inclusionAI/Ring-lite-rl-data huggingface dataset:
6. There are 900 three-digit numbers $(100,101, \cdots, 999)$. These three-digit numbers are printed on cards, with ten three-digit numbers on each card. Some cards, when flipped, still show three-digit numbers, such as 198, which when flipped becomes 861 (1 is still considered 1 when flipped); some cards do not, such as 531, which when flipped shows I $E$. Therefore, some cards can be used in two ways. Thus, at most, _ fewer cards can be printed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Given nine different numbers, how many different values of third-order determinants can be formed using all of them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10080}
Determine whether the sequence \( \frac{e^{n^2}}{n^n} \) converges or diverges as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
令 $I=\sqrt{\frac{2}{\pi}} \int_{-\infty}^{+\infty} e^{x-10 \cosh (2 x)} \mathrm{d} x$, 则 $[|\ln I|]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
8. (10 points) In a regular truncated quadrilateral pyramid $A B C D A_{1} B_{1} C_{1} D_{1}$, the section $A B C_{1} D_{1}$ is drawn. Find the ratio of the volumes of the resulting polyhedra $\frac{V_{A B C D D_{1} C_{1}}}{V_{A_{1} B_{1} C_{1} D_{1} A B}}$, if $A B: A_{1} B_{1}=3$. Round the answer to the hundredths.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.2}
In the polar coordinate system, the minimum distance from a point on the circle $p=2$ to the line $p(\cos \theta+ \sqrt {3}\sin \theta)=6$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set of epsilon numbers (\(\epsilon\)-numbers) less than \(\omega_1\), the first uncountable ordinal. Is this set countable or uncountable?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Find the closed form of the generating function for the series: $x^1+x^4 + x^9 +x^{16} +x^{25} + x^{36} +x^{49} +x^{64}+....$ where the exponents are the sequence of squares of positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} \left( \vartheta_3(0,x) - 1 \right)}
7. Petya is playing a computer game called "Pile of Stones." Initially, there are 16 stones in the pile. Players take turns taking 1, 2, 3, or 4 stones from the pile. The player who takes the last stone wins. Petya is playing for the first time and therefore takes a random number of stones each time, without violating the rules of the game. The computer plays according to the following algorithm: on each turn, it takes as many stones as needed to be in the most advantageous position. Petya always goes first. What is the probability that Petya will win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{256}}
Given a $n \times 1$ matrix $u$ such that $u^Tu=1$, determine the matrix $A = I - 2uu^T$. Is it true that $A = -I$ for all such matrices $u$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A \neq -I}
A3. Determine the number of pairs of integers $(m, n)$ such that

$$
\sqrt{n+\sqrt{2016}}+\sqrt{m-\sqrt{2016}} \in \mathbb{Q}
$$

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let O be the origin (0, 0) and C the line segment defined by \( (x, y) : x \in [1, 3], y = 1 \). Let K be the curve consisting of points \( P \) such that for some point \( Q \) on \( C \), \( P \) lies on the line segment \( OQ \) and the distance \( PQ = 0.01 \). Determine whether the length of the curve \( K \) is greater than or less than 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Less than 2}
Let the universal set U = {1,2,3,4,5,6,7}, set A = {1,3,5}, and set B = {2,3,6}. Find the result of the operation A ∪ (∁U B). Express your answer as a set of elements, separated by commas and enclosed in curly brackets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{1,3,4,5,7}}
Let $ABC$ be a non-equilateral triangle with integer sides. Let $D$ and $E$ be respectively the mid-points of $BC$ and $CA$; let $G$ be the centroid of $\Delta ABC$. Suppose, $D$, $C$, $E$, $G$ are concyclic. Find the least possible perimeter of $\Delta ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
在 100 和 1000000 之间有多少个整数, 其数字之和等于 5 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{246}
Given a recursive sequence $\left\{x_{n}\right\}$ that satisfies $x_{n+1}=x_{n}^{2}-4x_{n}$ for $n=1, 2, \cdots$, if for any initial term $x_{1} \in \mathbf{R}$ such that $x_{1} \neq 0$, there is always a term $x_{k} \geqslant b$ in the sequence, then the largest integer not exceeding $(2b-3)^{2}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
The Palace Museum is holding four exhibitions during the May Day holiday: "Opera Culture Exhibition," "Ming Dynasty Imperial Cellar Porcelain Exhibition," "Historical Green Landscape Painting Exhibition," and "Zhao Mengfu Calligraphy and Painting Exhibition." A student decides to visit one of them in the morning and one in the afternoon on May Day, with the condition that they must visit at least one painting exhibition. Calculate the total number of different visit plans.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
For any positive integer $n$, define
$$
S(n)=\left[\frac{n}{10^{[\lg n]}}\right]+10\left(n-10^{[\lg n]}\left[\frac{n}{10^{[\operatorname{Ig} n]}}\right]\right) \text {. }
$$

