inclusionAI/Ring-lite-rl-data huggingface dataset:
Given polynomial functions \( P, Q, R: \mathbb{C} \rightarrow \mathbb{C} \) with maximum degree 2, and complex numbers \( a, b, c \) such that:
\[
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(b) & Q(b) & R(b) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} = 1,
\]
calculate the following sum of determinants:
\[
\begin{vmatrix}
P(1) & Q(1) & R(1) \\
P(b) & Q(b) & R(b) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} +
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(1) & Q(1) & R(1) \\
P(c) & Q(c) & R(c) \\
\end{vmatrix} +
\begin{vmatrix}
P(a) & Q(a) & R(a) \\
P(b) & Q(b) & R(b) \\
P(1) & Q(1) & R(1) \\
\end{vmatrix}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the number of subsets \( S \) of \(\{1,2, \ldots 6\}\) satisfying the following conditions:
- \( S \) is non-empty.
- No subset of \( S \) has the property that the sum of its elements is 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
A regular octahedron [ABCDEF] is positioned such that its vertices lie on the coordinate axes. The face [ABC] is numbered with the number 1. Number the remaining seven faces with the numbers 2 to 8, using a different number for each face. How many different ways can the remaining faces be numbered such that at least three of the faces concurrent at vertex A have odd numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1872}
Let $N$ be the number of positive divisors of $2010^{2010}$ that end in the digit 2. What is the remainder when $N$ is divided by 2010?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
In a cube \( ABCD A_{1} B_{1} C_{1} D_{1} \), consider two planes \( \alpha \) and \( \beta \):
- Plane \( \alpha \) is perpendicular to the line \( A_{1} C_{1} \).
- Plane \( \beta \) is parallel to the line \( C D_{1} \).

