inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the sequence  $\{y_n\}_{n=1}^{\infty}$  defined by  $y_1=y_2=1$  and
\[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\]
find all integers  $k$  such that every term of the sequence is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k = 3}
Determine whether the integral \( \int_{0}^{\infty} \frac{1}{e^x + e^{-x}} \, dx \) converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Let $a, b, c$ be positive integers such that \( \gcd(a, b) + \text{lcm}(a, b) = \gcd(a, c) + \text{lcm}(a, c). \) Determine whether it necessarily follows that \( b = c. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b = c}
Let $a$, $b$, and $c$ be three prime numbers such that $a + b = c$. If the average of two of the three primes is four less than four times the fourth power of the last, find the second-largest of the three primes.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Determine the cardinality of $\mathbb{R}^\infty$. Is it the same as $\beth_1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beth_1}
Given that a normal vector of the straight line $l$ is $\overrightarrow{n} = (1, -\sqrt{3})$, find the size of the inclination angle of this straight line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
The hypotenuse of a right triangle is $10$ inches and the radius of the inscribed circle is $1$ inch. The perimeter of the triangle in inches is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$22$}
Let $R$ be a commutative ring with unity and $I$ be an ideal of $R$. Determine whether $\text{Hom}_R(R/I, R/I) \cong R/I$ as $R$-modules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R/I}
Suppose $H$ is an indefinite quaternion algebra over $\mathbb{Q}$. Determine whether there are infinitely many quadratic fields that can be embedded into $H$. Answer 'finite' or 'infinite'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{infinite}
Planes \(\alpha\) and \(\beta\) intersect. Points \(\mathrm{M}\) and \(\mathrm{N}\) do not lie on these planes. A perpendicular is dropped from point \(\mathrm{M}\) to plane \(\alpha\) and from point \(\mathrm{N}\) to plane \(\beta\), and these perpendiculars lie in the same plane. If a perpendicular is then dropped from point \(\mathrm{M}\) to plane \(\beta\) and from point \(\mathrm{N}\) to plane \(\alpha\), must these perpendiculars also lie in the same plane? Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
A circular plot with a radius of 30 feet requires a light placed directly above its center at a height that maximizes illumination at the plot's edge. The intensity of illumination, $I$, is directly proportional to the cosine of the angle of incidence, $\theta$, and inversely proportional to the square of the distance from the light source. If $h$ is the height of the light source, express the intensity $I$ in terms of $h$ and determine the height $h$ that maximizes the intensity at the edge of the plot.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15\sqrt{2}}
4. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system

$$
\left\{\begin{array}{l}
y=-b-x^{2}, \\
x^{2}+y^{2}+8 a^{2}=4+4 a(x+y)
\end{array}\right.
$$

has at least one solution $(x ; y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b\leq2\sqrt{2}+\frac{1}{4}}
Bobby received his weekly paycheck today. Apart from his usual salary of $450 per week, he also received a bonus this week, which varies depending on the company's performance. If the company's performance rate this week was 12%, he receives that percentage as a bonus. The actual paycheck is reduced because of taxes, insurance payments, and fees removed from his pay before his employer writes the check. The government removes 1/3 in federal taxes, 8% in state taxes, and 5% in local taxes from his total income (salary and bonus combined). The health insurance removes $50 per week, the life insurance removes $20 per week, and the city charges him a fee of $10 per week for parking. Additionally, Bobby contributes 3% of his salary to his retirement fund, and his employer matches 50% of his contribution. What is the final amount left in Bobby's paycheck, in dollars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$176.98}
已知 $x^{2}=4 y, M(2,2)$, 过 $M$ 点的直线交抛物线于 $A, B$ 两点, 过 $A, B$ 两点作抛物线的切线交与点 $P$, 求 $S_{\triangle A B P}$ 的最小值。请给出该最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
6. Points $A_{1}, \ldots, A_{12}$ are the vertices of a regular 12-gon. How many different 11-segment open broken lines without self-intersections with vertices at these points exist? Broken lines that can be transformed into each other by rotation are considered the same.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Given a general triangle \(ABC\) with points \(K, L, M, N, U\) on its sides:

- Point \(K\) is the midpoint of side \(AC\).
- Point \(U\) is the midpoint of side \(BC\).
- Points \(L\) and \(M\) lie on segments \(CK\) and \(CU\) respectively, such that \(LM \parallel KU\).
- Point \(N\) lies on segment \(AB\) such that \(|AN| : |AB| = 3 : 7\).
- The ratio of the areas of polygons \(UMLK\) and \(MLKNU\) is 3 : 7.

