inclusionAI/Ring-lite-rl-data huggingface dataset:
A magician and their assistant are planning to perform the following trick. A spectator writes a sequence of $N$ digits on a board. The magician's assistant covers two adjacent digits with a black circle. Then the magician enters. Their task is to guess both of the covered digits (and the order in which they are arranged). For what minimum $N$ can the magician and the assistant agree in advance to guarantee that the trick will always succeed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
A $3 \times 3 \times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $45^\circ$ angle with the horizontal plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
By solving the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\), find the sum of its integer solutions within the interval \([-25, 25]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{290}
Define \( ||x|| \) as the distance from \( x \) to the nearest integer. Find \( \lim_{n \to \infty} \frac{1}{n} \int \limits_{1}^{n} ||\frac{n}{x}|| \, dx \). You may assume that \( \prod_{n=1}^{\infty} \frac{2n}{2n-1} \cdot \frac{2n}{2n+1} = \frac{\pi}{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln \left( \frac{4}{\pi} \right)}
Let $a$ and $b$ be positive integers such that
$$
\frac{a^{2}+ab+b^{2}}{ab-1}=k \quad \left(k \in \mathbf{Z}_{+}\right).
$$

Find the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Carter has a jar with 35 green M&Ms, 25 red M&Ms, 10 blue M&Ms, and 15 orange M&Ms. He eats 20 of the green M&Ms and 8 red M&Ms. Then, his sister comes and eats half of the remaining red M&Ms, 5 blue M&Ms, and adds 14 yellow M&Ms. After that, their friend Alex comes over and adds 8 purple M&Ms, eats 7 orange M&Ms, and 3 yellow M&Ms. Later, their cousin eats twice the number of remaining blue M&Ms and adds 10 brown M&Ms. If Carter picks an M&M at random now, what is the percentage chance he'll get a green M&M?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
There are 250 jellybeans in a glass jar. Mrs. Copper's kindergarten class normally has 24 kids, but 2 children called in sick and stayed home that day. Each sick child would have eaten 7 jellybeans. The 22 remaining children who attended school are split into two groups. The first group, with 12 children, eat 5 jellybeans each, while the second group, with 10 children, eat 4 jellybeans each. How many jellybeans are still left in the jar?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Find the center of symmetry of the curve y = sin(2x) + cos(2x). Express your answer as an ordered pair (x, y), where x is in terms of pi.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{3\pi}{8}, 0\right)}
If the equation \( x^2 + y^2 - 2ax - 2y + 3 - a = 0 \) represents a circle with its center in the second quadrant, then the range of values for \( a \) is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -2)}
On an $8 \times 8$ chessboard, there are 16 rooks, each placed in a different square. What is the minimum number of pairs of rooks that can attack each other (a pair of rooks can attack each other if they are in the same row or column and there are no other rooks between them)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
给定正整数 $n$ 和正数 $M$ 。对于满足条件 $a_{1}^{2}+a_{n+1}^{2} \leqslant M$ 的所有等差数列 $a_{1}, a_{2}, a_{3}, \cdots$, 试求 $S=a_{n+1}+a_{n+2}+\cdots+a_{2 n+1}$ 的最大值.写出$M=10,n=5$时$S$的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given points $A(-3,0)$, $B(-1,-2)$, if there are exactly two points $M$, $N$ on the circle $(x-2)^{2}+y^{2}=r^{2}(r > 0)$ such that the areas of $\triangle MAB$ and $\triangle NAB$ are both $4$, then the range of $r$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{( \frac { \sqrt {2}}{2}, \frac {9 \sqrt {2}}{2})}
A school has four teachers - A, B, C, and D - available to teach elective classes during two periods on Thursday afternoon. Teachers A and B can each teach up to two classes, while teachers C and D can each teach up to one class. Each period must have exactly two teachers teaching. Calculate the total number of different teaching arrangements possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
