inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the largest value of \( 3k \) such that the equation \( \sqrt{x^{2}-k} + 2 \sqrt{x^{3}-1} = x \) has a real root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $f_1$ and $f_2$ be holomorphic functions. Suppose $f_2$ has a zero of order $k$ at $z'_0 = f_1(z_0)$. If $f_1(z)-z_0'$ has a zero of order $m$ at $z_0$, and $f_2(z)-(z-z_0')^k$ has a zero of order $n$ at $z_0'$, determine the order of the zero of $g = f_2 \circ f_1$ at $z_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{mk}
On each side of an equilateral triangle with side length  $n$  units, where  $n$  is an integer,  $1 \leq n \leq 100$ , consider  $n-1$  points that divide the side into  $n$  equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of  $n$  for which it is possible to turn all coins tail up after a finite number of moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
An isosceles triangle has two exterior angles in the ratio of 1:4. Find the measures of its three interior angles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140^	ext{\circ}, 20^	ext{\circ}, 20^	ext{\circ}}
Determine the automorphism group of the graph \( \Gamma \), where \( \Gamma \) is defined as follows: Let \( S_6 \) be the symmetric group on six letters, and let \( X \) denote the conjugacy class containing \((12)(34)\). The vertex set of \( \Gamma \) is \( X \), and the edges are precisely the 2-element subsets of \( X \) that commute as elements of \( S_6 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_6}
If 5a + some b = 70, where a and b are positive integers, what is the largest possible value of g for which exactly one pair of integers (a, b) makes the equation true?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
The smallest three positive proper divisors of an integer n are $d_1 < d_2 < d_3$ and they satisfy $d_1 + d_2 + d_3 = 57$. Find the sum of the possible values of $d_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Given a cube $A B C D A 1 B 1 C 1 D 1$ with edge length $a$. On the rays $C 1 C$, $C 1 B 1$, and $C 1 D 1$, segments $C 1 M$, $C 1 N$, and $C 1 K$ are respectively laid out, each equal to $\frac{5}{2} a$. Construct the section of this cube by the plane passing through points $M$, $N$, and $K$, and find the area of the resulting section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^2 \sqrt{3}}{8}}
The diagram shows eight small squares. Six of these squares are to be shaded so that the shaded squares form the net of a cube. In how many different ways can this be done? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Boys and girls are standing in a circle (there are both), a total of 20 children. It is known that each boy's neighbor in the clockwise direction is a child in a blue T-shirt, and each girl's neighbor in the counterclockwise direction is a child in a red T-shirt. Can you uniquely determine how many boys are in the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $O$ be a fixed point on a plane, and let $A$, $B$, and $C$ be three non-collinear points on the same plane. A moving point $P$ satisfies the equation $\overrightarrow{OP} = \overrightarrow{OA} + λ(\frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} + \frac{\overrightarrow{AC}}{|\overrightarrow{AC}|}), λ ∈ [0, +∞)$. Determine which center of $\triangle ABC$ the trajectory of $P$ will always pass through.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Incenter}
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least  $10$  matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Let $ABCD$ be a rhombus with angle $\angle A = 60^\circ$. Let $E$ be a point, different from $D$, on the line $AD$. The lines $CE$ and $AB$ intersect at $F$. The lines $DF$ and $BE$ intersect at $M$. Determine the angle $\angle BMD$ as a function of the position of $E$ on $AD.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Points  $A, B$  and  $C$  lie on the same line so that  $CA = AB$ . Square  $ABDE$  and the equilateral triangle  $CFA$ , are constructed  on the same side of line  $CB$ . Find the acute angle between straight lines  $CE$  and  $BF$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75^\circ}
$ AA_{3}$  and  $ BB_{3}$  are altitudes of acute-angled  $ \triangle ABC$ . Points  $ A_{1}$  and  $ B_{1}$  are second points of intersection lines  $ AA_{3}$  and  $ BB_{3}$  with circumcircle of  $ \triangle ABC$  respectively.  $ A_{2}$  and  $ B_{2}$  are points on  $ BC$  and  $ AC$  respectively.  $ A_{1}A_{2}\parallel AC$ ,  $ B_{1}B_{2}\parallel BC$ . Point  $ M$  is midpoint of  $ A_{2}B_{2}$ .  $ \angle BCA \equal{} x$ . Find  $ \angle A_{3}MB_{3}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x}
The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["39"]}
Evaluate the sum
$\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)$
$=3\cdot\frac{\ln2}{2}-\frac{\ln3}{3}-\frac{\ln4}{4}-\frac{\ln5}{5}+3\cdot\frac{\ln6}{6}-\frac{\ln7}{7}-\frac{\ln8}{8}-\frac{\ln9}{9}+3\cdot\frac{\ln10}{10}-\cdots.$
(As usual, $lnx$ denotes the natural logarithm of x.)
