inclusionAI/Ring-lite-rl-data huggingface dataset:
A set  $D$  of positive integers is called *indifferent* if there are at least two integers in the set, and for any two distinct elements  $x,y\in D$ , their positive difference  $|x-y|$  is also in  $D$ . Let  $M(x)$  be the smallest size of an indifferent set whose largest element is  $x$ . Compute the sum  $M(2)+M(3)+\dots+M(100)$ .

*Proposed by Yannick Yao*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1257}
Given $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(x,4)$, if the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute, then the range of real number $x$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-8,2\right)\cup \left(2,+\infty \right)}
In a group of 10 basketball teams which includes 2 strong teams, the teams are randomly split into two equal groups for a competition. What is the probability that the 2 strong teams do not end up in the same group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{9}}
We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into  $n$  sets of equal weight. What is the maximal  $n$  for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2886}
Given the cryptarithm: ЖАЛО + ЛОЖА = ОСЕНЬ. Identical letters represent identical digits, and different letters represent different digits. Determine the value of the letter А.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the quadratic function f(x) = ax^2 + bx + c satisfying 2a + c/2 > b and c < 0, determine an interval containing a zero of the function f(x). Express your answer as an interval (lower bound, upper bound) using parentheses.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 0)}
Donggyu wants to complete the diagram so that each box contains a positive integer and each box in the top two rows contains the product of the integers in the two boxes below it. He wants the integer in the top box to be 720. How many different values can the integer \( n \) take? Express your answer as a single integer count.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A sphere is placed inside a dihedral angle of 60°, and the sphere touches the two half-planes of the dihedral angle at points A and B, respectively. The spherical distance between points A and B is $2\pi$ cm. Find the radius of the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
6. The password setting for a certain lock involves assigning one of the two numbers, 0 or 1, to each vertex of a regular $n$-sided polygon $A_{1} A_{2} \cdots A_{n}$, and coloring each vertex either red or blue, such that for any two adjacent vertices, at least one of the number or color is the same. How many different password settings are there for this lock?
(2010, National High School Mathematics League Competition)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{n}+(-1)^{n}+2}
记 $S=\left\{(x, y, z, w) \in \mathbb{Z}^{4} \mid 0 \leq x, y, z, w \leq 18, x^{2}+y^{2} \equiv z^{2}+w^{2}(\bmod 19)\right\}$,则 S 中有 $\qquad$个元素。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7201}
Find all $a\in\mathbb{C}$ such that $f=X^4 −(a+4)X^3 +(4a+5)X^2 −(5a+2)X +2a$ has $a$ as a double root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let the curve $y = x^n + 1$ ($n \in \mathbb{N}^*$) have a tangent line at the point $(1,1)$, and let the x-coordinate of the point where this tangent line intersects the x-axis be $x_n$.  
Define $a_n = \lg x_n$, then the value of $a_1 + a_2 + \ldots + a_{99}$ is \_\_\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
The milk level in a rectangular box measuring 58 feet by some feet is to be lowered by 6 inches. To do this, 5437.5 gallons of milk must be removed. What is the width of the rectangular box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Suppose  $a$ ,  $b$ ,  $c$ , and  $d$  are positive real numbers which satisfy the system of equations \[\begin{aligned} a^2+b^2+c^2+d^2 &= 762,  ab+cd &= 260,  ac+bd &= 365,  ad+bc &= 244. \end{aligned}\] Compute  $abcd.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14400}
Given the function $f(x)=2\sin x\cos x+4\cos ^{2}x-1$, if real numbers $a$, $b$, $c$ satisfy $af(x)-bf(x+c)=3$ for any real number $x$, then $3a^{2}+2b+\cos c=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{4}}
Let $f_{1}(x)=\sqrt{1-x}$, and for integers $n \geq 2$, let \[f_{n}(x)=f_{n-1}\left(\sqrt{n^2 - x}\right).\]Let $N$ be the largest value of $n$ for which the domain of $f_n$ is nonempty. For this value of $N,$ the domain of $f_N$ consists of a single point $\{c\}.$ Compute $c.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-231}
Let $n$ be a positive integer. Determine the smallest positive integer $k$ such that for any colouring of the cells of a $2n\times k$ table with $n$ colours there are two rows and two columns which intersect in four squares of the same colour.