In the diagram, \(\triangle PQS\) is equilateral. Also, \(\triangle PQR\) and \(\triangle PSR\) are isosceles with \(PQ = PR = PS\). If \(\angle RPQ = \angle RPS\), find the measure of \(\angle QRS\). Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Given a continuous function $f(x)$ on $[0,1]$ with $f(0) = 0$, $f(1) = 1$, and $\int_0^1 [f'(x)]^2 \sqrt{1+x^2} \,dx = \frac{1}{\ln(1+\sqrt{2})}$, determine $\int_0^1 \frac{f(x)}{\sqrt{1+x^2}} \,dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{2} \cdot \ln(1 + \sqrt{2})\)}
Given circles $C\_1:x^{2}+y^{2}=1$ and $C\_2:x^{2}+y^{2}-8x+12=0$, find the length of their common tangent lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{15}}
A bagel is cut into sectors. After making 10 cuts, how many pieces are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6$, $DB = 2$, and the ratio $\frac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5$. Then, if $OA$ is the length of the circumradius, then $OA^2$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given that $a$, $b$, $c$ are the sides opposite to angles A, B, C respectively in $\triangle ABC$, and vectors $\overrightarrow {m}$ = (cosA, sinA), $\overrightarrow {n}$ = (1, $\sqrt {3}$), if $\overrightarrow {m} || \overrightarrow {n}$ and $a \cos B + b \cos A = c \sin C$, then angle B is equal to ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {\pi}{6}}
Along a straight alley, 100 lanterns are placed at equal intervals, numbered sequentially from 1 to 100. Simultaneously, Petya and Vasya start walking towards each other from opposite ends of the alley with different constant speeds (Petya starts from the first lantern, and Vasya starts from the hundredth lantern). When Petya reached the 22nd lantern, Vasya was at the 88th lantern. At which lantern will they meet? If the meeting occurs between two lanterns, indicate the smaller number of the two.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Aitana and Jayda went to Silverlake Flea market to buy some handmade crafts. Jayda bought items from 3 different stalls, spending $400, $120, and $250 respectively. Aitana, on the other hand, spent 2/5 times more than what Jayda spent at each stall. On their purchases at the first and second stalls, they were offered different discounts: Jayda received a discount of 5% for the first stall and Aitana received a 10% discount on the second stall. In addition, after their shopping spree, they realized that there was a sales tax of 10% applied to the final prices of all their purchases. Because they were from Canada, they also had to consider the exchange rate of 1 US Dollar to 1.25 Canadian Dollars. Calculate the total amount of money, in Canadian dollars, they spent together after accounting for all discounts, taxes, and exchange rates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$2490.40}
Jake wrote six consecutive numbers on six white pieces of paper, one number on each piece. He stuck these bits of paper onto the top and bottom of three coins. Then he tossed these three coins three times. On the first toss, he saw the numbers 6, 7, and 8 and then colored them red. On the second toss, the sum of the numbers he saw was 23, and on the third toss, the sum was 17. What was the sum of the numbers on the remaining three white pieces of paper?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
If the function $f(x)=2x^2 - \ln x$ is not monotonic on a subinterval $(k-1,k+1)$ within its domain, determine the range of the real number $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[1, \frac{3}{2}\right)}
Two buildings in a residential area are 220 meters apart. If 10 trees are planted at equal intervals between them, what is the distance between the 1st tree and the 6th tree?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100 \text{ meters}}
Anders encounters the expression \(\sqrt{15!}\) and attempts to simplify it into the form \(a \sqrt{b}\) where \(a\) and \(b\) are positive integers. The sum of all possible distinct values of \(ab\) can be expressed in the form \(q \cdot 15!\) for some rational number \(q\). Find \(q\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given the function $f(x)={a}^{2-x}-\frac{3}{4}$ ($a \gt 0$ and $a\neq 1$), whose graph always passes through a fixed point $A$, if the graph of the power function $y=g\left(x\right)$ also passes through this point, then $g(\frac{1}{2})=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
There are  $24$  different pencils,  $4$  different colors, and  $6$  pencils of each color. They were given to  $6$  children in such a way that each got  $4$  pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.

