inclusionAI/Ring-lite-rl-data huggingface dataset:
Point \( O \) is inside an equilateral triangle \( \triangle ABC \). Given that \( OB = 3 \), \( OA = 4 \), and \( OC = 5 \), find the difference in the areas of \( \triangle AOC \) and \( \triangle BOC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7 \sqrt{3}}{4}}
A player starts a game with $i$ tokens, and there are $n$ tokens in total. The game lasts for a maximum of $t$ rounds. In each round, the player wins one token with probability $\frac{1}{3}$ or loses one token with probability $\frac{2}{3}$. The game ends when one of the following conditions is met: the player runs out of tokens, the player collects all $n$ tokens, or $t$ rounds are completed. At the end of the game, the player's prize is calculated as $prize = 2^{result}$, where $result$ is the number of tokens the player has at the end. Calculate the expected value of the prize, $E[prize]$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^i}
Problem 5. (20 points) A point is randomly thrown onto the segment $[8 ; 13]$ and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-2 k-35\right) x^{2}+(3 k-9) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6}
Calculate the mass of a bullet-shaped object formed by a cylinder with a curved top. The cylinder has a radius of \(a\) and is situated with its base on the \((x, y)\)-plane, with the origin at the center of the base. The curved top is described by the equation \(z = 4a - \frac{x^2}{a} + \frac{y^2}{a}\). The density of the object is given by \(D \cdot \left(\frac{\rho^2}{a^2} + 1 \right)\), where \(a\) and \(D\) are constants. Determine the mass of the object using appropriate integration techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi D a^3}
Calculate the mass of the hemisphere with radius 1, centered along the z-axis such that the south pole is at the origin, using the mass function \( p(x,y,z) = 1-z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
It is known that
$$
\sqrt{9-8 \sin 50^{\circ}}=a+b \sin c^{\circ}
$$
for exactly one set of positive integers \((a, b, c)\), where \(0 < c < 90\). Find the value of \(\frac{b+c}{a}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given that \( n \) is a fixed positive integer, for \( -1 \leq x_{i} \leq 1 \) (where \( i = 1, 2, \cdots, 2n \)), find the maximum value of
\[ 
\sum_{1 \leq r < s \leq 2n} (s - r - n) x_{r} x_{s}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n-1)}
Let \( p \) be a prime number and \( f(x) = ax^2 + bx + c \) be a quadratic polynomial with integer coefficients such that \( 0 < a, b, c \leq p \). Suppose \( f(x) \) is divisible by \( p \) for every positive integer \( x \). Determine all possible values of \( a + b + c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3p}
What is the value of \( \sum_{1 \le i < j \le 10}(i+j) \) for odd \( i+j \) minus \( \sum_{1 \le i < j \le 10}(i+j) \) for even \( i+j \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Given that the cubic equation \(x^3 + bx^2 + cx + 1 = 0\) has only one real root \(\alpha\), with \(b < c\), find the value of the expression:
\[2\tan^{-1}(\csc \alpha) + \tan^{-1}(2\sin \alpha\sec^2\alpha)\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Find the smallest natural number \( n \) such that in any set of \( n \) integers, there exist two numbers whose sum or difference is divisible by \( 2k + 1 \), where \( k \) is a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k + 2}
In all the cells of a 4x8 table, except for the corner cells, non-negative numbers are placed such that the sum of the numbers in each cross of five cells is at most 8. What is the maximum possible sum of the numbers in the entire table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Evaluate the limit \(\lim\limits_{n \to \infty} \frac{\sqrt[n]{a_1a_2\cdots a_n}}{a_n},\) where \(a_n\) is the largest natural number \(k\) such that \(k! \leq \alpha^n\), and \(\alpha > 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Today is $12/14/24,$ which is of the form $ab/ac/bc$ for not necessarily distinct digits $a$, $b$, and $c$. Find the number of other dates in the $21$st century that can also be written in this form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
110
}
Find the probability that $Y > X/3$ given that the joint pdf of $X$ and $Y$ is:

$$f(x, y) = \frac {1}{2\pi} e^{-\sqrt{x^2 + y^2}}\,; \quad x,y\in \mathbb{R}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Evaluate the double integral \( \iint_E f(x,y) \, dx \, dy \) where \( f(x,y)=\begin{cases}(x-1/2)^{-3} &\text{if } 0<y<|x-1/2| \\ 0 &\text{else} \end{cases} \) and \( E=[0,1]\times[0,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that $\cos \alpha =\dfrac{\sqrt{5}}{5}$ and $\sin (\alpha -\beta )=\dfrac{\sqrt{10}}{10}$, calculate the value of $\cos \beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{2}}{2}}
Given that $A$, $B$, and $C$ are the three internal angles of $\triangle ABC$, if $\cos (B+C)= \frac {1}{2}$, then $A=$ _____ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {2\pi}{3}}
Evaluate the line integral \( \int_C (z^2, xz, 2xy) \cdot dr \), where \( C \) is the curve formed by the intersection of the surface \( z = 1 - y^2, z \ge 0 \) and the plane \( 2x + 3z = 6 \), oriented anti-clockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Calculate the surface integral of the vector field \( \vec{F} = x\vec{i} + y^3\vec{j} + \vec{k} \) over the plane passing through the points \((0,0,2)\), \((0,1,0)\), and \((2,1,0)\). The equation of the plane is \( 2y + z = 2 \). Use the unit outward normal vector \( \hat{n} = \frac{2\vec{j} + \vec{k}}{\sqrt{5}} \). Compute the integral \( \int\int_{S} \vec{F} \cdot \hat{n} \, dS \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{5}}
Find the smallest real number \( p\left( { \leq  1}\right) \) that satisfies the following
 condition.
 (Condition) For real numbers \( {x}_{1},{x}_{2},\ldots ,{x}_{2024},{y}_{1},{y}_{2},\ldots ,{y}_{2024} \) if

 - \( 0 \leq  {x}_{1} \leq  {x}_{2} \leq  \cdots  \leq  {x}_{2024} \leq  1 \) ,
 - \({0 \leq  {y}_{1} \leq  {y}_{2} \leq  \cdots  \leq  {y}_{2024}} \leq  1 \) ,
 - \( \mathop{\sum }\limits_{{i = 1}}^{2024}{x}_{i} = \mathop{\sum }\limits_{{i = 1}}^{2024}{y}_{i} = {2024p} \) ,