 The original answer is in $(\log m)^{n}$format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find four positive integers such that the sums of the numbers taken three at a time are 6, 7, 8, and 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 2, 3, 4}
Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}$. If $AM = 180$, find $LP$.
[asy] import markers; defaultpen(fontsize(8)); size(300); pair A=(0,0), B=(30*sqrt(331),0), C, K, L, M, P; C = intersectionpoints(Circle(A,450), Circle(B,300))[0]; K =  midpoint(A--C); L = (3*B+2*A)/5; P = extension(B,K,C,L); M = 2*K-P; draw(A--B--C--cycle); draw(C--L);draw(B--M--A); markangle(n=1,radius=15,A,C,L,marker(markinterval(stickframe(n=1),true))); markangle(n=1,radius=15,L,C,B,marker(markinterval(stickframe(n=1),true))); dot(A^^B^^C^^K^^L^^M^^P); label("$A$",A,(-1,-1));label("$B$",B,(1,-1));label("$C$",C,(1,1)); label("$K$",K,(0,2));label("$L$",L,(0,-2));label("$M$",M,(-1,1)); label("$P$",P,(1,1)); label("$180$",(A+M)/2,(-1,0));label("$180$",(P+C)/2,(-1,0));label("$225$",(A+K)/2,(0,2));label("$225$",(K+C)/2,(0,2)); label("$300$",(B+C)/2,(1,1)); [/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{072}
Let $a$, $b$, and $c$ be positive integers with $a\ge$ $b\ge$ $c$ such that $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$. What is $a$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{253}
Identical gases are in two thermally insulated vessels with volumes $V_{1}=1$ L and $V_{2}=2$ L. The gas pressures are $p_{1}=2$ atm and $p_{2}=3$ atm, and their temperatures are $T_{1}=300$ K and $T_{2}=400$ K, respectively. The gases are mixed. Determine the temperature that will be established in the vessels after mixing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{369 K}
On the number line, consider the point $x$ that corresponds to the value $10$. Consider $24$ distinct integer points $y_1, y_2, \ldots, y_{24}$ on the number line such that for all $k$ where $1 \leq k \leq 12$, $y_{2k-1}$ is the reflection of $y_{2k}$ across $x$. Find the minimum possible value of 
\[\sum_{n=1}^{24} (|y_n-1| + |y_n+1|).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{494}
Using eight $2 \times 1$ small rectangles as shown in the diagram, a $4 \times 4$ square can be formed. If a formed square pattern is considered the same as another if it can be rotated to match, the question is: How many different square patterns can be formed such that it has only one diagonal as its axis of symmetry?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that point $P$ moves on the circle $x^{2}+(y-2)^{2}=1$, and point $Q$ moves on the ellipse $\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of the distance $PQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{6}}{2} + 1}
Given a circle of radius 1, starting from a fixed point, points are placed along the circumference at curved distances of 0, 1, 2, ..., respectively. Determine the minimum number of these points needed to ensure that at least two of them are less than a distance of \( \frac{1}{5} \) apart.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the principal value of \( \left(\left(\frac{1+\sqrt{3}i}{2}\right)^{-3}\right)^{1-i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-e^{\pi}}
Let \( f(x)=e^{x+1}-1 \) and \( g(x)=100|f(x)|-\sum_{k=1}^n|f(x^k)| \), where \( n \in \mathbb{N} \). It is given that \( g(x) \) is a differentiable function over the complete real numbers. Find the sum of all possible values of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Determine the number of distinct $k$-dimensional subspaces in an $n$-dimensional vector space $V$ over a finite field with $q$ elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dbinom{n}{k}_q}
Let  $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$ , determine the value of  $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$ , where  $n\ge 2$  . ( $[x]$  denotes the greatest integer not exceeding x)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8n^3 - 3n^2 + 13n - 6}{6}}
Let $x,$ $y,$ $z$ be positive real numbers.  Find the minimum value of
\[\frac{(1 + 5z)(4z + 3x)(5x + 6y)(y + 18)}{xyz}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5120}
In  $\triangle ABC$ , point  $D$  lies on side  $AC$  such that  $\angle ABD=\angle C$ . Point  $E$  lies on side  $AB$  such that  $BE=DE$ .  $M$  is the midpoint of segment  $CD$ . Point  $H$  is the foot of the perpendicular from  $A$  to  $DE$ . Given  $AH=2-\sqrt{3}$  and  $AB=1$ , find the size of  $\angle AME$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
For how many natural numbers \( n \) not exceeding 600 are the triples of numbers

\[ \left\lfloor \frac{n}{2} \right\rfloor, \left\lfloor \frac{n}{3} \right\rfloor, \left\lfloor \frac{n}{5} \right\rfloor \quad\text{and}\quad \left\lfloor \frac{n+1}{2} \right\rfloor, \left\lfloor \frac{n+1}{3} \right\rfloor, \left\lfloor \frac{n+1}{5} \right\rfloor \]

distinct? As always, \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{440}
