inclusionAI/Ring-lite-rl-data huggingface dataset:
Compute $\sum_{n=2009}^{\infty} \frac{1}{\binom{n}{2009}}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2009}{2008}}
Let $n$ be the answer to this problem. Find the minimum number of colors needed to color the divisors of $(n-24)$! such that no two distinct divisors $s, t$ of the same color satisfy $s \mid t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
A car insurance company has 2,500 policyholders. The expected claim paid to a policyholder during a year is $1,000$ with a standard deviation of $900$. What premium should the company charge each policyholder to ensure that, with a probability of 0.999, the total premium income will cover the total cost of the claims?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1055.62}
89. In how many ways can 30 different books be arranged into three stacks so that each stack contains 10 books?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30!:(10!\cdot10!\cdot10!\cdot3!)}
Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test, she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Juan bought T-shirts for his employees. He bought shirts for men and women. Women's t-shirts are $5 cheaper than men's t-shirts of the same color. His company has 2 sectors, one in white t-shirts and the other in black t-shirts. He paid a certain amount for white men's t-shirts and $18 for black men's t-shirts. The 2 sectors have the same number of men and women, with a total of 40 employees. He spent a total of $660 on buying t-shirts. How much did he pay for white men's t-shirts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$20}
Let \( S_1, S_2, \ldots, S_{10} \) be the first ten terms of an arithmetic progression (A.P.) consisting of positive integers. If \( S_1 + S_2 + \ldots + S_{10} = 55 \) and \( \left(S_{10} - S_{8}\right) + \left(S_{9} - S_{7}\right) + \ldots + \left(S_{3} - S_{1}\right) = d \), find \( d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
已知集合 $S=\{(-1,0),(1,0),(0,1),(0,-1)\}$, 甲虫第一天在原点 $O(0,0)$, 第 $n+1$ 天从第 $n$天的位置出发沿向量 $\frac{1}{4^{n}} \boldsymbol{v}$ 移动, 其中 $\boldsymbol{v} \in S$, 用 $S_{n}$ 表示第 $n$ 天甲虫可能在多少个不同的位置上，则 $S_{2023}=4^a$ ，请问$a=\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022}
In triangle $ABC, AB=13, BC=15,$ and $CA = 14.$ Point $D$ is on $\overline{BC}$ with $CD=6.$ Point $E$ is on $\overline{BC}$ such that $\angle BAE\cong \angle CAD.$ Given that $BE=\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, find $q.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{463}
Let \( m \) and \( n \) be positive integers satisfying
\[ m n^{2} + 876 = 4 m n + 217 n. \]

Find the sum of all possible values of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93}
A person walks at 15 km/hr instead of a slower speed, and as a result, he would have walked 20 km more. The actual distance traveled by him is 40 km. What is the slower speed he could have walked at?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
3. In a convex quadrilateral $A B C D: A B=A C=A D=B D$ and $\angle B A C=\angle C B D$. Find $\angle A C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
On the side AB of triangle ABC with a $100^{\circ}$ angle at vertex C, points P and Q are taken such that $AP = BC$ and $BQ = AC$. Let M, N, and K be the midpoints of segments AB, CP, and CQ respectively. Find the angle $NMK$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
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.)