then the inequality \( \mathop{\sum }\limits_{{i = 1}}^{{2024}}{x}_{i}\left( {{y}_{{2025} - i} - {y}_{{2024} - i}}\right)  \geq  1 - p \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$p = \frac{\sqrt{5} - 1}{2}$}
In the city of Fuchs, Ostap Bender organized a distribution of elephants to people. There were present 28 trades union members and 37 non-members. Ostap distributed the elephants so that the shares of all trades union members were equal, and the shares of non-members were also equal. It turned out that there existed only one such distribution (of all elephants). What greatest number of elephants could O. Bender have had?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2072}
How many positive integers with at most $n$ digits are there whose square ends in the same $n$ digits (possibly padded with some leading zeros)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Transform the complex function \( f(x,y) = (x^3 - 3xy^2) + i(3x^2y - y^3) + iC \), where \( u(x,y) \) and \( v(x,y) \) are harmonic conjugates, into the form \( f(z) \) using the substitutions \( x = \frac{z + \overline{z}}{2} \) and \( y = \frac{z - \overline{z}}{2i} \). Express the function in terms of \( z \) only, ensuring that no \( \overline{z} \) terms remain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z^3 + iC}
If 260 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the limit: $$ \lim_{n \to \infty} \int_0^n \frac{dx}{1 + n^2 \cos^2 x} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider two distinct positive integers  $a$  and  $b$  having integer arithmetic, geometric and harmonic means. Find the minimal value of  $|a-b|$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
4. Given 100 lines on a plane, let $T$ denote the set of right-angled triangles formed by some three of these lines. Find the maximum value of $|T|$. (Supplied by Zou Jin)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62500}
The base three representation of $x$ is 
\[ 12112211122211112222. \]
What is the first digit (on the left) of the base nine representation of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that point $P$ is an intersection point of the ellipse $\frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1} > 0)$ and the hyperbola $\frac{x^{2}}{a_{2}^{2}} - \frac{y^{2}}{b_{2}^{2}} = 1 (a_{2} > 0, b_{2} > 0)$, $F_{1}$, $F_{2}$ are the common foci of the ellipse and hyperbola, $e_{1}$, $e_{2}$ are the eccentricities of the ellipse and hyperbola respectively, and $\angle F_{1}PF_{2} = \frac{2\pi}{3}$, find the maximum value of $\frac{1}{e_{1}} + \frac{1}{e_{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{3}}{3}}
Azar and Carl play a game of tic-tac-toe. Azar places an \(X\) in one of the boxes in a \(3\)-by-\(3\) array of boxes, then Carl places an \(O\) in one of the remaining boxes. After that, Azar places an \(X\) in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third \(O\). How many ways can the board look after the game is over?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{148}
Given that I is the incenter of triangle ABC, AC = 2, BC = 3, and AB = 4. If vector AI is expressed as a linear combination of vectors AB and AC, find the value of x + y, where x and y are the coefficients of vectors AB and AC, respectively. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Determine the number of finite dimensional Lie algebras of vector fields on the real line $\mathbf{R}$ up to diffeomorphisms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that each of the numbers n, n+1, n+2, n+3 is divisible by its sum of digits in its decimal representation, and the number in the ones column of n is 8, how many different values can the tens column of n have? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find a basis for the image of the linear transformation $T:\mathbb{R}^4 \rightarrow \mathbb{R}^3$ defined by $T(a,b,c,d) = (a+b, c-d, a+c)$, given that a basis for the kernel of $T$ is $\{(1,-1,0,0), (0,0,1,-1)\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{(1,0,1), (0,1,1)}}
Determine the maximum value of the function \( g(u,v) = \frac{u\sin(\log u) - v\sin(\log v)}{v-u} \) for \( 0 < u < v \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Determine the monotonically decreasing interval of the function $f(x)=\frac{e^x}{x^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2]}
Calculate the surface area of the segment of a sphere enclosed by a cylinder with center at (0, 2) and radius 2, using a double integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi}
A train 550 m long is running with a certain speed. In 29.997600191984645 seconds, it will pass a man who is running at 6 km/hr in the direction opposite to that in which the train is going. What is the speed of the train in km/hr?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
The ages of Patrick and Michael are in the ratio of 3 : 5, and the ages of another pair of people are in the same ratio. The sum of their ages is 196, and the difference between the ages of Patrick and the last person (Monica) is 64. What is the ratio of the ages of the second pair of people (Michael and Monica)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:5}
黑板上写着从1开始的n个连续正整数，擦去其中的一个数后，其余各数的平是36$\frac { 2 } { 5 }$，则擦去的数是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the maximum possible real value of the number k, such that
$$(a+b+c)\left(\frac{1}{a+b}+\frac{1}{c+b}+\frac{1}{a+c}-k\right) \geq k$$
for all real numbers $a, b, c \geq 0$ with $a+b+c=ab+bc+ca$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1
}
令 G 为三维空间中保持某个给定正方体的旋转构成的群. 设 G 的共轭类为 C_1, C_2, \cdots, C_n，满足 |C_1|\geq |C_2|\geq\cdots\geq|C_n|，求 |C_1|+|C_2|.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
设正整数 $a, b, c, d$ 满足 $a>b>c>d$, 且
$$
a b+b c+c a+d^{2} \mid(a+b)(b+c)(c+a)
$$
求 $a b+b c+c a+d^{2}$ 的素因子个数的最小值。(素因子的个数计重数，如 12 的素因子个数为 3 )。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit: $$\lim_{n \to \infty} \left( \frac{\ln(n+1)^{n+1}}{\ln n^n} \right)^n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
A train of a certain length crosses an electric pole at a speed of 122 km/hr. It takes 4.425875438161669 seconds to cross the pole. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Let \(a\), \(b\), \(c\), and \(d\) be distinct positive integers such that \(a+b\), \(a+c\), and \(a+d\) are all odd and are all squares. Let \(L\) be the least possible value of \(a + b + c + d\). What is the value of \(10L\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{670}
$JKLM$ is a square and $PQRS$ is a rectangle. If $JK$ is parallel to $PQ$, $JK = 8$ and $PS = 2$, then the total area of the shaded regions is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Suppose there is an arithmetic sequence $\{a_n\}$ with common difference $d$ where $a_1=1$, and $a_1, a_2$, and $a_4$ form a geometric sequence. Let another sequence $b_n$ be defined as $b_n = \frac{1}{a_n^2 - 1}$. Find the sum of the first $n$ terms of the sequence $b_n$, denoted as $S_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{2(n+1)}}
Evaluate the integral: $$\int_0^\infty \frac{e^{2x}}{\left(e^{2x}\cdot\frac{1-x}{1+x}\right)^2+1}\frac{x^2}{(1+x)^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{8}}
Let \( OP \) be the diameter of the circle \( \Omega \), and \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of the circle \( \Omega \) intersects the second circle at point \( A \). Find the length of segment \( AB \) if \( BD \cdot BC = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
If a positive integer can be expressed as the difference of squares of two consecutive even numbers, then this positive integer is called a "mysterious number". The sum of all "mysterious numbers" between 1 and 200 is     .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
Suppose  $\alpha,\beta,\gamma\in\{-2,3\}$  are chosen such that
\[M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy\]
is finite and positive (note:  $\mathbb{R}_{\ge0}$  is the set of nonnegative real numbers). What is the sum of the possible values of  $M$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{2}}
Let $f(x)=x^3+7x^2+9x+10$. Which value of $p$ satisfies the statement
\[
f(a) \equiv f(b) \    (\text{mod } p) \Rightarrow a \equiv b \  (\text{mod } p)
\]
for every integer $a,b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11
}
Determine the asymptotic behavior of the expression \( \frac{\Gamma(n+\alpha+1)}{\Gamma(n+2)} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^{\alpha - 1}}
Evaluate the integral \( \int_C \cot^2 z \, dz \) where \( C \) is the positively oriented circle \( |z| = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral: \[ \int_{0}^{4\pi}\frac{x|\sin x|}{1 + |\cos x|} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi \ln 2}
On the side  $ AB$  of a cyclic quadrilateral  $ ABCD$  there is a point  $ X$  such that diagonal  $ BD$  bisects  $ CX$  and diagonal  $ AC$  bisects  $ DX$ . What is the minimum possible value of  $ AB\over CD$ ?