Then among the positive integers $1,2, \cdots, 5000$, the number of positive integers $n$ that satisfy $S(S(n))=n$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
Find the determinant of a matrix similar to $A_n$, where the entries of the form $(i, i+3)$ are equal to $5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
100. In a regular truncated quadrilateral pyramid with lateral edges $A A_{1}, B B_{1}, C C_{1}, D D_{1}$, the side of the upper base $A_{1} B_{1} C_{1} D_{1}$ is 1, and the side of the lower base is 7. A plane passing through the edge $B_{1} C_{1}$ perpendicular to the plane $A D_{1} C$ divides the pyramid into two parts of equal volume. Find the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{38\sqrt{5}}{5}}
Determine the values of the real parameter $a$, such that the equation
\[ \sin 2x \sin 4x - \sin x \sin 3x = a \]
has a unique solution in the interval $[0, \pi)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
求满足 $\sin x _ { 1 } + \sin x _ { 2 } + \cdots + \sin x _ { n } = 0 ,$

$\sin x _ { 1 } + 2 \sin x _ { 2 } + \cdots + n \sin x _ { n } = 1 0 0$

的n的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
There are  $15$  people, including Petruk, Gareng, and Bagong, which will be partitioned into  $6$  groups, randomly, that consists of  $3, 3, 3, 2, 2$ , and  $2$  people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/455}
Let $X = \cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7} + \cdots + \cos\frac{2006\pi}{7} + \cos\frac{2008\pi}{7}$. Compute $\Big|\lfloor 2008 X \rfloor \Big|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1004}
**Problem 2**
Determine all pairs  $(n, m)$  of positive integers satisfying the equation  $$ 5^n = 6m^2 + 1\ .  $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 2)}
The sequence $\{a_n\}$ satisfies $a_{n+1}+(-1)^{n}a_{n}=2n-1$. Find the sum of the first $60$ terms of $\{a_n\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1830}
Several people played a single-round robin tournament in table tennis. At the end of the tournament, it turned out that for any four participants, there are two who scored the same number of points in the games among these four participants. What is the maximum number of tennis players who could have participated in this tournament? In table tennis, there are no draws, one point is awarded for a win, and zero points for a loss.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 - 2\sqrt{3}}
Find the smallest possible value of

$$
x y+y z+z x+\frac{1}{x}+\frac{2}{y}+\frac{5}{z},
$$

for positive real numbers $x, y$ and $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sqrt[3]{36}}
求所有的正整数 k，使得对任意满足 k(ab+bc+ca)>5(a^2+b^2+c^2) 的正实数 a,b,c，都存在三边长分别为 a,b,c 的三角形.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let the sequence  $(a_{n})$  be deﬁned by  $a_{1} = t$  and  $a_{n+1} = 4a_{n}(1 - a_{n})$  for  $n \geq 1$ . How many possible values of t are there, if  $a_{1998} = 0$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1996} + 1}
Compute the number of positive integers $n \leq 1000$ such that $\text{lcm}(n, 9)$ is a perfect square. (Recall that $\text{lcm}$ denotes the least common multiple.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Kamal's salary was decreased by 50%, then increased by 30%, and finally decreased by 20%. What is the overall percentage change in his salary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48\%}
Determine the optimal step size \(\nabla t\) for the Forward-Backward Splitting algorithm, which minimizes the sum of two convex functions \(f(x)\) and \(g(x)\). The algorithm iteration is given by:
\[x_{t+1}=\arg\min_u\left\{\frac{1}{2}\|u-(w_t-\nabla t \partial f(w_t))\|^2+\nabla tg(u) \right\}\] where \(f(x)\) and \(g(x)\) are convex functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{L}}
In triangle \(ABC\), side \(AC = 42\). The angle bisector \(CL\) is divided by the point of intersection of the angle bisectors of the triangle in the ratio \(2:1\) from the vertex. Find the length of side \(AB\) if the radius of the circle inscribed in triangle \(ABC\) is 14.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
已知函数 $f(x)=|x+1|+|3 x+a|$, 若 $f(x)$ 的最小值为 1 .
若 $a>0$, 且 $m, n$ 均为正实数, 且满足 $m+n=\frac{a}{2}$, 求 $m^{2}+n^{2}$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{9}{2}$}
Each cell of a 5x5 table is painted in one of several colors. Lada shuffled the rows of the table so that no row remained in its original place. Then Lera shuffled the columns so that no column remained in its original place. Surprisingly, the girls noticed that the resulting table was the same as the original one. What is the maximum number of different colors that can be used to paint this table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine whether the sum of the differences between the positive even and odd integer values of the Riemann zeta function equals \( \frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Assume $H$ is a Hilbert space. Given the weak convergences $u_m \rightharpoonup u$ in $L^2(0,T;H)$ and $u_m' \rightharpoonup v$ in $L^2(0,T;H^*)$, determine whether it follows that $v = u'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{v = u'}
Let $f$ be a continuous function on the interval $[0,1]$ mapping to $\mathbb{R}$. Define the linear operator $T$ by \[ T(f)(x) = \begin{cases} f(0), & \text{if } x = 0, \\ \frac{1}{x}\int_0^x f(t) \, dt, & \text{if } 0 < x \leq 1. \end{cases} \] Determine whether \[ \lim_{n \to \infty} \| T^n(f) - f(0) \|_{\infty} = 0. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For what value(s) of $a$ does the limit $\lim_{x\to 0}\frac{a^{\cos x}-a^{\cot x}}{\cot x -\cos x}$ equal $-1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{e}\)}
Given the function
$$
f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}\left(k \in \mathbf{Z}_{+}\right),
$$

