inclusionAI/Ring-lite-rl-data huggingface dataset:
Once, Carlson and Winnie the Pooh competed in the speed of eating honey and jam. Carlson, an expert in jam, eats a jar of jam in 2 minutes, while Winnie the Pooh takes a full 7 minutes to finish a jar of jam. Meanwhile, Winnie the Pooh can finish a pot of honey in 3 minutes, but Carlson requires 5 minutes to do the same. In total, they had 10 jars of jam and 10 pots of honey. For how many minutes did they eat everything, given that they started and finished eating at the same time? (Each eats a pot of honey or a jar of jam completely.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the expected profit from a game where a fair die with $n$ sides is rolled at each step, and you can choose to stop and keep the last roll's value as your profit or pay $1$ to continue playing. Find an asymptotic formula for the expectation as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - \sqrt{2n}}
Given that $α$ is in the interval $(0,π)$ and $\cos(α - \frac{π}{6}) = \frac{3}{5}.$ Find $\cos α$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos α = \frac{3\sqrt{3} - 4}{10}}
Two natural numbers, $p$ and $q$, do not end in zero. The product of any pair, p and q, is a power of 10(that is, $10, 100, 1000, 10 000$ , ...). If $p>q$, the last digit of $p - q$ cannot be
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
A square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed - a large one and two small ones. Calculate the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Find the minimal polynomial of the linear operator \( T \) on the vector space \( V = M_{n \times n}(R) \), where \( T(A) = A^t \) for each matrix \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - 1}
Let $x_i \in [-11, 5]$ for $i = 1, 2, \ldots, 2014$, and suppose $x_1 + x_2 + \cdots + x_{2014} = 0$. Determine the maximum value of $x_1^2 + x_2^2 + \cdots + x_{2014}^2$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110710}
After exploring the mountain country, she went to the famous beach on the continent. There, she collected 820 seashells in various colors including red, green, blue, yellow, and purple. The number of shells of each color follows the Fibonacci sequence. If she has 144 red shells and 89 green shells, and knowing that blue shells are the least numerous, how many blue, yellow, and purple shells does she have in total, given that the number of each color is a unique term in the Fibonacci sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665}
Find the number of $7$-tuples $(n_1,\ldots,n_7)$ of integers such that \[ \sum_{i=1}^7 n_i^6 = 96957. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2688}
Find the Gini index for the probability density function f(x) = 1 on [0, 1] and f(x) = 0 elsewhere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
Barbara, Edward, Abhinav, and Alex took turns writing this test. Working alone, they could finish it in  $10$ ,  $9$ ,  $11$ , and  $12$  days, respectively. If only one person works on the test per day, and nobody works on it unless everyone else has spent at least as many days working on it, how many days (an integer) did it take to write this test?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find a tight bound for the recurrence relation: \[ T(n) = T\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + T\left(\left\lfloor \frac{n}{4} \right\rfloor \right) + T\left(\left\lfloor \frac{n}{8} \right\rfloor \right) + n \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Suppose  $28$  objects are placed along a circle at equal distances. In how many ways can  $3$  objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2268}
Given the sets $A=\{x\mid -2 < x < 4\}$ and $B=\{x\mid x < a\}$, and their union $A\cup B=\{x\mid x < 4\}$, determine the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 4]}
8.5 On an island, there live 25 people: knights, liars, and tricksters. Knights always tell the truth, liars always lie, and tricksters answer the questions posed to them in turn, alternating between truth and lies. All the islanders were asked three questions: "Are you a knight?", "Are you a trickster?", "Are you a liar?" (the questions were asked in the exact order specified). To the first question, 21 people answered affirmatively, to the second - 17 people, to the third - 6 people. How many knights live on this island?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
There are three video game systems: the Paystation, the WHAT, and the ZBoz2 \pi, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2 \pi, and Dirac owns a ZBoz2 \pi and a Paystation. A store sells 4 different games for the Paystation, 6 different games for the WHAT, and 10 different games for the ZBoz2 \pi. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys 3 random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{25}}