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Answer: \boxed{ 2n^2 - n + 1 }
Evaluate the integral: $$\int_{0}^{2\pi} e^{i\cos\theta}\cos\theta \, d\theta$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i J_1(1)}
Find the simple interest on Rs. 69,000 at 16 2/3% per year for a certain period. The simple interest is Rs. 8,625. What is the duration of the investment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Oleg writes a real number above each column of a blank $50 \times 50$ grid (outside the grid) and a real number to the left of each row (outside the grid). It is known that all 100 real numbers he writes are distinct, and there are exactly 50 rational numbers and 50 irrational numbers among them. Then, Oleg writes the sum of the number to the left of the row and the number above the column in each cell of the grid (creating an addition table). What is the maximum number of rational numbers that can appear in this grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
Determine the order of the Taylor polynomial needed to approximate $f(x) = \cos x$ centered at $x = 0$ with an error of at most $10^{-8}$ for $x = \frac{\pi}{4}$. Use the remainder formula for Taylor series with $c = \frac{\pi}{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A permutation $ (a_1, a_2, a_3, a_4, a_5)$ of $ (1, 2, 3, 4, 5)$ is heavy-tailed if $ a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
4. In a sequence of English letter strings, the first string $a_{1}=A$, the second string $a_{2}=B$, and each subsequent string $a_{n}(n \geq 3)$ is formed by appending the reverse of $a_{n-2}$ to $a_{n-1}$. For example, $a_{3}=a_{2} \overline{a_{1}}=B A$ (we use $\overline{a_{i}}$ to denote the reverse of $a_{i}$, which is the string read from right to left, such as $\overline{A B B}=B B A$, $\overline{A A B A}=A B A A$), $a_{4}=a_{3} \bar{a}_{2}=B A B$, $a_{5}=a_{4} \overline{a_{3}}=B A B A B$, $a_{6}=a_{5} \overline{a_{4}}=B A B A B B A B$. Therefore, among the first 1000 strings in this sequence, there are $\qquad$ palindromic strings (a palindromic string is one that reads the same from left to right as from right to left, such as $A B A$, $A A B A A$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{667}
In trapezoid $ ABCD$, $ AB \parallel CD$, $ \angle CAB < 90^\circ$, $ AB = 5$, $ CD = 3$, $ AC = 15$. What are the sum of different integer values of possible $ BD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Given the function f(x) = sin(ωx + φ) where ω > 0 and |φ| ≤ π/2, we are told that x = -π/4 is a zero of f(x), and x = π/4 is the axis of symmetry of the graph of y = f(x). Additionally, f(x) is monotonic on the interval (π/18, 5π/36). Find the maximum value of ω. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of independent events defined on $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{P}(A_n)<1$ for all $n$ and $\mathbb{P}(\bigcup_{n=1}^{\infty}A_n)=1.$ Find the value of $\mathbb{P}(\bigcap_{i=1}^{\infty}\bigcup_{n=i}^{\infty}A_n).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A convex quadrilateral \( {ABCD} \) with area2002contains a point \( P \) in its interior such that \( {PA} = {24} \) , \( {PB} = {32},{PC} = {28} \) ,
 and \( {PD} = {45} \) . FInd the perimeter of \( {ABCD} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( 4\left( {{36} + \sqrt{113}}\right) \)}
In square \(ABCD\), an isosceles triangle \(AEF\) is inscribed such that point \(E\) lies on side \(BC\) and point \(F\) lies on side \(CD\), and \(AE = AF\). The tangent of angle \(AEF\) is 3. Find the cosine of angle \(FAD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{5}}{5}}
Let $a > 0$. If the inequality $22 < ax < 222$ holds for precisely $10$ positive integers $x$, find how many positive integers satisfy the inequality $222 < ax < 2022$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Find the sum of all numbers $x$ satisfying $x+25/x = 10.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
65. Suppose you have forgotten one digit of a phone number you need and dial it at random. What is the probability that you will have to make no more than two calls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2}
Given \( x \) satisfies \(\log _{5 x} 2 x = \log _{625 x} 8 x\), find the value of \(\log _{2} x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\ln 5}{2 \ln 2 - 3 \ln 5}}
Alyssa went to 11 soccer games this year but missed 12. She also watched 8 games online. Last year, she went to 13 games in person. For each in-person game, she paid $20 for a ticket. This year, she spent $120 on a streaming subscription to watch games online. Next year, she plans to go to 15 games in person and watch 10 games online. Two of her friends will join her at 5 of the games next year, and they will pay her back for their tickets. If a streaming subscription is expected to cost $150 next year, how much will Alyssa spend in total on attending and streaming soccer games over the three years, considering the amount her friends pay for their tickets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$850}
How many positive five-digit integers are there that have the product of their five digits equal to $900$?

