inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the set $I = \{x \mid 1 < x < 5, x \in \mathbb{N}\}$ and set $A = \{2, 3\}$, calculate $C_{I} \setminus A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{4\}}
■ Example 2 For non-negative real numbers $x_{1}$, $x_{2}, \cdots, x_{n}$ satisfying $x_{1}+x_{2}+\cdots+x_{n}=1$, find the maximum value of $\sum_{j=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$. (40th IMO China National Team Selection Exam Question)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
There is a fence that consists of  $n$  planks arranged in a line. Each plank is painted with one of the available  $100$  colors. Suppose that for any two distinct colors  $i$  and  $j$ , there is a plank with color  $i$  located to the left of a (not necessarily adjacent) plank with color  $j$ . Determine the minimum possible value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
Jackson, Brandon, and Meagan all invest in the stock market in different currencies. Jackson invests 500 USD, Brandon invests 600 CAD, and Meagan invests 400,000 JPY. Jackson's investment quadruples in value, Brandon's is reduced to 20% of its initial value, and Meagan's investment increases by 50%.

During this time, the exchange rate changes: 1 USD becomes equivalent to 1.2 CAD, and 1 USD becomes equivalent to 110 JPY. Considering these changes in exchange rates as well, how much more money in USD does Jackson have than Brandon and Meagan combined now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3554.55}
Triangle $ABC$ has $\angle A = 90^\circ$, $\angle C = 30^\circ$, and $AC = 12$. Let the circumcircle of this triangle be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle CAD = 60^\circ$. Define points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively.

Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$ intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c} - \sqrt{d})$ for positive integers $a, b, c, d$, where $a, b$ are relatively prime, and the sum of $a, b, c, d$ is minimal.

Find $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
(11) (15 points) Divide each side of the equilateral $\triangle A B C$ with side length 3 into three equal parts, and draw lines parallel to the other two sides through each division point. The 10 points formed by the intersections of the sides of $\triangle A B C$ and these parallel lines are called grid points. If $n$ grid points are chosen from these 10 grid points, there will definitely be three grid points that can form an isosceles triangle (including equilateral triangles). Find the minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be the roots in the complex plane of the polynomial \[ P(z) = \sum_{k=0}^{2019} b_k z^k. \] Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy \[ 1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019^{-1/2019}}
What is the radical conjugate of $2\sqrt{7} - 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\sqrt7-1}
Michelle has a word with $2^{n}$ letters, where a word can consist of letters from any alphabet. Michelle performs a switcheroo on the word as follows: for each $k=0,1, \ldots, n-1$, she switches the first $2^{k}$ letters of the word with the next $2^{k}$ letters of the word. In terms of $n$, what is the minimum positive integer $m$ such that after Michelle performs the switcheroo operation $m$ times on any word of length $2^{n}$, she will receive her original word?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n}}
Two vertical chords are drawn in a circle, dividing the circle into 3 distinct regions. Two horizontal chords are added in such a way that there are now 9 regions in the circle. A fifth chord is added that does not lie on top of one of the previous four chords. The maximum possible number of resulting regions is \( M \) and the minimum possible number of resulting regions is \( m \). What is \( M^{2} + m^{2} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{296}
Find all real values of $x$ that satisfy \(\frac{1}{x(x+1)} - \frac{1}{(x+1)(x+2)} < \frac{1}{4}\). (Give your answer in interval notation.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -2) \cup (-1, 0) \cup (2, \infty)}
Xiaoming bought 4 apples, 6 pears, and 8 peaches, while Xiaohong bought 5 apples, 7 pears, and 6 peaches. Over the next 18 days, each of them eats one fruit per day. On three days, both of them eat apples; on two days, both eat pears; and on three days, one eats an apple while the other eats a pear. How many days do both of them eat peaches?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a pair of standard $8$-sided dice is rolled once. The sum of the numbers rolled, if it is a prime number, determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{64}}
Two distinct natural numbers end in 5 zeros and have exactly 42 divisors. Find their sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{700000}
How many real solutions are there to the equation
\[ |||| x|-2|-2|-2|=|||| x|-3|-3|-3| ? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $V$ be a finite-dimensional vector space over the complex numbers $\mathbb{C}$, and let $T: V \rightarrow V$ be a linear transformation with a minimal polynomial of $x^{13}(x-1)$. What is the minimal polynomial of $T^4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^5 - x^4}
In a linear regression model given by \( Y = XB + U \), where \( U \) is the vector of error terms normally distributed with mean 0 and variance \( \sigma^2 \), and all classical assumptions are satisfied, determine the variance of \( Y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sigma^2 I}
An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The figure is created by arranging four squares, each with a side length of 5 units, on each side of a central square with a side length of 20 units. Calculate the perimeter of the entire figure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
# Problem 5.

