inclusionAI/Ring-lite-rl-data huggingface dataset:
Two linear functions \( f(x) \) and \( g(x) \) satisfy the properties that for all \( x \),
- \( f(x) + g(x) = 2 \)
- \( f(f(x)) = g(g(x)) \)
and \( f(0) = 2022 \). Compute \( f(1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find four distinct positive integers $a, b, c, d$ less than $15$ which are invertible modulo $15$. Calculate the remainder when $(abc + abd + acd + bcd)(abcd)^{-1}$ is divided by $15$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
What is the units (i.e., rightmost) digit of \[ \left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The increasing sequence of positive integers $a_1, a_2, a_3, \dots$ is defined by the rule
\[a_{n + 2} = a_{n + 1} + a_n\]
for all $n \ge 1.$ If $a_7 = 210$, then find $a_8.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{340}
Problem 3. In a certain company, there are 100 shareholders, and any 66 of them collectively own at least $50\%$ of the company's shares. What is the largest percentage of all shares that one shareholder can own? (The percentage of shares in the company owned by a shareholder can be non-integer.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
If there exist two different points on the graph of the function $y=f(x)$ such that the tangents to the function's graph at these two points are perpendicular to each other, then the function $y=f(x)$ is said to have property $T$. If the function $g(x)=ax-\frac{c}{2}+bsinxcosx+c{cos^2}x$ has property $T$, where $a$, $b$, and $c$ are real numbers and satisfy $b^{2}+c^{2}=1$, then the range of the real number $a+b+c$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\sqrt{2}, \sqrt{2}]}
Given a set of linearly independent vectors in $\mathbb{R}^3$: $\left\{a, b, c\right\}$. Determine the value of $\lambda$ such that the dimension of the subspace generated by the vectors $2a - 3b, (\lambda - 1)b - 2c, 3c - a, \lambda c - b$ is 2. In other words, find $\lambda$ for which there are only two linearly independent vectors in the span of the given vectors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that $f(x)$ is an even function defined on $\mathbb{R}$ and is differentiable, its derivative is $f′(x)$. When $x < 0$, we have $2f(x)+xf′(x) > x^{2}$. Find the solution set of the inequality $(x+2014)^{2}f(x+2014)-4f(-2) < 0$. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2016,-2012)}
Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the parabola y^2 = 4x whose directrix intersects the x-axis at point P, draw line l through point P with the slope k (k > 0), intersecting the parabola at points A and B. Let F be the focus of the parabola. If |FB| = 2|FA|, calculate the length of segment AB. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{17}}{2}}
For any set \( S \), let \( |S| \) represent the number of elements in set \( S \) and let \( n(S) \) represent the number of subsets of set \( S \). If \( A \), \( B \), and \( C \) are three finite sets such that:

(1) \( |A|=|B|=2016 \);
(2) \( n(A) + n(B) + n(C) = n(A \cup B \cup C) \),

then the maximum value of \( |A \cap B \cap C| \) is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
记 $Q$ 为 $1,2, \cdots, 100$ 的若干个排列组成的集合，且满足对于任意的 $1 \leqslant a 、 b \leqslant 100$ ， $a \neq b$ ，至多存在一个 $\sigma \in Q$ ，使得在 $\sigma$ 中 $a$的下一个数恰为 $b$ 。求集合 $Q$ 的元素个数的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
How many times are the hands of a clock at right angle in 2 days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88}
Two particles are placed at opposite vertices of the same face of a cube. If they are on the same edge at some time t, then they remain in the same position at time t+1. If not, one of the particles is chosen with equal probability and at any time t+1, it goes to one of the three neighboring vertices with probability 1/3. What is the expected value of the minimum time they are in the same edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Among the 100 natural numbers 1, 3, 5, 7, …, 199, select several numbers so that none of the selected numbers is an integer multiple of another. The maximum number of such numbers that can be selected is     .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Paco had 26 salty cookies and 17 sweet cookies. He ate some sweet cookies and 9 salty cookies. Paco had 17 salty cookies left. How many sweet cookies did Paco eat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
A trainee cook took two buckets of unpeeled potatoes and peeled everything in an hour. Meanwhile, $25\%$ of the potatoes went into peels. How much time did it take for him to collect exactly one bucket of peeled potatoes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40 	ext{ minutes}}
In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2+10$, $10+2$, and $3+2+2+5$ are three such ways.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
设f是定义在｛0,1，···，2005｝上的函数，取值于非负整数，对任意满足定义域的变量x，都有f(2x+1)=f(2x),f(3x+1)=f(3x),f(5x+1)=f(5x).问：这个函数最多能取到多少个函数值？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{536}
For constant $k>1$, 2 points $X,\ Y$ move on the part of the first quadrant of the line, which passes through $A(1,\ 0)$ and is perpendicular to the $x$ axis, satisfying $AY=kAX$. Let a circle with radius 1 centered on the origin $O(0,\ 0)$ intersect with line segments $OX,\ OY$ at $P,\ Q$ respectively. Express the maximum area of $\triangle{OPQ}$ in terms of $k$.