(Karl Czakler)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
Let \(p\) be a prime number, and consider a group \((G, \cdot)\) with \(p^3\) elements. The center of the group, \(\operatorname{Z}(G)\), has \(p\) elements. Determine the number of distinct centralisers \(\operatorname{C}(x)\) for elements \(x \in G\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + 2}
Find the almost sure limit of $\frac{X_t}{t}$ as $t \to +\infty$, where $X_{t} = \inf(n:\xi_{1}+\ldots+\xi_{n}> t)$ and $\{\xi_{i}\}$ are i.i.d. random variables with $\xi_{i}\sim \mathrm{Exp}(\lambda)$. What is this limit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda}
Find a positive real number $R$ such that for all complex numbers $z$ with $|z| > R$, the inequality $|z^{10} + 100z + 9| > 1$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that in the expansion of $(1-2x)^{n} (n \in \mathbb{N^*})$, the coefficient of $x^{3}$ is $-80$, find the sum of all the binomial coefficients in the expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["32"]}
设实数 $k, l, m$ 满足：函数 $y=(x+1)\left(x^{2}+k x+l\right)$ 的图像有对称中心 $(1,0)$ ，且与函数 $y=x^{3}+m$ 的图像有公共点, 则 $k+l+m$ 的取值范围是$\left(-\infty, \frac{a}{b}\right]$，求$a+b$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Evaluate the limit \( \lim_{x\to 3} \left(\dfrac{\sqrt{2x+3}-x}{\sqrt{x+1}-x+1} \right)^{ \left(\dfrac{x-1-\sqrt{x^2-5}}{x^2-5x+6}\right)} \) and express it in the form \(\dfrac{a\sqrt{b}}{c}\), where \(a,b,c \in \mathbb{N}\). Find the least value of \(a^2+b^2+c^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimeter of the largest rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{6}}
Consider  $\triangle \natural\flat\sharp$ .  Let  $\flat\sharp$ ,  $\sharp\natural$  and  $\natural\flat$  be the answers to problems  $4$ ,  $5$ , and  $6$ , respectively.  If the incircle of  $\triangle \natural\flat\sharp$  touches  $\natural\flat$  at  $\odot$ , find  $\flat\odot$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
Let $\mathcal{P}$ be a regular 17 -gon; we draw in the $\left(\begin{array}{c}17 \\ 2\end{array}\right)$ diagonals and sides of $\mathcal{P}$ and paint each side or diagonal one of eight different colors. Suppose that there is no triangle (with vertices among vertices of $\mathcal{P}$ ) whose three edges all have the same color. What is the maximum possible number of triangles, all of whose edges have different colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{544}
Stuart has drawn a pair of concentric circles. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the small one. If $m\angle ABC=60^\circ$, then how many segments will he draw before returning to his starting point at $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $29x + y = 2035$. Find the number of such distinct triangles whose area is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1225}
A rook has toured a chessboard, visiting each square at least once. What is the minimum number of turns it could have made in the process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Compute the double integral \( \iint_{M} y \, dx \, dy \) where \( M \) is the region defined by \( y \geq 0 \), \( x^2 + y^2 \leq 1 \), and \( (x - 1)^2 + y^2 \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{24}}
Determine the range of values for $a$ such that the polynomial \( p(x) = x^4 + ax^3 + bx^2 + ax + 1 \) has complex (non-real) roots, given that \( x = 1 \) is a root and \( b = -2(a+1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, 0)}
Given $A(-2,2)$, $B$ is a point on the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, and $F$ is the left focus of the ellipse. Find the coordinates of $B$ when $|AB|+\frac{5}{3}|BF|$ reaches its minimum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \left( -\frac{5}{2} \sqrt{3}, 2 \right) }
Given an isosceles triangle \(XYZ\) with \(XY = YZ\) and an angle at the vertex equal to \(96^{\circ}\). Point \(O\) is located inside triangle \(XYZ\) such that \(\angle OZX = 30^{\circ}\) and \(\angle OXZ = 18^{\circ}\). Find the measure of angle \(\angle YOX\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
On a calculator's keyboard, there are digits from 0 to 9 and signs for two operations. Initially, the display shows the number 0. You can press any keys. The calculator performs actions in the sequence of key presses. If an operation sign is pressed several times in a row, the calculator remembers only the last press. A distracted Scientist pressed a lot of buttons in a random sequence. Approximately find the probability that the result of the resulting sequence of actions is an odd number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
A cobbler can mend 3 pairs of shoes in an hour on Mondays, Tuesdays, and Wednesdays. On Thursdays, his work pace slows down, and he can only mend 2 pairs of shoes per hour. He takes a 30-minute break each day from Monday to Thursday. On Friday, he works from 8am to 11am, during which he mends shoes at a rate of 4 pairs per hour. How many pairs of shoes can the cobbler mend in a week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{94}
1. When $x^{2}$ was added to the quadratic trinomial $f(x)$, its maximum value increased by $\frac{27}{2}$, and when $4 x^{2}$ was subtracted from it, its maximum value decreased by 9. How will the maximum value of $f(x)$ change if $2 x^{2}$ is subtracted from it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{4}}
We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday.  On the first Monday the number we record is ten and a half.  On every Tuesday and every Thursday the number we record is one third of what it was on the previous workday.  On every Monday, Wednesday, and Friday the number we record is double what it was on the previous workday.  This will go on forever. What is the sum of all of the numbers we will record?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Find the smallest natural number \( k \) such that for some natural number \( a \), greater than 500,000, and some natural number \( b \), the equation \(\frac{1}{a} + \frac{1}{a+k} = \frac{1}{b}\) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
Find the product of all values of $d$ such that $x^{3} + 2x^{2} + 3x + 4 = 0$ and $x^{2} + dx + 3 = 0$ have a common root.