Determine the length ratio of segments \(LM\) and \(KU\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Determine if there exists a solution in positive integers for the system of equations: \[ \begin{array}{l} x_{1}^{2}+x_{2}^{2}+\cdots+x_{1985}^{2}=y^{3}, \\ x_{1}^{3}+x_{2}^{3}+\cdots+x_{1985}^{3}=z^{2} \end{array} \] where $x_{i} \neq x_{j}$ for $i \neq j$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Suppose $\cos R = \frac{5}{13}$ in the triangle below. If $RS = 13$, find the length of $QS$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{QS = 12}
If $A$ is a matrix with entries in the quaternions and $q$ is a quaternion, determine whether $(qA)^T = q A^T$ or $(qA)^T = \overline{q}A^T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(qA)^T = q A^T}
Determine the values of \(\alpha\) for which the sequence \(f_n(x) = n^\alpha X_{B_{1/n}}\) converges pointwise in \(\mathbb{R}^3\). For those values of \(\alpha\), find the pointwise limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha \leq 0}
Everyday Isaac wakes up at 7 in the morning and goes to sleep at a certain time in the night. Isaac sleeps for 8 hours in a day. At what time does Isaac go to sleep at night?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11:00}
Determine whether the limit \( \lim_{x \to y^-} \sum_{\{n : r_n \in (x, y)\}} \frac{1}{2^n} \) is equal to zero, where \( \{r_n\}_{n=1}^\infty \) is an enumeration of the rational numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A shop offers an initial discount of $25\%$ on all items. A week later, an additional discount of $15\%$ is applied to the already discounted prices. Finally, an extra $10\%$ discount is applied to the new sale price on a special sale day. The shop advertises that the cumulative discount equals $50\%$ off the original prices. Compute the actual total discount and find the percentage difference between the shop's claim and the true discount.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.375\%}
Each artist in the creative collective "Patience and Labor" has their own working schedule. Six of them paint one picture every two days, another eight of them paint one picture every three days, and the rest never paint pictures. From September 22 to September 26, they painted a total of 30 pictures. How many pictures will they paint on September 27?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $\{a_n\}$ be a sequence such that $a_n \geq 0$ for all $n$, $\{a_n\}$ is monotonically decreasing, and $\sum_{n=1}^\infty a_n$ converges. Determine if it is true that as $n\rightarrow\infty$, $$n\log n\;a_n\rightarrow 0$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( M = \left\{(x, y) \mid \tan(\pi y) + \sin^2(\pi x) = 0 \right\} \) and \( N = \left\{(x, y) \mid x^2 + y^2 \leq 2 \right\} \). Calculate the number of points in the intersection \( M \cap N \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the domain of the function $g(z, \bar{z}) = f(x, y)$, where $z = x + iy$ and $\bar{z} = x - iy$. Is the domain $g: \mathbb{C} \to \mathbb{C}$ or $g: \mathbb{C}^2 \to \mathbb{C}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{C}}
## Problem B3

$\mathrm{k}$ is the positive root of the equation $\mathrm{x}^{2}-1998 \mathrm{x}-1=0$. Define the sequence $\mathrm{x}_{0}, \mathrm{x}_{1}, \mathrm{x}_{2}, \ldots$ by $\mathrm{x}_{0}$ $=1, \mathrm{x}_{\mathrm{n}+1}=\left[\mathrm{k} \mathrm{x}_{\mathrm{n}}\right]$. Find the remainder when $\mathrm{x}_{1998}$ is divided by 1998.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. Find the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{359}
Evaluate the limit \(\lim_{a\rightarrow \infty} \int_{-a}^a \cos(x+b)\,dx\) and determine the values of \(b\) for which it exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The equation \[\frac{x}{x+1} + \frac{x}{x+2} = kx\]has exactly two complex roots. Find all possible complex values for $k.$

Enter all the possible values, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0,\tfrac32, 2i, -2i}
We say an integer  $n$  is naoish if  $n \geq 90$  and the second-to-last digit of  $n$  (in decimal notation) is equal to  $9$ . For example,  $10798$ ,  $1999$  and  $90$  are naoish, whereas  $9900$ ,  $2009$  and  $9$  are not. Nino expresses 2020 as a sum:
    \[
    2020=n_{1}+n_{2}+\ldots+n_{k}
    \]
    where each of the  $n_{j}$  is naoish.