Identify the smallest group that is not isomorphic to a cyclic group, a direct product of cyclic groups, or a semi-direct product of cyclic groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q_8}
Let $C \subset \mathbb{P}_\mathbb{C}^2$ be a projective curve, and $p \in C$ a point of multiplicity $m$. If $\mathcal{O}_p(C)$ is the local ring of $C$ at $p$, and $\mathfrak{m} \subset \mathcal{O}_p(C)$ is its maximal ideal, find $\text{dim}_\mathbb{C} \mathfrak{m}^k/\mathfrak{m}^{k+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min(k + 1, m)}
Find the probability assigned by the measure $\mu$ to the union of the top and left edges of the unit square $[0,1]^2$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $k \ll n$ denote that $k < n$ and $k \mid n$. Let $f: \{1, 2, \ldots, 2013\} \rightarrow \{1, 2, \ldots, M\}$ be such that, if $n \leq 2013$ and $k \ll n$, then $f(k) \ll f(n)$. What is the least possible value of $M$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given the identity $(x^3+px^2+2x-5)^{19}(x^2+qx-41)^{8}(x^4-x^3+x-7)^6 = x^{97} + 391x^{96} + a_{95}x^{95} + \cdots + a_1x + a_0$, where $p, q, a_{95}, \cdots, a_0$ are integers, find the smallest possible value of $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given an isosceles right triangle \(ABC\) with hypotenuse \(AB\). Point \(M\) is the midpoint of side \(BC\). A point \(K\) is chosen on the smaller arc \(AC\) of the circumcircle of triangle \(ABC\). Point \(H\) is the foot of the perpendicular dropped from \(K\) to line \(AB\). Find the angle \(\angle CAK\), given that \(KH = BM\) and lines \(MH\) and \(CK\) are parallel.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.5}
Given that $\binom{21}{13}=20349$, $\binom{21}{14}=11628$, and $\binom{23}{15}=490314$, find $\binom{22}{15}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{458337}
10. $1991^{2000}$ divided by $10^{6}$, the remainder is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{880001}
Given the line $l: \lambda x-y-\lambda +1=0$ and the circle $C: x^{2}+y^{2}-4y=0$, calculate the minimum value of $|AB|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
A man walks at a rate of 10 mph. After every ten miles, he rests for a certain amount of time. He takes 332 minutes to walk 50 miles. How many minutes does he rest after every ten miles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A right cylindrical oil tank is $15$ feet tall and its circular bases have diameters of $4$ feet each. When the tank is lying flat on its side (not on one of the circular ends), the oil inside is $3$ feet deep. How deep, in feet, would the oil have been if the tank had been standing upright on one of its bases? Express your answer as a decimal to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the exact value of the series
\[
\frac{1}{3 + 1} + \frac{2}{3^2 + 1} + \frac{4}{3^4 + 1} + \frac{8}{3^8 + 1} + \frac{16}{3^{16} + 1} + \dotsb.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
China Mathematical Olympiad 2018 Q6
Given the positive integer $n ,k$ $(n>k)$ and $ a_1,a_2,\cdots ,a_n\in (k-1,k)$ ,if positive number $x_1,x_2,\cdots ,x_n$ satisfying:For any set $\mathbb{I} \subseteq  \{1,2,\cdots,n\}$ ,$|\mathbb{I} |=k$,have $\sum_{i\in \mathbb{I}  }x_i\le \sum_{i\in \mathbb{I}  }a_i$ , find the maximum value of $x_1x_2\cdots  x_n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ a_1 a_2 \cdots a_n }
Determine the Galois group of the polynomial \( f(x) = x^6 - 9 \) over \( \mathbb{Q} \), given that the splitting fields of its factors \( x^3 - 3 \) and \( x^3 + 3 \) are identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
What angle is formed by the hour and minute hands at 4:12?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime, while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
The lateral edge of a regular triangular prism is equal to the height of the base, and the area of the cross-section passing through this lateral edge and the height of the base is equal to \( Q \). Determine the volume of the prism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Q \sqrt{\frac{Q}{3}}}
In a convex hexagon $ABCDEF$, let $P$, $Q$, $R$, $S$, $T$, and $U$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The areas of triangles $ABR$, $BCS$, $CDT$, $DEU$, $EFP$, and $FAQ$ are given as $12$, $34$, $56$, $12$, $34$, and $56$, respectively. Find the area of hexagon $ABCDEF$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{204}
# Problem 10. (2 points)

