inclusionAI/Ring-lite-rl-data huggingface dataset:
Given a $2018 \times 4$ grid, tint the cells with red and blue such that each row and each column has an equal number of red and blue cells. Determine the number of ways, $M$, to tint the grid with the mentioned requirement and find $M \pmod{2018}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and the area of $ABC$ is $S$. It is given that $(a^2 + b^2)\tan C = 8S$, and $\sin A \cos B = 2\cos A \sin B$. Find the value of $\cos A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{30}}{15}}
$ 500 will become $ 1000 in 5 years find the rate of interest ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
"Drawing a pie to satisfy hunger" is a ______ event.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{impossible}}
Julia's garden has a 3:7 ratio of tulips to daisies. She currently has 35 daisies. She plans to add 30 more daisies and wants to plant additional tulips to maintain the original ratio. How many tulips will she have after this addition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
In an alphabet with $n>1$ letters, a word is defined as any finite sequence of letters where any two consecutive letters are different. A word is called "good" if it is not possible to delete all but four letters from it to obtain a sequence of the form $a a b b$, where $a$ and $b$ are different letters. Find the maximum possible number of letters in a "good" word.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n+1}
The sum of the coefficients of the quadratic equation $5x(1-x) = 3$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The math scores of a high school's college entrance examination, denoted as $\xi$, approximately follow a normal distribution $N(100, 5^2)$. Given that $P(\xi < 110) = 0.98$, find the value of $P(90 < \xi < 100)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.48}
Compute $\lim_{x \to -1} f(x)$ for a function $f: \mathbb R \to \mathbb R$ given by the following limit equation:

$$4 = \lim_{x \to -1} \left( \frac{f(x)+2}{x+1} - \frac{x}{x^2-1} \right) \tag{1}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{2}}
求不定方程 $x^{2}+615=2^{y}$ 的正整数解的组数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the digits 1, 2, 3, 4, and 5, create a five-digit number without repetition, with 5 not in the hundred's place, and neither 2 nor 4 in the unit's or ten-thousand's place, and calculate the total number of such five-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Place five different files, labeled A, B, C, D, and E, into seven drawers numbered consecutively from 1 to 7. Each drawer can contain at most one type of file. If files A and B must be placed in adjacent drawers, and files C and D must also be placed in adjacent drawers, then find the total number of different ways to place the files into the drawers that meet these conditions. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Given the sets:
\[ M=\left\{(x, y) \mid \tan(\pi y) + \sin^2(\pi x) = 0 \right\}, \]
\[ N=\left\{(x, y) \mid x^{2}+y^{2} \leq 2\right\}, \]

