inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider a square on a Cartesian plane with vertices at \((0, 0), (0, 65), (65, 65),\) and \((65, 0)\). Determine the number of ways to select two grid points inside this square (excluding its boundary) such that at least one of these points lies on the line \(y = x\) or \(y = 65 - x\), and neither point lies on any line parallel to the coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500032}
Find the number of cyclic subgroups of order 10 in the group \( \mathbb{Z}_{30} \oplus \mathbb{Z}_{120} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
In a school, there are three classes, each with \( n \) students, and all \( 3n \) students have different heights. These students are grouped into \( n \) groups, with each group consisting of 3 students from different classes. The tallest student in each group is called the "tall one". It is known that no matter how the groups are formed, each class has at least 10 "tall ones". What is the smallest possible value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the Euler-Lagrange equation for the energy functional \[ E[u] = \frac{1}{2}\int_{\Omega} \|\nabla^3 u \|^2 \, dx \] where \( \Omega \subseteq \mathbb{R}^n \) and \( u: \Omega \to \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Delta^3 u = 0}
There are 2004 boxes on a table, each containing one ball. Some of the balls are white, and their number is even. You can point to any two boxes and ask if there is at least one white ball among them. What is the minimum number of questions required to guarantee determining at least one box that contains a white ball?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2003}
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 7, 8, 9\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1728}
Given the recursive sequence defined by \( a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n} \) with the initial condition \( a_1 = 1 \), evaluate \( \left \lfloor a_{2015} \right \rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
${\tiny ■} $She rolled a $6$ exactly three times;
${\tiny ■} $The product of her first three rolls is the same as the product of her last three rolls.
How many distinct sequences of six rolls could Kara have rolled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
72
}
Evaluate the integral \( \int_{\mathbb{C}} \frac{e^{-\frac{(x-im)^2}{2}}}{\sqrt{2\pi}} \, dx \) when \( m \) is replaced with \( im \), where \( i \) is the imaginary unit and \( m \) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \int_0^4 g'(\sqrt{x}) \, dx \), where the function \( f(x) \) is defined as \( f(x) = \begin{cases} x^x \ln x + k & : \ (x \geq 1) \\ f(2-x) & : \ (x<1) \end{cases} \) and satisfies \( \int_0^1 f(x) \, dx = 5 \). The function \( g(x) \) is given by \( g(x) = \int_0^x f(t) \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A and B start a business, with A investing the total capital of Rs. 5000, on the condition that B pays A interest @ 10% per annum on his half of the capital. A is a working partner and receives Rs. 500 per month from the total profit and any profit remaining is equally shared by both of them. At the end of the year, it was found that the income of A is twice that of B. What is the total profit for the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18750}
Find the area of the region bounded by the polar curve \( r^2 = a^2 \cos(2\theta) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2}
对 $\mathbb{R}^{100}$ 中的点 $x=\left(x_{1}, x_{2}, \cdots, x_{100}\right)$ 和 $y=\left(y_{1}, y_{2}, \cdots, y_{100}\right)$ 定义函数 $f(x,y)=\sqrt{\sum_{i=1}^{100}\left(x_{i}-y_{i}\right)^{2}}$. 当 $x \neq y$ 时，定义矩阵 $A_{x, y}=\left(a_{i j}\right)$ ，其中 $a_{i j}=$ $\frac{\partial}{\partial x_{i}} \frac{\partial}{\partial y_{j}} f(x, y)(i, j=1,2, \cdots, n)$. 则 $A_{x, y}$ 秩的最小可能值与最大可能值之和为 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
If there exist positive real numbers $x$ and $y$ satisfying $x\ln\frac{y}{x}-y{e^x}+x({x+1})≥0$, then the maximum value of $y$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{e}}
已知四面体$D-ABC$中，$AC=BC=AD=BD=1$，求$D-ABC$体积的最大值。答案的形式为\frac{k\sqrt{m}}{n}，请给出k + m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
A vessel of capacity 100 litres is fully filled with pure milk. Fifteen litres of milk is removed from the vessel and replaced with water. This process of removing 15 litres of the solution and replacing it with water is repeated three more times. Determine the quantity of pure milk remaining in the final milk solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.2}
Let \( k \) be a positive integer such that for any positive numbers \( a, b, c \) satisfying the inequality \( k(ab + bc + ca) > 5(a^2 + b^2 + c^2) \), there must exist a triangle with side lengths \( a, b, \) and \( c \). Find the minimum value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In triangle \(ABC\), angle \(C\) is \(60^\circ\) and the radius of the circumcircle of this triangle is \(2\sqrt{3}\).

