inclusionAI/Ring-lite-rl-data huggingface dataset:
9. Let $A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}$, and $A_{3} B_{3} C_{3}$ be three triangles in the plane. For $1 \leq i \leq 3$, let $D_{i}, E_{i}$, and $F_{i}$ be the midpoints of $B_{i} C_{i}, A_{i} C_{i}$, and $A_{i} B_{i}$, respectively. Furthermore, for $1 \leq i \leq 3$ let $G_{i}$ be the centroid of $A_{i} B_{i} C_{i}$.
Suppose that the areas of the triangles $A_{1} A_{2} A_{3}, B_{1} B_{2} B_{3}, C_{1} C_{2} C_{3}, D_{1} D_{2} D_{3}, E_{1} E_{2} E_{3}$, and $F_{1} F_{2} F_{3}$ are $2,3,4,20,21$, and 2020 , respectively. Compute the largest possible area of $G_{1} G_{2} G_{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{917}
Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x^2-2xy)e^{-(3x^2+2y^2+z^2+2xy)}dxdydz$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10010}
Of the 500 soccer balls that Nova went to inflate, 65 percent had holes in them and could not inflate, 25% of the remaining balls were overinflated and exploded, and 10% had manufacturing defects preventing proper inflation. How many balls were inflated successfully and could be used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
A graph has 8 points and no squares (a square is 4 distinct points A, B, C, D with A and C both joined to B and D). What is the largest number of edges it can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Find the value of $f(n) = \sum\limits_{i > j \ge 0} {{}^{n + 1}{C_i}{}^n{C_j}} $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2n}}
Find the total length of the intervals on the number line where the inequalities $|x|<1$ and $\sin \log _{3}|x|>0$ hold.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3^{\pi} + 1}}
Given that all the edges of a regular tetrahedron are $\sqrt{2}$, and all four vertices lie on the same sphere, the surface area of this sphere is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi}
In a movie theater, five friends took seats numbered 1 to 5 (with seat number 1 being the leftmost seat). During the movie, Anya left to buy popcorn. When she returned, she found that Varya had moved two seats to the right, Galya had moved one seat to the left, and Diana and Ella had switched their seats, leaving the end seat for Anya. In which seat was Anya sitting before she left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the covariance between the following processes: $X_t=\int_0^t(B_s+s)ds$ and $Y_t=\int_0^t(s+1)dB_s$, where $B$ is a one-dimensional Brownian Motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $\theta$ be a constant number such that $0\leq \theta \leq \pi.$
Evaluate 

