inclusionAI/Ring-lite-rl-data huggingface dataset:
What is the minimum number of Sportloto (6 out of 49) tickets required to guarantee that at least one number is guessed correctly on at least one of the tickets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the limit of the sequence $(S_n)$, where $S_n = \sum_{i=1}^{3n-1} \frac{1}{\sqrt{n^2+i}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Flights are arranged between 13 countries. For $k \ge 2$, the sequence $A_1, A_2, \ldots, A_k$ is said to be a cycle if there exists a flight from $A_1$ to $A_2$, from $A_2$ to $A_3$, $\ldots$, from $A_{k-1}$ to $A_k$, and from $A_k$ to $A_1$. Find the smallest possible number of flights such that, regardless of how the flights are arranged, there exists a cycle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
A towel, made of a fabric blend containing 60% cotton and 40% polyester, when bleached, experiences different shrinkage rates in its length and breadth. The shrinkage rates for each material are as follows:

- Cotton: length decreases by 35%, breadth decreases by 45%
- Polyester: length decreases by 25%, breadth decreases by 30%

Determine the overall percentage decrease in the area of the towel after bleaching.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57.91\%}
A positive integer $N$ greater than $1$ is described as special if, in its base-$8$ and base-$9$ representations, both the leading and ending digit of $N$ are equal to $1$. What is the smallest special integer in decimal representation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{793}
Problem 1

Find the largest integer $k$ having the following property: for any real numbers $x_{1}, x_{2}, \ldots, x_{2024}$ such that

$$
x_{1}^{2}=\left(x_{1}+x_{2}\right)^{2}=\cdots=\left(x_{1}+x_{2}+\cdots+x_{2024}\right)^{2}
$$

there are at least $k$ of them that are all equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{675}
Given a function f(x) = ax^3 - 3x^2 + 1, if f(-a), f(a), and f(3a) form an arithmetic sequence with a non-zero common difference, determine the number of tangent lines to the curve y = f(x) that can be drawn from the origin. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There are exactly $N$ distinct rational numbers $k$ such that $|k| < 200$ and \[5x^2 + kx + 12 = 0\] has at least one integer solution for $x$. What is the value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{78}
In an opaque bag, there are $2$ yellow balls, $3$ red balls, and $5$ white balls. They are identical except for their colors. What is the probability of drawing exactly one red ball from the bag?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{10}}
Given the parabola y^2 = 4x, points A and B on the parabola, and angle AFB = 2π/3, find the maximum value of |MM'|/|AB|, where M is the midpoint of chord AB and M' is its projection onto the directrix l. Express your answer as a single value, including necessary mathematical notation (e.g., square root, fraction).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
In how many different ways can four couples sit around a circular table such that no couple sits next to each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1488}
Given that three cultural courses (Chinese, Mathematics, and Foreign Language) and three other arts courses are randomly scheduled in six periods, find the probability that no two adjacent cultural courses are separated by more than one arts course.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{5}}
We are given that a point $(x,y,z)$ in $\mathbb{R}^3$ satisfies the following equations:

$$x\cos\alpha-y\sin\alpha+z =1+\cos\beta$$
$$x\sin\alpha+y\cos\alpha+z =1-\sin\beta$$
$$x\cos(\alpha+\beta)-y\sin(\alpha+\beta)+z=2$$

