inclusionAI/Ring-lite-rl-data huggingface dataset:
Suppose that a sequence \( {a}_{0},{a}_{1},\ldots \) of real numbers is defined by \( {a}_{0} = 1 \) and
 \( {a}_{n} = \left\{  \begin{array}{ll} {a}_{n - 1}{a}_{0} + {a}_{n - 3}{a}_{2} + \cdots  + {a}_{0}{a}_{n - 1} & \text{ if }n\text{ odd } \\  {a}_{n - 1}{a}_{1} + {a}_{n - 3}{a}_{3} + \cdots  + {a}_{1}{a}_{n - 1} & \text{ if }n\text{ even } \end{array}\right. \)
 for \( n \geq  1 \) . There is a positive real number \( r \) such that
 \( {a}_{0} + {a}_{1}r + {a}_{2}{r}^{2} + {a}_{3}{r}^{3} + \cdots  = \frac{5}{4}. \)
 If \( r \) can be written in the form \( \frac{a\sqrt{b} - c}{d} \) for positive integers \( a,b,c,d \) such that \( b \) is not divisible by the square of any prime and
 \( \gcd \left( {a,c,d}\right)  = 1 \) ,then compute \( a + b + c + d \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1923}
An acute triangle $ABC$ is inscribed in a unit circle. Points $P$ and $Q$ are interior points on sides $AB$ and $AC$ respectively, such that $\frac{AP}{PB} = 2$ and $\frac{AQ}{QC} = 1$, and the points $P, O, Q$ are collinear. Determine the range of possible lengths for $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{3}, 2)}
In a group of 43 children, each child throws exactly one snowball at another child. The throwing pattern is as follows:

- The first child throws a snowball at the child who threw a snowball at the second child.
- The second child throws a snowball at the child who threw a snowball at the third child.
- The forty-third child throws a snowball at the child who threw a snowball at the first child.

Determine the number of the child who threw a snowball at the third child.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
已知一个圆柱形水杯的质量为 $a \mathrm{~g}$, 其重心在圆柱轴的中点处 (杯底厚度及质量均忽略不计，且水杯直立放置）。质量为 $b \mathrm{~g}$ 的水恰好装满水杯，装满水后的水杯的重心还在圆柱轴的中点处. 若 $b=3 a$, 求装入半杯水的水杯的重心到水杯底面的距离与水杯高的比值，化为最简分数的形式后求分子加分母的值是多少。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
What is the local value of a digit in the product of the face value of 7 and the local value of 6 in 7098060, if the local value of 6 in the product is 6000?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6000}
Given the force field $\bar{F} (x, y, z) = (x − yz)\bar{i} + (y − xz)\bar{j} +(x(1−y) +z^2)\bar{k}$, calculate the work done in moving a particle along the closed path formed by the line segments connecting the points $(0, 0, 0), (1, 1, 1), (1, 1, 0), (0, 0, 1)$ in that order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the maximum possible dimension of a subspace $W$ in $\mathbb{R}^5$ that does not include the standard basis vectors $e_2$, $e_4$, and $e_5$? Given that $W$ includes $e_1$ and $e_3$, which are linearly independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $P$ be a polyhedron where every face is a regular polygon, and every edge has length 1. Each vertex of $P$ is incident to two regular hexagons and one square. Choose a vertex $V$ of the polyhedron. Find the volume of the set of all points contained in $P$ that are closer to $V$ than to any other vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{3}}
Evaluate the limit \( \lim_{n \rightarrow \infty} \int_{\gamma} \frac{1}{(z+a)^2} \cot(\pi z) \, dz \), where \( \gamma \) is the path defined by \( \gamma=[n+\frac{1}{2}+ni,-n-\frac{1}{2}+ni,-n-\frac{1}{2} -ni, n+\frac{1}{2}-ni,n+\frac{1}{2}+ni] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the minimum number of comparisons required to determine if a given number $x$ is present in a matrix $B$, where the matrix satisfies the condition $B_{ij} \le B_{kl}$ for all $i \ge k$ and $j \le l$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 1}
In square \(ABCD\), point \(M\) divides the diagonal \(AC\) in the ratio \(MC : AM = 1 : 4\). A line through \(M\) divides the square into two regions with areas in the ratio \(1 : 11\). Determine the ratio in which this line divides the perimeter of the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{19}}
Determine the minimum number of button presses required to try all possible 5-digit passwords on a 10-key keyboard without pressing enter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100004}
