the value of a_2. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
Given a sphere with a radius of 4 and two points \( A \) and \( B \) on its surface, where \( AB = 4\sqrt{2} \), and the center of the sphere is \( O \). If a moving point \( C \) on the sphere satisfies that the dihedral angle \( C-AB-O \) is \( 60^\circ \), find the radius of the circumscribed sphere of the tetrahedron \( OABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{6}}{3}}
Given are 100 different positive integers. We call a pair of numbers "good" if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
11 Let $f(x)=a x^{3}+b x+c$ (where $a, b, c$ are real numbers), when $0 \leqslant x \leqslant 1$, $0 \leqslant f(x) \leqslant 1$. Find the maximum possible value of $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{3}}{2}}
设 $a, b \in \mathbb{N}_+$，且满足 $\frac{1}{a}-\frac{1}{b}=\frac{1}{2021}$，则所有正整数对 $(a, b)$ 的个数为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Dani wrote the integers from 1 to \( N \). She used the digit 1 fifteen times. She used the digit 2 fourteen times.
What is \( N \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
In a certain year, a specific date was never a Sunday in any month. Determine this date.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Two trains of equal length are running on parallel inclined tracks in the same direction, with Train A moving at 42 km/hr and Train B moving at 36 km/hr. The incline of the tracks exerts an additional constant deceleration on both trains. Train A has a higher weight than Train B, and wind resistance is acting against both trains. Taking into account the deceleration due to the incline, the weight difference between the trains, and wind resistance, the faster Train A still manages to pass the slower Train B in 36 seconds. Determine the length of each train.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given a sphere of radius 1 with four non-coplanar points \( A, B, C, D \), and \( AB = CD = x \), \( BC = DA = y \), \( CA = BD = z \), what is the value of \( x^{2} + y^{2} + z^{2} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are 28 garden gnomes in a yard. Three-fourths of them have red hats, and the rest have blue hats. Half the garden gnomes have big noses instead of small noses. If six gnomes with blue hats have big noses, and the rest of the gnomes are divided equally into those with striped shirts or with polka-dotted shirts, how many garden gnomes with red hats, small noses and striped shirts are present in the yard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
在核酸检测中,"k合1"混采核酸检测是指:先将k个人的样本混合在一起进行1次检测，如果这k个人都没有感染新冠病毒，则检测结果为阴性，得到每人的检测结果都为阴性，检测结束；如果这k个人中有人感染新冠病毒，则检测结果为阳性，此时需对每人再进行1次检测，得到每人的检测结果，检测结束.现对100人进行核酸检测，假设其中只有2人感染新冠病毒，并假设每次检测结果准确.
$将这100人随机分成10组，每组10人，且对每组都采用“10合1”混采核酸检测. 已知感染新冠病毒的2人分在同一组的概率为\frac{1}{11}. 设X是检测的总次数，求X的数学期望E(X).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{320}{11}$}
Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x \), \( y \), and \( z \) such that \( x \mid y^{3} \) and \( y \mid z^{3} \) and \( z \mid x^{3} \), it always holds that \( x y z \mid (x+y+z)^{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Jeremy has two pieces of wood: one is 90 inches and the other is some length. He wants to cut both pieces of wood into smaller pieces so that all these pieces are the same length. He cuts both pieces of wood into 18-inch pieces. How long is the second piece of wood?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4, \ldots, n\}$ into two sets, at least one of these sets contains three (not necessarily distinct) numbers $a, b, c$ for which $ab = c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
In acute triangle $A B C$, let $D, E$, and $F$ be the feet of the altitudes from $A, B$, and $C$ respectively, and let $L, M$, and $N$ be the midpoints of $B C, C A$, and $A B$, respectively. Lines $D E$ and $N L$ intersect at $X$, lines $D F$ and $L M$ intersect at $Y$, and lines $X Y$ and $B C$ intersect at $Z$. Find $\frac{Z B}{Z C}$ in terms of $A B, A C$, and $B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{A B^{2}}{A C^{2}}}
We have two regular pyramids with $2n$ sides $(n \geq 2)$ (base: regular polygon, sides: equal). Their congruent bases are aligned to form a convex polyhedron with triangular faces. What is the maximum number of sides a planar cross-section of the resulting polyhedron can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(n+1)}
Let $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and extend $AB$ to $F$ such that $AB: BF = 1:3$. Let $G(32,24)$ be the centroid of the triangle $ABC$ and $K$ be the centroid of the triangle $DEF$. Find the length $GK$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
10. A cylindrical container with a base radius of $1 \mathrm{~cm}$ contains four solid iron balls, each with a radius of $\frac{1}{2} \mathrm{~cm}$. The four balls are pairwise tangent, and the two balls at the bottom are tangent to the bottom of the container. Water is now poured into the container until the water level just covers all the iron balls. The amount of water needed is $\qquad$ $\mathrm{cm}^{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi(\frac{1}{3}+\frac{\sqrt{2}}{2})}