The segments of two lines, enclosed between two parallel planes, are in the ratio $5: 9$, and the acute angles between these lines and one of the planes are in the ratio $2: 1$. Find the cosine of the smaller angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9}
Using 2 colors to paint the 4 small squares on a $2 \times 2$ chessboard, how many different coloring schemes are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the sum of the infinite series: \[ 2^2 + \frac{3^2}{2!} + \frac{4^2}{3!} + \cdots \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5e - 1}
A hot air balloon with 200 balloons is blowing up. After about half an hour, 1/5 of the total number of balloons in the hot air balloon have blown up. The wind speed then increases, creating a variation in the rate at which balloons blow up. In the next hour, 30% of the remaining balloons blow up. Due to different manufacturing quality, 10% of the balloons still intact have double durability and are less likely to blow up in the same conditions. At the end of the second hour, twice the number of balloons that had already blown up, excluding the more durable ones, also blow up. How many balloons in the hot air balloon remain intact?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given an integer \( n \geq 2 \), find the smallest positive real number \( c \) such that for any complex numbers \( z_1, z_2, \ldots, z_n \), the following inequality holds:
$$
\left|\sum_{i=1}^{n} z_{i}\right| + c \sum_{1 \leq i < j \leq n} \left|z_{i} - z_{j}\right| \geq \sum_{i=1}^{n} \left|z_{i}\right|.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{n}}
What is the minimum possible cardinality of a finite set $X$ such that there are precisely seven open sets in a topology on $X$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? The original answer is in the format $\frac{\sqrt{m}}{n}$, please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, calculate the value of $y$ in degrees, rounding to one decimal place if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67.5}
It rained for 5 days out of a week (which has 7 days). In how many ways can it rain on exactly 3 consecutive days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the range of $a-b$ if the equation $x^2+ax+b-2=0$ has one root in the interval $(-\infty, -1)$ and another root in the interval $(-1, 0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, +\infty)}
Example 25 (2004 National Women's Mathematical Olympiad Problem) A deck of tri-color cards, totaling 32 cards, includes 10 cards of each of the red, yellow, and blue colors, numbered $1, 2, \cdots, 10$; there are also a big joker and a small joker, each numbered 0. From this deck, several cards are drawn, and then the score is calculated according to the following rule: each card numbered $k$ is counted as $2^{k}$ points; if their total score is 2004, then these cards are called a "good card group." Try to find the number of "good card groups."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006009}
Find the conditional probability that a standard Gaussian random variable $x$ is greater than $0$ given that $x$ is greater than another independent standard Gaussian random variable $y$. Express your answer in terms of standard Gaussian distribution properties.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
The circle constructed on the larger lateral side $AB$ of the right trapezoid $ABCD$ as its diameter intersects the base $AD$ at its midpoint. It is known that $AB=10, CD=6$. Find the midline of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the number of ways to distribute the elements of the set \([n]\) into four subsets \(A_1, A_2, A_3, A_4\) such that \(A_1 \cap A_2 \cap A_3 \cap A_4 = \emptyset\) and \(A_1 \cup A_2 \cup A_3 \cup A_4 = [n]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14^n}
Calculate the following limit involving the Gamma function: 

$$ \lim_{r \to \infty}\sqrt{\frac{r}{2}} \cdot \frac{\Gamma\left(\frac{r - 1}{2}\right)}{\Gamma\left(\frac{r}{2}\right)} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \int_{0}^{\infty}\frac{\sin (7\sqrt{x})}{\sqrt{x}}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by 2 and add 1 or he may multiply the stored value by 4 and add 3. The first player to make the stored value exceed \(2^{100}\) wins. The stored value starts at 1 and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move? 