    What is the smallest positive number  $k$  for which Nino can do this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $(b_n)$ be a sequence of real numbers defined as $b_0 = \frac{1}{2023}$ and $b_{n+1} = b_n - \arctan(\tan^2 b_n)$ for $n \geq 0$. Determine the value of $\lim_{n \to \infty} (\sqrt{n} \cdot b_n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the additive upper density \( \operatorname{\delta}_{(u, +)}(A_i) \) for the sets \( A_i = \{n \in \mathbb{N} : L(n) \sim_i c_i \sqrt{n} \} \) where \( i \in \{1, 2\} \), \( L(n) = \sum_{i=1}^{n} \lambda(i) \), and \( \lambda(n) = (-1)^{\Omega(n)} \) with \( \Omega(n) \) being the number of prime divisors of \( n \) (counting with multiplicity).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the ring of convergence of the Laurent series for \( f(z) = \frac{1}{e^z - 1} \) centered at 0 is the exterior of the unit disk \(|z| \geq 1\). If so, what is the radius of convergence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?
$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 + 10\sqrt{3}}
Determine whether the cardinal characteristic \( \mathfrak{sd} \), defined as the minimum \( \kappa \) such that player I has a winning strategy in the game \( d(\kappa) \), is equal to \( \omega_1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega_1}
Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic. Assume that $X$ has rational singularities, meaning there exists a resolution of singularities $f:Y\to X$ such that for all $i>0$, $R^if_*\mathcal{O}_Y=0$ and $R^if_*\omega_Y=0$. Determine if $f_*\omega_Y=\omega_X$, i.e., if the trace is an isomorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f_*\omega_Y = \omega_X}
Determine if the function \( Z(s) = \prod_{p}\frac{1}{1+p^{-s}} = \sum_{k=1}^{\infty}\frac{(-1)^{\Omega(k)}}{k^s} \), where \( \Omega(k) \) is the number of distinct prime factors of \( k \), can be analytically continued. Specifically, find the value of \( Z(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The city of Neverreturn has $N$ bus stops numbered $1, 2, \\cdots , N.$ Each bus route is one-way and has only two stops, the beginning and the end. The route network is such that departing from any stop one cannot return to it using city buses. When the mayor notices a route going from a stop with a greater number to a stop with a lesser number, he orders to exchange the number plates of its beginning and its end. Can the plate changing go on infinitely?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
no

}
There is a polynomial $P$ such that for every real number $x$,
\[
  x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).
\]
When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{511}
用一个平面去截棱长为a的正方体，得截面是一个四边形，这个四边形面积的取值范围为。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$( 0 , \sqrt { 2 } a ^ { 2 } ]$.}
求所有的正整数$x，y$，使得$( x ^ { 2 } +y ) ( y ^ { 2 } + x )$是一个素数的5次幂．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(2,5)$和$(5,2)$}
if the least common addition of two prime numbers x and y is 36 , where x < y , then the value of 4 x + y is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Given the parabola $C:x^{2}=4y$, a line passing through the point $M(0,2)$ intersects $C$ at points $A$ and $B$. A line parallel to the $y$-axis is drawn through point $B$ and intersects line $AO$ at point $D$. An arbitrary tangent line $l$ (not including the $x$-axis) of the parabola $C$ intersects the line $y=2$ at point $N_{1}$ and intersects the line $y=y_{D}$ at point $N_{2}$. Calculate the value of $|MN_{2}|^{2}-|MN_{1}|^{2}=\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{The answer is } 8.}
Find the mass percentage of Cl in a mixture containing 3 moles of NaClO and 2 moles of NaClO2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43.85\%}
Inside a large rectangle of size $A \times B$ there is a smaller rectangle of size $a \times b$ (see the figure).

Find the difference between the total area of the yellow quadrilaterals and the total area of the green quadrilaterals, given that $A=20$, $B=30$, $a=4$, $b=7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Part of an 'n-pointed regular star' is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_1,A_2,\cdots,A_n$ are congruent, and angles $B_1,B_2,\cdots,B_n$ are congruent. If the acute angle at $A_1$ is $10^\circ$ less than the acute angle at $B_1$, then find the value of $n$. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given the following four propositions:

① Two lines that do not have common points are parallel;

② Two lines that are perpendicular to each other intersect;

③ Lines that are neither parallel nor intersecting are skew lines;

④ Two lines that are not in the same plane are skew lines.

