inclusionAI/Ring-lite-rl-data huggingface dataset:
Let \( a \) and \( b \) be nonnegative real numbers such that
\[ \cos (ax + b) = \cos 31x \]
for all integers \( x \). Find the smallest possible value of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
The letter T is formed by placing a $2\:\text{inch} \times 6\:\text{inch}$ rectangle vertically and a $2\:\text{inch} \times 4\:\text{inch}$ rectangle horizontally across the top center of the vertical rectangle. What is the perimeter of the T, in inches?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Using the method of Lagrange Multipliers, find the maximum value of $a + b + c + abc$ given that $0< a,b,c < 1$ and $ab + bc + ca = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{10\sqrt3}{9}\)}
袋中装有 2 个红球、 3 个白球、 4 个黄球，从中任取 4 个球，则其中三种颜色的球都有的概率是 $\qquad$。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{11}
Mr. and Mrs. Seventh have 7 children, all born on April 1, actually within six consecutive April 10. This year, for their birthdays, Mrs. Seventh made a cake with candles for each one - the number of candles equal to the number of years of each. João Seventh, the child who likes Mathematics the most, noticed that this year the total number of candles is double what it was 2 years ago and that there are 2 more cakes. How many candles will be lit this year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116}
What is the greatest positive integer $m$ such that $n^2(1+n^2-n^4) \equiv 1 \pmod{2^m}$ for all odd integers $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given a $p\times q$ matrix $A$ of rank $\alpha$ and an $r\times s$ matrix $B$ of rank $\beta$, consider the set $V$ of all $q\times r$ matrices $C$ such that $ACB=0_{p\times s}$. Determine the dimension of the vector space $V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{rq - \alpha\beta}
Farmer John has $ 5$ cows, $ 4$ pigs, and $ 7$ horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100800}
Given that point A has coordinates (4, 3), and F is the focus of the parabola $y^2=4x$, if point P moves on the parabola, then when $|PA|+|PF|$ takes the minimum value, the coordinates of point P are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{9}{4}, 3 \right)}
Cyclic quadrilateral $ABCD$ satisfies $\angle ADC = 2 \cdot \angle BAD = 80^\circ$ and $\overline{BC} = \overline{CD}$. Let the angle bisector of $\angle BCD$ meet $AD$ at $P$. What is the measure, in degrees, of $\angle BPD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
From the set $\{1,2,3, \cdots, 100\}$, find the subset with the largest number of elements such that no element is double any other element in the subset.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
A stack contains 300 cards: 100 white, 100 black, and 100 red. For each white card, the number of black cards below it is counted; for each black card, the number of red cards below it is counted; and for each red card, the number of white cards below it is counted. Find the maximum possible value of the sum of these 300 counts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20000}
Points \( A, B, C \) lie on a circle \(\omega\) such that \( BC \) is a diameter. \( AB \) is extended past \( B \) to point \( B' \) and \( AC \) is extended past \( C \) to point \( C' \) such that line \( B'C' \) is parallel to \( BC \) and tangent to \(\omega\) at point \( D \). If \( B'D = 4 \) and \( C'D = 6 \), compute \( BC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{24}{5}}
Given two congruent circles $\odot A$ and $\odot B$ intersecting at points $E$ and $F$, with each circle passing through the center of the other, let line $AB$ intersect the two circles at points $C$ and $D$. Determine the measure of angle $\angle CED$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Given the hyperbola $x^{2}-y^{2}=a^{2}$ ($a \gt 0$) with left and right foci $F_{1}$ and $F_{2}$ respectively, and the circle $F_{2}$: $(x-c)^{2}+y^{2}=1$ intersecting the hyperbola and a line $x-y=0$ at point $M$, calculate the area of $\triangle F_{1}MF_{2}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the least positive integer $k$ such that, in every convex 101 -gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4900}
If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$. If $n$ is a positive integer, let $S(n)$ be defined by
\[
  S(n) 
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor} 
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Evaluate the limit $\lim_{n \rightarrow \infty} \left(\prod_{k=1}^{n} \sin \frac{k}{n}\right)^{\frac{1}{n}}$ using the fact that $\lim_{n \rightarrow \infty} \frac{n!}{n^n} = \frac{1}{e}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{e}$}
The sequence $\{a_i\}_{i \ge 1}$ is defined by $a_1 = 1$ and \[ a_n = \lfloor a_{n-1} + \sqrt{a_{n-1}} \rfloor \] for all $n \ge 2$. Compute the eighth perfect square in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16384}
Given two circles $C_1: x^2 + y^2 = m^2$ and $C_2: x^2 + y^2 - 2x - 4y - 20 = 0$ have exactly two common tangents, the range of real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5-\sqrt{5}, 5+\sqrt{5})}
Three volleyballs with a radius of 18 lie on a horizontal floor, each pair touching one another. A tennis ball with a radius of 6 is placed on top of them, touching all three volleyballs. Find the distance from the top of the tennis ball to the floor. (All balls are spherical in shape.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)\) with the right focus at \(F(c, 0)\), there exists a line \(l\) passing through point \(F\) that intersects the ellipse at points \(A\) and \(B\) such that \(OA \perp OB\). Find the range of the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[\frac{\sqrt{5} - 1}{2}, 1\right)}
Given the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) with the left vertex \(A\) and the right focus \(F\), let \(P\) be an arbitrary point on the hyperbola in the first quadrant. If it is always true that \(\angle PFA = 2 \angle FAP\), find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A regular triangle and a regular nonagon are coplanar and share a common side $\overline{AD}$. What is the degree measure of the exterior angle $BAC$?

