inclusionAI/Ring-lite-rl-data huggingface dataset:
已知$\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {a}、\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {b}$是平面内两个互相垂直的单位向量，且此平面内另一向量$\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {c}$在满足$\left( 3\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {a}+\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {c} \right)\left( 4\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {b}-\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {c} \right)=0$，均能使$\left| \overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {c}-\overset{\lower0.5em\hbox{\smash{\scriptscriptstyle\rightharpoonup}}} {b} \right|\le k$成立 ，则$k$的最小值是_________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{5+\sqrt{13}}{2}$}
If $f(x) = g(x^2)$ where $g(x)$ is a twice-differentiable function, and $f''(0) = 4$, what is $g''(0)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are 2009 numbers arranged in a circle, each of which is either 1 or -1, and not all numbers are the same. Consider all possible consecutive groups of ten numbers. Compute the product of the numbers in each group of ten and sum these products. What is the maximum possible sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005}
A bowl of fruit holds 18 peaches. Some of the peaches are ripe and two more ripen every day, but on the third day three are eaten. After five days, there are 7 more ripe peaches than unripe peaches in the bowl. How many peaches were ripe initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the cardinality of the set \( \{ (x,y) \in \mathbb{R}^2 \; | \; y > x > 0, \; x^x = y^y \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
If points $A(x_{1}$,$-1)$, $B(x_{2}$,$3)$, and $C(x_{3}$,$5)$ are all on the graph of the inverse proportion function $y=-\frac{4}{x}$, then the relationship between $x_{1}$, $x_{2}$, and $x_{3}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_{2} < x_{3} < x_{1}}
Calculate the surface area of the sphere defined by \(x^2 + y^2 + z^2 = 1\) that lies above the disc \((x-1/2)^2 + y^2 \le 1/4\) in the xy-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi - 2}
Burattino got on a train. After travelling half of the total distance, he fell asleep and slept until there was only half of the distance he slept left to travel. What fraction of the total journey did Burattino travel awake?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
The lateral sides \(AB\) and \(CD\) of trapezoid \(ABCD\) are equal to 2 and 3, and the angles at vertices \(A\) and \(D\) are acute. The angle bisectors of angles \(A\) and \(B\) of the trapezoid intersect at point \(M\), and the angle bisectors of angles \(C\) and \(D\) intersect at point \(N\).

The length of segment \(MN\) is 4. Find the radius of the circle inscribed in triangle \(ABD\) if the area of trapezoid \(ABCD\) is \(\frac{26\sqrt{2}}{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{16\sqrt{2}}{15 + \sqrt{129}} }
A car radiator has a 6-liter capacity. If the liquid in the radiator is a certain percentage of antifreeze, 1 liter of liquid must be replaced with 100% antifreeze to bring the mixture up to a 50% solution. What is the initial percentage of antifreeze in the radiator?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
On a rectangular sheet of paper, a picture is drawn in the shape of a "cross" formed by two rectangles $ABCD$ and $EFGH$, where the sides are parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.5}
Determine the maximum area of a $1000 \times 1000$ grid of unit squares that can be covered by eels, where an eel is a polyomino formed by a path of unit squares making two turns in opposite directions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000000}
James injured his ankle and decides to slowly start working back up to his previous running goals and then surpass them.  Before the injury, he was able to run 100 miles per week.  He wants to get up to 20% more than that total in 280 days and each week he will increase miles walked in the week by the same amount.  How many miles does he need to add per week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The sides of a triangle have the combined equation \(x^2 - 3y^2 - 2xy + 8y - 4 = 0.\) The third side, which is variable, always passes through the point \((-5, -1)\). If the range of values of the slope of the third line, such that the origin is an interior point of the triangle, lies in the interval \((a, b)\), find the value of \(a + \frac{1}{b^2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A car travels due east at a speed of $\frac{5}{4}$ miles per minute on a straight road. Simultaneously, a circular storm with a 51-mile radius moves south at $\frac{1}{2}$ mile per minute. Initially, the center of the storm is 110 miles due north of the car. Calculate the average of the times, $t_1$ and $t_2$, when the car enters and leaves the storm respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{880}{29}}
A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{20}$}
