inclusionAI/Ring-lite-rl-data huggingface dataset:
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. Evaluate
\[
\sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi^2}{16}}
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \dfrac{3}{16}\]}
Given the circle \( M: 2x^{2} + 2y^{2} - 8x - 8y - 1 = 0 \), and the line \( l: x + y - 9 = 0 \), let \( A \) be a point on \( l \). Construct \(\triangle ABC\) such that \(\angle BAC = \frac{\pi}{4}\), with side \( AB \) passing through the center of the circle \( M \) and points \( B \) and \( C \) lying on the circle \( M \). What is the range of the y-coordinate of point \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[3, 6]}
If the equation $\sin ^{2}x+2\sin x-1+m=0$ with respect to $x$ has a solution, then the range of the real number $m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\leqslant m\leqslant 2}
In a hexagon, how many different triangles whose vertices are vertices of the hexagon can be formed by drawing diagonals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{6}{3}}
If $3(\sin ^{5}\theta +\cos ^{5}2\theta ) \gt 5(\sin ^{3}\theta +\cos ^{3}2\theta ),\theta \in \left[0,2\pi \right)$, then the range of $\theta$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{7\pi}{6}, \frac{11\pi}{6})}
在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}=2 \overrightarrow{B C} \cdot \overrightarrow{B A}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$, 则 $\triangle A B C$ 最大角的正弦值为。原始答案为$\frac{m\sqrt{n}}{k}$的格式，其中$m,n,k$均为正整数，请给出$m + n+k$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let \( S \) be the part of the sphere \( x^2+y^2+z^2=25 \) which lies above the plane \( z=4 \). Orient \( S \) by upward pointing normal vectors. Evaluate \( \int\!\!\!\int_{S}\text{curl}F\,dS \) where \( F=yze^xi+(x+ze^x)j+yzk \). (Hint: Consider a different surface.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\pi}
A pack of 20 cards is shuffled using the "interlacing" method, where the original order 1, 2, 3, ..., 20 becomes 1, 11, 2, 12, ..., 10, 20 after one shuffle. Determine how many times this shuffle must be repeated for the cards to return to their original order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Example 6.21. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from 0.85 in absolute value does not exceed 0.01?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11475}
A subset $M$ of $\{1, 2, . . . , 2006\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$, $x+ y$ does not divide $z$. Determine the largest possible size of $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 1004 }
Determine the angle of inclination between two lines that lie in the second bisector plane, which is located in the second and fourth quadrants and forms equal angles with the two image planes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Let $S = \{1, 2, \ldots, 8\}$. How many ways are there to select two disjoint subsets of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6561}
A set S contains triangles whose sides have integer lengths less than 7, and no two elements of S are congruent or similar. Calculate the largest number of elements that S can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Consider the stochastic process defined by:
\[ \tilde{W}(t) := W(t) + \int_{0}^{t} e^{W(s)}\mathbb{1}_{\{|W(s)\le 1\}}\,ds \quad \forall \, t \in [0,T] \]
where \( W(t) \) is a Wiener process. For the stopping time \( \sigma := \inf\{t \ge 0 : W(t) = 1 - \int_{0}^{t} e^{W(s)}\mathbb{1}_{\{|W(s)\le 1\}}\,ds \} \), compute the expectation:
\[ \mathbb{E}_{\mathcal{Q}}\left[e^{-\sigma/2} \right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1}}
A cylinder with a radius of $20 \text{ cm}$ touches the ground along the line $e$. Perpendicular to the line $e$, a $50 \text{ cm}$ long rod is placed resting against the cylinder, such that the end of the rod on the ground is $40 \text{ cm}$ away from the line $e$. How high is the other end of the rod?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40 \, \text{cm}}
Huahua is writing letters to Yuanyuan with a pen. When she finishes the 3rd pen refill, she is working on the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up. If Huahua uses the same amount of ink for each letter, how many pen refills does she need to write 16 letters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Find the value of the following limit:
$$\lim_{n\to\infty} \frac {\cos 1 \cdot \arccos \frac{1}{n}+\cos\frac {1}{2} \cdot \arccos \frac{1}{(n-1)}+ \cdots +\cos \frac{1}{n} \cdot \arccos{1}}{n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{2}\)}
Two types of shapes formed by unit squares (each with an area of 3) are to be placed into an $8 \times 14$ large rectangular grid, with the condition that no two shapes share a common point. What is the maximum number of these shapes that can be placed into the $8 \times 14$ rectangular grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Consider a function \( f \) that assigns a number \( f(\boldsymbol{v}) \) to each vector \( \boldsymbol{v} \) in space, originating from a common point \( O \). The function satisfies the condition that for any vectors \( \boldsymbol{u}, \boldsymbol{v} \) and any scalars \( \alpha, \beta \), the value \( f(\alpha \boldsymbol{u} + \beta \boldsymbol{v}) \) does not exceed at least one of the values \( f(\boldsymbol{u}) \) or \( f(\boldsymbol{v}) \). Determine the maximum number of distinct values that the function \( f \) can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The function \( f(x) = \frac{x^{2}}{8} + x \cos x + \cos (2x) \) (for \( x \in \mathbf{R} \)) has a minimum value of ___
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Line segments connecting an internal point of a convex, unequal-sided $n$-gon with its vertices divide the $n$-gon into $n$ equal triangles.