P.S. for 10 grade gives same problem with  $40$  pencils,  $10$  of each color and  $10$  children.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Arrange the numbers $1, 2, 3, 4, 5, 6$ randomly in a row, denoted as $a, b, c, d, e, f$. The probability that $abc + def$ is an even number is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{10}}
Find the sum of the following expression, assuming $a_1, a_2, ..., a_n$ are distinct non-zero values:

$$\sum_{j=1}^n \frac{a_j^{n-2}}{f''(a_j)}$$

where $f(x) = (x-a_1)(x-a_2)...(x-a_n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The value of  $ 21!$  is  $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$ , where  $ a$ ,  $ b$ , and  $ c$  are digits.  What is the value of  $ 100a \plus{} 10b \plus{} c$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{709}
设 $a, b$ 为不超过 12 的正整数，满足：存在常数 $C$ ，使得 $a^{n}+b^{n+9} \equiv C(\bmod 13)$ 对任意正整数 $n$ 成立. 求所有满足条件的有序数对 $(a, b)$. 请提供所有满足条件的数对中所有整数的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Compute the area of the figure bounded by the parabola \( y = -x^2 + 6x - 5 \) and the coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given $(625^{\log_2 250})^{\frac{1}{4}}$, find the value of the expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Let $\sigma(n)$ be the sum of all positive divisors of a positive integer $n$. Evaluate $\left\lfloor\sqrt[3]{\sum_{n=1}^{2020} \frac{\sigma(n)}{n}}\right\rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
How many times do we need to flip a coin so that with an error probability of less than 0.01, we can state that the frequency of getting heads will be between 0.4 and 0.6?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{166}
Evaluate the limit: \[ \lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-y^{n-k}(1-y)^{k}| \] where \( x \in (0,\frac{1}{2}) \) and \( y \in (0,\frac{1}{2}] \). Specifically, find the value of the limit when \( y = \frac{1}{2} \): \[ \lim_{n\rightarrow \infty}\sum_{k=0}^{n}{n \choose k}|x^{n-k}(1-x)^{k}-\frac{1}{2^{n}}| \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Solve for the time \( t \) in the equation \( 318.45 = 288.45e^{-t/4.5} + 44.1t \) without using graphing methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.13}
Determine the range of \( \alpha \) for which the function \( f(x) = \begin{cases} x^\alpha \sin\left(\frac{1}{x}\right)\sin(\pi x), & x \neq 0 \\ 0, & x = 0 \end{cases} \) satisfies the conditions of Rolle's theorem on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, \infty)}
Compute the value of the series \( \sum_{n=0}^\infty \frac{\sin((2n+1)\phi)}{2n+1} \) for \( 0 < \phi < \pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
In a 50x50 grid, each cell contains a number representing the count of 1x16 rectangles (both vertical and horizontal) for which this cell is an endpoint. Determine the number of cells that contain numbers greater than or equal to 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1600}
Let \( S_0=\{A,B\} \), where \( A,B \) are subsets of some set \( M \). Define \( S_{i+1} \) inductively to be the collection of subsets of \( M \) that are of the form \( C \cup D, C \cap D, M \backslash C \) where \( C,D \in S_i \). Determine the largest cardinality of \( \bigcup S_i \) as \( i \rightarrow \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the argument of the complex number \( w = \left( 1 - \sqrt{3}i \right)^{2i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2}
Determine the number of ring homomorphisms from the ring \( \mathbb{Z}_{16} \) to the ring \( \mathbb{Z}_{20} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: $$\lim_{x\to 0} \left( \frac{1+x\cdot4^x}{1+x\cdot5^x} \right)^{\frac{1}{x^2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{5}}
Given a set, such as \( \{1, 7, 4, 90, 23\} \), and a value \( A \), determine the number of subsets of the set whose sum is a multiple of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the value(s) of $x$ for which the matrices $A$ and $B$ are similar, given that both matrices share the characteristic polynomial $(\lambda - 1)^2(\lambda - 2)$, where
\[ A = \begin{pmatrix} 3&0&-1 \\ -2&1&1 \\ 2&0&0 \end{pmatrix}, \quad B = \begin{pmatrix} 1&x&0 \\ 0&1&0 \\ -1&x&2 \end{pmatrix}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the correlation coefficient between the minimum and maximum order statistics, \(X_{(1)}\) and \(X_{(n)}\), from a uniform distribution on the interval \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n}}
Find a irreducible fraction with denominator not greater than 2005, that is closest to $\frac{9}{25}$ but is not $\frac{9}{25}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{716}{1989}}
Find the limit of the sequence of functions \( \frac{\epsilon}{\epsilon^2+x^2} \) as \( \epsilon \to 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \delta(x)}
Determine the number of permutations \( s \) in the symmetric group \( S_{10} \) such that \( bs = sa \), where \( a = (135)(2468) \) and \( b = (13)(24568) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the Jacobian matrix of the function \( f(X) = X^T x \), where \( X \) is an \( n \times m \) matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_m \otimes x^\top}