If the mass of the gun is 2 kg, the mass of the bullet is 20 gm, and the velocity of the bullet after firing is 200 m/s, then what is the velocity of the gun after firing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\,\text{m/s}}
Reflect a square in sequence across each of its four sides. How many different transformations result from the sequence of four reflections?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( M \) be a real vector space of \( 2 \times 3 \) matrices with real entries. Define a linear transformation \( T: M \rightarrow M \) as follows:
\[ T\left( \begin{pmatrix} x_{1} & x_{2} & x_{3} \\ x_{4} & x_{5} & x_{6} \end{pmatrix} \right) = \begin{pmatrix} -x_{6} & x_{4} & x_{1} \\ x_{3} & x_{5} & x_{2} \end{pmatrix} \]
Determine the determinant of the linear transformation \( T \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Calculate the limit of the sum of binomial coefficients:
$$S_{n}=\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1} \right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the complex numbers \(a = \cos(q) + i\sin(q)\), \(b = \cos(r) + i\sin(r)\), and \(c = \cos(e) + i\sin(e)\), if \(\frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1\), find the value of \(\cos(r - e) \cos(e - q) + \cos(q - r)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Oleg drew an empty $50 \times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the sum of the numbers written next to its row and its column ("addition table"). What is the maximum number of sums in this table that could turn out to be rational numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
There are eight seats at a round table. Six adults $A_1, \ldots, A_6$ and two children sit around the table. The two children are not allowed to six next to each other. All the seating configurations where the children are not seated next to each other are equally likely. What is the probability that the adults $A_1$ and $A_2$ end up sitting next to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4/15}
Opal won $100.00 betting on a horse race. She put some portion of her winnings into savings and bet the other portion of her winnings. This time, she made a 60% profit and again, put half of her earnings into savings. She put $90 into her savings. What is the ratio of the amount she put into savings to the amount she bet after winning the horse race?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:1}
In a vast oak tree, numerous species of birds are perched on its branches. There are twice as many cardinals as bluebirds, and the number of goldfinches is equal to the product of bluebirds and swallows. The number of sparrows is half the sum of cardinals and goldfinches, while the number of robins is 2 less than the quotient of bluebirds divided by swallows. If there are 12 swallows, which is half as many as the number of bluebirds, what is the total number of birds perched on the oak tree?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540}
Find the constant \(a\) such that the primitive function \(F(x)\) of \(f(x) = x^3 - 4a^2x\) has a minimum value of 0 and satisfies \(F(2) = 4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{197}
For how many sets of natural numbers \((x, y, z)\) with \(1 < x < y < z\) is \((x-1)(y-1)(z-1)\) a factor of \(xyz-1\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Prince Accelerated invited Princess Slow on a journey to his castle. After waiting for a long time, he set out to meet her. After two days of traveling, he met her at one-fifth of her journey. Together, they continued traveling twice as fast as the princess traveled alone. They arrived at the prince's castle on the second Saturday after they met. On which day did they meet, given that the princess started her journey from her castle on a Friday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Wednesday}
