inclusionAI/Ring-lite-rl-data huggingface dataset:
$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数, $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left\{x_{i 3}, x_{i 4}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right.\right.$, 求 x 大于 y 的概率.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Using squares of side 1, a stair-like figure is formed in stages following a specific pattern. For example, the first stage uses 1 square, the second stage uses 5 squares, and so on. Determine the last stage for which the corresponding figure uses less than 2014 squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
A line whose $y$-intercept is $(0,8)$ intersects the ellipse $4x^2 + 25y^2 = 100.$ Find all possible slopes of this line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -\sqrt{2.4}] \cup [\sqrt{2.4}, \infty)}
Grandma tells Xiao Ming, "There are 53 Saturdays in 2016." The clever Xiao Ming immediately tells Grandma: The New Year's Day of 2017 must be on a \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Monday}
A square contains \( K \) points (\( K > 2 \)). What is the minimum number of triangles needed to divide the square so that each triangle contains at most one point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K+1}
Let $S$ be the set of all nondegenerate triangles formed from the vertices of a regular octagon with side length 1. Find the ratio of the largest area of any triangle in $S$ to the smallest area of any triangle in $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3+2 \sqrt{2}}
Find all possible three-digit numbers that can be obtained by removing three digits from the number 112277. Sum them and write the result as the answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1159}
10,11

During an interview, ten people were offered a test consisting of several questions. It is known that any five people together answered all the questions (that is, at least one of the five gave the correct answer to each question), but any four did not. What is the minimum number of questions for which this could have been the case?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
$n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in the clockwise direction. If the sum of all numbers is $278$, find how many different values $n$ can take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Triangle $ABC$ has side lengths $AB = 20$, $AC = 14$, and $BC = 22$. The median from $B$ intersects $AC$ at $M$, and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\frac{p}{q} = \frac{[AMPN]}{[ABC]}$ for positive integers $p$ and $q$ that are coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{331}
In triangle $ABC$, $\angle C = \frac{\pi}{3}$, and let $\angle BAC = \theta$. If there exists a point $M$ on line segment $BC$ (distinct from points $B$ and $C$) such that $\triangle BAM$ is folded over line $AM$ to obtain $\triangle B'A'M$, and the condition $AB' \perp CM$ is satisfied, then determine the range of values for $\theta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{\pi}{6}, \frac{2\pi}{3} \right)}
The smallest number which, when diminished by 16, is divisible by some numbers, including 4, 8, and 10, and the result is 136. What is the second divisor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \( a_{1} = 3 \), and for \( n \geq 1 \), let \( a_{n+1} = (n+1) a_{n} - n \). Find the smallest \( m \geq 2005 \) such that \( a_{m+1} - 1 \mid a_{m}^{2} - 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2010}
What is the least positive integer such that the product of its digits is $9!$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2578899}
Given a quadratic function $f(x) = x^2 + ax + 5$, it holds that $f(t) = f(-4-t)$ for any $t$ and the function has a maximum value of 5 and a minimum value of 1 on the closed interval $[m, 0]$. Determine the range of values for $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4 \leq m \leq -2}
Let $n \ge 2$ be an integer. Find the largest integer $k \ge 1$ with the following property: for any $k$ matrices: $A_1,A_2,...,A_k \in \mathcal{M}_n(\mathbb{C})$, if $I_n-A_1A_2...A_k$ is invertible, then so is $I_n-A_{\pi(1)}A_{\pi(2)}...A_{\pi(k)}$ for every permutation $\pi \in S_k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Anna joined coaching on the first day of the year. Anna took coaching every day without fail till Anna discontinued on September 4. How many days Anna took coaching, if the year is not a leap year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{247}
Given \(x\) and \(y\) are real numbers, and:
\[
z_1 = x + (y + 2)i,
\]
\[
z_2 = (x - 2) + yi,
\]
with the condition \(\left|z_1\right| + \left|z_2\right| = 4\),