*Proposed by Miles Yamner and Andrew Wu*

(Note: wording changed from original to clarify)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1225}
In $\triangle ABC$, if $b \cos C + c \sin B = 0$, then $\angle C = \_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{4}}
Triangle $ABC$ has $AB=2 \cdot AC$. Let $D$ and $E$ be on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\triangle CFE$ is equilateral. What is $\angle ACB$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^{\circ}}
Mario is once again on a quest to save Princess Peach. Mario enters Peach's castle and finds himself in a room with 4 doors. This room is the first in a sequence of 6 indistinguishable rooms. In each room, 1 door leads to the next room in the sequence (or, for the last room, Bowser's level), while the other 3 doors lead to the first room. Now what is the expected number of doors through which Mario will pass before he reaches Bowser's level?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5460}
An equilateral triangle with side length \( n \) is divided into \( n^2 \) smaller equilateral triangles each with side length 1 by lines parallel to its sides. This creates a network of nodes connected by line segments of length 1. What is the maximum number of segments that can be chosen such that no three chosen segments form a triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n+1)}
The minimum value of $ω$ such that the function $y=A\sin ωx (A > 0,ω > 0)$ has at least 25 minimum values in the closed interval $[0,1]$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49.5π}
Assume it takes 1 minute to wash a small kettle, 15 minutes to boil water, 3 minutes to wash teacups, 2 minutes to prepare and place tea leaves, and another 1 minute to steep the tea. What is the minimum number of minutes required to be able to drink tea, considering that some tasks can be performed simultaneously? Express your answer as a single integer representing the minimum number of minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given vectors $a=(1,2)$, $b=(-2,-4)$, and $|c|=\sqrt{5}$. If $(a+b) \cdot c=\frac{5}{2}$, then the angle between vector $a$ and $c$ is \_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^{\circ}}
Compare the sizes of $-1$ and $-2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 > -2}
Find the sum of the smallest and largest numbers that are made up of 1, 2, and 5 and are greater than 100 and less than 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{646}
Let  $[n]$  denote the set of integers  $\left\{ 1, 2, \ldots, n \right\}$ . We randomly choose a function  $f:[n] \to [n]$ , out of the  $n^n$  possible functions. We also choose an integer  $a$  uniformly at random from  $[n]$ . Find the probability that there exist positive integers  $b, c \geq 1$  such that  $f^b(1) = a$  and  $f^c(a) = 1$ . ( $f^k(x)$  denotes the result of applying  $f$  to  $x$   $k$  times.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{n}}
Bored in an infinitely long class, Evan jots down a fraction whose numerator and denominator are both $70$-character strings, as follows:
\[ r = \frac{\text{loooloolloolloololllloloollollolllloollloloolooololooolololooooollllol}}
{\text{lolooloolollollolloooooloooloololloolllooollololoooollllooolollloloool}}. \]
If $o=2013$ and $l=\frac{1}{50}$, find $\lceil roll \rceil$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Fill in the 3x3 grid with 9 different natural numbers such that for each row, the sum of the first two numbers equals the third number, and for each column, the sum of the top two numbers equals the bottom number. What is the smallest possible value for the number in the bottom right corner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that point $P$ is a moving point on the curve $y= \frac {3-e^{x}}{e^{x}+1}$, find the minimum value of the slant angle $\alpha$ of the tangent line at point $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{4}}
A circle $U$ has a circumference of $18\pi$ inches, and segment $AB$ is a diameter. If the measure of angle $UAV$ is $45^{\circ}$, what is the length, in inches, of segment $AV$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\sqrt{2 - \sqrt{2}}}
In a regular tetrahedron \( S-ABC \) with side length \( a \), \( E \) and \( F \) are the midpoints of \( SA \) and \( BC \) respectively. Find the angle between the skew lines \( BE \) and \( SF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos \left( \frac{2}{3} \right)}
Compute $$\lim _{A \rightarrow+\infty} \frac{1}{A} \int_{1}^{A} A^{\frac{1}{x}} \mathrm{~d} x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute $\tan \left(\frac{\pi}{7}\right) \tan \left(\frac{2 \pi}{7}\right) \tan \left(\frac{3 \pi}{7}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{7}}
Danny collects bottle caps and wrappers. He found some bottle caps and 18 wrappers at the park. Now he has 67 wrappers and 35 bottle caps in his collection, and he has 32 more wrappers than bottle caps. How many bottle caps did he find at the park?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
For how many three-element sets (order doesn't matter) of positive integers $\{a,b,c\}$ is it true that $a\times b\times c=2310$?  (The positive integers $a,$ $b,$ and $c$ are distinct.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In a football championship, 16 teams participated. Each team played against every other team once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is called successful if it earned at least half of the maximum possible points. What is the maximum number of successful teams that could be in the tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Thirty-six players are playing a game: they take turns drawing a random card from a 36-card deck. If a player draws the ace of diamonds, they win. If they draw any other card, they return it to the deck, and the next player takes their turn. This continues in a cycle: first player, then the second, and so on. If the ace of diamonds is not drawn in the first round, the players continue drawing cards in the same order in the second round, and so on. This process repeats until someone draws the ace of diamonds.