Find the minimum possible angle between planes \( \alpha \) and \( \beta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Given the function \( w = z \tan^{-1}\left(\frac{x}{y}\right) \), calculate the value of \( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of subgroups of order \(p^2\) in the group \(\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + p + 1}
Calculate the Cramer-Rao lower bound for the parameter $\lambda$ of the exponential distribution, $f(x) = \lambda e^{-\lambda x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\lambda^2}{n}\)}
Through vertex $C$ of square $ABCD$, a line passes, intersecting diagonal $BD$ at point $K$ and the perpendicular bisector of side $AB$ at point $M$ ($M$ lies between $C$ and $K$). Find the angle $\angle DCK$ if $\angle AKB = \angle AMB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15^\circ}
Evaluate the limit for $0<p<1$:
$$\lim_{n\rightarrow \infty}\frac{(p+1)\sum\limits_{k=1}^n k^p \log k-n^{p+1}\log(n)}{n^{p+1}(p+1)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{-1}{(p+1)^2}\)}
Through the midpoint $D$ of the base of an isosceles triangle, a line is drawn at an angle of $30^{\circ}$ to this base, on which the angle $\angle ACB$ intercepts a segment $EF$. It is known that $ED = 6$ and $FD = 4$. Find the height of the triangle drawn to the base.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Calculate the Gaussian curvature of a surface with the metric \( ds^2 = du^2 + \cosh^2(u) dv^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given a convex quadrilateral ABCD with side lengths AB=8, BC=4, CD=DA=6, express the range of angle A using an inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < \angle A < 90^\circ}
Evaluate the integral \( \int x^2 \, d\mu \) where \( C \) is the Cantor set and \( \mu \) is the uniform measure on \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
From a container having pure milk, 30% is replaced by water. This process is repeated five times. At the end of the fifth operation, what is the composition of the mixture in the container?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83.193\%}
Linda spent 3 / 4 of her savings on furniture and the rest on a TV and a refrigerator. The TV cost her $210, and the refrigerator cost her 20% more than the TV. Additionally, she got a 7% discount on the furniture and had to pay a 6% tax on the TV. What were her original savings before making these purchases?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1898.40}
Given two independent random variables $x_i$ and $x_j$ sampled from the same distribution $X$ with finite moments, find the expected value of their product, $\mathbb{E}[x_ix_j]$. Assume $x_i$ and $x_j$ are independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\mathbb{E}[X])^2}
Given that a ( a - 6 ) equals some number and b ( b - 6 ) equals the same number, where a ≠ b , the sum of a and b is 6. What is the value of the number that both equations equal to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
Evaluate the improper integral \( \int_0^\infty \frac{\ln(1/x)}{1-x^2}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi^2}{4}}
For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in a row. Find the remainder when $S(2015)$ is divided by 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12, what is the area of the equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the number of non-negative integer solutions to the equation \(x_1 + x_2 + x_3 + x_4 + x_5 = 45\) given the following conditions: \(x_1 + x_2 > 0\), \(x_2 + x_3 > 0\), \(x_3 + x_4 > 0\), and \(x_4 + x_5 > 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{205669}
Find the order of the group generated by all cycles of length $n$. For example, when $n=4$, the set of cycles is: \( G = \{(1234), (1243), (1342), (1324), (1432), (1423)\} \). Determine the order of the group generated by this set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Determine the maximum number of distinct real roots that the equation \( \sum_{i=1}^{n}a_{i}b_{i}^{x} + c = 0 \) can have, where \( a_{i}, b_{i}, c \) are real constants, \( b_{i} > 0 \), \( b_{i} \neq b_{j} \) for \( i \neq j \), and \( c \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Determine the number of real solutions to the equation \( x = 99 \sin(\pi x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
Carrie wants to take a trip to New York. She can get a 20% discount on a flight with Delta Airlines. She can also save 30% off an $1100 flight with United Airlines. She would save $90 by choosing the cheapest flight. What is the original price of the Delta Airlines flight?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$850}
In a society of 30 people, any two people are either friends or enemies, and everyone has exactly six enemies. How many ways are there to choose three people from the society so that any two of them are either all friends or all enemies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1990}
Let $p, q$ be two-digit numbers, neither of which are divisible by $10$. Let $r$ be the four-digit number formed by placing the digits of $p$ followed by the digits of $q$ in order. As $p$ and $q$ vary, a computer prints $r$ on the screen if $\gcd(p, q) = 1$ and $p + q$ divides $r$. Suppose that the largest number printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the values of $x$ for which the series \( \sum_{n=1}^{\infty}\frac{\sin(nx)}{n^\alpha} \) converges, given that \( \alpha > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle. Given that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$, find a simple expression for $d$ in terms of $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{ab}{c}}
In triangle \( ABC \), a point \( K \) is taken on side \( AB \), such that \( AK: BK = 1:2 \). Another point \( L \) is taken on side \( BC \), such that \( CL: BL = 2:1 \). Point \( Q \) is the intersection of lines \( AL \) and \( CK \). Find the area of triangle \( ABC \) if it is known that \( S_{BQC} = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{4}}
How many times must a fair die be rolled so that there is less than a 1% chance that the mean of the scores differs from 3.5 by more than 0.1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1936}
If $a$ , $b$ are integers and $s=a^3+b^3-60ab(a+b)\geq 2012$ , find the least possible value of $s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangement for the 100th stage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
In Prime Land, there are seven major cities, labeled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e., we take the indices modulo $7$. Al initially starts at city $C_0$.

Each minute for ten minutes, Al flips a fair coin. If the coin lands heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise, he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\frac{m}{1024}$, find $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{147}
Little Tiger places chess pieces on the grid points of a 19 × 19 Go board, forming a solid rectangular dot matrix. Then, by adding 45 more chess pieces, he transforms it into a larger solid rectangular dot matrix with one side unchanged. What is the maximum number of chess pieces that Little Tiger originally used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{285}
从一个正 9 边形的 9 个顶点中选 3 个使得它们是一个等腰三角形的三个顶点的方法数是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let  $ABC$  be a scalene triangle whose side lengths are positive integers. It is called *stable* if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle?