find the number of elements in the intersection \( M \cap N \). Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Five people participate in a relay game, each randomly choosing between two options: left path and right path. If more than 2 people choose the same path, each of them gets 1 point; if fewer than 3 people choose the same path, each of them gets 0 points. Let $\xi$ denote the score of a player named Xiaoqiang. Calculate the expected value of $\xi$, $E\xi$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{16}}
Given the expression
$$
\left(\left(\cdots\left(\left((x-2)^{2}-2\right)^{2}-2\right)^{2}-\cdots-2\right)^{2}-2\right)^{2},
$$
which contains $k$ nested sets of parentheses, remove the parentheses and combine like terms to obtain a polynomial. Find the coefficient of $x^{2}$ in this polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4^{k-1} \cdot \frac{4^{k}-1}{3}}
Find the unique pair of positive integers $(a, b)$ with $a<b$ for which $$\frac{2020-a}{a} \cdot \frac{2020-b}{b}=2$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(505,1212)}
An alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as $U$, $W$, $XW$, $ZZ$, $V$, $Y$, $ZV$, $ZW$, $ZY$, and $X$. Each letter represents a nonzero digit. Each letter represents the same digit everywhere it appears, and different letters represent different digits. Also, the alien is using a base other than base ten. The alien writes another number as $UZWX$. Compute this number (expressed in base ten, with the usual, human digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5390}
Given that $f(x)$ is an even function defined on $\mathbb{R}$, $g(x) = f(x) + x^2$, and $g(x)$ is monotonically increasing when $x \in (-\infty, 0]$, find the solution set of the inequality $f(x+1) - f(x-1) + 4x > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0)}
The integer $n$ has exactly six positive divisors, which are $1 < a < b < c < d < n$. Let $k = a - 1$. If the $k$-th divisor (according to the above ordering) of $n$ is equal to $(1 + a + b)b$, find the highest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
已知$f(x)$为定义在$\mathbf{R}$上的奇函数，当$x\leqslant0$时，
$$f(x)=\left|x+\frac{1}{2}\sin\alpha\right|+\left|x+\frac{3}{2}\sin\alpha\right|-2\sin\alpha$$
其中，$\alpha\in(-\pi,\pi)$。若对任意实数$x$，均有$f(x)\geqslant f(x+2)$，则$\alpha$的最小值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two parts such that with the two pieces of this stick and the two whole sticks, he could form the contour of a right triangle. How many square centimeters is the area of this triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
A coin with probability of heads p is being tossed repeatedly. Consider the 4 state Markov chain given by the results of the previous toss and the toss before that. Using Markov chain hitting time arguments, find the expected time we need to wait until we see two consecutive heads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac {1+p}{p^2}\)}
分解因式：$\left( xy+1 \right)\left( x+1 \right)\left( y+1 \right)+xy=$_______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(xy+x+1)(xy+y+1)}
Given $a \in \mathbb{R}$, let the set $S = \{x \mid 2ax^2 - x \leq 0\}$ and $T = \{x \mid 4ax^2 - 4a(1-2a)x + 1 \geq 0\}$. If $S \cup T = \mathbb{R}$ (where $\mathbb{R}$ is the set of all real numbers), then the range of values for the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 \leq a \leq 1}
Eighteen children solved a problem from the April issue of KöMaL. They want to discuss their solutions via email. What is the minimum number of emails they need to send to ensure that each of them learns about every solution? (Each email counts as many times as the number of people it is sent to.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Weiming Real Estate Company sold a house to Mr. Qian at a 5% discount off the list price. Three years later, Mr. Qian sold the house to Mr. Jin at a price 60% higher than the original list price. Considering the total inflation rate of 40% over three years, Mr. Qian actually made a profit at a rate of     % (rounded to one decimal place).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.3}
Given positive numbers \( a \) and \( b \) satisfying \( a + b = 1 \), find the minimum value of \( M = \sqrt{1 + 2a^2} + 2 \sqrt{\left(\frac{5}{12}\right)^2 + b^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{34}}{12}}
Three sportsmen called Primus, Secundus, and Tertius take part in a race every day. Primus wears the number '1' on his shirt, Secundus wears '2', and Tertius wears '3'.

On Saturday, Primus wins, Secundus is second, and Tertius is third. Using their shirt numbers this result is recorded as '123'.

On Sunday, Primus starts the race in the lead with Secundus in second. During Sunday's race:
- Primus and Secundus change places exactly 9 times,
- Secundus and Tertius change places exactly 10 times,
- Primus and Tertius change places exactly 11 times.

How will Sunday's result be recorded?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{231}
Given that $\cos(\pi+\theta) = -\frac{1}{2}$, find the value of $\tan(\theta - 9\pi)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Given the sequence $\{a_n\}$: $\frac{1}{2}$, $\frac{1}{3} + \frac{2}{3}$, $\frac{1}{4} + \frac{2}{4} + \frac{3}{4}$, $\frac{1}{5} + \frac{2}{5} + \frac{3}{5} + \frac{4}{5}$, ..., then the sum of the first $n$ terms of the sequence $b_n = \frac{1}{a_n a_{n+1}}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4n}{n+1}}
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. For example, her expression could be $4 + 3 \times 3$, with a value of 13. Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given that quadrilateral ABCD is a cyclic quadrilateral inscribed in circle O, with AC = sqrt(3)BD and angle ADC = 2*angle BAD. If AB*CD + BC*AD = 4*sqrt(3), calculate the radius of circle O. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \frac{1}{3}$, and
\[c_n = \frac{2 - c_{n-1}}{3c_{n-2}}\] for all $n \ge 3$. Find $c_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
What is the number of ways in which one can choose  $60$  unit squares from a  $11 \times 11$  chessboard such that no two chosen squares have a side in common?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
12 men are equivalent to as many women as are equal to 20 boys. All together, they earn a combined Rs. 300. What are the men's wages?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
There are $n$ students standing one behind the other in a circle, with heights $h_{1}<h_{2}<\ldots<h_{n}$. At any moment, if a student of height $h_{k}$ is directly behind a student of height less than or equal to $h_{k-2}$, they can choose to switch places with that student. Determine the maximum number of exchanges that can occur among all initial configurations and possible exchange choices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{n}{3}}
A pet store has 17 bird cages. Each cage contains a different number of parrots and parakeets. The first cage has 2 parrots and 7 parakeets. Each subsequent cage has one additional parrot and two additional parakeets more than the preceding cage. If 3 cages are temporarily empty due to cleaning, how many birds are there in total presently hosted in the cages of this pet store?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{399}
Find the positive integer $k$ such that the roots of $x^3 - 15x^2 + kx -1105$ are three distinct collinear points in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{271}
Let $p$ and $q$ be any two prime numbers greater than 100. The minimum value of the greatest common divisor of $p^2-1$ and $q^2-1$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The coefficient of the $x^2$ term in the expansion of $((a+x)(1 + \sqrt{x}))^5$ is $15$. Find the sum of all the coefficients in the expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\pi$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144\pi}
Find the minimum value of $\int_{0}^{1} (f'(x))^2 \, dx$ given that $f \in C^{(1)} [0,1]$, $f(1) = b$, and $\int_{0}^{1} f(x) \, dx = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3b^2}
Determine $x^2+y^2+z^2+w^2$ if