if for any real number \(a\), 
$$
\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbf{R}\},
$$

find the smallest value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
集合 M 为连续 n 个正整数构成的集合. 若 M 中所有元素的和恰为 2024，且 n 为偶数，则集合 M 中最小的元素为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119}
Determine if the function \( \sum_{n=1}^{\infty} \cos \frac{n \pi x}{l} \) is periodic. If it is periodic, find its period.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2l}
4. A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at point $(x, y)$, she takes a step based on the following rules:

(a) if $x+y$ is even she moves to either $(x+1, y)$ or $(x+1, y+1)$;

(b) if $x+y$ is odd she moves to either $(x, y+1)$ or $(x+1, y+1)$.

How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x, y)$ to $(x+1, y))$ ?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{11}{6}}
Hexagon $ABCDEF$ is divided into four rhombuses, $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown. Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are congruent, and each has area $\sqrt{2006}.$ Let $K$ be the area of rhombus $\mathcal{T}.$ Given that $K$ is a positive integer, find the number of possible values for $K.$



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89
}
Determine whether the series \( \sum_{n=0}^\infty \int_{[-1,1]^2} \left(1-\frac{x^2+y^2}{2}\right)^{n/2}\lambda_2(d(x,y)) \) is finite, where \( \lambda_2 \) is the 2-dimensional Lebesgue measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given the function $y=a^{x}+b (b > 0)$, its graph passes through point $P(1,3)$. Find the minimum value of $\frac{4}{a-1}+\frac{1}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
As shown in the figure, a rectangular prism with length, width, and height of 15 cm, 5 cm, and 4 cm respectively has a smaller rectangular prism cut out of it. The smaller prism has length, width, and height of \( y \) cm, 5 cm, and \( x \) cm respectively, where \( x \) and \( y \) are integers. If the remaining volume is 120 cm\(^3\), then what is \( x + y \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
8. Given $a_{k}$ as the number of integer terms in $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$. Then $\sum_{k=1}^{2018} a_{k}=$ $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4102}
Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is $7$?




Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ \frac{2}{11} \]}
Determine whether there exists a constant \( c > 0 \) such that, for any division of a pile of books between two couples \((b_1, g_1)\) and \((b_2, g_2)\), each person can receive at least \( c \) value, regardless of the values and number of books.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
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 \mathrm{\ and \ } a_6

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
}
12. (22 points) If integers $a$ and $b$ are neither coprime nor have a divisibility relationship, then $a$ and $b$ are called a "coalition" pair. Let $A$ be an $n$-element subset of the set $M=\{1,2, \cdots, 2014\}$, and any two numbers in $A$ are coalition pairs. Find the maximum value of $n$.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
Given a positive geometric sequence $\{a_n\}$, where $a_5a_{2n-5}=10^{2n}$ $(n\geqslant 3,n\in N^{*})$, find the value of the expression $\lg a_1+\lg a_2+\lg a_3+\ldots+\lg a_n$ when $n\geqslant 1$, $n\in N^{*}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {n(n+1)}{2}}
Call an integer  $k$  *debatable* if the number of odd factors of  $k$  is a power of two. What is the largest positive integer  $n$  such that there exists  $n$  consecutive debatable numbers? (Here, a power of two is defined to be any number of the form  $2^m$ , where  $m$  is a nonnegative integer.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Evaluate the following limit for $\alpha \in (0, 1)$, $a > 0$, and $\beta \in \mathbb{R} \setminus \{0\}$:

$$ \lim_{t\rightarrow \infty} \beta \, t^\alpha \sum_{k=0}^{+\infty}\dfrac{(-a\,t^\alpha)^k}{\Gamma(\alpha\,k+\alpha+1)} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\beta}{a}\)}
Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$.