*Proposed by Evan Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given the system of equations:
\[ x^2 - y^2 - u^3 + v^2 + 4 = 0 \]
\[ 2xy + y^2 - 2u^2 + 3v^4 + 8 = 0 \]
find \( \frac{\partial u}{\partial x} \) at the point \((x,y) = (2,-1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{32}}
Solve the equation \(\sqrt{x} = 1 + \ln(5 + x)\) for \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16.58}
There are $100$ countries participating in an olympiad. Suppose $n$ is a positive integer such that each of the $100$ countries is willing to communicate in exactly $n$ languages. If each set of $20$ countries can communicate in exactly one common language, and no language is common to all $100$ countries, what is the minimum possible value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A right circular cone has a base radius $R$; its height is $m=2R$. At what distance from the vertex must the cone be cut by a plane parallel to the base so that the total surface areas of the resulting parts are equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{R}{5}\sqrt{10(5+\sqrt{5})}}
(5 points) $2\lg5 \cdot 2\lg2 + e\ln3 =$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A box measuring some inches long by 20 inches wide by 10 inches deep is to be filled entirely with identical cubes. No space is to be left unfilled. The smallest number of cubes that can accomplish this objective is 56. What is the length of the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{280}
The base three representation of $x$ is $12112211122211112222$. The first digit (on the left) of the base nine representation of $x$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Suppose $Y_1, Y_2, \ldots$ are independent and identically distributed random variables, and $M$ is a random variable independent of the $Y_i$. Define $Z = \sum_{i=1}^M Y_i$. What is the expected value of $Z$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E[M] \cdot E[Y_1]}
Find the angle between two lines whose direction cosines \((l, m, n)\) satisfy the equations:

\[ l + m + n = 0 \]
\[ \frac{mn}{q-r} + \frac{nl}{r-p} + \frac{lm}{p-q} = 0 \]

where \(p\), \(q\), and \(r\) are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Find the value of $a$ such that the integral

$$\int \limits^{\infty }_{0} \frac{dx}{a^2 + \left(x - \frac{1}{x}\right)^2} = \frac{\pi}{5050}$$

holds true, given that $a^2 \geq 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2525}
Let $f$ be a function such that $$ \sqrt {x - \sqrt { x + f(x) } } = f(x) , $$for $x > 1$. In that domain, $f(x)$ has the form $\frac{a+\sqrt{cx+d}}{b},$ where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
11.5. In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, two diagonals of the lateral faces $A_{1} C_{1}$ and $C_{1} D$ are drawn. Find the angle between them, given that the diagonal $A_{1} C_{1}$ is equal in length to one of the edges of the parallelepiped, and the diagonal $C_{1} D$ forms an angle of 30 degrees with this same edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos\frac{\sqrt{3}}{6}}
在 $\triangle A B C$ 中, $\sin ^{2} A+\sin ^{2} C=2018 \sin ^{2} B$, 则 $\frac{(\tan A+\tan C) \tan ^{2} B}{\tan A+\tan B+\tan C}=$ $\qquad$.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{2019}
On a similar one-way single-lane highway, cars travel at the same speed and obey a modified safety rule: the distance from the back of the car ahead to the front of the car behind is exactly two car lengths for each 20 kilometers per hour of speed or fraction thereof. Assuming each car is 5 meters long and that cars can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye in one hour. Find the quotient when $N$ is divided by 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Calculate the limit \( \lim_{n \to \infty} -n \mathbb{1}_{Z \geq 1 - \frac{1}{n}} \), where \( Z \) is a random variable on the sample space \( \Omega = (0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given an arithmetic sequence $\{a_{n}\}(a_{n} \gt 0)$, where $S_{n}$ is the sum of its first $n$ terms. If $\frac{1}{3}$, $S_{3}+1$, and $S_{9}$ form a geometric sequence, then the minimum value of $\frac{S_{6}}{S_{3}}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A cube with a side length of 1 meter was cut into smaller cubes with a side length of 1 centimeter and arranged in a straight line. What is the length of the resulting line?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
In the base of a regular triangular prism is a triangle \( ABC \) with a side length of \( a \). Points \( A_{1} \), \( B_{1} \), and \( C_{1} \) are taken on the lateral edges, and their distances from the base plane are \( \frac{a}{2} \), \( a \), and \( \frac{3a}{2} \) respectively. Find the angle between the planes \( ABC \) and \( A_{1} B_{1} C_{1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^{\circ}}
Evaluate the integral \( \int_{S^2} \frac{dS}{((x-a)^2 +y^2+z^2)^{1/2}} \), where \( a > 1 \) and \( S \) is the unit sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{a}}
Two positive numbers are in a certain ratio. The second number is 20% more than the first. If the ratio is 1 / 6 : 1 / 5, what is the second number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the maximum value of \( n \) such that for every subset \( A \subseteq N_n \) with \(|A| \geq 10\), the inequality \( \max_{\substack{a, b \in A \\ a \neq b}} |a-b| \leq 10 \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
10. Betya and Vasya are playing a game. Betya selects 100 (possibly the same) non-negative numbers $x_{1}, x_{2}, \cdots, x_{100}$, whose sum equals 1. Vasya then divides them into 50 pairs, calculates the product of the two numbers in each pair, and writes the largest of these products on the blackboard. Betya wants the number written on the blackboard to be as large as possible, while Vasya wants it to be as small as possible. What will the number written on the blackboard be under the correct strategy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{396}}
A convex pentagon  $P=ABCDE$  is inscribed in a circle of radius  $1$ . Find the maximum area of  $P$  subject to the condition that the chords  $AC$  and  $BD$  are perpendicular.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \frac{3\sqrt{3}}{4}}
Among the numbers from 1 to 1000, how many are divisible by 4 and do not contain the digit 4 in their representation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
Given that the function f(x-1) is an even function and the graph of the function f(x) is symmetric about the point (1,0), and f(x) = ax - 1 for x in [-1,1], calculate f(2022).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a total weight of 263 grams using only these given weights?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
The probability of selecting a first-grade product is 0.65, and the probability of selecting a second-grade product is 0.3. Calculate the probability of selecting a defective product. Express your answer as a decimal value between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.05}
Let point $M(x_{0},2-x_{0})$ be given, and suppose there exists a point $N$ on the circle $O:x^{2}+y^{2}=1$ such that $\angle OMN=30^{\circ}$. Then, the range of the real number $x_{0}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,2]}
Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{304}
Question 224, Given a positive integer $n(n \geq 2)$, choose $m$ different numbers from $1, 2, \ldots, 3n$. Among these, there must be four pairwise distinct numbers $a, b, c, d$, satisfying $a=b+c+d$. Find the minimum value of $m$.