For what smallest value of $n$ is this possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A company sells a brand of cars in two locations, A and B, with profits (in ten thousand yuan) of $L_1 = -x^2 + 21x$ and $L_2 = 2x$, respectively, where $x$ is the sales volume (in units). If the company sells a total of 15 units in both locations, calculate the maximum profit it can achieve. Express your answer in ten thousand yuan.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Calculate the derivative \( \frac{d|a|^2}{da} \), where \( a \) is a complex number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\overline{a}}
Two trains of length 90 m and 95 m are a certain distance apart. They start moving towards each other on parallel tracks, at speeds 64 kmph and 92 kmph. The trains meet after 5.768769267689355 seconds. What is the initial distance between the two trains?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
A car has four tires, each with an exponential lifetime averaging 5000 miles, and one spare tire with an exponential lifetime averaging 1000 miles. When a primary tire fails, the spare is used. Calculate the expected number of miles the car can travel before any of the remaining three primary tires or the spare fails.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1875}
Find the 257th derivative of $e^{-t} \sin t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(2^{128}e^{-t}(\cos t - \sin t)\)}
Given \( a, b \in \mathbf{R}^{*} \), find the minimum value of \( y = \frac{a}{\sin^{3/2} \theta} + \frac{b}{\cos^{3/2} \theta} \) for \( \theta \in \left(0, \frac{\pi}{2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( a^{4/7} + b^{4/7} \right)^{7/4}}
The prduct of a few primes is ten times as much as the sum of the primes. What are these (not necessarily distinct) primes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{${2, 3, 5, 5}$}
If the line $3x+4y-3=0$ is parallel to the line $6x+my+14=0$, then the distance between them is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For positive integers $m$ and $n$, the decimal representation for the fraction $\frac{m}{n}$ begins with $0.711$ followed by other digits. Find the least possible value for $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
7. In the Cartesian coordinate system $x O y$, $F$ is the focus of the parabola $\Gamma: y^{2}=2 p x(p>0)$. Point $B$ is on the $x$-axis and to the right of point $F$. Point $A$ is on $\Gamma$, and $|A F|=$ $|B F|$. The second intersection points of lines $A F$ and $A B$ with $\Gamma$ are $M$ and $N$, respectively. If $\angle A M N=90^{\circ}$, then the slope of line $A F$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Find the coefficient of \(x^4\) in the expansion of \(\frac{(x-2)^2}{(x^2-1)(x^2-4)}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{8}}
In the Cartesian coordinate system, the coordinates of the point $A(2,3)$ with respect to the $x$-axis are ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(2,-3\right)}
14. For each integer $n \geqslant 2$, determine the maximum value of the sum $a_{0}+a_{1}+a_{2}+\cdots+a_{n}$ satisfying the conditions $a_{0}=1, a_{i} \geqslant a_{i+1}+a_{i+2}, i=0$. $1, \cdots, n-2$ (where $a_{0}, a_{1}, \cdots a_{n}$ are non-negative).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{F_{n+1}}{F_{n-1}}}
One dimension of a cube is tripled, another is decreased by `a/2`, and the third dimension remains unchanged. The volume gap between the new solid and the original cube is equal to `2a^2`. Calculate the volume of the original cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Compute the integral of the differential form \( \omega = -x_1^2 x_2 \,dx_1 + x_1 x_2 \log(1+e^{2x_2})\,dx_2 \) along the unit circle \( \Gamma = \{x \in \mathbb{R}^2 \mid x_1^2 + x_2^2 = 1\} \) in the trigonometric direction. Use Green's theorem if necessary. What is the value of the integral?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the triple integral \( \iiint_V (3x^{2}+2y+z) \, dxdydz \) over the region defined by the inequalities:
\[
\begin{array}{c}
|x-y| \le 1 \\
|y-z| \le 1 \\
|x+z| \le 1
\end{array}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
令 f(x)=[x]+[2x]+[3x]+[4x]+[5x]，当 x 取遍 [0,50] 中的实数时，求 f(x) 能取到的整数值的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501}
Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the remainder when the determinant \( \begin{vmatrix} { 2014 }^{ 2014 } & { 2015 }^{ 2015 } & { 2016 }^{ 2016 } \\ { 2017 }^{ 2017 } & { 2018 }^{ 2018 } & { 2019 }^{ 2019 } \\ { 2020 }^{ 2020 } & { 2021 }^{ 2021 } & { 2022 }^{ 2022 } \end{vmatrix} \) is divided by 5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$.