(Optimal play means that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Alex runs a successful family bakery and is planning to expand the operation by opening two new branches. Each new branch requires an initial investment of $15,000. To fund the expansion, Alex decides to reinvest 40% of the bakery's monthly profit of $10,000 into the new branches. Additionally, Alex has saved $5,000 specifically for this expansion. After covering the initial investments for both branches, how much money will Alex have left from the reinvested profits and savings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of distinct prime divisors of the integer \(5685858885855807765856785858569666876865656567858576786786785^{22}\) given that it has 6,436,343 divisors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A $1.4 \mathrm{~m}$ long rod has $3 \mathrm{~kg}$ masses at its ends. Where should the rod be pivoted so that, when released from a horizontal position, the mass on the left side passes under the pivot at a speed of $1.6 \mathrm{~m} / \mathrm{s}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8}
A train of a certain length moving at a speed of 72 kmph takes 79.99360051195904 seconds to cross another train of length 500 m, moving at 36 kmph in the same direction. What is the length of the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{299.94}
Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ with an eccentricity of $\frac{\sqrt{2}}{2}$, a line $MA$ is drawn through a point $M$ on the ellipse, intersecting the ellipse at points $A$ and $B$ with slopes $k_{1}$ and $k_{2}$, respectively. If points $A$ and $B$ are symmetric about the origin, find the value of $k_{1} \cdot k_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
Jake has 84 more apples and some fewer peaches than Steven. Steven has 13 peaches and 52 apples. How many fewer peaches does Jake have compared to Steven?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Three equally spaced parallel lines intersect with a circle, resulting in three chords of lengths 38, 38, 34 units. Calculate the distance between two adjacent parallel chords. Express your answer in units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, if $2a_3-a_7^2+2a=0$, and $\{b_n\}$ is a geometric sequence with $b_7=a_7$, then $b_6b_8=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{306}
In how many ways can the set \( \{a_{1}, a_{2}, \ldots, a_{9}, b_{1}, b_{2}, \ldots, b_{9}, c_{1}, c_{2}, \ldots, c_{9}\} \) be partitioned into 9 subsets of the form \( \{ a_{i}, b_{j}, c_{k} \} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(9!)^2}
Given a sequence $\{a_{n}\}$ that satisfies $\frac{a_1}{2}+\frac{a_2}{2^2}+⋯+\frac{a_n}{2^n}=n(n≤N^*)$, and $b_n=λ(a_n-1)-n^2+4n$. If the sequence $\{b_{n}\}$ is a monotonically increasing sequence, then the range of $\lambda$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{3}{8}, +\infty\right)}
Problem 11.5. In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 people, are competing against each other. During the tournament, each schoolchild must play against each student exactly once, and each person must play no more than one game per day. The number of games played on different days may vary.