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The translation maintains the original format and line breaks as requested.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n+2}
Zaslavsky A.A.

The number of the current Olympiad (70) is formed by the last digits of the year of its holding, written in reverse order. How many more times will such a situation occur in this millennium?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( x, y \) be nonnegative integers such that \( x + 2y \) is a multiple of 5, \( x + y \) is a multiple of 3, and \( 2x + y \geq 99 \). Find the minimum possible value of \( 7x + 5y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{366}.}
In how many ways can 4 purple balls and 4 green balls be placed into a $4 \times 4$ grid such that every row and column contains one purple ball and one green ball? Only one ball may be placed in each box, and rotations and reflections of a single configuration are considered different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Four points \( A, O, B, O' \) are aligned in this order on a line. Let \( C \) be the circle centered at \( O \) with radius 2015, and \( C' \) be the circle centered at \( O' \) with radius 2016. Suppose that \( A \) and \( B \) are the intersection points of two common tangents to the two circles. Calculate \( AB \) given that \( AB \) is an integer \( < 10^7 \) and that \( AO \) and \( AO' \) are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8124480}
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
$\begin{array}{cccccc}&A&B&B&C&B\ +&B&C&A&D&A\ \hline &D&B&D&D&D\end{array}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
From the vertex of the obtuse angle \( A \) of triangle \( ABC \), the height \( AD \) is dropped. A circle with center \( D \) and radius \( DA \) is drawn, which intersects the sides \( AB \) and \( AC \) again at points \( M \) and \( N \) respectively. Find \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{mc}{n}}
Determine the limiting velocity of a spherical object increasing in size due to accumulation of material, given that the rate of increase in radius is proportional to the radius and neglecting air resistance.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{g}{3k}\)}
Let $(W_t)_{t\ge0}$ be a Standard Brownian Motion. Compute the conditional probability $$P\left(\inf_{t\in [1,2]} W_t < 0 ~ \middle| ~ W_1 >0,~ W_2 >0\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}