Assume that before the game starts, the players place bets, and the winner takes all. How should the players' bets be related so that the game is fair, meaning the expected winnings for all players are equal to each other (i.e., equal to zero taking into account the amount each player bet)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{35}{36}}
Find all the solutions to
\[\frac{1}{x^2 + 11x - 8} + \frac{1}{x^2 + 2x - 8} + \frac{1}{x^2 - 13x - 8} = 0.\]Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8,1,-1,-8}
A car traveling at a certain constant speed takes 10 seconds longer to travel 1 km than it would take to travel 1 km at a certain speed. The car is traveling at 65.45454545454545 km/h. What is the speed it is being compared to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Randomly number 960 people as 1, 2, ..., 960, and use systematic sampling to select 32 people for a survey. If the number drawn by simple random sampling in the first group is 9, then the number of people to be drawn among those numbered within [450,750] is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Tabby is training for a triathlon. She swims at a speed of 1 mile per hour. She runs at a speed of 8 miles per hour. What is her average speed for these two events?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
The distance between cities $A$ and $B$ is $30$ km. A bus departs from $A$ and makes a stop every $5$ km for $2$ minutes. The bus travels between stops at a speed of $80$ km/h. Simultaneously with the departure of the bus from $A$, a cyclist leaves $B$ to meet the bus, traveling at a speed of $27$ km/h. How far from $A$ will the cyclist meet the bus?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Suppose that $x, y$, and $z$ are complex numbers of equal magnitude that satisfy $$x+y+z=-\frac{\sqrt{3}}{2}-i \sqrt{5}$$ and $$x y z=\sqrt{3}+i \sqrt{5}.$$ If $x=x_{1}+i x_{2}, y=y_{1}+i y_{2}$, and $z=z_{1}+i z_{2}$ for real $x_{1}, x_{2}, y_{1}, y_{2}, z_{1}$, and $z_{2}$, then $$\left(x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}\right)^{2}$$ can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1516}
Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{33}}
How many positive integers $k$ are there such that $$\frac{k}{2013}(a+b)=\operatorname{lcm}(a, b)$$ has a solution in positive integers $(a, b)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
Given the hyperbola $$\frac {x^{2}}{12}- \frac {y^{2}}{4}=1$$ with its right focus as F. If a line passing through point F intersects the right branch of the hyperbola at only one point, then the range of the slope of this line is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[- \frac { \sqrt {3}}{3}, \frac { \sqrt {3}}{3}]}
The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that:

1. The union of any three of these subsets is the entire set $X$.
2. The union of any two of them contains at most $1979$ members.

What is the largest possible value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
The base of the pyramid \( T ABCD \) is a trapezoid \( ABCD \) with the length of the shorter base \( BC \) equal to \( 4 \sqrt{11} \). The diagonals of the trapezoid \( AC \) and \( BD \) intersect at point \( O \). The ratio of the areas of triangles \( AOB \) and \( BOC \) is \( 3:2 \). The area of triangle \( DOC \) is \( 132 \sqrt{\frac{2}{25}} \). All the lateral edges of the pyramid \( T ABCD \) are inclined to the base at an angle of \( 30^\circ \). Find the volume of the pyramid \( T BMNC \), where points \( M \) and \( N \) are the midpoints of the edges \( TA \) and \( TD \) respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105.875}
Calculate the flux of the vector field $\vec F = \langle \sin(xyz), x^2y, z^2e^{x/5} \rangle$ through the surface $S$ defined by the elliptical cylinder $4y^2 + z^2 = 4$ for $x \in [-2, 2]$. The surface $S$ includes both the top and bottom caps.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32π/3}
In the geometric sequence $\{a_n\}$, if $a_2a_5= -\frac{3}{4}$ and $a_2+a_3+a_4+a_5= \frac{5}{4}$, calculate $\frac{1}{a_2}+ \frac{1}{a_3}+ \frac{1}{a_4}+ \frac{1}{a_5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{3}}
What is the expected number of points on a circle when adding points uniformly until three of them form an acute angled triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In the result obtained by subtracting 101011 from 10000000000, how many times does the digit 9 appear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Alín has 8 keys on a keyring. The keys are indistinguishable by sight and both sides of each key are the same. To be able to tell the keys apart, Alín puts a colored cap on each one. How many colors does Alín need?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle  $ABC$  with  $AB < AC$ , let  $H$  be the orthocenter and  $O$  be the circumcenter. Given that the midpoint of  $OH$  lies on  $BC$ ,  $BC = 1$ , and the perimeter of  $ABC$  is 6, find the area of  $ABC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{7}}
Let \( x, y, \) and \( z \) be real numbers such that \(\frac{4}{x} + \frac{2}{y} + \frac{1}{z} = 1\). Find the minimum of \( x + 8y + 4z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
The nineteenth question: Find the smallest real number $\lambda$ such that $\sum_{i=1}^{100}\left(a_{i}-a_{i+1}\right)^{2} \leq \lambda\left(100-\left(\sum_{i=1}^{100} a_{i}\right)\right)$ holds for any real numbers $a_{1} 、 a_{2} 、 \ldots 、 a_{100}$ satisfying $\sum_{i=1}^{100} a_{i}^{2}=100$. Here $a_{101}=a_{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Problem 4. A circle with radius 4 is inscribed in trapezoid $ABCD$, touching the base $AB$ at point $M$. Find the area of the trapezoid if $BM=16$ and $CD=3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
A health center research suggests that for a person with height $h$ (in meters), their ideal weight $W$ (in kg) should follow the formula $W=22h^2$ (kg). It is defined that a weight within ±10% of the ideal weight is considered standard weight; exceeding 10% but not more than 20% is considered slightly overweight; exceeding 20% is considered obese. Both slightly overweight and obese are considered overweight conditions. For a person with height $h$ and weight $W$, the necessary and sufficient condition for being overweight is $W > ch^2 + dh + e$. Therefore, $(c, d, e)=$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(24.2, 0, 0)}
If circular arcs $AC$ and $BC$ have centers at $B$ and $A$, respectively, then there exists a circle tangent to both $\overarc {AC}$ and $\overarc{BC}$, and to $\overline{AB}$. If the length of $\overarc{BC}$ is $12$, find the circumference of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Tetrahedron $A B C D$ has side lengths $A B=6, B D=6 \sqrt{2}, B C=10, A C=8, C D=10$, and $A D=6$. The distance from vertex $A$ to face $B C D$ can be written as $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive integers, $b$ is square-free, and $\operatorname{gcd}(a, c)=1$. Find $100 a+10 b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2851}
Given a line $l$ passes through the foci of the ellipse $\frac {y^{2}}{2}+x^{2}=1$ and intersects the ellipse at points P and Q. The perpendicular bisector of segment PQ intersects the x-axis at point M. The maximum area of $\triangle MPQ$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac {3 \\sqrt {6}}{8}"]}
Evaluate the limit: $$\lim_{x\to 0} \dfrac{\tan(\sin(\tan x)) - \sin(\tan(\sin x))}{ \tan(\tan x) - \tan(\sin x)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
To test the quality of a certain product, it was decided to use the random number table method to draw 5 samples from 300 products for inspection. The products are numbered from 000, 001, 002, ..., to 299. The following are the 7th and 8th rows of the random number table. If we start from the 5th number in the 7th row and read to the right, the second sample number among the 5 obtained is:

7th row: 84 42 17 53 31 57 24 55 06 88 77 04 74 47 67 21 76 33 50 25 83 92 12 06 76  
8th row: 63 01 63 78 59 16 95 55 67 19 98 10 50 71 75 12 86 73 58 07 44 39 52 38 79
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{057}
已知方程$αcosx+bsinx+c=0$在0&lt;x&lt;π上有两个不同的根α,β，则sin(α+β)=
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { 2 a b } { a ^ { 2 } + b ^ { 2 } }$}
In $\triangle ABC$, the sides $a$, $b$, and $c$ correspond to angles $A$, $B$, and $C$ respectively. If $2\sin ^{2}B+3\sin ^{2}C=2\sin A\sin B\sin C+\sin ^{2}A$, find the value of $\tan A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
A 98 x 98 chessboard has the squares colored alternately black and white in the usual way. A move consists of selecting a rectangular subset of the squares (with boundary parallel to the sides of the board) and changing their color. What is the smallest number of moves required to make all the squares black?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
What is the maximum number of consecutive positive integers in which all numbers have prime factorizations with only odd powers of their prime factors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the definite integral:

$$
\int_{0}^{2 \pi} \sin ^{2}\left(\frac{x}{4}\right) \cos ^{6}\left(\frac{x}{4}\right) d x
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{5\\pi}{64}"]}
99 dwarfs stand in a circle, some of them wear hats. There are no adjacent dwarfs in hats and no dwarfs in hats with exactly 48 dwarfs standing between them. What is the maximal possible number of dwarfs in hats?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Seven thousand twenty-two can be written as
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7022}
Paula has 20 candies to be given out to her ten friends. She needs to buy six additional candies so she can give an equal number of candies to her friends. How many candies will each of her friends get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a fixed circle $\odot P$ with a radius of 1, the distance from the center $P$ to a fixed line $l$ is 2. Point $Q$ is a moving point on $l$, and circle $\odot Q$ is externally tangent to circle $\odot P$. Circle $\odot Q$ intersects $l$ at points $M$ and $N$. For any diameter $MN$, there is always a fixed point $A$ on the plane such that the angle $\angle MAN$ is a constant value. Find the degree measure of $\angle MAN$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Let $a,b,c>0$. Find the maximum value of:
$$P= \frac{2ab+3b^2}{(a+3b)^2}+\frac{2bc+3c^2}{(b+3c)^2}+\frac{2ca+a^2}{(c+3a)^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15/16}
Six musicians gathered at a chamber music festival. At each scheduled concert, some of the musicians played while the others listened as members of the audience. What is the least number of such concerts which would need to be scheduled so that for every two musicians, each must play for the other in some concert?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}