\[ \int_0^{2\pi} \sin 8x|\sin (x-\theta)|\ dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-\\frac{4}{63} \\sin 8\\theta"]}
In Vila Par, all the truth coins weigh an even quantity of grams and the false coins weigh an odd quantity of grams. The eletronic device only gives the parity of the weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, determine the least $k$, such that, there exists a strategy that allows to identify the two false coins using the eletronic device, at most, $k$ times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Find the sum placed on simple interest (SI) if the SI on that sum for 3 years at different increasing interest rates for each year (8% for the first year, 12% for the second year, and a 5% increase on the previous year's rate for the third year) is half the compound interest (CI) on Rs. 4000 for 2 years at 10% per annum, given that the interest rate increases by 2% every 6 months during the 2-year period and compounded semi-annually.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1546.93}
The function \( f(x) \) has a domain of \( \mathbf{R} \). For any \( x \in \mathbf{R} \) and \( y \neq 0 \), \( f(x+y)=f\left(x y-\frac{x}{y}\right) \), and \( f(x) \) is a periodic function. Find one of its positive periods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{5}}{2}}
What is the result of evaluating the following integral using the orthogonality of Hermite polynomials: $$\int_{-\infty}^\infty H_{n+1}\left(x+\frac{x_0}{2}\right)H_n\left(x+\frac{x_0}{2}\right)e^{^{-\frac{x^2}{2}}}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The math teacher of the ninth grade teaches an average of 6 tutoring classes per month. If the classes are taught by female teachers, each female teacher should teach 15 classes per month. Let $a$ be the number of classes taught by each male teacher if only male teachers are teaching. Using the information given, calculate the value of $a$. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
There are $n$ schools in a city. The $i$th school dispatches $C_i$ students to watch a football match, where $1 \leq C_i \leq 39$ and $1 \leq i \leq n$. The total number of students is $\sum_{i=1}^{n} C_{i} = 1990$. Each line contains 199 seats, but students from the same school must sit in the same line. How many lines of seats are needed to ensure all students have a seat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let C be the boundary of the quarter-circle x^2 + y^2 = r^2, x >= 0, y >= 0, traversed in the clockwise direction. Calculate the line integral ∫ (x^2 - y^2) dx + (x^2 + y^2) dy.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4r^3/3}
12. What is the sum of all possible $\binom{i}{j}$ subject to the restrictions that $i \geq 10$, $j \geq 0$, and $i+j \leq 20$? Count different $i, j$ that yield the same value separately - for example, count both $\binom{10}{1}$ and $\binom{10}{9}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27633}
Let \( ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) and \( E \) be the midpoints of segments \( AB \) and \( AC \), respectively. Suppose that there exists a point \( F \) on ray \( \overrightarrow{DE} \) outside of \( ABC \) such that triangle \( BFA \) is similar to triangle \( ABC \). Compute \( \frac{AB}{BC} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Nylah has her living room lights on a timer. Each evening, the timer switches the lights on at exactly 7:00 p.m., 7:30 p.m., 8:00 p.m., 8:30 p.m., or 9:00 p.m. Later in the evening, the timer switches the lights off at any random time between 11 p.m. and 1 a.m. On a given night, Nylah's lights are on for \( t \) hours. Find the probability that \( 4 < t < 5 \). Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{20}}
Two sheets of 11 cm x 8 cm paper are placed on top of each other, forming an overlapping 8 cm x 8 cm square in the center. Calculate the area of rectangle WXYZ formed by the overlapping sheets. Express your answer in square centimeters (cm^2) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Given \( -1 < a < 0 \) and \( \theta = \arcsin a \), what is the solution set for the inequality \( \sin x < a \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{(2n-1)π - θ < x < 2nπ + θ, n ∈ Z}}
Given an arithmetic-geometric sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $\frac{S_6}{S_3} = -\frac{19}{8}$ and $a_4 - a_2 = -\frac{15}{8}$. Find the value of $a_3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{4}}
The monotonic increasing interval of the function $y=a^{x^{2}-3x+2}$ ($a>1$) is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[\frac{3}{2}, +\infty\right)}
Given $a_{1}$ in the sequence $\{a_{n}\}$, and the function $f(x)=\frac{1}{3}{x^3}-{a_{n+1}}sinx+({{a_n}+2})x+1$ has a unique zero for its derivative, and the function $g\left(x\right)=8x+\sin \left(\pi x\right)-\cos \left(\pi x\right)$ with $g(a_{1})+g(a_{2})+\ldots +g(a_{9})=18$. Find $a_{5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Let $x_1,$ $x_2,$ $x_3,$ $\dots,$ $x_{100}$ be positive real numbers such that $x_1^2 + x_2^2 + x_3^2 + \dots + x_{100}^2 = 1.$  Find the minimum value of
\[\frac{x_1}{1 - x_1^2} + \frac{x_2}{1 - x_2^2} + \frac{x_3}{1 - x_3^2} + \dots + \frac{x_{100}}{1 - x_{100}^2}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \sqrt{3}}{2}}
Given the sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) that satisfies \( a_{1}=a_{2}=1, a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Furthermore, it is given that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( a_{1} + a_{2} + \cdots + a_{100} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
在 $\triangle A B C$ 中, $\sin A=\frac{\sqrt{2}}{2}$, 求 $\cos B+\sqrt{2} \cos C$ 的取值范围，请提供该取值范围端点之和的整数部分。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Three people are playing table tennis, with the player who loses a match giving up their place to the player who did not participate in that match. As a result, it turned out that the first player played 10 matches, and the second player played 21 matches. How many matches did the third player play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
In the number $2016 * * * * 02 *$, you need to replace each of the 5 asterisks with any of the digits $0, 2, 4, 6, 7, 8$ (digits may repeat) so that the resulting 11-digit number is divisible by 6. How many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2160}
Lee can make 24 cookies with four cups of flour. If the ratio of flour to sugar needed is 2:1 and he has 3 cups of sugar available, how many cookies can he make?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product \(\begin{pmatrix}1-\frac{1}{10}\end{pmatrix}\begin{pmatrix}1-\frac{1}{10^2}\end{pmatrix}\cdots\begin{pmatrix}1-\frac{1}{10^{99}}\end{pmatrix}.\) The original answer is a decimal number with five decimal places, please provide the value as an integer by removing the decimal point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89001}
The total age of a and b is 15 years more than the total age of b and c. Additionally, the total age of a and d is 12 years more than the total age of c and d. If a is 3 years older than d, how many years younger is c than a?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A frog is located at the origin of the Cartesian coordinate system. Every second, the frog jumps either horizontally or vertically to one of the four adjacent points with integer coordinates. Determine the number of different points where the frog can be located after 2015 seconds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4064256}
Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Let \( M \) be a set consisting of a finite number of positive integers, and suppose \( M = A_{1} \cup A_{2} \cup \cdots \cup A_{20} = B_{1} \cup B_{2} \cup \cdots \cup B_{20} \), where \( A_{i} \neq \varnothing, B_{i} \neq \varnothing \) for \( i = 1, 2, \ldots, 20 \). For any \( 1 \leq i < j \leq 20 \), we have \( A_{i} \cap A_{j} = \varnothing \) and \( B_{i} \cap B_{j} = \varnothing \).