The sum of a geometric progression consisting of four natural numbers is 1417. Find the second largest of these numbers.

If there are multiple possible answers, list them in ascending order separated by commas or semicolons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Let $g(x)$ be a continuous function for all real values of $x$ that satisfies the following equation:

$$\int^x_1 g(t)dt =\int^0_{x} t^3g(t)dt +\frac{x^{18}}{9}+\frac{x^8}{4}+b$$

Find the value of $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{13}{36}}
Translate the function $f(x)=1-2\sqrt{3}\cos^2{x}-(\sin{x}-\cos{x})^2$ by shifting its graph to the left by $\dfrac{\pi}{3}$ units to obtain the graph of the function $y=g(x)$. If $x\in\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$, then the interval of monotonic increase for the function $g(x)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[-\frac{5\pi}{12}, \frac{\pi}{12}\right]}
Determine the dimension of the ideal \( I = \langle x+1 \rangle / \langle x^9+1 \rangle \) in the ring \( R = \mathbb{Z}_2[x]/\langle x^9+1 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
10. Given $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} \in \mathbf{N}$ and satisfy $a_{1}+a_{2}+a_{3}=b_{1}+b_{2}+b_{3}$
$$
\begin{array}{c}
a_{1} a_{2} a_{3}=b_{1} b_{2} b_{3} \\
a_{1} a_{2}+a_{1} a_{3}+a_{2} a_{3} \neq b_{1} b_{2}+b_{1} b_{3}+b_{2} b_{3}
\end{array}
$$

then the minimum value of the maximum number among $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
求最大的实数 $k$, 使得对于任意的正数 $a 、 b$ ,均有  
$$  
(a+b)(a b+1)(b+1) \geqslant k a b^{2}.  
$$  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{31}
In the triangle \( \triangle ABC \), \( AB = AC \), \( \angle ABC = 40^\circ \) and the point \( D \) is on \( AC \) such that \( BD \) is the angle bisector of \( \angle ABC \). If \( BD \) is extended to the point \( E \) such that \( DE = AD \), find the measure of \( \angle ECA \) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40^\circ}
Given the parabola $y^{2}=4x$ and the circle $E:(x-4)^{2}+y^{2}=12$. Let $O$ be the origin. The line passing through the center of circle $E$ intersects the circle at points $A$ and $B$. The lines $OA$ and $OB$ intersect the parabola at points $P$ and $Q$ (points $P$ and $Q$ are not coincident with point $O$). Let $S_{1}$ be the area of triangle $OAB$ and $S_{2}$ be the area of triangle $OPQ$. Find the maximum value of $\frac{S_{1}}{S_{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{16}}
Sergei and Lena have several chocolate bars, each weighing no more than 100 grams. No matter how they divide these chocolate bars, the total weight of the chocolate bars for one of them will not exceed 100 grams. What is the greatest possible total weight of all the chocolate bars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $n$ between 1 and 25. Betül draws $n$ balls from the bag one by one such that no ball is put back after it is drawn. If exactly 2 of the $n$ balls are blue and the second blue ball is drawn at the $n^{th}$ order, Ahmet wins; otherwise, Betül wins. To increase the possibility of winning, what integer should Ahmet choose?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A regular tetrahedron has two spheres, one inscribed within it and the other circumscribed around it. Between each face of the tetrahedron and the circumscribed sphere, there are four smaller spheres centered on each face. Given a point \( P \) within the circumscribed sphere, the probability that \( P \) lies within one of the five smaller spheres is closest to:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2}
Find the smallest positive integer $n$ such that the last 30 digits of the decimal representation of $n!(n+1)!(2n+1)! - 1$ are all 9s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
若 $x_{1}^{4}+x_{2}^{4}+\cdots+x_{n}^{4}=799$ 有正整数解，则正整数 $n$ 的最小值为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Suppose $X$ and $Y$ are independent and identically distributed (i.i.d.) random variables, each following a normal distribution $N(0,1)$. Calculate the conditional probability $P(X > 0 \mid X + Y > 0)$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Myrtle’s 3 hens lay 3 eggs a day. She was gone for a certain number of days and told her neighbor to take as many as they would like. The neighbor took 12 eggs. Once home, Myrtle collected the remaining eggs, dropping 5 on the way into her house. Myrtle has 46 eggs. How many days was Myrtle gone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Evaluate the integral \[ I = \iint_{[0,1]^2} \frac{dx \, dy}{(1+x^2+y^2)^{3/2}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
Estimate the upper bound of the following integral using the ML lemma, without evaluating it:

$$\left|\int_{Γ}\frac{z^{\frac{3}{2}}}{z^2 + 4}dz\right|$$

where $Γ$ is the semicircular arc {$z ∈ C : |z| = 2,\quad 0 ≤ Arg(z) ≤ π$} described in the clockwise direction. Consider the principal branch of $z^{3/2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi\sqrt 2}{2}\)}
Compute the limit \( \displaystyle\lim_{x \to +\infty} f(x) \), where \( f(x) = \frac{1}{\Gamma(x)}\int_x^{+\infty}t^{x-1}e^{-t}dt \) for all \( x \in \Bbb R_+^* \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Given a closed, smooth, oriented Riemannian manifold $([M]^n, g)$, with two closed, smooth, oriented submanifolds $[S]^k$ and $[T]^{n-k}$ intersecting transversely, let $\omega$ be a $k$-form on $M$ such that $\oint_{[S]} \omega = r \ne 0$. If $\eta$ is the Hodge star of $\omega$, determine $\oint_{[T]} \eta$ given that the $\mathbb{Z}\pi_1(M)$-intersection number of $S$ and $T$ is $\pm 1$, where $\pi = \pi_1(M)$. Provide your answer as a single value or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm r}
In $\triangle ABC$, given $C= \sqrt {2}$, $\angle B= \frac {\pi}{4}$, and $b=2$, find $\angle A=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7\pi}{12}}
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is selected. On one side of \(AO\), points \(B\) and \(C\) are chosen on the circle such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\), if the radius of the circle is 10 and \(\cos \alpha = \frac{2}{5}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are 1000 rooms in a row along a long corridor. Initially the first room contains 1000 people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different
rooms will have people in them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 61 }
Calculate the area bounded by the inverse of the function \( f(x) = x^3 + 3x + 2 \) between \( x = -2 \) and \( x = 6 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Find the value of the limit: \[ \lim_{n\to \infty }\left(\sqrt[\leftroot{-2}\uproot{2}n+1]{(n+1)!}-\sqrt[\leftroot{-2}\uproot{2}n]{n!}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
The function $f(x)=k\ln x-x^{2}+3x (k\neq 0)$ has extreme values on the interval $(0,\frac{3}{2})$. The range of real number $k$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\frac{9}{8}, 0)}
As shown in Figure 2, let $E$ and $F$ be the midpoints of sides $AD$ and $BC$ respectively of rectangle $ABCD$. Points $M$ and $N$ are on segments $AB$ and $CF$ respectively. Line segment $MN$ intersects $AF$ at point $P$, with $MN = 12$, $AD = 2AB$, and $PE \perp MN$. Find the length of $PE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The numbers 1, 2, 3, and 4 are assigned, in some order, to the variables p, q, r, and s. What is the maximum possible value of p^q + r^s? Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{83}
Given that $2^{2004}$ is a $604$-[[digit]] number whose first digit is $1$, how many [[element]]s of the [[set]] $S = \{2^0,2^1,2^2,\ldots ,2^{2003}\}$ have a first digit of $4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{195}
Find the coefficient of \(x^{25}\) in the expansion of \((1 + x^2 + x^4 + x^7)^{10}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34750}
Two cyclists set off simultaneously towards each other from points $A$ and $B$ and met 70 km from $A$. Continuing with the same speeds, they reached points $A$ and $B$ and turned back. The second time they met 90 km from $B$. Find the distance from $A$ to $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120 \text{ km}}
Evaluate the surface integral \(\iint_{S} xz \, dy \, dz\) where \(S = \{(x,y,z) \mid x \geq 0, y \geq 0, z \geq 0, 2x + 2y + z = 2\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is  $social$  if it knows at least  $20$  other persons and at least  $2$  of those  $20$  know each other. For a person, we say that it is  $shy$  if it doesn't know at least  $20$  other persons and at least  $2$  of those  $20$  don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any  $social$  nor  $shy$  persons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given a random variable $Y=\frac{c(X_1-X_2)}{\sqrt{(X_3)^2+(X_4)^2}}$, where $X_1$, $X_2$, $X_3$ and $X_4$ are independent standard normally distributed variables, find the value of $c$ such that $Y$ follows a t-distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given circle $C: x^2 + y^2 = 1$, and point $P(x_0, y_0)$ is on the line $l: 3x + 2y - 4 = 0$. If there always exist two different points $A, B$ on circle $C$ such that $\vec{OA} + \vec{OB} = \vec{OP}$, then the range of values for $x_0$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(0, \frac{24}{13}\right)}
Solve the system of equations for \((x, y, z)\), where \(\{x\}\) denotes the fractional part of \(x\):

\[\begin{cases} x + \lfloor y \rfloor + \{z\} = 12.5\\ \{x\} + y + \lfloor z \rfloor = 16.2\\ \lfloor x \rfloor + \{y\} + z = 15.7 \end{cases}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(6.5, 6.7, 9.0)}