A man saves a certain portion of his income during a year and spends the remaining portion on his personal expenses. Next year, his income increases by 25% and inflation causes his expenses to increase by 10%. In addition to this, he receives a tax rebate equal to 5% of his previous year's income, which he adds to his savings. As a result, his savings increase by 100%. If his total expenditure over the two years is double his expenditure in the first year, taking into account inflation, and assuming his tax rate remains constant, what percentage of his income in the first year did he save?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\%}
What is the probability that the sum of the squares of four random numbers chosen uniformly from the interval [0, 1] is less than one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi^2}{32}$}
If the line $y=kx+1$ intersects the circle $x^2+y^2+kx+my-4=0$ at points M and N, and M and N are symmetric with respect to the line $x+y-1=0$, then the value of $k-m$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $x,y,z>0$ such that: $x+y+z=1$. Using Lagrange multipliers, find the minimum value of the expression:

$$P=(\frac{x+1}{x})^3.\frac{z^2}{z^2+1}+(\frac{y+1}{y})^3.\frac{x^2}{x^2+1}+(\frac{z+1}{z})^3.\frac{y^2}{y^2+1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{96}{5}\)}
Convex quadrilateral ABCD is given, and lines are extended from points A, B, C, and D to points B', C', D', and A' respectively, such that AB = BB' = 6, BC = CC' = 7, CD = DD' = 8, and DA = AA' = 9. If the area of quadrilateral ABCD is 10, calculate the area of quadrilateral A'B'C'D'. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Consider the Burgers equation \( u_t + uu_x = 0 \) with the initial condition
\[ u(x,0) = \phi(x) = \begin{cases} 
      2 & x \leq \frac{\pi}{2} \\
      \sin(x) + 1 & \frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \\
      0 & x \geq \frac{3\pi}{2} 
   \end{cases} \].
Determine the breaking time \( t_B \), which is the time at which the characteristic curves intersect.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
从正九边形中任取三个顶点构成三角形, 则正九边形的中心在三角形内的概率为 $\qquad$.  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{19}
Given point M(-2, 1), let N(x0, 1). If there exists a point P on the circle ⊙O: x² + y² = 1 such that ∠MNP = 60°, determine the range of x0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\frac{\sqrt{3}}{3}, \sqrt{3}]}
Line segment $\overline{AB}$ is a diameter of a circle with $AB = 36$. Point $C$, not equal to $A$ or $B$, lies on the circle in such a manner that $\overline{AC}$ subtends a central angle less than $180^\circ$. As point $C$ moves within these restrictions, what is the area of the region traced by the centroid (center of mass) of $\triangle ABC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18\pi}
Evaluate the following limit: \lim _{n \rightarrow \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos ^{2}\left\{\frac{\pi}{2 n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\} d x_{1} d x_{2} \cdots d x_{n}. The original answer is in the format $\frac{m}{n}$, please give the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the parabola $y^{2}=4x$, a line with a slope of $\frac{\pi}{4}$ intersects the parabola at points $P$ and $Q$, and $O$ is the origin of the coordinate system. Find the area of triangle $POQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Evaluate the integral \( \int_{\Gamma} \pi e^{\pi\bar{z}} \, dz \) where \( \Gamma \) is the square with vertices at \( 0, 1, 1+i, i \) oriented anticlockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4(e^{\pi} - 1)}
Isabel has $204 in her piggy bank. She spent 40% of the amount on a toy for herself. Then, with the remaining money, she spent 50% on a book for her brother and saved 20% for a charity donation. After that, she spent 30% of the remaining amount on a gift for her friend and decided to save 10% of what was left for her future expenses. How much money, in dollars, was left over in her piggy bank after all transactions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$23.13}
Determine the limit (if it exists) of the sequence $\{a_n\}$ defined by $a_1 = 1$ and the recursion formula $a_{n+1} = \frac{3n-1}{3n} a_n + \frac{1}{n^2}$ for all $n \geq 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The inclination angle of the line $\sqrt{3}x+y-1=0$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}}
Given a positive integer  $n$ , consider a triangular array with entries  $a_{ij}$  where  $i$  ranges from  $1$  to  $n$  and  $j$  ranges from  $1$  to  $n-i+1$ . The entries of the array are all either  $0$  or  $1$ , and, for all  $i > 1$  and any associated  $j$  ,  $a_{ij}$  is  $0$  if  $a_{i-1,j} = a_{i-1,j+1}$ , and  $a_{ij}$  is  $1$  otherwise. Let  $S$  denote the set of binary sequences of length  $n$ , and define a map  $f \colon S \to S$  via  $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$ . Determine the number of fixed points of  $f$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\left\lceil \frac{n}{2} \right\rceil}}
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given a sample with a sample size of $7$, an average of $5$, and a variance of $2$. If a new data point of $5$ is added to the sample, what will be the variance of the sample?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{4}}