Among these, the correct propositions are __________. (Fill in the serial numbers)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 4}
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations $ax+by=1\,$ $x^2+y^2=50\,$ has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. How many such ordered pairs $(a,b)\,$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$72$}
A truck with a width of $2.7$ meters needs to pass through a semi-circular tunnel with a radius of $4.5$ meters (double lanes, no violations allowed). What is the maximum height of the truck's flat-top canopy from the ground that will allow it to pass through the tunnel? Express your answer in meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.6}
A square \( C \) is completely covered with an integer number of unit squares, without overlaps. If one places inside \( C \) as many squares of area 2 as possible, with sides parallel to the sides of \( C \), it is possible to cover eight-ninths of the area of the square. Determine all possible dimensions of such squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 6, 9, 12, 15, 18, 21, 24}
Call a positive integer $n \geq 2$ "junk" if there exist two distinct $n$-digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that:
- $a_1 + a_2 = b_1 + b_2$,
- $a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1}$ for all $2 \leq i \leq n-1$,
- $a_{n-1} + a_n = b_{n-1} + b_n$.

Find the number of junk positive integers less than or equal to $2016$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
A permutation of the set of positive integers $[n] = \{1, 2, \ldots, n\}$ is a sequence $(a_1, a_2, \ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$ . Let $P(n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1\leq k\leq n$ . Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\boxed{4489}\)}
Determine the smallest positive integer \(n\) for which there exists positive real numbers \(a\) and \(b\) such that
\[(a + 3bi)^n = (a - 3bi)^n,\]
and compute \(\frac{b}{a}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
The median \( A M \) of triangle \( A B C \) is equal to half of side \( B C \). The angle between \( A M \) and altitude \( A H \) is \( 40^\circ \). Find the angles of triangle \( A B C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ, 25^\circ, 65^\circ}
As shown in the figure, three circles intersect creating seven regions. Fill integers 0 through 6 into the seven regions such that the sum of the four numbers in each circle is equal. What is the maximum possible sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. What is the number of square units in the area of the fourth square?

[Diagram similar to the original, with different labeled areas]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{138}
Team A was tasked with installing 66 air conditioners in Community A, and Team B was tasked with installing 60 air conditioners in Community B. Both teams started and finished their tasks at the same time. If Team A installs 2 more air conditioners per day than Team B, calculate the daily installation numbers of Team A and Team B. Express your answer as two whole numbers, representing the daily installation rates for Team A and Team B, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22, 20}
A square was cut into four equal rectangles and then arranged to form a large letter 'P' as shown in the diagram, with a perimeter of 56. What is the perimeter of the original square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
There is a polynomial $P$ such that for every real number $x$,
\[
  x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).
\]When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{341}
In the diagram, eight circles, each with a radius of 1, are drawn inside a rectangle. Four of the circles are tangent to two sides of the rectangle and to two other circles. The other four circles are tangent to one side of the rectangle and to three other circles. A region has been shaded, consisting of three spaces (each space bounded by a different set of four circles) and four of the circles themselves. Calculate the area of this shaded region. Express your answer as a single number, rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15.14}
In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{341}
例3 设$f ( x ) = a x ^ { 2 } + b x + c ( a > 0 )$，方程f(x)=x的两个根是$x _ { 1 }$和$x _ { 2 }$，且$x _ { 1 } > 0$,$x _ { 2 } - x _ { 1 } > \frac { 1 } { a } .$.又若$0 < t < x _ { 1 }$，试比较$f(t)$与$x _ { 1 }$的大小.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$当0 < t < x _ { 1 }时，有f ( t ) > x _ { 1 }$.}
For each $t\geqslant0$, let $S_{t}$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$ f(1)-2 f(2 / 3)+f(1 / 3)\geqslant t[f(2 / 3)-2 f(1 / 3)+f(0)] . $$ Develop necessary and sufficient conditions on $t$ for $S_{t}$ to be closed under multiplication. Please provide the sum of the endpoints of the interval for which the condition holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
设$A=100101102103\cdot \cdot \cdot 798799$为2100位的正整数，其由100到799的三位数顺序连接而成，则$A$被126除的余数为______。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
The length of edge AB is 51 units, and the lengths of the other five edges of the tetrahedron ABCD measure 10, 20, 25, 34, and 45 units. Determine the length of edge CD.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}