At some point in the tournament, the organizer noticed that the schedule for the next day can be arranged in exactly 1 way with 15 games, and in $N$ ways with 1 game (the order of the games in the schedule does not matter, only who plays against whom). Find the maximum possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Josh had some money at first. He spent $1.75 on a drink, and then spent another amount. After spending, he had $6 left. How much money did Josh spend after buying the drink?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$7.75}
Let  $n$  be a positive integer. For a permutation  $a_1, a_2, \dots, a_n$  of the numbers  $1, 2, \dots, n$  we define  $$ b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j $$  We say that the permutation  $a_1, a_2, \dots, a_n$  is *guadiana* if the sequence  $b_1, b_2, \dots, b_n$  does not contain two consecutive equal terms. How many guadiana permutations exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Evaluate the limit of the double integral: \[ \lim_{n\rightarrow \infty}\int_{-n}^{n}\int_{-n}^{n}\sin(x^2+y^2)\,dx\,dy \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Given there are 2, 1, 3, and 4 paths leading to the top of the mountain from the east, west, south, and north sides, respectively, calculate the maximum number of ways to ascend from one side and descend from any other side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the number of solutions to the equation \(x_1 + x_2 + x_3 + \ldots + x_{10} = 27\) where each \(x_i\) is an integer satisfying \(1 \leq x_i \leq 4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100110}
Given the function $f(x)=({x}^{3}-x)(\frac{1}{2}-\frac{1}{{e}^{x}+1})$, the solution set of the inequality $f\left(x\right)\ln \left(x+2\right)\leqslant 0$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2,1]}
Point \( A \) lies on the line \( y = \frac{15}{8} x - 4 \), and point \( B \) on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47/32}
Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{972538}
In the triangular pyramid \( PABC \), the sums of the three plane angles at each of the vertices \( A \) and \( B \) are equal to \( 180^\circ \), and \( PC = AB \). Inside the pyramid, a certain point \( D \) is taken such that the sum of the distances from \( D \) to the three lateral faces \( PAB \), \( PAC \), and \( PBC \) is 7. Find the distance from the center of the circumscribed sphere to the face \( PAB \), given that the volumes of the pyramids \( PABC \) and \( BABC \) are in the ratio \( 8: 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the maximum number of distinct real roots that a polynomial $P$ can have if it satisfies the following property: the product of any two distinct roots of $P$ is also a root of $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the integers 2, 2, 5, 5, 8, and 9 on six cards, determine the number of integers from 1 to 31 that cannot be obtained as sums by choosing any number of these cards. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the number of solutions to:
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval $(0,50 \pi)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given that \( A \) is an antisymmetric matrix and \( B \) is a symmetric matrix, find the condition under which \((A - B)^2\) is antisymmetric.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^2 + B^2 = 0}
Evaluate the limit of the integral over the semicircle as $R \to \infty$:

$$\lim_{R\to\infty}\int_{\Gamma_R}\frac{e^{izt}}{\pi\cos(iz)}dz$$

where $t > 0$ and $\Gamma_R$ is the upper semicircle of radius $R$ in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly two red points such that the distance from the white point to each of these red points is $2023$. Find the minimum number of red points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Sets $A$, $B$, and $C$, depicted in the Venn diagram, are such that the total number of elements in set $A$ is three times the total number of elements in set $B$. Their intersection has 1200 elements, and altogether, there are 4200 elements in the union of $A$, $B$, and $C$. If set $C$ intersects only with set $A$ adding 300 more elements to the union, how many elements are in set $A$?

[asy]
label("$A$", (2,67));
label("$B$", (80,67));
label("$C$", (41,10));
draw(Circle((30,45), 22));
draw(Circle((58, 45), 22));
draw(Circle((44, 27), 22));
label("1200", (44, 45));
label("300", (44, 27));
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3825}
Let $n$ be a positive integer. Find as many zeros as possible for the last digits of the following expression: $1^n + 2^n + 3^n + 4^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Malcolm has 240 followers on Instagram and 500 followers on Facebook. The number of followers he has on Twitter is half the number of followers he has on Instagram and Facebook combined. Meanwhile, the number of followers he has on TikTok is 3 times the number of followers he has on Twitter, and he has 510 more followers on Youtube than he has on TikTok. Additionally, he has 120 followers on Pinterest which is double the number of his Snapchat followers.

Over the next month, he gains 15% more followers on Instagram, 20% more followers on Facebook, loses 12 followers on Twitter, gains 10% more followers on TikTok, and loses 5% followers on Snapchat. During the same period, Malcolm gains 8% more followers on Youtube and a fixed number of 20 new followers on Pinterest. After this one month period, what is the total number of followers Malcolm has on all his social media platforms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4402}