Knot must travel from point \( K \) to point \( T \) on a figure where \( OK = 2 \) and \( OT = 4 \). Before reaching \( T \), Knot must first reach both solid lines depicted in the figure in order to collect the two pieces of the Lineforce. What is the minimal distance Knot must travel to accomplish this?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{5}}
From the positive integers, $m, m+1, \dots, m+n$, only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$. Find the maximum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
An empty fuel tank with a capacity of 300 gallons was filled partially with three different types of fuel: fuel A, fuel B, and fuel C. Fuel A contains 10% ethanol by volume, fuel B contains 15% ethanol by volume, and fuel C contains 20% ethanol by volume. If the total volume of ethanol in the full fuel tank is 'x' gallons, where 30 <= x <= 60, find an equation that represents the relationship between the volumes of fuel A, B, and C added to the tank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
How many ways are there to choose distinct positive integers  $a, b, c, d$  dividing  $15^6$  such that none of  $a, b, c,$  or  $d$  divide each other? (Order does not matter.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1225}
Given the expansion of $(a+b)^n$, if the fourth term has the maximum binomial coefficient, find all possible values of $n$. Express your answer as a list of values separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5, 6, 7}
Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{167}
已知正整数n不超过2000，并且能表示成不少于60个连续正整数之和，那么，这样的n的个数是 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 }
There are 20 points, each pair of adjacent points are equally spaced. By connecting four points with straight lines, you can form a square. Using this method, you can form _ squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $P$ and $Q$ be points on line $l$ with $P Q=12$. Two circles, $\omega$ and $\Omega$, are both tangent to $l$ at $P$ and are externally tangent to each other. A line through $Q$ intersects $\omega$ at $A$ and $B$, with $A$ closer to $Q$ than $B$, such that $A B=10$. Similarly, another line through $Q$ intersects $\Omega$ at $C$ and $D$, with $C$ closer to $Q$ than $D$, such that $C D=7$. Find the ratio $A D / B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{9}}
A square was cut into four equal rectangles and used to form a large letter П, as shown in the figure, 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}
What is the mass percentage of Ca in a compound if the mass percentage of Ca in Calcium carbonate is the same as in that compound?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40.04\%}
Compute the largest possible number of distinct real solutions for  $x$  to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where  $a$ ,  $b$ , and  $c$  are real numbers.

[i]Proposed by Tristan Shin
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In triangle $ABC$, if it is known that $\angle A : \angle B : \angle C = 2 : 3 : 5$, then triangle $ABC$ is a _______ triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{right-angled}}
For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[ \sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)}, \] where $N=6^{1999}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
As illustrated, to set up a "gesture password" on a mobile phone in a $2 \times 2$ grid containing 9 points, start at one point and consecutively connect multiple points with line segments. Each segment cannot pass through any unused points, and the endpoints of each segment cannot both be previously used points. If a person's gesture password starts at the center point and uses only three points, how many connection methods are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
In the figure below, parallelograms $ABCD$ and $BFEC$ have areas $1234\, \text{cm}^2$ and $2804\, \text{cm}^2$, respectively. Points $M$ and $N$ are chosen on sides $AD$ and $FE$, respectively, so that segment $MN$ passes through $B$. Find the area of $\triangle MNC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Let  $ABC$  be a triangle with  $\angle BAC = 90^\circ$ . Construct the square  $BDEC$  such as  $A$  and the square are at opposite sides of  $BC$ . Let the angle bisector of  $\angle BAC$  cut the sides  $[BC]$  and  $[DE]$  at  $F$  and  $G$ , respectively. If  $|AB|=24$  and  $|AC|=10$ , calculate the area of quadrilateral  $BDGF$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The area of quadrilateral BDGF is 338.}
试求出具有如下性质的最小的正整数：它可以表示为2002个各位数字和相等的正整数之和，又可以表示为2003个各位数字之和相等的正整数之和．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10010}
Let  $ n\geq 3 $  be an integer and let  $ x_1,x_2,\ldots,x_{n-1} $  be nonnegative integers such that 

\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n  x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}

Find the minimal value of  $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n(n-1)}
Determine the maximum number of bishops that we can place in a  $8 \times 8$  chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop. 

Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the  $2$  main diagonals and the ones parallel to those ones.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
$A$ rectangle has sides of length $a$ and $36$. $A$ hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length $36$. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of $24$, the hexagon has the same area as the original rectangle. Find $a^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
In right triangle  $ ABC$  with right angle at  $ C$ ,  $ \angle BAC < 45$  degrees and  $ AB \equal{} 4$ .  Point  $ P$  on  $ AB$  is chosen such that  $ \angle APC \equal{} 2\angle ACP$  and  $ CP \equal{} 1$ .  The ratio  $ \frac{AP}{BP}$  can be represented in the form  $ p \plus{} q\sqrt{r}$ , where  $ p,q,r$  are positive integers and  $ r$  is not divisible by the square of any prime.  Find  $ p\plus{}q\plus{}r$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Rory uses four identical standard dice to build a solid. Whenever two dice touch, the numbers on the touching faces are the same. The numbers on some faces of the solid are shown. Given that the numbers on opposite faces of a standard die add up to 7, what number is written on the face marked with a question mark? Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let  $ n > 1, n \in \mathbb{Z}$  and  $ B \equal{}\{1,2,\ldots, 2^n\}.$  A subset  $ A$  of  $ B$  is called weird if it contains exactly one of the distinct elements  $ x,y \in B$  such that the sum of  $ x$  and  $ y$  is a power of two. How many weird subsets does  $ B$  have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{n+1} }
The incircle  $\Gamma$  of a scalene triangle  $ABC$  touches  $BC$  at  $D, CA$  at  $E$  and  $AB$  at  $F$ . Let  $r_A$  be the radius of the circle inside  $ABC$  which is tangent to  $\Gamma$  and the sides  $AB$  and  $AC$ . Define  $r_B$  and  $r_C$  similarly. If  $r_A = 16, r_B = 25$  and  $r_C = 36$ , determine the radius of  $\Gamma$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
计算 $\frac{3^3\div3(3)+3}{\frac{3}{3}}+3!.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Evaluate the improper integral by converting to polar coordinates: $$\int_{0}^\infty \int_{-\infty}^{\infty} \exp[-((y^2+z^2)/2)]\,dy\,dz$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
The complex number \( \left(\sqrt{\frac{1}{2} + \frac{\sqrt{3}}{2}i}\right)^{1/3} \) can be expressed in polar form as \( r(\cos x + i\sin x) \) with \( r < 0 \). Find the smallest positive value of \( x \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Cassandra sets her watch to the correct time at noon. At the actual time of $ \text{1: \!\,00 PM}$, she notices that her watch reads $ \text{12: \!\,57}$ and $ 36$ seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch ﬁrst reads $ \text{10: \!\,00 PM}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10:25 PM
}
On a planet,far,far away,the Yaliens have defined: \( x \) "equals" \( y \) if and only if
 \( \left| {x - y}\right|  \leq  3 \) . Let \( S \) be a set of positive integers. What is the smallest possible number
 of elements in \( S \) such that,for any positive integer \( r \) ,where \( 1 \leq  r \leq  {2024},r \) "equals"
 some element in \( S \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
290
}
Determine the number of partitions of the natural numbers into countably many countable sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the radius of convergence for the differential equation \((t^3-3t^2+3t)y''+4(t-1)y'+y = 0\) at \(t = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the expected number of points in the interval \([0,1]\) for the sine point process, which is a determinantal point process with kernel \(K(x,y)=\frac{\sin \pi(x-y)}{\pi(x-y)}\) when \(x\neq y\) and \(1\) when \(x=y\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}