Suppose that starting at $t=0$, we introduce bulbs to light a room one by one, immediately replacing any failed bulb. Each new bulb is chosen randomly, with equal probability of being either type A or type B. The lifetime of a type-A bulb follows an exponential distribution with a rate of 1 ($f_{X_A}(x) = e^{-x}, x \geq 0$), while a type-B bulb has an exponential distribution with a rate of 3 ($f_{X_B}(x) = 3e^{-3x}, x \geq 0$). What is the probability that the combined lifetime of the first two type-B bulbs exceeds the lifetime of the first type-A bulb?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7/16}
Let \( V \) be a vector space of functions from \([0,1]\) to \( \mathbb{R} \). Determine the maximal possible dimension of a subspace \( U \subseteq V \) consisting of monotone functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
设双曲线 $C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=k(a>2, k>0)$, 椭圆 $C_{2}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$, 若 $C_{2}$ 的短轴长与 $C_{1}$ 的实轴长的比值等于 $C_{2}$ 的离心率，则求 $C_{1}$ 在 $C_{2}$ 的一条准线上截得线段的长。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
将正整数 1,2,\cdots,100 填入 10\times10 方格表中，每个小方格恰好填 1 个数，要求每行从左到右 10 个数依次递减，对 1\leq i\leq 10，记第 i 行的 10 个数之和为 S_i. 设 n\in\{1,2,\cdots,10\} 满足：存在一种填法，使得 S_1, S_2, \cdots, S_{10} 均大于第 n 列上的 10 个数之和，求 n 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a sports club with 150 members, 65 play badminton, 80 play tennis, 60 play basketball, 20 play both badminton and tennis, 15 play both badminton and basketball, 25 play both tennis and basketball, and 10 play all three sports. Furthermore, 12 members don't participate in any of these sports. How many members play only one of badminton, tennis or basketball?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{115}
In a 3D cubic grid of size $8 \times 8 \times 8$, how many straight lines can be formed that contain exactly 8 symbols in a row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{244}
Determine the condition on the real numbers \(a\) and \(b\) such that the series \[ S = \frac{a}{b} + \frac{a(a+1)}{b(b+1)} + \frac{a(a+1)(a+2)}{b(b+1)(b+2)} + \ldots \] converges. The sequence of terms is given by \[ u_n = \frac{\prod_{i=0}^{i=n-1} (a+i)}{\prod_{j=0}^{j=n-1} (b+j)}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b - a > 1}
Find the analytical solution of the equation \( x(\ln(\ln(9)) - \ln(\ln(x))) = \ln(8) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Compare the values of $a = 211!$ and $b = 106^{211}$. Determine which is greater.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b}
In an inscribed quadrilateral $ABCD$, given $|AB|=a$ and $|AD|=b$, where $a > b$. Find the side $|BC|$, given that $BC$, $CD$, and $AD$ are tangent to a circle whose center is on $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a - b}
Let \( {ABC} \) be a triangle with circumcenter \( O \) ,incenter \( I \) ,and circumcircle \( \Gamma \) . It is known that \( {AB} = 7,{BC} = 8,{CA} = 9 \) . Let
 \( M \) denote the midpoint of major arc \( \overset{⏜}{BAC} \) of \( \Gamma \) ,and let \( D \) denote the intersection of \( \Gamma \) with the circumcircle of \( \bigtriangleup {IMO} \) (other
 than \( M \) ). Let \( E \) denote the reflection of \( D \) over line \( {IO} \) . Find the integer closest to \( {1000} \cdot  \frac{BE}{CE} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
467}
Suppose lines $p$ and $q$ in the first quadrant are defined by the equations $y = -2x + 8$ and $y = -3x + 9$, respectively. What is the probability that a point randomly selected in the 1st quadrant and below $p$ will fall between $p$ and $q$? Express your answer as a decimal to the nearest hundredth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.16}
Determine the maximum number of squares a chess piece, which can move 8 or 9 squares horizontally or vertically without revisiting any square, can cover on a $15 \times 15$ board. The piece may start from any square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{113}
In an acute-angled, non-isosceles triangle $ABC$, the altitudes $AA'$ and $BB'$ intersect at point $H$, and the medians of triangle $AHB$ intersect at point $M$. The line $CM$ bisects segment $A'B'$ into two equal parts. Find the angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1\times1$ squares of area $2010\times2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$ where $\frac{a}{b}$ is a reduced fraction, find $a+b$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6029