A circle $\omega$ is circumscribed around triangle $ABC$. Tangents to the circle at points $A$ and $B$ intersect at point $K$. Point $M$ is the midpoint of side $AC$. A line passing through point $K$ parallel to $AC$ intersects side $BC$ at point $L$. Find the angle $AML$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Given that $F(-c,0)$ is the left focus of the hyperbola $\frac{{{x}^{2}}}{{{a}^{2}}}- \frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0,b > 0)$, $A$ is the left vertex, and the line perpendicular to $AF$ intersects the hyperbola at points $B$ and $C$. The perpendicular lines $AB$ and $AC$ intersect at point $D$. If the distance from $D$ to the line $BC$ is $3(a+c)$, calculate the eccentricity of the hyperbola. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a tetrahedron \( A B C D \) such that \( A B = C D = 2a \), \( A C = B D = B C = A D = \sqrt{10} \), find the range of values for \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < a < \sqrt{5}}
设实数 x_1,x_2,\cdots,x_{1997} 满足：(1) -\df{1}{\sqrt{3}}\leq x_i\leq\sqrt{3},i=1,2,\cdots,1997；(2) x_1+x_2+\cdots+x_{1997}=-318\sqrt{3}.求 x_1^{12}+x_2^{12}+\cdots+x_{1997}^{12} 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{189548}
You can arrange 15 balls in the shape of a triangle, but you cannot arrange 96 balls in the shape of a square (missing one ball). Out of how many balls, not exceeding 50, can you arrange them both in the shape of a triangle and a square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
From one point on a circle, two chords with lengths of 9 cm and 17 cm are drawn. Find the radius of the circle if the distance between the midpoints of these chords is 5 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 \frac{5}{8} \text{ cm}}
In a square, there are $K$ points ($K>2$). What is the smallest number of triangles needed to divide the square so that each triangle contains no more than one point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K+1}
In a row of 10 chairs, one of which is broken and cannot be used, Mary and James randomly select their seats. What is the probability that they do not sit next to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{9}}
A pyramid of non-negative integers is constructed as follows:

1. The first row consists of only $0$.
2. The second row consists of $1$ and $1$.
3. The $n^{th}$ row (for $n > 2$) is an array of $n$ integers, where the leftmost and rightmost elements are equal to $n - 1$, and the interior numbers are equal to the sum of two adjacent numbers from the $(n - 1)^{th}$ row.

Let $S_n$ be the sum of numbers in the $n^{th}$ row. Determine the remainder when dividing $S_{2018}$ by $2018$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i). $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{673}
Sherry is waiting for a train. Every minute, there is a $75 \%$ chance that a train will arrive. However, she is engrossed in her game of sudoku, so even if a train arrives she has a $75 \%$ chance of not noticing it (and hence missing the train). What is the probability that Sherry catches the train in the next five minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\left(\frac{13}{16}\right)^{5}}
The probability of randomly selecting 5 different numbers from the set {1, 2, ..., 20} such that at least two of them are consecutive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{232}{323}}
12. Using the digits $1,2,3, \cdots 9$, without repetition, form 3 three-digit numbers such that the sum of the first two equals the third, for example: $245+736=981$. The minimum value of the sum of these three three-digit numbers is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{918}
Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by the square of a prime. Find $a+b+c+d+e$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{850}
In the cells of a $3 \times 3$ square are the numbers $1, 2, 3, \ldots, 9$. It is known that any two consecutive numbers are located in adjacent (side-by-side) cells. What number can be in the central cell if the sum of the numbers in the corner cells is $18$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given any \( n \) points \( P_{1}, P_{2}, \cdots, P_{n} \) on a plane, where no three points are collinear, each point \( P_{i} \) (\(1 \leqslant i \leqslant n \)) is colored either red or blue. Let \( S \) be the set of triangles with vertices from \(\{ P_{1}, P_{2}, \cdots, P_{n} \}\) that satisfy the property: for any two line segments \( P_{i} P_{j} \) and \( P_{u} P_{v} \), the number of triangles in \( S \) that have \( P_{i} P_{j} \) as an edge is the same as the number of triangles that have \( P_{u} P_{v} \) as an edge. Find the smallest \( n \) such that \( S \) always contains at least two triangles, each of whose vertices are all the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
\(\left(\left(\frac{\sqrt{a}-1}{\sqrt{a}+1}\right)^{-1} \cdot\left(\frac{\sqrt{a}-1}{\sqrt{a}+1}\right)^{1 / 2}-\sqrt{a-1} \cdot(\sqrt{a}+1)^{-1}\right)^{-2} \cdot \frac{1}{a^{2 / 3}+a^{1 / 3}+1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt[3]{a}-1}{4}}
How can we calculate the probability that a number $n$ is prime if it passes the Fermat test $x$ times, excluding Carmichael numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1 - \left(\frac{1}{2}\right)^x$}
Find the limit of the following expression as $n$ approaches infinity: $\lim_{n \rightarrow \infty} \left( \sin \frac{1}{n} \cdot \sin \frac{2}{n} \cdot \ldots \cdot \sin 1 \right)^{\frac{1}{n}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^(-1)}