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Answer: \boxed{1}
抛物线 $y^{2}=2 p\left(x-\frac{p}{2}\right)(p>0)$ 上动点 $A$ 到点 $B(3,0)$ 的距离的最小值记为 $d(p)$ ，满足 $d(p)=2$ 的所有实数 $p$ 的和为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the largest constant $C>0$ satisfying the following property:

For any 2023-tuple $\left(a_{1}, a_{2}, \ldots, a_{2023}\right)$ of strictly positive real numbers that are pairwise distinct, the following inequality holds:

$$
\frac{a_{1}}{\left|a_{2}-a_{3}\right|}+\frac{a_{2}}{\left|a_{3}-a_{4}\right|}+\ldots+\frac{a_{2023}}{\left|a_{1}-a_{2}\right|}>C
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Find the smallest positive integer $n$ such that $$\underbrace{2^{2^{2^{2}}}}_{n 2^{\prime} s}>\underbrace{((\cdots((100!)!)!\cdots)!)!}_{100 \text { factorials }}$$
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Answer: \boxed{104}
A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better ranked team always wins, and the worse ranked team is eliminated. Let $p$ be the probability that the second best ranked team is eliminated in the last round. Compute $\lfloor 2021 p \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{674}
Five consecutive integers are represented by p, q, r, s, and t. The two integers in the leftmost circle add to 63, and the two integers in the rightmost circle add to 57. Find the value of r. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A person crosses a 3,500 meters long street in 9.5 minutes. What is their speed in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.74}
9. Given that $\triangle A B C$ is an acute triangle, $b=2 c$, and $\sin B-\sin (A+B)=2 \sin C \cos A$. Then the range of $(\cos B+\sin B)^{2}+$ $\sin 2 C$ is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+\frac{\sqrt{3}}{2}}
求满足下列条件的大于 5 的最小奇数 a：存在正整数 m_1, n_1, m_2, n_2，使得 a=m_1^2+n_1^2，a^2=m_2^2+n_2^2，且 m_1-n_1=m_2-n_2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{261}
Let  $ABC$  be a triangle with  $AB=5$ ,  $BC=6$ ,  $CA=7$ . Let  $D$  be a point on ray  $AB$  beyond  $B$  such that  $BD=7$ ,  $E$  be a point on ray  $BC$  beyond  $C$  such that  $CE=5$ , and  $F$  be a point on ray  $CA$  beyond  $A$  such that  $AF=6$ . Compute the area of the circumcircle of  $DEF$ .

*Proposed by James Lin.*
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Answer: \boxed{\frac{251}{3} \pi}
Given that (1+ex)<sup>2019</sup>=a<sub>0</sub>+a<sub>1</sub>x+a<sub>2</sub>x<sup>2</sup>+……+a<sub>2019</sub>x<sup>2019</sup>, find the value of:
- $$\frac {a_{1}}{e}$$+ $$\frac {a_{2}}{e^{2}}$$\- $$\frac {a_{3}}{e^{3}}$$+ $$\frac {a_{4}}{e^{4}}$$\-……- $$\frac {a_{2019}}{e^{2019}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
What is the maximum number of cells on an $8 \times 8$ chessboard that can be cut by a single straight line?
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Answer: \boxed{15}
If the line $y = -2x + k$ forms a triangle with the coordinate axes and the area of this triangle is 4, then the value of $k$ is.
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Answer: \boxed{\pm 4}
Determine the number of invertible elements in the quotient ring \( \mathbb{Z}[i]/(220+55i)\mathbb{Z}[i] \).
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Answer: \boxed{30720}
Given the equations \(xy + x + y = 44\) and \(x^2y + xy^2 = 448\), find the value of \(x^2 + y^2\).
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Answer: \boxed{200}
A certain organism starts with 4 cells. Each cell splits into two cells at the end of three days. However, at the end of each 3-day period, 10% of the cells die immediately after splitting. This process continues for a total of 9 days. How many cells are there at the end of the $9^\text{th}$ day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Given the function \(f_X(x) = \begin{cases}4x^3 & 0<x<1\\ 0 & \text{otherwise}\end{cases}\) and the transformation \(y = -\log(x)\), find the cumulative distribution function (CDF) \(F_Y(y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - e^{-4y}}