[i]Proposed by Isabella Grabski[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
There are 4 numbers written on the board. Vasya multiplied the first of these numbers by \(\sin \alpha\), the second by \(\cos \alpha\), the third by \(\operatorname{tg} \alpha\), and the fourth by \(\operatorname{ctg} \alpha\) (for some angle \(\alpha\)) and obtained a set of the same 4 numbers (possibly written in a different order). What is the maximum number of distinct numbers that could be written on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What is the units digit of the expression 7 ^ y + 6, where y is a positive integer, and the units digit of the expression is 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In the convex quadrilateral \(ABCD\), \(\angle ABC=60^\circ\), \(\angle BAD=\angle BCD=90^\circ\), \(AB=2\), \(CD=1\), and the diagonals \(AC\) and \(BD\) intersect at point \(O\). Find \(\sin \angle AOB\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15 + 6\sqrt{3}}{26}}
Given the following two propositions:  
Proposition $p$: The quadratic equation with real coefficients $x^{2}+mx+2=0$ has imaginary roots;  
Proposition $q$: For the equation $2x^{2}-4(m-1)x+m^{2}+7=0$ ($m\in\mathbb{R}$), the sum of the moduli of its two imaginary roots does not exceed $4\sqrt{2}$,  
If both propositions $p$ and $q$ are true, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,2\sqrt{2})}
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3432}
设函数 $f(x)=a x^{2}+b x+c \quad(a \neq 0)$ 满足
$|f(0)| \leq 2, \quad|f(2)| \leq 2, \quad|f(-2)| \leq 2$,
求当 $x \in[-2,2]$ 时 $y=|f(x)|$ 的最大值.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{7}
The perpendicular bisectors of the sides of triangle $PQR$ meet its circumcircle at points $P',$ $Q',$ and $R',$ respectively. If the perimeter of triangle $PQR$ is 30 and the radius of the circumcircle is 7, then find the area of hexagon $PQ'RP'QR'.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Let $a > 1$ and $x > 1$ satisfy $\log_a(\log_a(\log_a 2) + \log_a 24 - 128) = 128$ and $\log_a(\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{896}
The random variable $X$ follows a normal distribution $N(1, 4)$, where the mean $\mu = 1$ and the variance $\sigma^2 = 4$. Given that $P(X \geq 2) = 0.2$, calculate the probability $P(0 \leq X \leq 1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3}
Compute the integral: $$\int_{0}^{1}\frac{x^2}{x+\sqrt{1-x^2}}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/4}
设四面体四个面的面积分别为 $S_{1}, S_{2}, S_{3}, S_{4}$，它们的最大值为 $S$，记 $\lambda=\left(\sum_{i=1}^{4} S_{i}\right) / S$。找出 $\lambda$ 一定满足的范围。答案的形式为 $a < \lambda \leqslant b$，请给出 $a + b$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the number of integers $D$ such that whenever $a$ and $b$ are both real numbers with $-1 / 4<a, b<1 / 4$, then $\left|a^{2}-D b^{2}\right|<1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $ABC$ be a triangle such that $AB=\sqrt{10}$, $BC=4$, and $CA=3\sqrt{2}$. Circle $\omega$ has diameter $BC$, with center at $O$. Extend the altitude from $A$ to $BC$ to hit $\omega$ at $P$ and $P'$, where $AP < AP'$. Suppose line $P'O$ intersects $AC$ at $X$. Given that $PX$ can be expressed as $m\sqrt{n}-\sqrt{p}$, where $n$ and $p$ are squarefree, find $m+n+p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If $13$ married couples attended, how many handshakes were there among these $26$ people?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$234$}
In ten boxes, there are balls in the following quantities: 1, 3, 5, ..., 19 balls. Two friends, Peter and Vasily, take turns taking one ball from any box, with Peter going first. The player who loses is the one after whose turn the number of balls in any two boxes becomes the same (including possibly zero). Determine which player can guarantee a win regardless of the opponent's moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Vasily}