Where $\alpha,\beta\in(0,2\pi)$. We need to find the minimum value, $M$, of $x^2+y^2+z^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
将函数 $y=\sqrt{4+6 x-x^{2}}-2(x \in[0,6])$ 的图象逆时针方向旋转 $\theta(0 \leqslant \theta \leqslant \alpha)$, 得到曲线 C。若对于每一个旋转角 $\theta$ ，曲线 C 都是一个函数的图象，则 $\alpha$ 的最大值为多少？答案的形式为\arctan \frac{k}{m}，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A school needs to arrange 6 intern teachers into three classes, A, B, and C, with 2 teachers per class. Given that Teacher A cannot be arranged into Class A, and Teachers B and C cannot be arranged into the same class, calculate the total number of possible arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
The base of the right prism $A B C A_{1} B_{1} C_{1}$ is an isosceles triangle $A B C$, where $A B=B C=5$, $\angle A B C=2 \arcsin 3 / 5$. A plane perpendicular to the line $A_{1} C$ intersects the edges $A C$ and $A_{1} C_{1}$ at points $D$ and $E$ respectively, such that $A D=1 / 3 A C, E C_{1}=1 / 3 A_{1} C_{1}$. Find the area of the section of the prism by this plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{40}{3}}
Given the function f(x) = (1 ⊕ x) + (2 ⊕ x), where the operation ⊕ is defined as follows: when a ≥ b, a ⊕ b = a; when a < b, a ⊕ b = b^2, find the maximum value of f(x) on the domain [-2, 3].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
24. The width of a rectangle is called the length of its shortest side *). In how many different ways can a rectangle of width 3 be cut out from a square sheet of paper consisting of 100 cells? (Cuts must be made only along the cell boundaries.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8^3}
8. Let $A$ be any set of 7 real numbers, and let the set $B=\{x y \mid x, y \in A$ and $x \neq y\}$, denote the number of elements in set $B$ as $|B|$, then $|B|_{\text {max }}+|B|_{\text {min }}=$ $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let $\alpha$ be an element in an algebraic closure of $GF(64)$ such that $\alpha^4=\alpha+1$. For which $r\in \mathbb{N}$ is $GF(64)$ adjoined $\alpha$ isomorphic to $GF(2^r)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Point $P$ is $9$ units from the center of a circle with a radius of $15$. How many different chords of the circle contain $P$ and have integer lengths?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
已知函数 f:\bR\rightarrow\bR，使得任取实数 x,y,z 都有f(xy)+f(xz)-2f(x)f(yz)\geq\df{1}{2},则 [1\cdot f(1)]+[2\cdot f(2)]+\cdots+[2022\cdot f(2022)]=__________.（其中 [x] 表示不大于 x 的最大整数）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1022121}
In triangle \(ABC\), point \(D\) lies on side \(AC\) and point \(E\) lies on side \(BC\). The measures of angles \(ABD\), \(BAE\), \(CAE\), and \(CBD\) are \(30^\circ\), \(60^\circ\), \(20^\circ\), and \(30^\circ\) respectively.

Determine the measure of angle \(AED\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20^\circ}
设实数 $x_{1}, x_{2}, \cdots, x_{21}$ 满足 $0 \leq x_{i} \leq 1(1,2, \cdots, 21)$ ，求 $\sum_{i=1}^{21} \sum_{k=1}^{21}\left|x_{i}-x_{k}\right|$ 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{220}
John buys three types of flags: Flag A, Flag B, and Flag C. 

Flag A has 30 stripes, with the first stripe being red, half of the remaining stripes being red, and the rest being blue.

Flag B has 45 stripes, with every third stripe being red, every fourth stripe being blue, and the rest being white.

Flag C has 60 stripes, with a repeating pattern of two red stripes, followed by three blue stripes, and four white stripes.

John buys 20 Flag As, 30 Flag Bs, and 40 Flag Cs. How many red stripes are there overall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1310}
Given the integral sequence $U_n = \int_0^{\frac{\pi}{2}} x\sin^n x \, dx$, determine the value of $\frac{100U_{10} - 1}{U_8}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
There is number  $1$  on the blackboard initially. The first step is to erase  $1$  and write two nonnegative reals whose sum is  $1$ . Call the smaller number of the two  $L_2$ . 
For integer  $k \ge 2$ , the  ${k}$  the step is to erase a number on the blackboard arbitrarily and write two nonnegative reals whose sum is the number erased just now. Call the smallest number of the  $k+1$  on the blackboard  $L_{k+1}$ . 
Find the maximum of  $L_2+L_3+\cdots+L_{2024}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1405}{256}}
A $2 \times 3$ rectangle has vertices at $(0,0),(2,0),(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (The side originally on the $x$-axis is now back on the $x$-axis.) Find the area of the region above the $x$-axis and below the curve traced out by the point whose initial position is $(1,1)$. The original answer is in the form of \(\frac{m\pi}{n} + k\), please provide the value of \(m + n + k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass  $a$  and  $b$  and entangles them; this process destroys the omon with mass  $a$ , preserves the one with mass  $b$ , and creates a new omon whose mass is  $\frac 12 (a+b)$ . The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than  $1000$ . What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?

