inclusionAI/Ring-lite-rl-data huggingface dataset:
已知向量 $\boldsymbol{a} 、 \boldsymbol{b} 、 \boldsymbol{c}$ 满足 $|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=4$, $\boldsymbol{a} \cdot \boldsymbol{b}=8$. 则 $|3 \boldsymbol{c}-\boldsymbol{a}|+\left|\frac{1}{4} \boldsymbol{c}-\boldsymbol{b}\right|$ 的最小值的$\sqrt{133}$倍是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{133}
There are 11 foreign language translators, among which there are 5 English translators, 4 Japanese translators, and the other 2 are proficient in both English and Japanese. From these, 8 people are selected to form two translation teams, with 4 people translating English and the other 4 translating Japanese, allowing these two teams to work simultaneously. How many different rosters can be made for such an arrangement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{185}
Heinz produces tomato puree by boiling tomato juice. The tomato puree has a certain percentage of water while the tomato juice has more water. Heinz obtains 2.5 litres of tomato puree from 20 litres of tomato juice. What is the percentage of water in the tomato juice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87.5\%}
Calculate the value of: $$(\frac {3}{2})^{- \frac {1}{3}} - \frac {1}{3} \times (- \frac {7}{6})^{0} + 8^{ \frac {1}{4}} \times \sqrt[4]{2} - \sqrt {(- \frac {2}{3})^{ \frac {2}{3}}}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {5}{3}}
Find the sum $\displaystyle \sum_{k = 1}^{2023}\dfrac1{1+\tan^2 \left(\dfrac{k\pi}{2\cdot 2024}\right)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1011}
At a robot racing competition, a certain number of mechanisms were presented. The robots were paired to race the same distance. The protocol recorded the differences in the finishing times between the winner and the loser in each of the races. All the differences were distinct: 1 sec, 2 sec, 3 sec, 4 sec, 5 sec, 6 sec, 7 sec, 8 sec, 9 sec, 13 sec. It is known that during the races, each robot competed against each other robot exactly once, and each robot always ran at the same speed. Determine the time of the slowest mechanism if the best time to complete the distance was 50 seconds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Let \( D \) be a point inside the acute triangle \( \triangle ABC \) such that 
\[
\angle ADB = \angle ACB + 90^\circ
\]
and \( AC \cdot BD = AD \cdot BC \). Find the value of \(\frac{AB \cdot CD}{AC \cdot BD}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Esmeralda chooses a divisor $d > 1$ of $n$, erases $n$, and writes $n + d$. If the initial number on the board is $2022$, what is the largest composite number that Esmeralda will never be able to write on the blackboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2023}
An equilateral triangle is inscribed in a circle. A smaller equilateral triangle has one vertex coinciding with a vertex of the larger triangle and another vertex on the midpoint of a side of the larger triangle. What percent of the area of the larger triangle is the area of the smaller triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
Find the $2019^{th}$ strictly positive integer $n$ such that $\binom{2n}{n}$ is not divisible by $5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37805}
Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ respectively, and $a=2$, $b=\sqrt{6}$, $A+C=2B$, find $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Create a six-digit number using the digits 1, 2, 3, 4, 5, and 6 without repetition, such that the digit 1 is not at either end. Additionally, out of the three even digits 2, 4, and 6, there must be exactly one pair of adjacent even digits. How many such six-digit numbers can be formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
On each OMA lottery ticket there is a  $9$ -digit number that only uses the digits  $1, 2$  and  $3$  (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the  $9$  figures, then they are of different colors. Bill  $122222222$  is red,  $222222222$  is green, what color is bill  $123123123$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{red}
Find the number of real solutions $(x,y)$ to the system of equations:
$$
\begin{cases}
\sin(x^2-y) = 0 \\
|x|+|y|=2\pi
\end{cases}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Three bags of jelly beans contain 26, 28, and 30 beans.  The ratios of yellow beans to all beans in each of these bags are $50\%$, $25\%$, and $20\%$, respectively.  All three bags of candy are dumped into one bowl.  Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31\%}
Given a pyramid $P-ABC$ where $PA=PB=2PC=2$, and $\triangle ABC$ is an equilateral triangle with side length $\sqrt{3}$, the radius of the circumscribed sphere of the pyramid $P-ABC$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\dfrac{\\sqrt{5}}{2}"]}
There exist constants $a_1,$ $a_2,$ $a_3,$ $a_4,$ $a_5$ such that
\[\cos^5 \theta = a_1 \cos \theta + a_2 \cos 2 \theta + a_3 \cos 3 \theta + a_4 \cos 4 \theta + a_5 \cos 5 \theta\]for all angles $\theta.$  Find $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{63}{128}}