Given $\triangle ABC$ with $\angle A = 15^{\circ}$, let $M$ be the midpoint of $BC$ and let $E$ and $F$ be points on ray $BA$ and $CA$ respectively such that $BE = BM = CF$. Let $R_1$ be the radius of $(MEF)$ and $R_2$ be the radius of $(AEF)$. If $\frac{R_1^2}{R_2^2} = a - \sqrt{b + \sqrt{c}}$ where $a, b, c$ are integers, find $a^{b^{c}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Big Joe is the tallest player on the basketball team. He is one foot taller than Ben, who is one foot taller than Larry, who is one foot taller than Frank, who is taller than Pepe. Pepe is 4 feet six inches tall, and Big Joe is 8 feet tall. How much taller is Frank than Pepe, in feet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
On a  $300 \times 300$  board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook.  At what least  $k$ , it is possible to state that there is at least one rook in each  $k\times k$  square ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
Find $G'(0)$ if \[ G(x) = \int_{\cos x}^{x^2}\left( \int_0^{t^2}\sqrt{1+s^4} \ ds \right) dt. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the largest positive integer $k$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties:

- The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and
- It is possible to color the edges of $\mathcal{P}$ with $k$ colors such that for every color $c$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the color $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Ann and Max play a game on a $100 \times 100$ board. Ann writes an integer from 1 to 10,000 in each square of the board, using each number exactly once. Max starts by placing a token on a square in the leftmost column and moves it to the rightmost column, moving to adjacent squares by side or vertex. Max pays Ann the number of coins equal to the number in each visited square. If both players use their best strategies, how much money will Max pay Ann?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{495100}
已知点 A,F_1,F_2 分别为椭圆 \df{x^2}{4}+\df{y^2}{3}=1 的左顶点和左、右焦点. 过点 A 作斜率为 k 的直线交椭圆于另一点 B，连接 BF_2 并延长交椭圆于点 C. 若 F_1C\bot AB，则 k^2 的值为__________.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{25}
Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Let $x$ be the minimal root of the equation $x^2 - 4x + 2 = 0$. Find the first two digits after the decimal point of the number $\{x + x^2 + \cdots + x^{20}\}$, where $\{a\}$ denotes the fractional part of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
有 $n$ 个人，已知他们中的任意两人至多通电话一次，他们中的任意 $n-2$ 个人之间通电话的次数相等, 都是 $3 k$ 次, 其中 $k$ 是自然数, 求 $n$ 的可能值的和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that $\sin \alpha = \frac{4\sqrt{3}}{7}$ and $\cos(\alpha + \beta) = -\frac{11}{14}$, where both $\alpha$ and $\beta$ are acute angles, find the value of $\cos \beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{10}}{5}}
A square $OPQR$ is inscribed in $\triangle ABC$. The areas of $\triangle AOR$, $\triangle BOP$, and $\triangle CRQ$ are given as $S_{1}=1$, $S_{2}=3$, and $S_{3}=1$, respectively. Find the side length of the square $OPQR$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6423
}
Calculate the line integral $$\int_C(x-y)dz$$ where curve $C$ is the intersection of the paraboloid $z=x^2+y^2$ and the plane $z-2x-2y=1$, oriented counterclockwise when viewed from the point $(0,0,10)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12π}
8 identical books are to be divided into three piles. There are a total of $\boxed{\text{answer}}$ different ways to do this.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The cubic polynomial
\[8x^3 - 3x^2 - 3x - 1 = 0\]has a real root of the form $\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},$ where $a,$ $b,$ and $c$ are positive integers.  Find $a + b + c.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Find the divergence of the vector

$$
\mathbf{a}=\varphi(r) \mathbf{r}^{0}=\frac{\varphi(r)}{r} \mathbf{r}
$$

where \( r = |r| \) is the distance from the origin to the variable point \( M(x, y, z) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \frac{\varphi(r)}{r} + \varphi'(r)}
Among five points $A, B, C, D, E$ in space, any four points are non-coplanar and the distances between any two points are distinct. For each tetrahedron formed by selecting four out of the five points, the shortest edge is colored black. After all the edges have been colored, the number of black segments emanating from a point is called the "wish number" of that point. What is the maximum value of the wish number among these points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral: \[ \int_{0}^{2\pi} \frac{\cos(3\theta)}{5-4\cos(\theta)}d\theta \] without using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Find the sum of the series:
$$\sum_{n=2}^\infty\frac{({2}^{n})(n-1)(n!)}{(2n)!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2}$}
Let $Z_1, Z_2, \ldots, Z_7$ be independent standard normal random variables, each distributed as $N(0,1)$. Calculate the probability that the following inequality holds:
\[ P(Z_1^2 + Z_2^2 + Z_3^2 \leq 4.943(Z_4^2 + Z_5^2 + Z_6^2 + Z_7^2)) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.95}
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_1,\ldots,n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^{4}=B^{7}=ABA^{-1}B=1,A^{2}\neq1$, and $B\neq1$.