A point \(D\) is taken on the side \(AB\) such that \(AD = 2DB\) and \(CD = 2\sqrt{2}\). Find the area of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
A lady made 3 round doilies with radii of 2, 3, and 10 inches, respectively. She placed them on a round table so that each doily touches the two others and the edge of the table. What is the radius of the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad \(N\), \(0<N<10\), it will jump to pad \(N-1\) with probability \(\frac{N}{10}\) and to pad \(N+1\) with probability \(1-\frac{N}{10}\). Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? Provide your answer as a fraction in simplest form, \(\frac{m}{n}\), and give the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{209}
Let $X = \{1, 2, \ldots, n\}$. Determine the number of ways to choose subsets $A$, $B$, and $C$ such that $A \cap B = \{1\}$, $B \cap C = \{2\}$, $A \cap C = \{3\}$, and $A$, $B$, and $C$ form a partition of $X$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For which sets $X$ is the composition operation $f \circ g$ on the set of all bijections $f: X \rightarrow X$ commutative?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|X| \leq 2}
Determine the genus of the subset \( S = \{[x_0, x_1, x_2, x_3] \in \mathbb{RP}^3 : x_0^2 + x_1^2 - x_2^2 - x_3^2 = 0\} \), which is an embedded submanifold of \( \mathbb{RP}^3 \) with codimension 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the function \( f : [-\pi, \pi] \to [-\pi,\pi] \) defined by \( f(x)= \frac{\sin(x)}{2} \). This function is a contraction. Using the fixed point iteration \( x_{k+1}= f(x_{k}) \) starting from \( x_{0}=\frac{\pi}{2} \), determine the minimum number of iterations \( k \) required such that the error satisfies \(|x_{k}-x^{*}| \leq \frac{1}{1024}\), where \( x^{*} \) is the fixed point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find all functions \( f: \mathbb{Z} \to \mathbb{Z} \) such that for all integers \( k \), the equation \( f(f(k+1)+3) = k \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(k) = k - 2}
If the square root of a positive number $m$ is $x+1$ and $5+2x$, then the value of $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
There are 3 valves in a pool system. With all 3 valves open, the pool is filled with water in 36 minutes. The first valve alone would fill the pool in 3 hours, while the second valve would fill the pool in 4 hours when functioning separately. If the third valve emits 30 cubic meters more water per minute than the first valve and 10 cubic meters more water per minute than the second valve, and there's a constant leakage of 20 cubic meters of water per minute from the pool, what is the capacity W of the pool?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{800}
In rectangle $PQRS$, $PQ = 150$. Let $T$ be the midpoint of $\overline{PS}$. Given that line $PT$ and line $QT$ are perpendicular, find the greatest integer less than $PS$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{212}
Suppose we calculate the Taylor polynomial of the $n^{\text{th}}$ degree centered at $a=0$ for the function $\sin(x)$. Determine the smallest integer $n$ such that the residual error $R_{n}f(x)$ is less than or equal to $0.01$ for all $x \in [0, \frac{1}{3}]$, and provide the value of this $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the geometric mean of 1 and 9. Express your answer in the form ±n, where n is a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±3}
4*. Find all such positive integers $k$ for which 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}
设有理数 $r=\frac{p}{q} \in(0,1)$, 其中 $p, q$ 为互素的正整数, 且 $p q$ 整除 3600. 这样的有理数 $r$ 的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Given the function \( y = \sqrt{a x^2 + b x + c} \) (where \(a, b, c \in \mathbb{R}\) and \(a < 0\)), the domain is \( D \). If the points \( (s, f(t)) \) (where \( s, t \in D \)) form a square, then the real number \( a \) equals ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
In a parallelepiped $A B C D A_1 B_1 C_1 D_1$, points $M$, $L$, and $K$ are located on edges $A D$, $A_1 D_1$, and $B_1 C_1$, respectively. Given that $B_1 K = \frac{1}{3} A_1 L$, $A M = \frac{1}{2} A_1 L$, and $K L = 2$, determine the length of the segment by which the plane $K L M$ intersects the parallelogram $A B C D$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
On the side \( BC \) of triangle \( ABC \), points \( A_1 \) and \( A_2 \) are marked such that \( BA_1 = 6 \), \( A_1A_2 = 8 \), and \( CA_2 = 4 \). On the side \( AC \), points \( B_1 \) and \( B_2 \) are marked such that \( AB_1 = 9 \) and \( CB_2 = 6 \). Segments \( AA_1 \) and \( BB_1 \) intersect at point \( K \), and segments \( AA_2 \) and \( BB_2 \) intersect at point \( L \). Points \( K \), \( L \), and \( C \) lie on the same line. Find \( B_1B_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The pentagon $ABCDE$ is inscribed in a circle. Points $M, Q, N,$ and $P$ are the bases of the perpendiculars dropped from vertex $E$ to the sides $AB, BC, CD$ (or their extensions) and diagonal $AD$ respectively. It is known that $|EP|=d$, and the ratio of the area of triangle $MQE$ to the area of triangle $PNE$ is $k$. Find $|EM|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d \sqrt{k}}