Four people went to a bookstore to buy books. Each person bought 4 different books, and each pair of people had exactly 2 books in common. How many different books did these 4 people buy at least?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the Cartesian coordinate system, the slope angle of the line $y=- \frac{ \sqrt{3}}{3}x+1$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{6}}
In the rectangular coordinate system, points (8, 0) and a point (x, 0) both lie on circle C. The maximum possible value of the radius of C is 8. What is the value of x?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
If line $l_1: (2m+1)x - 4y + 3m = 0$ is parallel to line $l_2: x + (m+5)y - 3m = 0$, determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{9}{2}}
The expression  $\circ \ 1\  \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012$  is written on a blackboard. Catherine places a  $+$  sign or a  $-$  sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
计算 $2+\sqrt{3}$ 和 $2-\sqrt{3}$ 的等比中项是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
In triangle $\triangle ABC$, $(3\overrightarrow{AB}+2\overrightarrow{AC})\cdot \overrightarrow{BC}=0$, and for $t\in R$, the minimum value of $|\overrightarrow{BA}-t\overrightarrow{BC}|$ is $\frac{6}{5}|BC|$, then $\angle BAC=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
The inclination angle of the line $\sqrt{3}x + y - 1 = 0$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}}
Given non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying the inequality $x_{1} + x_{2} + \cdots + x_{n} \leqslant \frac{1}{2}$, find the minimum value of $f\left(x_{1}, x_{2}, \cdots, x_{n}\right) = \prod_{i=1}^{n}\left(1 - x_{i}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
For $a > 0$, find the minimum value of $\int_{0}^{\frac{1}{a}}(a^{3}+4x-a^{5}x^{2})e^{ax}\ dx.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
When you turn on a computer, a certain bit (0/1) in its memory is initialized to the value 0. Suppose that starting from when you turn on your computer the bit is rewritten at random instants, on average λ times per millisecond, and each rewrite results in a 1 independently with probability 𝑝 (or a 0 is written with probability (1 − 𝑝) when a rewrite occurs). What is the probability that after t milliseconds the bit stores the value 0?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-\lambda t} + (1 - e^{-\lambda t})(1 - p)}
Given that $\sin(a + \frac{\pi}{4}) = \sqrt{2}(\sin \alpha + 2\cos \alpha)$, determine the value of $\sin 2\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{5}}
Suppose that point $D$ lies on side $B C$ of triangle $A B C$ such that $A D$ bisects $\angle B A C$, and let $\ell$ denote the line through $A$ perpendicular to $A D$. If the distances from $B$ and $C$ to $\ell$ are 5 and 6 , respectively, compute $A D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{60}{11}}
When a tree tends to fall, forest rangers often use two wooden sticks to support the tree. This is an application of the triangle's ______ in real life.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{stability}}
Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\leq i < j < k\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
What is the maximum $k$-dimensional volume ($V_{n,k}$) of the projection of an $n$-dimensional cube ($C_n$) with side length 1 onto a $k$-dimensional plane ($P_k$), where $k \leq n$? Specifically, focus on the case where $k = n - 1$. Provide a formula for $V_{n,k}$ when $k = n - 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{n}}
For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique <b>alternating sum</b> is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply $5$. Find the sum of all such alternating sums for $n=7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$448$}
If 1000 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
How many ways to fill the board $ 4\times 4$ by nonnegative integers, such that sum of the numbers of each row and each column is 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2008}
记$$F(x)=16 \int_{\frac{\pi^{2}}{4}+\cos \left(3 x+\frac{\pi}{2}\right)}^{\frac{25 \pi^{2}}{4}+2 x^{2}} \sin \sqrt{t} d t$$$F^{\prime}(0)$ 是 $F(x)$ 在 $x=0$ 处的导数. 求取整 $\left[F^{\prime}(0)\right]$ 的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Nelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the peak of its trajectory between bounces. On any given bounce, there is an $80\%$ chance that the next peak occurs at $\frac13$ the height of the previous peak and a $20\%$ chance that the next peak occurs at $3$ times the height of the previous peak (where the first peak is at $12$ feet). If Nelson can only reach $4$ feet into the air and will catch the ball as soon as possible, the probability that Nelson catches the ball after exactly $13$ bounces is $2^a \times 3^b  \times 5^c  \times 7^d  \times 11^e$ for integers $a, b, c, d$, and $e$. Find $|a| + |b| + |c| + |d| + |e|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given the function f(x) = 1/2 * e^(2x) + (a-e) * e^x - a*e + b, where a, b are real numbers and e is the base of the natural logarithm, and it has a maximum value at x = 1, find the range of a. Express your answer as an inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a < -e}
Compute the definite integral:

$$
\int_{\pi / 4}^{\arccos (1 / \sqrt{26})} \frac{d x}{(6 - \tan x) \sin 2 x}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\ln 5}{6}}