For an entire month of June, Owen purchased 2 burgers each day. If each burger costs 12 dollars and there was a special deal where Owen could buy 2 burgers and get 1 free, how much money did Owen save in June?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$360$}
Suppose \( x_0, x_1, x_2, \cdots \) is a sequence of numbers such that \( x_0 = 1000 \), and
\[ x_n = -\frac{1000}{n}\left(x_0 + x_1 + x_2 + \cdots + x_{n-1}\right) \]
for all \( n \geq 1 \). Find the value of
\[ \frac{1}{2^{2}} x_0 + \frac{1}{2} x_1 + x_2 + 2 x_3 + 2^2 x_4 + \cdots + 2^{997} x_{999} + 2^{998} x_{1000} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Let \( a, b \), and \( c \) be complex numbers such that \( |a|=|b|=|c|=1 \). If
\[ \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}=1 \]
as well, determine the product of all possible values of \( |a+b+c| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the work done by the vector field \( F = (x^2 + y^2)\mathbf{i} + (x^2 - y^2)\mathbf{j} \) along the closed curve formed by the line segments from \((0, 0)\) to \((2, 0)\), from \((2, 0)\) to \((2, 2)\), and from \((2, 2)\) to \((0, 0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
Julian has 80 Facebook friends and 150 Instagram followers. If 60% of his Facebook friends are boys and 40% are girls, and 70% of his Instagram followers are girls while the rest are boys. Meanwhile, Boyd has twice as many friends who are girls on Facebook than Julian and three times the boys among his Instagram followers compared to Julian. Boyd's total number of friends and followers across these two social media platforms are 200. What percentage of Boyd’s friends on both platforms are boys?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68\%}
Evaluate the line integral \( \int_C ze^{\sqrt{x^2+y^2}} \, \mathrm{ds} \) where \( C \) is the curve of intersection of the sphere \( x^2+y^2+z^2=a^2 \) and the plane \( x+y=0 \), with \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the partial derivative \( \frac{\partial c^{*}}{\partial c} \) where \( c \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \((X_n)_{n>1}\) be a sequence of independent random variables such that \(X_n \sim \text{Bern}(1-\frac{1}{n^a})\) where \(a>0\) is a constant. Define \(Y_n=\prod_{j=1}^n X_j.\) Determine the values of \(a\) for which \(E[Y_n^2]\) converges to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \leq 1}
Given a natural number \( n \), determine the largest number \( k \) such that for any \( n \) rays in space, no two of which have common points, it is possible to mark \( k \) points on these rays that lie on the same sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $(X_{n})_{n\geq1}$ be a sequence of i.i.d. random variables such that $\mathbb{P}(X_{n} = -1) = \frac{1}{3}$ and $\mathbb{P}(X_{n} = 1) = \frac{2}{3}$. Let $S_{0} = 0$ and for $n\geq 1$ let $S_{n} = \sum_{k=1}^{n} X_{k}$. Determine the value of $\mathbb{P}(\liminf S_{n} = -\infty)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the values of \( n \) for which an \( n \times n \) board can be completely covered by T-shaped tetrominoes without overlapping or extending off the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \equiv 0 \pmod{4}}
There is an oak tree growing by the road from Yolkino to Palkino, and it is twice as close to Yolkino as it is to Palkino. Fedya, traveling at a constant (and positive) speed from Yolkino to Palkino, was twice as close to the oak tree as he was to Yolkino at 12:00. At 12:40, it turned out again that Fedya was twice as close to the oak tree as he was to Yolkino. When will Fedya arrive in Palkino?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13:10}
Tall Tuna has twice as many tuna fish as Jerk Tuna but also has 30% mackerel in addition. Swell Tuna has fifty percent more tuna fish than Tall Tuna and has 25% more mackerel than Jerk Tuna's mackerel count. If Jerk Tuna has one hundred forty-four tuna fish and eighty mackerel, how many fish (tuna and mackerel) do they have together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1148}
The binomial expansion of $\left(\frac{\sqrt{x+1}}{2 \sqrt[4]{x}}\right)^{n}$ is arranged in descending powers of $x$. If the coefficients of the first two terms form an arithmetic sequence, how many terms in this expansion have integer exponents of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
John carves a giant marble rectangular prism 8 meters tall with a 2-meter square base.  It has a density of 2700 kg per cubic meter.  How much does it weigh?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86400}
The 110th anniversary celebration of Guiyang No.1 High School was held on September 30, 2016. During the celebration, 2 volunteers from the 10th grade and 4 volunteers from the 11th grade were randomly selected to serve at Gate 1 to welcome the alumni. The probability that at least one of the selected volunteers is from the 10th grade is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {3}{5}}