*Proposed by Michael Kural*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Ms. Lopez deposits $140 in an account that pays interest, compounded semiannually. The interest rate for the first six months is 20%, but it increases by 2% per thousand dollars increment in the account balance after the first semester. Additionally, she withdraws $40 at the end of the first six months. How much money will there be in the account at the end of one year, considering the variable interest rates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$125.4}
In a court-mandated traffic class, there are a certain number of drunk drivers, 3 less than 7 times that many speeders, twice the number of drunk drivers who are seatbelt violators, and 5 more than half the number of speeders who are texting drivers. If there are 180 students total, how many drunk drivers are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Joey is standing at a corner of a rectangular field. He walks the perimeter of the field 5 times. How many metres does Joey walk?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1500 \, \text{m}}
What is the largest integer $n$ such that $3^n$ divides the product of all elements in the set $P$, where $P$ consists of positive integer triplets $(a, b, c)$ satisfying $a^2 + b^2 = c^2$ and $3$ divides $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Assuming \(a\neq3\), \(b\neq4\), and \(c\neq5\), find the integer value of the following expression:
\[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}\]
Please provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given two circles. The first circle is inscribed in triangle $A B C$, the second touches side $A C$ and the extensions of sides $A B$ and $B C$. It is known that these circles touch each other, the product of their radii is 20, and the angle $B A C$ is $\arccos \frac{2}{3}$. Find the perimeter of triangle $A B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\sqrt{5}}
Lisa is a cat-loving gamer who has a collection of 15 cat-themed video games. She decided to buy more games to add to her collection. She found a gaming store that has a special offer: buy 2 cat-themed games and get the 3rd one free. If Lisa buys 6 cat-themed games, how many games in total will she have in her collection after taking advantage of the store's offer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,200 \pi).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Given that \( f(x) \) is a function defined on \( \mathbb{R} \), and for any \( x, y \in \mathbb{R} \), the following functional equation holds:
\[ f(x) + f(y) = 2 f\left(\frac{x+y}{2}\right) f\left(\frac{x-y}{2}\right) \]
Additionally, \( f\left( \frac{\pi}{4} \right) = 0 \), \( f(x) \) is not identically zero, and \( f(0) > 0 \). Find the period of \( f(x) \). Express your answer as a single mathematical expression or value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S, S'$, and $n$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(S/S')^{\frac{1}{2}n}$}
Determine the area of a triangle if two of its sides are 35 cm and 14 cm, and the bisector of the angle between them is 12 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{235.2 \text{ cm}^2}
The convex pentagon ABCDE has all sides of equal length, and angles A and B are both 90 degrees. What is the degree measure of angle E? Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Let $G$ be a complete graph on $25$ vertices with each edge colored red or blue. If there are no red triangles, what is the largest possible $m$ so that you are guaranteed to have a blue copy of $K_m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
设 $A, B, C, D$ 是空间四个不共面的点, 以 $\frac{1}{2}$ 的概率在每对点之前连一条边, 任意两对点之间是否连边是相互独立的, 则 $A, B$ 可用空间折线（一条或若干条边组成的）连接的概率为 $\qquad$。
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}
After folding the long rope in half 6 times, calculate the number of segments the rope will be cut into.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
In a convex polygon with 1992 sides, the minimum number of interior angles that are not acute is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1989}
5. (20 points) Two mutually perpendicular rays, propagating in the same vertical plane, fall from air onto a horizontal glass surface. The refractive index of the glass $n=1.6$. The angle of refraction for the first ray $\beta=30^{\circ}$. Determine the angle between the refracted rays.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
A circle can be inscribed in a trapezoid with bases of lengths 3 and 5, and a circle can also be circumscribed about it. Calculate the area of the pentagon formed by the radii of the inscribed circle that are perpendicular to the legs of the trapezoid, its smaller base, and the corresponding segments of the legs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \sqrt{15}}{2}}
How many distinct products, which are divisible by 10 (order does not matter), can be formed from the numbers $2, 3, 5, 7, 9$? The numbers in the product do not repeat!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
At Jefferson Summer Camp, $60\%$ of the children play soccer, $30\%$ of the children swim, and $40\%$ of the soccer players swim. To the nearest whole percent, what percent of the non-swimmers play soccer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$51\%$}
Compute the number of positive real numbers $x$ that satisfy $\left(3 \cdot 2^{\left\lfloor\log _{2} x\right\rfloor}-x\right)^{16}=2022 x^{13}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the volume of a regular pyramid whose base is a regular pentagon and whose lateral faces are equilateral triangles with side length \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a^{3}(5+\sqrt{5})}{24}}