$\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$$\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$$\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$$\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
3. A student wrote a program for recoloring a pixel into one of 128 different colors. These colors he numbered with natural numbers from 1 to 128, and the primary colors received the following numbers: white color - number 1, red - 5, orange - 13, yellow - 21, green - 45, blue - 75, indigo - 87, violet - 91, black - 128. If the initial color of the pixel has a number \( n \leq 19 \), then the student's program recolors it to the color with the number \( n+4 \), and if the initial color of the pixel has a number \( n \geq 20 \), then the pixel is recolored to the color with the number \( |129-2n| \). Initially, the pixel was red. The student applied his program 2019 times sequentially. What color did the pixel end up as a result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
In an investment, the amount increases by a certain percentage of itself each year, with interest compounding annually. In the first year, the investment increases by 1/9th of its value, and in the second year, it increases by 1/6th of its value. If the present value is Rs. 64,000, how much will the investment be worth after two years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{82,962.96}
The diagonal \( AC \) of the inscribed quadrilateral \( ABCD \) is the diameter of the circumscribed circle \( \omega \). From point \( D \), a line is drawn perpendicular to the segment \( BC \), and it intersects the circle \( \omega \) again at point \( E \). Find the ratio of the areas of triangle \( BCD \) and quadrilateral \( ABEC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Two concentric circles have radii of 1 and 2 units, respectively. How small can the area of a cyclic quadrilateral be, which is inscribed in the larger circle and contains the smaller circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{3}}
The leading coefficient $a$ in the quadratic polynomial $P(x) = ax^2 + bx + c$ is greater than $100$. Determine the maximum number of integer values for $x$ such that $|P(x)| < 50$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
当 $\frac{\pi}{4} \leq x \leq \frac{\pi}{2}$ 时, $y=\sin ^{2} x+\sqrt{3} \sin x \cos x$ 的取值范围是 $[a, \frac{c}{b}]$。请给出$a + b+c$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given $60 < x < 84$ and $28 < y < 33$, find the range of $x - y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27 < x - y < 56}
Right triangle DEF has leg lengths DE = 18 and EF = 24. If the foot of the altitude from vertex E to hypotenuse DF is F', then find the number of line segments with integer length that can be drawn from vertex E to a point on hypotenuse DF.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A point $Q$ is chosen inside $\triangle DEF$ such that lines drawn through $Q$ parallel to $\triangle DEF$'s sides decompose it into three smaller triangles $u_1$, $u_2$, and $u_3$, which have areas $3$, $12$, and $15$ respectively. Determine the area of $\triangle DEF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
给定正整数 $n(n \geqslant 3)$ 。对于 $1,2, \cdots, n$的任意一个排列 $P=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$, 若 $i<j<k$, 则称 $x_{j}$ 介于 $x_{i}$ 和 $x_{k}$ 之间（如在排列（ $1,3,2,4$ ）中，3 介于 1 和 4 之间， 4 不介于 1 和 2 之间）。设集合 $S=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$的每个元素 $P_{i}$ 是 $1,2, \cdots, n$ 的排列. 已知 \{1,2, \cdots, n\} 的任意三个不同数中都有一个数，它在每个 $P_{i}(1 \leqslant i \leqslant m)$ 中都不介于另外两个数之间。求 $m$ 的最大值。请给出当 $n = 5$ 时，$m$ 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A couple agreed to split their expenses equally, including the salary of their house help. While doing her job, the house help had an accident; thus, she underwent a medical procedure which cost her $128. The couple decided to pay half of the medical expenses and the other half will be deducted from her $160 salary. If the husband paid for the entire half of the hospital bill, how much does he still need to pay so that they split the expenses for the house help equally?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
In triangle \(ABC\), the radii of the circumscribed and inscribed circles are given as \(R\) and \(r\) respectively. Let \(A_1, B_1, C_1\) be the points where the angle bisectors of triangle \(ABC\) intersect the circumscribed circle. Find the ratio of the areas of triangles \(ABC\) and \(A_1B_1C_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2r}{R}}