[asy]
for(int i=0; i <=9; ++i) {
draw(dir(360*i/9+90)--dir(360*(i+1)/9+90));
}
pair A = dir(360*5/9+90);
pair D = dir(360*6/9+90);
pair C = A+dir(-30)*(D-A);
pair B = dir(360*4/9+90);

draw(A--C--D);

label("$A$",A,NE);
label("$B$",B,W);
label("$C$",C,S);
label("$D$",D,NE);

[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160^\circ}
Codger is a sloth with a certain number of feet. He has a challenging time buying shoes because the stores only sell the shoes in pairs. He already owns the set of shoes he is wearing. He needs to buy 6 pairs of shoes to have 5 complete sets of shoes. How many feet does Codger have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A bus, a truck, and a car are driving in the same direction on a straight road at constant speeds. At a certain moment, the bus is in front, the car is behind, and the truck is exactly in the middle between the bus and the car. After 10 minutes, the car catches up with the truck; another 5 minutes later, the car catches up with the bus. How many more minutes will it take for the truck to catch up with the bus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Points $A$, $B$, $Q$, $D$, and $C$ lie on the circle shown and the measures of arcs $BQ$ and $QD$ are $42^\circ$ and $38^\circ$, respectively.  Find the sum of the measures of angles $P$ and $Q$, in degrees.

[asy]
import graph;

unitsize(2 cm);

pair A, B, C, D, P, Q;

A = dir(160);
B = dir(45);
C = dir(190);
D = dir(-30);
P = extension(A,B,C,D);
Q = dir(0);

draw(Circle((0,0),1));
draw(B--P--D);
draw(A--Q--C);

label("$A$", A, NW);
label("$B$", B, NE);
label("$C$", C, SW);
label("$D$", D, SE);
label("$P$", P, W);
label("$Q$", Q, E);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{2^{2 n}} \binom{2 n}{n}\)}
On the side \( CD \) of the trapezoid \( ABCD \) with \( AD \parallel BC \), a point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is dropped to the segment \( BM \). It is known that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
When the roots of the polynomial 

\[P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \cdot \cdot \cdot (x-10)^{10}\]