The least possible value of the expression e = (x - 1)(x - 3)(x - 4)(x - 6) + some constant for real values of x is 1. What is the missing constant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-0.5625}
If the quadratic inequality $2kx^{2}+kx- \frac {3}{8} < 0$ holds for all real numbers $x$, find the range of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3 < k < 0}
Solve the system in positive numbers:

$$
\begin{cases}x^{y} & =z \\ y^{z} & =x \\ z^{x} & =y\end{cases}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = y = z = 1}
Determine the number of elements \( \alpha \in \mathbb{F}_{3^5} \) such that \( \mathbb{F}_{3}(\alpha) = \mathbb{F}_{3^5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Determine the general form of the nth derivative of the function $$g(x)=\frac{x^7}{(1-x^2)^{2}}$$ at x = 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the circles $S_1: x^2 + y^2 + 4y - 1 = 0$, $S_2: x^2 + y^2 + 6x + y + 8 = 0$, and $S_3: x^2 + y^2 - 4x - 4y - 37 = 0$, which are mutually tangent, let $P_1, P_2, P_3$ be the points of contact and $C_1, C_2, C_3$ be their centers. Determine the ratio of the area of triangle $P_1P_2P_3$ to the area of triangle $C_1C_2C_3$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
The number of edges of a convex polyhedron is 99. What is the greatest number of edges a plane, not passing through its vertices, can intersect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Determine the number of homomorphisms from the dihedral group $D_{21}$ to the dihedral group $D_{15}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Using Rouché's theorem, determine the number of zeros of the polynomial \( f(z) = z^{47} - z^{23} + 2z^{11} - z^5 + 4z^2 + 1 \) within the disk \( D(0, \frac{2}{3}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a moving large circle $\odot O$ tangent externally to a fixed small circle $\odot O_{1}$ with radius 3 at point $P$, $AB$ is the common external tangent of the two circles with $A$ and $B$ as the points of tangency. A line $l$ parallel to $AB$ is tangent to $\odot O_{1}$ at point $C$ and intersects $\odot O$ at points $D$ and $E$. Find $C D \cdot C E = \quad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Francis and Kiera had breakfast at a cafe. Muffins cost $2 each, fruit cups cost $3 each, and coffee costs $1.50 per cup. There is also a 10% discount for customers who order a combination of 2 muffins and 1 fruit cup. Francis had 2 muffins, 2 fruit cups, and 1 coffee, while Kiera had 2 muffins, 1 fruit cup, and 2 cups of coffee. How much did their breakfast cost, taking into account any applicable discounts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$20.80}
A production line is checked daily by inspecting a sample of 100 units. Historically, the process has a yield of 80%, with a proportion defective of 20% and a standard deviation of 0.04 for the proportion defective. On a particular day, the sample contains 38 defective units. Assuming the prior distribution of the proportion defective, θ, is a Beta distribution, calculate the mean of the posterior distribution of θ for that day.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2905}
Find the number of solutions to the equation
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval \( (0, 150 \pi) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
If $O$ is the coordinate origin, and the line $l$ passing through point $P(1,1)$ intersects the graph of the function $f(x)=\frac{{2}^{x+1}}{{2}^{x}-2}$ at points $A$ and $B$, then $(\overrightarrow{OA}+\overrightarrow{OB})\cdot \overrightarrow{OP}=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
已知函数 $y=f(x)$ 的图像既关于点 $(1,1)$ 中心对称, 又关于直线 $x+y=0$轴对称。若 $x \in(0,1)$ 时, $f(x)=\log _{2}(x+1)$, 则 $f\left(\log _{2} 10\right)$ 的值为 $\qquad$.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{22}
Calculate the sum of the sequence $81 + 891 + 8991 + 89991 + \dots + 8\underbrace{99\dots99}_{n-1\text{ nines}} = \_\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^{n+1} - 9n - 10}
Given a sequence of events $\{A_n\}$ such that the covariance between any pair of distinct events satisfies $\operatorname{Cov}(I_{A_i}, I_{A_j}) \leq 0$ for all $i \neq j$, where $I_{A_k}$ denotes the indicator function for $A_k$. If $\sum P(A_n) = \infty$, determine $P[\limsup A_n]$. Provide your answer as a probability value or statement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the sign of the determinant of the matrix \(\begin{bmatrix} A & B \\ C & D \end{bmatrix}\) in terms of the determinant of \(AD - BC\), where \(A, B, C, D\) are commuting \(n \times n\) matrices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\det(AD - BC)}
Given the differentiable functions $z$ and $w$ of $x$ and $y$ that satisfy the equations:

$$xw^3 + yz^2 + z^3 = -1$$
$$zw^3 - xz^3 + y^2w = 1$$

Find the partial derivative of $z$ with respect to $x$, $\frac{\partial z}{\partial x}$, at the point $(x, y, z, w) = (1, -1, -1, 1)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{4}}
An isosceles trapezoid is circumscribed around a circle with radius $r$. The legs of the trapezoid form an angle $\alpha$ with the shorter base. Find the radius of the circle circumscribed around the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{r \sqrt{1 + \sin^2 \alpha}}{\sin^2 \alpha}}
On a tough week, Haji's mother sells goods worth $800, which is half the amount she sells on a good week. In addition to the revenue from selling goods, she also offers a 10% discount on purchases during tough weeks. She incurs a 5% tax on her total sales and has to spend $50 per week on transportation costs to get the goods. What's the total amount of money she makes (net profit) if she has 5 good weeks and 3 tough weeks?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$9252}
Mary has a sequence $m_{2}, m_{3}, m_{4}, \ldots$, such that for each $b \geq 2, m_{b}$ is the least positive integer $m$ for which none of the base-$b$ logarithms $\log _{b}(m), \log _{b}(m+1), \ldots, \log _{b}(m+2017)$ are integers. Find the largest number in her sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2188}
For a real number \( x \), find the maximum value of
\[
\frac{x^6}{x^{12} + 3x^8 - 6x^6 + 12x^4 + 36}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{18}}
Let  $ABCD$  be a trapezoid of bases  $AB$  and  $CD$  . Let  $O$  be the intersection point of the diagonals  $AC$  and  $BD$ . If the area of the triangle  $ABC$  is  $150$  and the area of the triangle  $ACD$  is  $120$ , calculate the area of the triangle  $BCO$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{200}{3}}
Xavier starts from point P towards point Q at a speed of 40 kmph and after a certain time interval increases his speed by 20 kmph. The distance between point P and point Q is 56 km. He takes 48 minutes to cover the distance. After how many minutes does he increase his speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A monkey is typing randomly on a typewriter with three keys: A, B, and C. Each key is pressed with equal probability, independently of previous presses. Let \( T \) be the random time at which the sequence "ABC" first appears in the typed sequence. What is the expected value of \( T \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Tanya and Vera are playing a game. Tanya has cards with numbers from 1 to 30. She arranges them in some order in a circle. For every two neighboring numbers, Vera calculates their difference by subtracting the smaller number from the larger one and writes down the resulting 30 numbers in her notebook. After that, Vera gives Tanya the number of candies equal to the smallest number written in the notebook. Tanya wants to arrange the cards to get as many candies as possible. What is the maximum number of candies Tanya can receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Calculate the work required to fill a hemispherical tank with water through a hole in its base. The tank has a radius of 2 feet and is positioned with its base circular. The water source is at the base, and the weight-density of water is 62.4 pounds per cubic foot. Provide the work done in foot-pounds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{249.6\pi}
Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if its endpoints divide the polygon into two parts, each containing an odd number of the polygon's sides. Each side of \( P \) is also considered a "good edge".

Given that 2003 non-intersecting diagonals divide \( P \) into several triangles, determine the maximum number of isosceles triangles, under this division, that have two "good edges".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
Given the sets $S=\{ x\mid | x-2 | > 3 \}$ and $T=\{ x\mid a < x < a+8 \}$, and their union $S\cup T=R$, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3,-1)}
In an acute-angled triangle \( \mathrm{ABC} \), angle \( \mathrm{B} = 75^\circ \). A point \( K \) is chosen on side \( \mathrm{AC} \). Circles are circumscribed around triangles \( \mathrm{ABK} \) and \( \mathrm{CBK} \), with centers \( \mathrm{O}_1 \) and \( \mathrm{O}_2 \) respectively. Find the radius of the circumcircle of triangle \( \mathrm{ABC} \) if the smallest possible length of segment \( \mathrm{O}_1 \mathrm{O}_2 \) is 2 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{6} - 2 \sqrt{2} \text{ cm}}
Determine the number of non-isomorphic groups of order 2013.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Diane wants to buy 3 packages of cookies for her friends. Each package of cookies costs 75 cents. There is also a 10% sales tax on the purchase. Diane currently has 55 cents. What's the difference between the total cost of the cookies (including sales tax) and the money Diane has?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1.93}