Daniel is 85 years old, Elise is 55 years old, and Ishaan is 15 years old. How many years will it take until Daniel is 3.5 times as old as Ishaan, and Elise is twice as old as Ishaan?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest value of $k$ for which, by a series of such moves, it is possible to achieve a configuration where each chip is shifted one position clockwise relative to its initial position?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\angle BAC = 120^{\circ}$ and the angle bisector of $\angle BAC$ intersects $BC$ at $D$, if $AD = 1$, then the minimum value of $b + 4c$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
On the reverse sides of 2005 cards, different numbers are written (one number on each card). In one query, you are allowed to point to any three cards and learn the set of numbers written on them. What is the minimum number of queries needed to determine the numbers written on each card?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
Joe's quiz scores were 88, 92, 95, 81, and 90, and then he took one more quiz and scored 87. What was his mean score after all six quizzes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88.83}
9 friends decided to divide the hotel bill evenly. The bill was a certain amount, and they paid $124.11 since 1 cent is the smallest unit. What was the original bill amount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1116.99}
Given that \( n \) is a positive integer, \( P \) is a prime number, and \( pn \) has exactly 8 positive divisors, arrange them in ascending order as \( 1=d_{1}<d_{2}< \cdots <d_{8}=pn \). Additionally, let \( d_{17p-d_{3}}=\left(d_{1}+d_{2}+d_{3}\right)\left(d_{3}+d_{4}+13p\right) \). Find \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
Evaluate the integral \( \int_0^\infty xI_0(2x)e^{-x^2}\,dx \), where \( I_0(x) \) is the modified Bessel function of the first kind, defined by \( I_0(x) = \frac{1}{\pi} \int_0^\pi e^{x\cos\theta}\,d\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e}{2}}
Allan brought 6 balloons and Jake brought 3 balloons to the park. Jake then bought 4 more balloons at the park. Meanwhile, Emily brought 5 balloons to the park. How many more balloons did Jake have than the combined total of Allan and Emily in the park?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( \int\limits_{0}^{\infty} {\frac{2}{3x^{4/3}} \sqrt{\frac{\lambda}{2\pi}} \exp \left( \frac{-\lambda (x^{2/3}-\mu)^2}{2\mu^2 x^{2/3}}  \right) dx} \) for \(\mu>0, \lambda>0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $x, y, z$ be positive real numbers satisfying $2 x y z=3 x^2+4 y^2+5 z^2$.
Find the minimum of the expression $P=3 x+2 y+z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Evaluate the sum of the infinite series: $$\sum_{k=5}^{\infty}\binom{k-1}{k-5}\frac{k^3}{2^{k}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1330}
Adam has a box with $15$ pool balls numbered from $1$ to $15$. He picks out $5$ of them and sorts them in increasing order. He then calculates the four differences between each pair of adjacent balls and finds that exactly two of these differences are equal to $1$. How many different selections of $5$ balls could he have drawn from the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{990}
Sara has an ice cream cone with every meal. The cone has a height of $2\sqrt2$ inches and the base of the cone has a diameter of $2$ inches. Ice cream protrudes from the top of the cone in a perfect hempisphere. Find the surface area of the ice cream cone, ice cream included, in square inches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$5\pi$

}
In the context of the ratio test for determining the radius of convergence of a power series \( \sum c_{n}z^n \), if \( \limsup \frac{c_{n}}{c_{n+1}} \neq \liminf \frac{c_{n}}{c_{n+1}} \), which should be considered: the lim sup or the lim inf?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\liminf}
Let $a$ and $b$ be two positive reals such that the following inequality
    \[ ax^3 + by^2 \geq xy - 1 \] is satisfied for any positive reals $x, y \geq 1$. Determine the smallest possible value of $a^2 + b$.

[i]Proposed by Fajar Yuliawan[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3\sqrt{3}}}