Find the maximum value of \(|x + y|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Evaluate the limit $$\lim\limits_{n\to\infty}(f(n+1)-f(n))$$ where $f(n)=|B_{2n}|^{1/2n}$ and $B_{2n}$ is the $(2n)$-th Bernoulli's number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/πe}
Given a natural number \( n \), consider the function
\[ f_n(x, y) = x^n + x^{n-1}y + x^{n-2}y^2 + \dots + xy^{n-1} + y^n \]
of two real variables. Determine the minimum number \( k \) for which there exist functions \( g_1, \dots, g_k \) and \( h_1, \dots, h_k \) of one real variable such that
\[ f_n(x, y) = \sum_{i=1}^k g_i(x)h_i(y). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}
11.1. In each of the four volleyball teams, there are six players, including a captain and a setter, and these are different people. In how many ways can a team of six players be formed from these four teams, such that there is at least one player from each team and there must be a pair of captain and setter from at least one team?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9720}
An isosceles trapezoid \(ABCD\) with the larger base \(AB\) is circumscribed around a circle that touches side \(BC\) at point \(M\). Segment \(AM\) intersects the circle at point \(N\). Find the ratio \(\frac{AB}{CD}\) if \(\frac{MN}{AN}=k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8-k}{k}}
The area of triangle $ABC$ is $2007$. One of its sides has length $18$, and the tangent of the angle opposite that side is $\frac{2007}{24832}$. When the altitude is dropped to the side of length $18$, it cuts that side into two segments. Find the sum of the squares of those two segments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{194}
In the election for the class president, Petya and Vasya competed. Over three hours, 27 students voted for one of the two candidates. During the first two hours, Petya received 9 more votes than Vasya. In the last two hours, Vasya received 9 more votes than Petya. In the end, Petya won. By the largest possible margin, how many more votes could Petya have received than Vasya?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\triangle A P F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2} + \sqrt{10}}
9. How many 9 -term sequences $a_{1}, \ldots, a_{9}$ of nonnegative integers are there such that
- $0 \leq a_{i}<i$ for all $i=1, \ldots, 9$; and
- there are no ordered triples $(i, j, k)$ with $1 \leq i<j<k \leq 9$, such that $a_{i}, a_{j}, a_{k}$ are all distinct?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule \[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\] where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2005}}
How many different patterns can be made by shading exactly two of the nine squares?  Patterns that can be matched by flips and/or turns are not considered different.  For example, the patterns shown below are not considered different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the proposition $p$: "The equation $x^{2}+mx+1=0$ has two real roots"; and the proposition $q$: "$\forall x \in \mathbb{R}$, $4x^{2}+4(m-2)x+1 \neq 0$", if $p \wedge q$ is false, and $\neg q$ is false, find the range of values for the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
在椭圆 $\Gamma$ 中, $F$ 为一个焦点, $A, B$ 为两个顶点. 若 $|F A|=3,|F B|=2$,求 $|A B|$ 的所有可能值。请提供所有可能值平方和的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
$a$ , $b$ , $c$  are real. What is the highest value of  $a+b+c$  if  $a^2+4b^2+9c^2-2a-12b+6c+2=0$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{17}{3}}
Given propositions $p$ and $q$:

- $p$: The function $f(x) = \frac{1}{3}x^3 - x^2 + (5 - a^2)x + a$ is increasing on $\mathbb{R}$.
- $q$: The function $g(x) = \frac{e^x}{x}$ is strictly increasing on $[a, +\infty)$.

If the proposition "$p \vee (\neg q)$" is true, then "$(\neg p) \vee q$" is also true. Find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \in (-\infty, -2) \cup [1, 2]}
Given a triangle \(ABC\) with \(\angle C = 120^{\circ}\). Point \(D\) is the foot of the perpendicular dropped from point \(C\) to side \(AB\); points \(E\) and \(F\) are the feet of the perpendiculars dropped from point \(D\) to sides \(AC\) and \(BC\) respectively. Find the perimeter of triangle \(ABC\) given that triangle \(EFC\) is isosceles and its area is \(\sqrt{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16 + 8\sqrt{3}}
Jeffrey rolls three fair six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \). Compute \( m+n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Matt did his homework for 150 minutes. He spent 20% of that time on math, 25% on science, 10% on history, and 15% on English. He had to spend at least 30 minutes on each remaining subject, and he could only spend the remaining time on other subjects. How much time did Matt spend on homework in other subjects?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Archie received his medical receipt from his doctor, and he needs to take two different types of antibiotics for a week. The first antibiotic, Antibiotic A, costs $3 per dose and he needs to take it twice a day on Monday, Wednesday, and Friday. The second antibiotic, Antibiotic B, costs $4.50 per dose and he needs to take it once a day on Tuesday, Thursday, Saturday, and Sunday. Additionally, there is a promotion available: if Archie buys 10 doses of Antibiotic A, he will get a 20% discount on those doses. How much money does Archie need to spend on the antibiotics for the week, considering the discount?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$36}
The largest divisor of $2,014,000,000$ is itself. What is its fifth-largest divisor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{251750000}
[The Pigeonhole Principle (continued).]