*Proposed by S. Berlov*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the value of the constant $c$ such that the random variable \( Y = \frac{c(X_1 - X_2)}{\sqrt{(X_3)^2 + (X_4)^2}} \), where \( X_1, X_2, X_3, \) and \( X_4 \) are independent standard normally distributed variables, follows a t-distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many parallelograms with sides 1 and 2, and angles $60^{\circ}$ and $120^{\circ}$ can be placed at most inside a regular hexagon with side length 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In an addition problem where the digits were written on cards, two cards were swapped, resulting in an incorrect expression: $37541 + 43839 = 80280$. Find the error and write the correct value of the sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81380}
Given that for every \(0 \leq k \leq n\), \(a_k \geq 0\) and \(\sum_{k=0}^n \frac{a_k}{2^k} < \delta\), find an upper bound for \(\sum_{k=0}^n a_k\) that is less than \(n2^n\delta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n \delta}
Determine the dimension of the symplectic group \( \text{Sp}(2n, F) \) when \( F \) is a finite field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(2n + 1)}
Given that the difference between two 3-digit numbers $\overline{x y z}$ and $\overline{z y x}$ is $700-c$, where $x>z$. If $d$ is the greatest value of $x+z$, find the value of $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
John took a test that was divided into four sections with varying numbers of questions, different success rates, and partial marks. 

In the first section, there were 50 questions, and John answered 82% of the questions correctly. For the remaining 18% questions, he earned half the marks by getting partial answers correct.

The second section had 60 questions, and John was able to answer 75% of the questions correctly. For the remaining questions, he earned one-third of the marks for providing partially correct answers.

The third section had 30 questions, out of which John answered 90% correctly. For the remaining 10% questions, he didn't receive any marks.

The final section had 40 questions, and John managed to secure 95% of the correct answers. He earned half the marks for the rest of the questions by partially answering them.

If there is a negative marking scheme in which 0.25 marks are deducted for every wrong answer in sections two and three only, how many total marks will John obtain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{157}
Let set \( A = \{1, 2, 3, \cdots, 1997\} \). For any 999-element subset \( X \) of \( A \), if there exist \( x, y \in X \) such that \( x < y \) and \( x \mid y \), then \( X \) is called a good subset. Find the largest natural number \( a \) (where \( a \in A \)) such that any 999-element subset containing \( a \) is a good subset.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665}
Determine the fair price of the option to regenerate your number after both numbers have been revealed in a game where each participant contributes 20 USD, and the participant with the higher number wins the entire 40 USD.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20}{3}}
A deck of cards has 54 cards with assigned points: Jokers = 0, Ace = 1, Jack = 11, Queen = 12, King = 13, and other cards receive their face values as points. What is the minimum number of cards you need to draw to ensure that among the drawn cards, there are always 2 cards whose point total equals 14? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
On February 1, it was $16.2^{\circ} \mathrm{C}$ outside Jacinta's house at 3:00 p.m. On February 2, it was $-3.6^{\circ} \mathrm{C}$ outside Jacinta's house at 2:00 a.m. If the temperature changed at a constant rate between these times, what was the rate at which the temperature decreased?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.8^{\circ} \mathrm{C} / \mathrm{h}}
Natural numbers, multiples of 3, are painted in two colors: red and blue, such that the sum of a blue and a red number is red, and the product of a blue and a red number is blue. In how many ways can the numbers be painted so that the number 546 is blue?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In triangle \(PQR\), the angle \(QRP\) is \(60^\circ\). Find the distance between the points of tangency on side \(QR\) of the inscribed circle with radius 2 and the circle with radius 3 that is tangent to the extensions of sides \(PQ\) and \(PR\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
记 \(F(x, y)=(x-y)^{2}+\left(\frac{x}{3}+\frac{3}{y}\right)^{2}(y \neq 0)\)。求 \(F(x, y)\) 的最小值。答案的形式为 \(\frac{k}{m}\)，请给出 \(k + m\) 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}