Find an integer $x$ such that $x^2 \equiv 7 \pmod{2579}$. If no such integer exists, state that 7 is not a quadratic residue modulo 2579.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
Determine the dual number that corresponds to the matrix \( \begin{pmatrix}0 & 0 \\1 & 0 \end{pmatrix} \) in the algebra of dual numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\epsilon}
Determine the limit of the sequence \( S(n) = \sum_{k=0}^{2n}{\frac{k}{k+n^2}} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
James's favorite basketball player got paid based on how many points he averaged each week. He gets $10,000 if he averages 30 or more points a game and $8,000 if he averages under 30 points a game. For his last week, in the first game he scored 30 points and in the second game he scored 28 points. In the third game he scored 32 points. In the fourth game he scored 34 points and in the fifth game he scored 26 points. How much does he get paid for the week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
Let $I$ be the ideal of $\mathbb{Z}[X]$ generated by $(X^2+2)(X+1)$ and $5$. Determine the number of elements $f \in \mathbb{Z}[X]/I$ such that $f^{18} = 1$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A rectangular prism has 4 green faces, 2 yellow faces, and 6 blue faces. What's the probability that when it is rolled, a blue face will be facing up?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
In a scalene triangle \(\bigtriangleup ABC\), the area is 45 square units. Points \(P_1\) and \(P_2\) lie on side \(AB\) such that \(AP_1 = P_1P_2 = BP_2\). Similarly, points \(Q_1\) and \(Q_2\) are on side \(AC\) with \(AQ_1 = Q_1Q_2 = CQ_2\). The intersection area of triangles \(BQ_1Q_2\) and \(CP_1P_2\) can be expressed as \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers. Determine the value of \(m+n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
In the acute-angled triangle \(ABC\), it is known that \(\sin (A+B)=\frac{3}{5}\), \(\sin (A-B)=\frac{1}{5}\), and \(AB=3\). Find the area of \(\triangle ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6 + 3\sqrt{6}}{2}}
Let $f$ be an integrable function on the interval $[0,1]$ such that $\lim_{x\to 0} f(x)=L$. Compute the limit \( \lim_{x\to 0^+} x\int_x^1 \frac{f(t)}{t^2}\, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L}
Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class.  Each year at the college one third of the freshman class drops out of school,  $40$  students in the sophomore class drop out of school, and one tenth of the junior class drops out of school.  Given that the college only admits new freshman students, and that it wants to begin each school year with  $3400$  students enrolled, how many students does it need to admit into the freshman class each year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5727}
Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 9 }
Evaluate the line integral \( \int_C \vec{F} \cdot d\vec{s} \) where \( \vec{F}(x,y,z) = \left\langle\frac{z^2}{x},\frac{z^2}{y},2z\ln(xy)\right\rangle \) and \( C \) is the path consisting of straight line segments from \( P = (1, 2, 1) \) to \( Q = (4, 1, 7) \) to \( R = (5, 11, 7) \), and then back to \( P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A sequence of numbers \(a_1, a_2, \cdots, a_n, \cdots\) is defined. Let \(S(a_i)\) be the sum of all the digits of \(a_i\). For example, \(S(22) = 2 + 2 = 4\). If \(a_1 = 2017\), \(a_2 = 22\), and \(a_n = S(a_{n-1}) + S(a_{n-2})\), what is the value of \(a_{2017}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Calculate the determinant of an $n \times n$ matrix $A$ where the diagonal elements alternate between -2 and -1, and all off-diagonal elements are -1. Specifically, find the determinant of matrix $A$ when $n=53$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There are 73 children standing in a circle. An evil Santa Claus walks around the circle in a clockwise direction and distributes candies. First, he gives one candy to the first child, then skips 1 child, gives one candy to the next child, skips 2 children, gives one candy to the next child, skips 3 children, and so on.

After distributing 2020 candies, he leaves. How many children did not receive any candies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
8. We will call two numbers almost equal to each other if they are equal to each other or differ from each other by no more than one. A rectangular grid with sides of natural numbers a and b is such that it is impossible to cut out a rectangle along the grid lines, the area of which is almost equal to half the area of the original rectangle. What is the smallest value that the number $|a-b|$ can take? (E. Molchanov, S. Berlov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The average age of a family of 5 members is 20 years. If the age of the youngest member is some years, and the average age of the family at the time of the birth of the youngest member was 12.5 years, what is the current age of the youngest member?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $T$ denote the 15-element set $\{10a+b : a,b \in \mathbb{Z}, 1 \le a < b \le 6\}$. Let $S$ be a subset of $T$ in which all six digits $1,2,\ldots ,6$ appear and in which no three elements together use all these six digits. Determine the largest possible size of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}