}
Consider the sequence of letters $Z\,\,W\,\,Y\,\,X\,\,V$. There are two ways to modify the sequence: we can either swap two adjacent letters or reverse the entire sequence. What is the least number of these changes we need to make in order to put the letters in alphabetical order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Jimmy Butler, a shooting guard, makes 30% of his three-point shot attempts. In any game, the number of three-point shots he attempts follows a Poisson distribution with a mean of 5. What is the probability that he makes exactly 2 three-point shots in a random game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.251}
Evaluate the integral \( \int_0^\infty \frac{1 - \cos(x\sqrt{e-1})}{xe^x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let \( f:[0,\infty)\to\mathbb{R} \) be a continuously differentiable function such that \( \lim _{x\to \infty} f'(x)=\ell \) for a real number \( \ell \). Find \( \lim _{x \to \infty} \frac{f(6x)}{x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\ell}
Find the minimal polynomial of $\beta^k$ over the field $L$, where $\beta$ is a root of the irreducible polynomial $x^p-b$ and $k \in \mathbb{N}$ divides $p$. Provide your answer in terms of $b$, $p$, and $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^{p/k} - b}
Evaluate the integral \( \int_{0}^{\pi/2}\cos(x)\ln(\tan(x))\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln 2}
Given that the ABC stock pays no dividend and is currently priced at $S_0 = \$10$, with the stock price at expiry $T$ going up to $uS_0$ with probability $0 < p < 1$ and down to $dS_0$ with probability $1 - p$, where $d < 1 < u$. Assume no arbitrage and a zero interest rate. A European put option with a strike price of $\$9$ is priced at $\$14/9$, and a European put option with a strike price of $\$8$ is priced at $\$8/9$. What is the fair value of a European call option with a strike price of $\$7$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{29}{9}}
Calculate the limit \( \lim_{n\to\infty}P\left(\sum^n_{i=1}X_i\ge\sum^n_{i=1}Y_i\right) \), where \( X_1, X_2, \ldots \) are independent and identically distributed with a uniform distribution over \((0,1)\), and \( Y_1, Y_2, \ldots \) are independent and identically distributed with density \( e^{-x}\cdot\mathbb{I}\{x\ge0\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Approximate the integral \( \int_{0}^{1}\frac{\sin(x)}{x}dx \) with an error less than \( 10^{-5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.946083}
Find the $x^6$ term in the numerator of the limit expression \(\lim_{x\to 0^+} \frac{-\tan^{-1}(2x^3)+\frac{(x+2)x^5}{x^2+3}+2x\sin^2(x)}{x e^{-1/x}-2x^2+\ln(\frac{x^2+1}{1-x^2})}\) by expanding \(\sin x\) up to order 3 and taking the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Determine the range of values for \(c \in \mathbb{R}^+\) such that the Banach fixed-point theorem's condition is satisfied for the function \(g(x_1, x_2) = \left(\frac{1}{6}(x_1x_2 + cx_2 - 1), \frac{1}{6}(x_1^2 - x_2 + 1)\right)\) on the set \(M = [-1,1]\times [-1,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 4)}
Find a nonzero function $f(x)$ in the span of \( \{x^2, x^3, x^4\} \) that satisfies the conditions \( \langle f, x^2 \rangle = 0 \) and \( \langle f, x^3 \rangle = 0 \), where the inner product is defined as \( \langle f, g \rangle = \int_{-1}^1 f(x)g(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7x^4 - 5x^2}
In the diagram, $ABCDEFGH$ is a cube with an edge length of $12 \text{ cm}$. An ant sits on the cube at vertex $A$. The ant can only walk along the edges of the cube, and cannot walk along any edge more than once. What is the greatest distance that the ant can walk before it cannot continue? Express your answer in centimeters as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Find the positive integer that is coprime with every term of the sequence \( \{a_{n}\} \) where the general term is given by \( a_{n} = 2^{n} + 3^{n} + 6^{n} - 1 \) for \( n \in \mathbf{Z}_{+} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle $ABC$, the median $CM$ is perpendicular to the angle bisector $AL$, and their lengths are in the ratio $\sqrt{2} : 1$. Find $\cos A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{17}}