从全体正整数1,2,3，···中划去3和4的倍数，但其中凡是5的倍数都保留（例如15,20,60，···等都保留），划完后，将剩下的数从小到大排成一个数列：$a _ { 1 } = 1$,$a _ { 2 } = 2 ,$ $a _ { 3 } = 5$, $a _ { 4 } = 7 ,$.$a _ { 5 } = 1 0 ,$,⋯,求a2011之值．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3350}
Find \(\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2 j + 2ij + ij^2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{4}}
Compute the argument of the sum:
\[ e^{2\pi i/40} + e^{6\pi i/40} + e^{10\pi i/40} + e^{14\pi i/40} + e^{18\pi i/40} + e^{22\pi i/40} + e^{26\pi i/40} + e^{30\pi i/40} + e^{34\pi i/40} + e^{38\pi i/40} \]
and express it in the form \( r e^{i \theta} \), where \( 0 \le \theta < 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
There are 3 boys and 3 girls arranged in a row. If it is required that any two boys and any two girls cannot be adjacent, and boy A and girl B must be adjacent, then find the total number of such arrangements. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given a square grid of size $2023 \times 2023$, with each cell colored in one of $n$ colors. It is known that for any six cells of the same color located in one row, there are no cells of the same color above the leftmost of these six cells and below the rightmost of these six cells. What is the smallest $n$ for which this configuration is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{338}
Fibonacci, an Italian mathematician, is known for the Fibonacci sequence, where the sequence $\{a_{n}\}$ satisfies $a_{1}=a_{2}=1$, and $a_{n+2}=a_{n+1}+a_{n}$, then the sequence $\{a_{n}\}$ is called the Fibonacci sequence. Given that the sequence $\{a_{n}\}$ is a Fibonacci sequence, and the sequence $\{b_{n}\}$ satisfies ${b_{n+3}}+{(-1)^{{a_n}}}{b_n}=n$, if the sum of the first $12$ terms of the sequence $\{b_{n}\}$ is $86$, then $b_{1}+b_{2}=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Initially, five variables are defined: $a_1 = 1, a_2 = 0, a_3 = 0, a_4 = 0, a_5 = 0$. On a turn, Evan can choose an integer $2 \le i \le 5$. Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2$, then now $a_1 = 1, a_2 = 0 + 1 = 1, a_3 = 0, a_4 = 0, a_5 = 0$. Find the minimum number of turns Evan needs to make $a_5$ exceed 1,000,000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
Find the limit of the expression as \(x \to -\infty\): \[ \frac{5}{\sqrt{x}\left( (1+\frac{9}{x^2})^{1/2}-(1+\frac{2}{x^2})^{1/2} \right)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Consider a Brownian motion $W_t$ and the Bachelier model $S_t = 1 + \mu t + \sigma W_t$ for the stock price $S_t$. Find the value of an option that pays $1(S_1 > 1)$. Use the risk-neutral valuation argument to determine the option's value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the definite integral \( \int_{-\infty}^{\infty} \frac{\sin^2(\pi Rx)}{R(\pi x)^2}dx \) for \( R > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For a function $h(t)$, it is given that $\int_{-\infty}^{\infty} h(t)e^{-j\omega t}dt=\frac{1}{\omega ^2+1}$. If $x(t)=\int_{-\infty}^{t}h(\tau)d\tau$, find the value of $\int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt$ at $\omega=1$. \(\boxed{?}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{j}{2}}
Determine one of the symmetry axes of the function $y = \cos 2x - \sin 2x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\pi}{8}}
Let \( a \in \mathbf{R} \) be a constant and consider the function \( f(x) = (a - x)|x| \). Suppose \( f(x) \) has an inverse function \( f^{-1}(x) \). Determine the range of values for the real number \( m \) such that the inequality \( f^{-1}(x^{2} + m) < f(x) \) holds for all \( x \in [-2,2] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(12, \infty)}
Given the prime numbers \( p \) and \( q \) satisfying \( p q + p^{2} + q^{2} = 199 \) where \( p < q \), find the value of the algebraic expression \(\frac{2}{p(p+1)}+\frac{2}{(p+1)(p+2)}+\cdots+\frac{2}{(q-1) q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{13}}
Determine the number of onto relations from the set \( A = \{1, 2, 3, \ldots, m\} \) to the set \( B = \{1, 2, 3, \ldots, n\} \), where the range of the relation is exactly \( B \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2^m - 1)^n}
In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as its diameter, intersects side \( BC \) at point \( K \). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \( BOC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
已知多项式 f(z)=z^3+az^2+bz+c(a,b,c\in\bZ) 的所有根的模均为 20 或 15，则这样的多项式的个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{572}
Consider a graph $G$ where each vertex represents a 3-element subset of the set $\{1, 2, 3, 4, 5, 6\}$. Two vertices, corresponding to subsets $A$ and $B$, are connected by an edge if they share exactly one element, and the cost of the edge is equal to the shared element. Determine the minimum cost of a spanning tree for this graph.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Calculate the quadratic variation of the process $Y_t = B_{2t}$, where $B_t$ is a Brownian motion with $\\langle B \\rangle_t = t$. Express the quadratic variation $\\langle Y \\rangle_t$ in terms of $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2t}