In $\triangle ABC$, $\angle A = 100^\circ$, $\angle B = 50^\circ$, $\angle C = 30^\circ$, $\overline{AH}$ is an altitude, and $\overline{BM}$ is a median. Then $\angle MHC=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Among the divisors of 72 and 120, you write duplicate numbers on paper in order. What is the 4th largest number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Ivan the Tsarevich is learning to shoot a bow. He placed 14 arrows in his quiver and went to the forest to shoot at cones. He hits a cone with a probability of 0.1, and for each hit cone, the Frog Princess gives him 3 additional arrows. Ivan shoots until he runs out of arrows. Find the expected number of shots that Ivan will take.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
There are five "equations" of the form \( x^{2} + \ldots x + \ldots = 0 \) written on the board. Two players take turns filling in the blanks with natural numbers from 1 to 10, with each number being used exactly once. The game ends when all numbers are filled in. The player who makes the first move aims to maximize the number of equations that have two distinct roots at the end of the game, while their opponent aims to minimize this number. What is the best possible outcome that the first player can achieve regardless of the second player's moves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
求最大的正整数 n，使得存在实数 x_1,x_2,\cdots,x_n，满足(1) 1\leq x_1,x_2,\cdots,x_n\leq 2022；(2) 对任意 1\leq i<j<k\leq n，x_i^2+x_j^2+x_k^2\geq2(x_ix_j+x_jx_k+x_kx_i).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
On a straight line lie $100$ points and another point outside the line. What is the maximum number of isosceles triangles that can be formed using these $101$ points as vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Find the smallest prime number that can be represented as the sum of two, three, four, five, and six distinct prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Have $b, c \in R$ satisfy $b \in (0, 1)$ and $c > 0$, then let $A,B$ denote the points of intersection of the line $y = bx+c$ with $y = |x|$, and let $O$ denote the origin of $R^2$. Let $f(b, c)$ denote the area of triangle $\vartriangle OAB$. Let $k_0 = \frac{1}{2022}$ , and for $n \ge 1$ let $k_n = k^2_{n-1}$. If the sum  $\sum^{\infty}_{n=1}f(k_n, k_{n-1})$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find the remainder when $p+q$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{484}
如果: (1) $a, b, c, d$ 都属于 $\{1,2,3,4\}$;
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;
(3) $a$ 是 $a, b, c, d$ 中的最小值,
那么，可以组成的不同的四位数 $\overline{a b c d}$ 的个数是 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Given an arithmetic-geometric sequence $\{a\_n\}$, let $S\_n$ denote the sum of its first $n$ terms. If $S\_2 = 3$ and $S\_4 = 15$, find the value of $S\_6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
In a company of 100 children, some children are friends (friendship is always mutual). It is known that if any one child is excluded, the remaining 99 children can be divided into 33 groups of three such that in each group all three children are mutual friends. Find the minimum possible number of pairs of children who are friends.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
Translate the graph of the function $f(x)=\sin x$ ($x\in[0,2\pi]$) to the right by $\dfrac{\pi}{3}$ units to obtain the graph of the function $g(x)$. The area of the enclosed shape formed by the graphs of $f(x)$ and $g(x)$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Today's date is written as 22.11.2015. How many other days in this year can be written using this same set of digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The average age of 10 men is increased by years when one of them, whose age is 21 years, and another man are replaced by two new men with an average age of 32. What is the age of the other man who was replaced?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.5}
An equilateral triangle is constructed externally on the side \(AB\) of triangle \(ABC\). Find the distance between its center and vertex \(C\), given \(AB=c\) and \(\angle C = 120^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{c \sqrt{3}}{3}}
Find the minimum value of the function \((\sqrt{2} \sin x - 3 \tan y)^{2} + (\sqrt{2} \cos x - 3 \cot y)^{2}\), where \(x, y \in \left(0, \frac{\pi}{2}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a $2 \times 2$ tile and seven dominoes ($2 \times 1$ tile), find the number of ways to tile a $2 \times 7$ rectangle using some of these tiles without leaving gaps or overlapping any two tiles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Let \( P \) be a 40-sided convex polygon. Find the number of triangles \( S \) formed by the vertices of \( P \) such that any two vertices of \( S \) are separated by at least two other vertices of \( P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7040}
Given the hyperbola $C$: $\frac{x^{2}}{4}-y^{2}=1$ with left and right foci $F\_1$ and $F\_2$ respectively, find the area of the quadrilateral $P\_1P\_2P\_3P\_4$ for a point $P$ on the hyperbola that satisfies $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8\sqrt{6}}{5}}
1. Each of the 4 colors has 6 pencils. Each of the six children received 4 pencils. For what smallest $k$ can we always choose $k$ children and decorate their pencils so that we have at least one in each color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}