are removed from the number line, what remains is the union of $11$ disjoint open intervals. Find the number of these intervals on which \( P(x) \) is positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
There are \( n \) colored balls, of which some are red and some are black. Out of every 50 balls, 49 are red, and thereafter, out of every 8 balls, 7 are red. Given that at least 90% of all the balls are red, what is the maximum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
In quadrilateral $\mathrm{ABCD}$, $\angle B=90^{\circ}$, and $A B=B C=5$, $C D=2 \sqrt{15}$, $D A=\sqrt{10}$. What is the measure of $\angle B A D$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135^{\circ}}
从圆 \(x^{2}+y^{2}=4\) 上的点向椭圆 \(C: \frac{x^{2}}{2}+y^{2}= 1\) 引切线，两个切点间的线段称为切点弦。求所有的切点弦所在的直线围成区域的面积。原始答案的形式为\(\frac{k\pi}{m}\)，请给出k + m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the Fibonacci sequence $\{a_n\}$, where each number from the third one is equal to the sum of the two preceding numbers, find the term of the Fibonacci sequence that corresponds to $\frac{{a_1}^2 + {a_2}^2 + {a_3}^2 + … + {a_{2017}}^2}{a_{2017}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
One day, School A bought 56 kilograms of fruit candy, each kilogram costing 8.06 yuan. Several days later, School B also needed to buy the same 56 kilograms of fruit candy, but happened to catch a promotional offer, reducing the price per kilogram by 0.56 yuan, and also offering an additional 5% of the same fruit candy for free with any purchase. How much less in yuan will School B spend compared to School A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51.36}
Given an arithmetic sequence $\{a_n\}$, the sum of the first $m$ terms is 30, and the sum of the first $2m$ terms is 100. Find the sum of the first $3m$ terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Evaluate the limit: \[ \lim_{n \to \infty} \int_{\epsilon}^1\frac {n+1}{2^{n+1}}\frac{(1+t)^{n+1}-(1-t)^{n+1}}{t}\,dt \] where \(\epsilon \in (0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Identify a non-cyclic group such that the intersection of all its non-trivial subgroups is a non-trivial subgroup.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q_8}
Evaluate the integral \( \iiint_A x^2 y^2 z \, dx \, dy \, dz \) where the region \( A \) is defined by \( A = \{ (x, y, z) \in \mathbb{R}^3 \mid 0 \leq z \leq x^2 + y^2, \ |x| + |y| = 1 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{450}}
Find the least significant digit of the product \( \prod\limits_{i=1}^{n/3} (i)(n-2i+1) \) for multiples of 3, where \( n \geq 9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the largest natural number $n$ such that for any subset of $A = \{1, 2, 3, \ldots, 2003\}$ with $n$ elements, the complementary set contains at least one pair of consecutive numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
A circle has a radius of 1. A rhombus and a triangle are described around this circle. Two sides of the triangle are parallel to the diagonals of the rhombus, and the third side is parallel to one of the sides of the rhombus and measures 5 units. Determine the side length of the rhombus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{12}}
Express the antipodal map as a composition of one, two, or three reflections. Determine the minimum number of reflections needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the double integral \( \iint_{D} xy\sqrt{1-x-y} \, dx \, dy \) where the domain is \( D=\{(x,y): x\ge0, y\ge0, x+y\le1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{16}{945}}
Determine the minimum number of states in a DFA that accepts the language \(L_n \subseteq \{0,1\}^*\) defined by:

\[L_n=\{s\in \{0,1\}^*\, |\,\, \text{for every substring } t \text{ of } s, \text{abs}(N_0(t)-N_1(t)) \leq n \}\]

where \(\text{abs}\) denotes the absolute value and \(N_0(t),N_1(t)\) denote the number of \(0\)'s and \(1\)'s in \(t\) respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(n+1)(n+2)}{2} + 1}
Determine the Lebesgue measure of the set \( C = \left(\bigcup_{n=1}^\infty \left(-\frac{1}{2^n}, \frac{1}{2^n}\right)\right) \cap \mathbb{R} \setminus \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of distinct elements in the quotient group \(\mathbb{R}/\mathbb{Q}\), where \(\mathbb{R}\) is the set of real numbers and \(\mathbb{Q}\) is the set of rational numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the value of the constant \( c \) such that the random variable \( Y = \frac{c(X_1 + X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \) follows a t-distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a one-kilometer race, the completion times of three individuals are modeled by random variables. Let $X_1$, $X_2$, and $X_3$ represent the completion times in minutes for each person, where:
- $X_1$ is uniformly distributed over [2.9, 3.1]
- $X_2$ is uniformly distributed over [2.7, 3.1]
- $X_3$ is uniformly distributed over [2.9, 3.3]