 find $\frac{AF}{FD}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{AF/FD = 24/11}
Evaluate the complex function \( f(x) = x^{2/3} \) at \( x = -1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2} + \frac{\sqrt{3}}{2}i}
Let $X = \{(x,y) : x^2 + y^2 < 9\}$ and $K = \{(x,y) : 4 \leq x^2 + y^2 \leq 5 \text{ or } 7 \leq x^2 + y^2 \leq 8\}$. Determine the number of relatively compact connected components of $X \setminus K$ within $X$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In unit square $A B C D$, points $E, F, G$ are chosen on side $B C, C D, D A$ respectively such that $A E$ is perpendicular to $E F$ and $E F$ is perpendicular to $F G$. Given that $G A=\frac{404}{1331}$, find all possible values of the length of $B E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{11}}
A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \sqrt{15}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{33}}
Find the maximum number of permutations of the set $\{1, 2, 3, \ldots, 2014\}$ such that for every two different numbers $a$ and $b$ in this set, there exists at least one permutation where $b$ comes immediately after $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
顺次连结圆$x ^ { 2 } + y ^ { 2 } = 9$与双曲线xy=3的交点，得到一个凸四边形.则此凸四边形的面积为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6 \sqrt { 5 }$ }
Find the number of pairs $(a,b)$ of natural numbers such that:
1. $b$ is a 3-digit number.
2. $a+1$ divides $b-1$.
3. $b$ divides $a^{2} + a + 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
If for any ${x}_{1},{x}_{2}∈[1,\frac{π}{2}]$, $x_{1} \lt x_{2}$, $\frac{{x}_{2}sin{x}_{1}-{x}_{1}sin{x}_{2}}{{x}_{1}-{x}_{2}}＞a$ always holds, then the maximum value of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Jill has some unit cubes which are all black, grey, or white. She uses 27 of them to build a 3 × 3 × 3 cube. She wants the surface to be exactly one-third black, one-third grey, and one-third white. The smallest possible number of black cubes she can use is \(X\) and the largest possible number of black cubes she can use is \(Y\). What is the value of \(Y - X\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\frac1{1985}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$32$}
Determine the set of accumulation points for the sequence \( A = \{ \sqrt{n} - \sqrt{m} : m, n \in \mathbb{N} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Given the six whole numbers 10-15, compute the largest possible value for the sum, S, of the three numbers on each side of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
A circle is circumscribed around an isosceles triangle whose two base angles are equal to $x^{\circ}$. Two points are chosen independently and randomly on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the sum of the largest and smallest possible value of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Find the smallest integer \( n_0 \) such that for the sequence \( a_n = \frac{n^2 - 3n + (-1)^n}{3n^2 - 7n + 5} \), the inequality \( |a_n - \frac{1}{3}| < 0.01 \) holds for all \( n > n_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
What is the smallest integer $n$ such that $n^2 - 14n + 45 > 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
若函数$f(x)=-{{x}^{2}}+a+2\ln x\text{ }$在$\left[ \frac{1}{{{e2}}},e \right]$上有两个不同的零点，则实数$a$的取值范围为_____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(1,\frac{1}{{{e}^{4}}}+4]$}
Find the greatest negative value of the expression \( x - y \) for all pairs of numbers \( (x, y) \) that satisfy the equation \( (1 - \operatorname{ctg} x)(1 + \operatorname{ctg} y) = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3\pi}{4}}