A scientist has a package of 10 pills for forgetfulness. Each time he takes a pill, he notes the remaining count. When only one pill is left, he orders a new package with instant delivery. If a package is empty, he discards it. What is the probability that at 10:00 AM on December 31, the scientist will have exactly two packages of pills in his cabinet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10}}
Given the function $f\left(x+ \frac {1}{2}\right)= \frac {2x^{4}+x^{2}\sin x+4}{x^{4}+2}$, calculate the value of $f\left( \frac {1}{2017}\right)+f\left( \frac {2}{2017}\right)+\ldots+f\left( \frac {2016}{2017}\right)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4032}
In a right triangle, the sine of the smaller angle is \(\frac{1}{3}\). A line is drawn perpendicular to the hypotenuse, dividing the triangle into two regions of equal area. Determine the ratio in which this line divides the hypotenuse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
The extension of the altitude $BH$ of triangle $ABC$ intersects its circumcircle at point $D$ (points $B$ and $D$ are on opposite sides of line $AC$). The degree measures of arcs $AD$ and $CD$ that do not contain point $B$ are $120^{\circ}$ and $90^{\circ}$, respectively. Determine the ratio in which segment $BD$ is divided by side $AC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:\sqrt{3}}
Determine the number of integer solutions for the equation \( \sum_{i=1}^n i a_i = n \), where \( a_i \in \{0, \dotsc, n\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p(n)}
Let $\|\|_1$ and $\|\|_2$ be norms on the space $C[0,1]$. Define a linear operator $S$ from $(C[0,1], \|\|_2)$ to $(C[0,1], \|\|_1)$ by:

$$S(f)(t)=\frac{d}{dt}f(t)$$

Determine the norm of the operator $S$. \( \|S\| \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
$\text{How many ordered integer pairs }(x,y)(0\leq x,y<31)\text{are there satisfying }(x^{2}-18)^{2}\equiv y^{2}({\mathrm{mod}}31)?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Express $7.\overline{123}$ as a common fraction in lowest terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2372}{333}}
Find the inverse of the polynomial \(2x^2 + x + 1\) in the field \(\mathbb{Z}_3[x]/(x^3 + x^2 + x + 2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x^2}
Provide an example of a function $f(x)$ such that the integral \( \int_{1}^{\infty} f(x^2) \, dx \) converges, but the integral \( \int_{1}^{\infty} f(x) \, dx \) diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{x}}
During the mathematics Olympiad, students solved three problems. Each task was evaluated with an integer number of points from $0$ to $7$. There is at most one problem for each pair of students, for which they got the same number of points. Determine the maximum number of students that could participate in the Olympics.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Compute the remainder when $1^{2547} + 2^{2547} + \ldots + 2547^{2547}$ is divided by $25$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider two functions, \( f \) and \( g \), defined on a real variable such that \( g(x) = \int_0^x f(y)(y-x)dy \) for all \( x \). If \( g \) is three times continuously differentiable, find the smallest integer \( n \) such that \( f \) must be \( n \) times continuously differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $ABCD$ be a convex quadrilateral with positive area such that every side has a positive integer length and $AC=BC=AD=25$. If $P_{\text{max}}$ and $P_{\text{min}}$ are the quadrilaterals with maximum and minimum possible perimeter, the ratio of the area of $P_{\text{max}}$ and $P_{\text{min}}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ for some positive integers $a,b,c$, where $a,c$ are relatively prime and $b$ is not divisible by the square of any integer. Find $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{585}