Assuming the completion times are independent, calculate the expected value of the latest completion time, $Y = \max\{X_1, X_2, X_3\}$, to the nearest 0.1 minute.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.1}
Evaluate the limit: $$A = \lim_{n \to \infty} \left(\binom{n}{0} \binom{n}{1} \binom{n}{2}\cdots \binom{n}{n}\right)^{1/((n)(n+1))}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Provide an example of a decreasing function \( f(x) \) such that \( \lim_{x\to +\infty} f(x) = e. \) The function \( f(x) \) should have a simple form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e + \dfrac{1}{x}}
Given the function \( f(z) = \sqrt[3]{z^2 - 1} \), where \( f(3) = 2 \), find the value of \( f(-3) \) by considering the analytic continuation of \( f \) along the semicircle \( |z| = 3 \) in the lower half-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 - i\sqrt{3}}
Find the smallest positive number \( \lambda \) such that for any sequence of real numbers \( \{x_n\} \) and for all positive integers \( m \), the inequality \( \frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i} \le \sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} \) holds, where \( \{y_n\} \) is defined by \( y_1 = x_1 \) and \( y_{n+1} = x_{n+1} - (\sum\limits_{i = 1}^{n} {x^2_i})^{\frac{1}{2}} \) for \( n \ge 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Trapezoid \( {ABCD} \) ,with \( {AB} \parallel  {CD} \) ,has side lengths \( {AB} = {11},{BC} = 8,{CD} = {19} \) ,and \( {DA} = 4 \) . Compute the area of
 the convex quadrilateral whose vertices are the circumcenters of \( \bigtriangleup  {ABC}, \bigtriangleup  {BCD}, \bigtriangleup  {CDA}, \) and \( \bigtriangleup  {DAB} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\sqrt{15}}
Let \( G \) be a non-abelian group of order \( p^3 \), where \( p \) is a prime. Determine the number of distinct conjugacy classes in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + p - 1}
Determine the minimum number of times a rectangular object with faces labeled A, B, C, and D must be tossed so that the probability of the ratio of the number of D's to the total number of tosses being between 0.15 and 0.35 is at least 0.95.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
Determine the range of values for $a > 0$ such that the curves $C_1: y = e^{6x}$ and $C_2: y = ax^2$ have exactly two common tangents.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(9e^2, \infty)}
Determine whether there is a closed-form expression for the integral: $$\frac{1}{2\pi i}\oint_{x=0} e^{\sqrt{z}(x+\frac{1}{x})} \frac{dx}{x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_0(2\sqrt{z})}
Given a geometric sequence with the first term a_1 = 1536 and the common ratio q = -1/2, find the value of n that maximizes the product of the first n terms, denoted as Π_n. Express your answer as an integer value of n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
All digits in the 6-digit natural numbers $a$ and $b$ are even, and any number between them contains at least one odd digit. Find the largest possible value of the difference $b-a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111112}
two numbers have a h . c . f of 21 and a product of two numbers is 138567 . find the l . c . m of the two numbers ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6603}
5. [5 points] On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at points $(0 ; 0),(0 ; 63),(63 ; 63)$, and $(63 ; 0)$. Find the number of ways to choose two grid nodes inside this square (not including its boundary) such that at least one of these nodes lies on one of the lines $y=x$ or $y=63-x$, but neither of the chosen nodes lies on any line parallel to any of the coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{453902}
On the parade ground, 2018 soldiers are lined up in a single row. The commander can order either all soldiers standing in even positions or all soldiers standing in odd positions to leave the formation. After this order, the soldiers remaining in formation close ranks into a single row. In how many ways can the commander issue a series of 8 orders so that there are exactly 7 soldiers left in formation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given the point \( P \) lies in the plane of the right triangle \( \triangle ABC \) with \( \angle BAC = 90^\circ \), and \( \angle CAP \) is an acute angle. Also given are the conditions:
\[ |\overrightarrow{AP}| = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AC} = 2, \quad \overrightarrow{AP} \cdot \overrightarrow{AB} = 1. \]

Find the value of \( \tan \angle CAP \) when \( |\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AP}| \) is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
David and Brenda are playing Scrabble. In the first turn, Brenda scores 18 points, and David scores 10 points. In the second turn, Brenda scores 25 points and David scores 35 points. In the third turn, Brenda is ahead by 22 points when she makes a 15-point play. David responds with a 32-point play. By how many points is Brenda now ahead?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
60. Given $\triangle A B C$. On the ray $B A$, take a point $D$ such that $|B D|=|B A|+|A C|$. Let $K$ and $M$ be two points on the rays $B A$ and $B C$ respectively such that the area of $\triangle B D M$ is equal to the area of $\triangle B C K$. Find $\widehat{B K M}$, if $\widehat{B A C}=\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha/2}
Find $a$ such that $83359654581036155008716649031639683153293510843035531 ^ a \equiv 1 \pmod{32003240081}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32003240080}
Find all such positive integers \( k \) that the number

\[ 
\underbrace{1 \ldots 1 \overbrace{2 \ldots 2}^{k}}_{2000} - \underbrace{2 \ldots 2}_{1001} 
\]

is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\angle DBA = 23^{\circ}$, and $\angle BDC = 46^{\circ}$.  The ratio $BC: AD$ is $9: 5$.  What is $CD$?

$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{5}}
已知 $f(x)=2 m x^{2}-2 m x-8 x+9, g(x)=m x-m$ ，对任意 $x \in \mathbf{R}, f(x)$ 与 $g(x)$ 至少有一个为正数，则实数 m 的取值范围是. 请给出该区间的两个端点之和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Compute the limit of the function:

\[
\lim _{x \rightarrow 0}\left(\frac{1+\sin x \cdot \cos \alpha x}{1+\sin x \cdot \cos \beta x}\right)^{\operatorname{ctg}^{3} x}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{\beta^2-\alpha^2}{2}}}
Grass on an entire meadow grows uniformly dense and fast. It is known that 70 cows would eat all the grass in 24 days, and 30 cows would eat it in 60 days. How many cows would eat all the grass in 96 days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}