In a photo studio, 20 birds flew in - 8 sparrows, 7 wagtails, and 5 woodpeckers. Each time the photographer clicks the camera shutter, one of the birds flies away (permanently). How many shots can the photographer take to be sure: he will have at least four birds of one species left, and at least three of another?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
"a > 0" is a \_\_\_\_\_\_\_\_\_\_\_ condition for "a^2 + a ≥ 0".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{sufficient}}
Given that $\sin(\alpha - 70^\circ) = \alpha$, find the value of $\cos(\alpha + 20^\circ)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\alpha}
Evaluate the following limit:
$$\lim_{x \to 0^{+}} \left[\frac{\exp(\sin x \log x) - \exp(x \log \sin x)}{x^3} + \frac{\log x}{6}\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}}
There are $n\geq 3$ cities in a country and between any two cities $A$ and $B$, there is either a one way road from $A$ to $B$, or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define $d(A,B)$ to be the minimum number of roads you must go through to go from city $A$ to $B$. Consider all possible ways to build the roads. Find the minimum possible average value of $d(A,B)$ over all possible ordered pairs of distinct cities in the country.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{3}{2} }
4. The numbers $x$ and $y$ are such that the equations $\sin y + \sin x + \cos 3x = 0$ and $\sin 2y - \sin 2x = \cos 4x + \cos 2x$ are satisfied. What is the greatest value that the sum $\cos y + \cos x$ can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+\frac{\sqrt{2+\sqrt{2}}}{2}}
Point $P_{1}(1, y_{1})$ and point $P_{2}(-3, y_{2})$ are two points on the graph of the linear function $y=-4x+3$. The relationship between $y_{1}$ and $y_{2}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y_1 < y_2}
A rhombus with a 60° angle is rotated about its center on a plane until it coincides with its original position. Find the minimum degree of rotation required for this to happen. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
There is a five-digit number that, when divided by each of the 12 natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13, gives different remainders. What is this five-digit number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83159}
Determine the order of the group presented by $$\langle a, b\mid a^3, b^3, [a, [b, a^{-1}]], [b, [b, a^{-1}]]\rangle.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Find the sum of the two smallest possible values of  $x^\circ$  (in degrees) that satisfy the following equation if  $x$  is greater than  $2017^\circ$ :   $$ \cos^59x+\cos^5x=32\cos^55x\cos^54x+5\cos^29x\cos^2x\left(\cos9x+\cos x\right). $$  *2017 CCA Math Bonanza Individual Round #10*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4064}
How many ways can one color the squares of a \(6 \times 6\) grid red and blue such that the number of red squares in each row and column is exactly 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67950}
Find an estimate for the sum $\sum_{1\leq x, y\leq n, x\neq y}\Lambda(x)^{2}\cdot \Lambda(y)^{2}$, where $\Lambda$ is the von-Mangoldt function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2\log^2n}
Sue borrowed 15 books and 6 movies. She returned 8 books. She then returned a third of the movies to the library and checked out 9 more books. How many movies and books does Sue now have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In how many ways can four married couples sit around a circular table such that no man sits next to his wife?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1488}
Suppose that $X_1,X_2,\ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S= \sum _{i= 1}^{k}X_{i}/ 2^{i}$, where $k$ is the least positive integer such that $X_k<X_{k+1}$, or $k=\infty$ if there is no such integer. Find the expected value of S.  The original answer is in $ae^{b / c}-d$ format, please provide the value of a+b+c+d.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given triangle \( ABC \). On the side \( AC \), which is the largest in the triangle, points \( M \) and \( N \) are marked such that \( AM = AB \) and \( CN = CB \). It turns out that angle \( NBM \) is three times smaller than angle \( ABC \). Find \( \angle ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
what is the least number . which should be added to 0.0320 to make it a perfect square ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9680}
A Princeton slot machine has $100$ pictures, each equally likely to occur. One is a picture of a tiger. Alice and Bob independently use the slot machine, and each repeatedly makes independent plays. Alice keeps playing until she sees a tiger, at which point she stops. Similarly, Bob keeps playing until he sees a tiger. Given that Bob plays twice as much as Alice, let the expected number of plays for Alice be $\tfrac{a}{b}$ with $a, b$ relatively prime positive integers. Find the remainder when $a + b$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{701}
The sequence $\{a_{n}\}_{n\geq 0}$ is defined by:

- $a_{0} = 20$
- $a_{1} = 100$
- $a_{n+2} = 4a_{n+1} + 5a_{n} + 20$ for $n = 0, 1, 2, \ldots$

Find the smallest positive integer $h$ satisfying $1998 \mid (a_{n+h} - a_{n})$ for all $n = 0, 1, 2, \ldots$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
mary passed a certain gas station on a highway while traveling west at a constant speed of 50 miles per hour . then , 15 minutes later , paul passed the same gas station while traveling west at a constant speed of 80 miles per hour . if both drivers maintained their speeds and both remained on the highway for at least 2 hours , how long after he passed the gas station did paul catch up with mary ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Consider an $8 \times 8$ grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e., it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
The number  $201212200619$  has a factor  $m$  such that  $6 \cdot 10^9 <m <6.5 \cdot 10^9$ . Find  $m$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6490716149}
A man has a certain number of small boxes to pack into parcels. If he packs 3 or 4 in a parcel, he is left with one over; if he packs some other number of boxes in a parcel, he is also left with one over; if he packs 7 in a parcel, none is left over. The number of boxes he may have to pack is 301. What is the third number of boxes that leaves one over when packed in a parcel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be all the irreducible proper fractions with a denominator of 60. Then \(\sum_{i=1}^{n}\left(\cos \frac{a_{i} \pi}{2}\right)^{2}\) equals what?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
在平面直角坐标系中，函数 $y=\frac{x+1}{|x|+1}$ 的图像上有三个不同的点位于直线 $l$ 上，且这三点的横坐标之和为 0。求 $l$ 的斜率的取值范围。原始答案格式为 $0<k<\frac{m}{n}$，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine the positive integers expressible in the form $\frac{x^2+y}{xy+1}$, for at least 2 pairs $(x,y)$ of positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The tax on a commodity is diminished by 19%, and its consumption increased by 15%. What is the effect on revenue in percentage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.85\%}
Find the largest number of distinct natural numbers, each less than 50, such that any two of them are coprime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given positive numbers \(a, b, c, d\) such that \(a + b + c + d = 1\), find the minimum value of \(M = \sqrt{a^{2}+\frac{1}{8a}}+\sqrt{b^{2}+\frac{1}{8b}}+\sqrt{c^{2}+\frac{1}{8c}}+\sqrt{d^{2}+\frac{1}{8d}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given \( n \) points on a plane (where \( n > 2 \)), no three of which are collinear, in how many different ways can this set of points be divided into two non-empty subsets such that the convex hulls of these subsets do not intersect?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} n (n - 1)}
Find the value of the product \(\left(1+2 \cos \frac{2 \pi}{7}\right)\left(1+2 \cos \frac{4 \pi}{7}\right) \cdots\left(1+2 \cos \frac{12 \pi}{7}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the function y = 2^x and the domain P = {1, 2, 3}, calculate the range of the function. Express your answer as a set of values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{2, 4, 8}}
What least number should be added to 29989, so that the sum is completely divisible by 73?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $(a_1,a_2,a_3,\ldots,a_{12})$ be a permutation of $(1,2,3,\ldots,12)$ for which  
  $a_1 > a_2 > a_3 > a_4 > a_5 > a_6$ and $a_6 < a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12}.$ 
An example of such a permutation is $(6,5,4,3,2,1,7,8,9,10,11,12).$ Find the number of such permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{462}
Let \( a, b, c, d, e \) be natural numbers with \( a < b < c < d < e \), and \( a + 2b + 3c + 4d + 5e = 300 \). Determine the maximum value of \( a + b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
$P Q R S$ is a square with side length $8$. $X$ is the midpoint of side $P Q$, and $Y$ and $Z$ are the midpoints of $X S$ and $X R$, respectively. Calculate the area of trapezoid $Y Z R S$. Express your answer in square units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}