Given that for any \( 1 \leq i \leq 20 \) and \( 1 \leq j \leq 20 \), if \( A_{i} \cap B_{j} = \varnothing \), then \( \left|A_{i} \cup B_{j}\right| \geq 18 \). Find the minimum number of elements in the set \( M \) (where \( |X| \) denotes the number of elements in set \( X \)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
How many positive integers $N$ are there such that for every multiple of $N$ denoted as $\overline{\mathrm{abc}}$, the numbers $\overline{\mathrm{bca}}$ and $\overline{\mathrm{cab}}$ are also multiples of $N$? Here, $a$, $b$, and $c$ are digits from 0 to 9, and numbers like 123 represent 123, 028 represents 28, and 007 represents 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
What is the maximum number of \(4 \times 1\) rectangles that can be placed inside a \(6 \times 6\) square (without breaking cell boundaries)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Four copper weights together weigh 40 kilograms; with their help, every whole number weight from 1 to 40 kilograms can be measured; what are the weights of these copper weights?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 3, 9, 27\}}
Given the function $f(x)=x^{5}-3x^{3}-6x^{2}+x-1$, calculate the value of $f(-2)$ using the Horner's method. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-35}
Find the limit
\[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{\pi}{2}$}
设实数 $a$ 使得不等式 $|2 x-a|+|3 x-2 a| \geqslant a^{2}$ 对任意实数 $x$ 恒成立，求满足条件的 $a$ 所组成的集合的区间端点。如果区间为 $[m, n] = \left[\frac{k}{l}, \frac{p}{q}\right]$，请给出k+l+p+q的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
[ Ratio of areas of similar triangles ]

In a right triangle, the sine of the smaller angle is $\frac{1}{3}$. A line perpendicular to the hypotenuse divides the triangle into two equal areas. In what ratio does this line divide the hypotenuse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
A person travels from P to Q at a certain speed and returns by increasing his speed by 30%. His average speed for both the trips is 34.5 km/hr. What is his initial speed from P to Q?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30.52}
Machiavelli added together $1+3i$ and $2-4i$. What number did he get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3-i}
Evaluate the limit using L'Hopital's rule: \[ \lim_{x \to 0} \log (\cot x)^{\tan x} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in \(\{(x, y) \mid 1 \leq x, y \leq 5\}\) blue such that the distance between any two blue points is not an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
How many ordered triples $(a, b, c)$ of integers with $-15 \le a, b, c \le 15$ are there such that the three equations $ax + by = c$, $bx + cy = a$, and $cx + ay = b$ correspond to lines that are distinct and concurrent?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
Determine the prime \( p \) such that the number of groups of order \( 31p^2 \) is maximized. Consider the cases where \( p = 31 \) and \( p \neq 31 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Let $n$ be a positive integer. How many polynomials $P$ with coefficients in the set $\{0,1,2,3\}$ are there such that $P(2)=n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lfloorn/2\rfloor+1}
In the prism \(A B C A_1 B_1 C_1\), the medians of the bases \(A B C\) and \(A_1 B_1 C_1\) intersect at points \(O\) and \(O_1\) respectively. A line is drawn through the midpoint of the segment \(O O_1\) parallel to the line \(C A_1\). Given that \(C A_1 = a\), find the length of the segment of this line that lies inside the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2a}{3}}
Compute the number of ordered triples of integers $(a, b, c)$ between $1$ and $12$, inclusive, such that, if \[ q = a + \frac{1}{b} - \frac{1}{b + \frac{1}{c}}, \] then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
In a certain sports competition, the scoring rules are: winning a match gives 3 points, drawing a match gives 1 point, and losing a match gives 0 points. If a particular athlete plays 15 matches and scores a total of 22 points, calculate the number of possible scenarios. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of real values of $x$ that satisfy the equation:

$$\left| 3 -x \right|^{ \log_7(x^2) - 7\log_x (49)} = (3-x)^3$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
An arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. Determine how many good arrangements exist in which the first and last numbers are 1, and the second number is the same as the third.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Two irrational numbers in the interval $(0,1)$ are chosen at random, and their decimal parts are considered up to 2014 places. The probability that the smaller number can be subtracted from the larger one without borrowing is expressed as $p^k$, where $p$ is a rational number and $k > 1000$ is a positive integer. Given that the first decimal digit of the larger number is greater than that of the smaller number, find the last three digits of $\lfloor{100(p+k+1)}\rfloor$. Provide your answer as a three-digit number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{455}
对于给定的正整数 $k$, 定义 $f_1(k)$ 为 $k$ 的数字和的平方, 并令

$$
f_{n+1}(k)=f_1\left(f_n(k)\right)
$$


求 $f_{1991}\left(2^{1990}\right)$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Find the smallest positive distance $d$ such that for any line in the plane not parallel to the $x$- or $y$-axis, there exists a lattice point whose distance to the line does not exceed $d$. Express your answer in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{2}}{4}}
Find a closed form expression for the sum $\sum^{n}_{k=0}{n\choose k} (-1)^k {n+m-k\choose r-k}$ using generating functions, where $n,m,r$ are natural numbers and $n \leq r \leq n+m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{m\choose r-n}}
Evaluate the integral \( \oint_{|z|=2} \frac{f^{\prime}(z)}{f(z)} \, dz \) for the function \( f(z)=\frac{2 z^{4}-7 z^{3}-2 z-i}{(z-i)^{2}\left(3 z^{2}-z+9\right)^{2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6\pi i}
Calculate the expectation of a random variable $Z$ with the cumulative distribution function given by: $$F_Z(z) = 1 - 0.2e^{-0.4z} + 0.7e^{-0.1z}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
Consider a cuboid with base edge lengths $a$ and $b$ and height $h$. Given that the base area $L = ab$ and the lateral area $S = 2h(a + b)$ are constant, determine the minimum value of the perimeter $P = 4(a + b + h)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\sqrt{L} + \dfrac{S}{\sqrt{L}}}
Let \( \phi \) be a continuous function on the interval \([0,b]\) such that \( \phi(x) > 0 \) for all \( x \in [0,b] \). Assume \( \phi(x) \sim A x^r \) for some constant \( A > 0 \) and \( r \geq 0 \). Define the function \( F(\varepsilon) = \int_0^b \frac{dx}{\varepsilon + \phi(x)} \). Find \( \lim_{\varepsilon \to 0} F(\varepsilon) \) for \( r = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Consider the system of equations:

\[ a + 2b + 3c + \ldots + 26z = 2020 \]
\[ b + 2c + 3d + \ldots + 26a = 2019 \]
\[ \vdots \]
\[ y + 2z + 3a + \ldots + 26x = 1996 \]
\[ z + 2a + 3b + \ldots + 26y = 1995 \]

where each equation is a rearrangement of the first equation with the variables cycling and the coefficients staying in place. Find the value of:

\[ z + 2y + 3x + \ldots + 26a \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1995}
It is known that $\int_1^2x^{-1}\arctan (1+x)\ dx = q\pi\ln(2)$ for some rational number $q.$ Determine $q.$ Here, $0\leq\arctan(x)<\frac{\pi}{2}$ for $0\leq x <\infty.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ \frac{3}{8}$}
At some time between 9:30 and 10 o'clock, the triangle determined by the minute hand and the hour hand is an isosceles triangle. If the two equal angles in this triangle are each twice as large as the third angle, what is the time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9:36}
Given the function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(f(x+y)) = f(x^2) + f(y^2) \), and \( f(f(2020)) = 1010 \), determine the value of \( f(2025) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{505}
We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is **kawaii** if \( T \) has the following properties:

1. \( T \) has at least two distinct elements;
2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \).

For example, the subset \( T = \{31, 71, 2024\}\) is **kawaii**, but \( T = \{5, 15, 75\}\) is not **kawaii** because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \).

What is the largest possible number of elements that a **kawaii** subset can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{675}
On Mary's birthday, her brother surprised her with $200. She decided to spend some of it on a shopping trip. She bought a new video game which originally cost $60, but she had a 15% discount coupon. Additionally, she spent 20% of the remaining amount on swimming goggles, which had an 8% sales tax added to the final price. 

Mary also decided to buy a sports jacket that had a 25% discount, and the jacket's original price was $80. She then bought a book with a 5% sales tax, spending 10% of her remaining money. Finally, she used a $20 gift card to buy a pair of stylish socks. 

How much money did Mary have left after her shopping trip?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$50.85}