Determine the intersection of the fields $\mathbb{F}_{2^{4}}$ and $\mathbb{F}_{2^{6}}$ within the algebraic closure $\overline{\mathbb{F}}_2$. Use the fact that the lattice of finite extensions of a finite field is isomorphic to the lattice of positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F}_{2^{2}}}
Find the range of values for $p$ such that the interval $(1,2)$ is between the roots of the equation $2^x + 2^{2-x} + p = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -5)}
On Monday a school library was attended by 5 students, on Tuesday, by 6, on Wednesday, by 4, on Thursday, by 8, and on Friday, by 7. None of them have attended the library two days running. What is the least possible number of students who visited the library during a week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
After a storm deposits 120 billion gallons of water into the city reservoir, the reservoir is 85% full. If the original contents of the reservoir totaled a certain amount, and the reservoir was approximately 55.00000000000001% full before the storm, how many gallons of water can the reservoir hold when it is full?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Last year Jessica paid $1000 for rent, $200 for food, and $100 for car insurance each month. This year her rent goes up by some percentage, food costs increase by 50%, and the cost of her car insurance triples because she was at fault in an accident. Jessica pays $7200 more for her expenses over the whole year compared to last year. By what percentage did Jessica's rent increase this year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\%}
An archipelago consists of an infinite number of islands along the southern shore of an endless sea. Each island is connected to the next by a bridge, and each island is also connected to the shore by a bridge. During a strong earthquake, each bridge independently has a probability of 0.5 of being destroyed. What is the probability that it will still be possible to travel from the first island to the shore using the remaining bridges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Define a  $\textit{tasty residue}$  of  $n$  to be an integer  $1<a<n$  such that there exists an integer  $m>1$  satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of  $2016$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{831}
Call a number orz if it is a positive integer less than $2024$. Call a number admitting if it can be expressed as $a^2-1$ where $a$ is a positive integer. Finally call a number muztaba if it has exactly $4$ positive integer factors. Find the number of muztaba admitting orz numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7
}
Suppose $x$ and $y$ are real numbers such that\[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\]The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$?

Proposed by David Altizio
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
803
}
Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \]for all integers $n\geq 0$, where $i = \sqrt{-1}$. What is
\[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
For a sequence $s = (s_1, s_2, . . . , s_n)$, define
$$F(s) =\sum^{n-1}_{i=1} (-1)^{i+1}(s_i - s_{i+1})^2.$$
Consider the sequence $S =\left(2^1, 2^2, . . . , 2^{1000}\right)$. Let $R$ be the sum of all $F(m)$ for all non-empty subsequences $m$ of $S$. Find the remainder when $R$ is divided by $1000$.

Note: A subsequence is a sequence that can be obtained from another sequence by deleting some non-negative number of values without changing the order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
Determine the total surface area of a cube if the distance between the non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1152}
Find the area of an isosceles trapezoid if its height is $h$ and the lateral side is seen from the center of the circumscribed circle at an angle of $60^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h^2\sqrt{3}}
Let \(\Gamma\) be a circle in the plane and \(S\) be a point on \(\Gamma\). Mario and Luigi drive around the circle \(\Gamma\) with their go-karts. They both start at \(S\) at the same time. They both drive for exactly 6 minutes at constant speed counterclockwise around the track. During these 6 minutes, Luigi makes exactly one lap around \(\Gamma\) while Mario, who is three times as fast, makes three laps.

While Mario and Luigi drive their go-karts, Princess Daisy positions herself such that she is always exactly in the middle of the chord between them. When she reaches a point she has already visited, she marks it with a banana.

How many points in the plane, apart from \(S\), are marked with a banana by the end of the race?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If the shortest distance from a point on the ellipse $\frac{y^2}{16} + \frac{x^2}{9} = 1$ to the line $y = x + m$ is $\sqrt{2}$, find the minimum value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}