 How many elements of $G$ are of the form $C^{2}$ with $C$ in $G$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Petya is playing a computer game called "Pile of Stones." Initially, there are 16 stones in the pile. Players take turns to take 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each time, while still following the rules of the game. The computer plays according to the following algorithm: on each turn, it takes as many stones as necessary to put itself in the most advantageous position. Petya always starts the game. What is the probability that Petya will win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{256}}
The numbers \(1, 2, \cdots, 9\) are randomly placed into a \(3 \times 3\) grid such that each cell contains exactly one number and each number is used exactly once. Calculate the probability that the sum of the numbers in each row and each column is odd. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{14}}
If $(x+1)^5 - x^5 = a + a_1 (x+4)^4 x + a_2 (x+1)^3 x^2 + a_3 (x+1)^2 x^3 + a_4 (x+1) x^4$, and $a_1$ (where $i=0,1,\ldots,4$) are constants, then find the value of $a_1 + a_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
For what smallest positive value of \(a\) is the inequality \(\frac{\sqrt[3]{\sin ^{2} x} - \sqrt[3]{\cos ^{2} x}}{\sqrt[3]{\tan ^{2} x} - \sqrt[3]{\cot ^{2} x}} < \frac{a}{2}\) satisfied for all permissible \(x \in \left(\frac{3 \pi}{2}, 2 \pi\right)\)? Round the answer to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.79}
已知实数 $x_{1}, x_{2}, \cdots, x_{2018}$ 满足：对任意的 $1 \leqslant i<j \leqslant 2018$, 均有 $x_{i}+x_{j} \geqslant(-1)^{i+j}$. 求 $\sum\limits_{i=1}^{2018} i x_{i}$ 的最小值.  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{2037173}
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac{1}{4}$ of the way to his destination, and in $45$ minutes he will be exactly $\frac{1}{3}$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from the point where he passes his school.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{245}
Given an equilateral triangle with vertices R, S, and T, and midpoints of its sides X, Y, and Z, how many noncongruent triangles can be formed using any three of these six points as vertices? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
18 people play in a tournament of 17 rounds. There are 9 games in each round and each person plays in one game in each round. Each person plays every other person just once in the tournament. What is the largest \( n \) such that however the tournament is arranged we can find 4 people amongst whom only one game has been played by the end of round \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Provide an example of a non-commutative ring that has only the trivial ideals \( \{0\} \) and the ring itself, but is not a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M_2(\mathbb{R})}
Find the maximum value of the integral $\int_0^1 (f(x))^3 dx$ if $|f(x)| \leq 1$ and $\int_0^1 f(x)dx = 0$. Express this maximum value as the fraction $\frac{p}{q}$, where $p$ and $q$ are coprime integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/4}
Find the maximum value of the parameter \( b \) for which the inequality \( b \sqrt{b}\left(x^{2}-10 x+25\right)+\frac{\sqrt{b}}{\left(x^{2}-10 x+25\right)} \leq \frac{1}{5} \cdot \sqrt[4]{b^{3}} \cdot \left| \sin \frac{\pi x}{10} \right| \) has at least one solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{10000}}