Evaluate the integral \( \int_M (x^2+y^2) \, \mathrm{d}V \) where \( M = \{(x, y) \in \mathbb{R}^2: 1 \leq x^2 - y^2 \leq 9, 2 \leq xy \leq 4, x, y \geq 0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the order of the quotient ring \( \mathbb{Z}_{10}[x] / \langle 5x^2 + 2x + 1 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $M$ be the number of ways to write $3050$ in the form $3050 = b_3 \cdot 10^3 + b_2 \cdot 10^2 + b_1 \cdot 10 + b_0$, where the $b_i$'s are integers, and $0 \le b_i \le 99$. Find $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{306}
Suppose that $X_1, X_2, \ldots, X_n$ are independent and identically distributed random variables with the characteristic function $\chi(t) = e^{-|t|^{1.9}}$. Determine the weak limit of $n^{-5/9} S_n$ as $n$ approaches infinity, where $S_n = \sum_{k=1}^n X_k$. Provide your answer in terms of a probability distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta_0}
Calculate the determinant of the following $n \times n$ matrix, where $n$ is a multiple of $3$:
\[\begin{pmatrix}
0 & 0 & 1 & & & & & & &\\
& & & 0 & 0 & 1 & & & &\\
& & & & & &\ddots\\
& & & & & & & 0 & 0 & 1\\
0 & 1 & 0 & & & & & & &\\
& & & 0 & 1 & 0 & & & &\\
& & & & & &\ddots\\
& & & & & & & 0 & 1 & 0\\
1 & 0 & 0 & & & & & & &\\
& & & 1 & 0 & 0 & & & &\\
& & & & & &\ddots\\
& & & & & & & 1 & 0 & 0
\end{pmatrix}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{n/3}}
Let \( n \) be a non-negative integer that ends with exactly \( k \) zeros in its decimal notation and is greater than \( 10^k \). Given that \( k = k(n) \geq 2 \), determine the number of different ways \( n \) can be expressed as the difference of two squares of non-negative integers, as a function of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k^2 - 1}
For a graph $G$ with $n \geq 1$ vertices, determine the maximum number of vertices in its block-cutpoint graph $BC(G)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 3}
Factorize the polynomial $x^9 - 1$ in the finite field $\mathbb{F}_3[x]$. Provide the complete factorization into irreducible polynomials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - 1)^9}
Convex quadrilateral $ABCD$ has $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Michael is celebrating his fifteenth birthday today. How many Sundays have there been in his lifetime?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{783}
Let $ABCD$ be a parallelogram with $\angle BAD < 90^\circ.$ A circle tangent to sides $\overline{DA},$ $\overline{AB},$ and $\overline{BC}$ intersects diagonal $\overline{AC}$ at points $P$ and $Q$ with $AP < AQ,$ as shown. Suppose that $AP=3,$ $PQ=9,$ and $QC=16.$ Then the area of $ABCD$ can be expressed in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
We repeatedly toss a coin until we get either three consecutive heads ($HHH$) or the sequence $HTH$ (where $H$ represents heads and $T$ represents tails). What is the probability that $HHH$ occurs before $HTH$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{5}}
Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, find the largest integer value of $x$ such that the area of this new triangle is less than 2004.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Into how many smallest non-overlapping tetrahedrons can a cube be divided?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
设正实数 a,b 满足 a+2b=1，令 I 为 \df{b^2+a+2}{ab} 的最小值，则 [I]=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
已知 $f(x)=e^{x}-m x$.
若 $x>0$ 时, 不等式 $(x-2) f(x)+m x^{2}+2>0$ 恒成立, 求实数 $m$ 的取值范围;
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left[\frac{1}{2},+\infty\right)$}
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
在边长为 8 的正方形 $A B C D$ 中, $M$ 是 $B C$ 边的中点, $N$ 是 $D A$ 边上一点, 且 $D N=3 N A$。若对于常数 $m$ ，在正方形的边上恰有 6 个不同的点 $P$ ，使 $\overrightarrow{P M} \cdot \overrightarrow{P N}=m$ ，则实数 $m$ 的取值范围是 \((a, b)\)。请计算a+b的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}