[i]2011 Tokyo Institute of Technology entrance exam, Problem 3[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} \left( \frac{k-1}{k+1} \right)}
In how many ways can we change the sign $*$ with $+$ or $-$, such that the following equation is true?
\[ 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 = 29 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80^\circ}
ABCD is a trapezoid with AB parallel to CD. M is the midpoint of AD, ∠MCB = 150°, BC = x, and MC = y. Find the area of trapezoid ABCD in terms of x and y.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{xy}{2}}
In the acute-angled triangle $ABC$, the angle $\angle BAC = 60^\circ$. A circle constructed with $BC$ as the diameter intersects side $AB$ at point $D$ and side $AC$ at point $E$. Determine the ratio of the area of quadrilateral $BDEC$ to the area of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
在正方体中随机取 3 条棱, 它们两两异面的概率为 $\qquad$.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{57}
Given the number $500 \ldots 005$ (with 80 zeros in between). We need to replace any two zeros with non-zero digits so that the resulting number is divisible by 165. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17280}
For a modified toothpick pattern, the first stage is constructed using 5 toothpicks. If each subsequent stage is formed by adding three more toothpicks than the previous stage, what is the total number of toothpicks needed for the $15^{th}$ stage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
已知正数 $a, b, c$, 满足 $(a+b-c)\left(\frac{1}{a}+\right.$ $\left.\frac{1}{b}-\frac{1}{c}\right)=3$, 求 $f=\left(a^{4}+b^{4}+c^{4}\right)\left(\frac{1}{a^{4}}+\frac{1}{b^{4}}+\frac{1}{c^{4}}\right)-240\sqrt{3}$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{417}
Using Cauchy's integral formula, evaluate the contour integral $$\oint_{\Gamma} \frac{\cos(z)+i\sin(z)}{(z^2+36)(z+2)}dz$$ where $\Gamma$ is the circle centered at the origin with radius $3$ traced counterclockwise. Note that the denominator cannot be factored further for partial fraction decomposition due to non-interior points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sin(2)+i\cos(2)}{20}\pi\)}
The quadrilateral $ABCD$ satisfies the following conditions: \( \angle ABC = \angle BCD = 150^{\circ} \). Additionally, \( AB = 18 \) and \( BC = 24 \). Equilateral triangles \( \triangle APB \), \( \triangle BQC \), and \( \triangle CRD \) are constructed outside the quadrilateral. If \( P(X) \) represents the perimeter of polygon \( X \), then the equation \( P(APQRD) = P(ABCD) + 32 \) holds. Determine the length of side \( CD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let \( P_{1} \) and \( P_{2} \) be any two different points on the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\), and let \( P \) be a variable point on the circle with diameter \( P_{1} P_{2} \). Find the maximum area of the circle with radius \( OP \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13 \pi}
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Draw a line $l$ passing through the right focus that intersects the ellipse at points $P$ and $Q$. What is the maximum area of the inscribed circle of triangle $\triangle F_{1} P Q$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9\pi}{16}}
A conveyor system produces on average 85% of first-class products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the frequency of first-class products from 0.85 in absolute magnitude does not exceed 0.01?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11475}
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? 

*(A. Golovanov)*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the minimal polynomial of $\sqrt{3} + \sqrt{5}$ over $\mathbb{Q}(\sqrt{5})$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - 2*sqrt(5)*x + 2}
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex on the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
In the first year, Grandma Molly created 4 statues and placed them on her front lawn. Each year after, she began creating sets of statues that were inspired by mathematical sequences. In the second year, she created two sets. The first set contained statues based on the Fibonacci sequence, placing 1, 1, and 2 statues on her lawn. The second set she placed was inspired by triangular numbers, adding 3 statues. However, that year a hail storm broke 2 statues from each set, which she threw away. In the third year, Molly added two more sets. One set contained 3 statues as the next number in the Fibonacci sequence and the other set contained 6 statues as the next triangular number. Unfortunately, another hail storm broke statues again. This time, 3 statues from the Fibonacci set and 4 from the triangular set were destroyed. For the fourth year, keeping to her tradition, she was planning to add two more sets again. The first set was supposed to have 5 statues, as the next Fibonacci number, and the other 10 statues, as the next triangular number, but a forecasted snow storm made her reconsider. So she decided to add a number of statues equal to the subtraction of the sum of the broken statues in previous years from the total number she was planning to add. After four years of such intricate work and adjustments for weather, how many turtle statues were left on her front lawn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{409}
The focus of the parabola \( y^{2}=2px \) \( (p>0) \) is point \( F \), and the directrix is line \( l \). Points \( A \) and \( B \) are on the parabola and satisfy \( \angle AFB=\frac{\pi}{3} \). Let \( M \) be the midpoint of segment \( AB \), and let \( N \) be the projection of \( M \) onto \( l \). Find the maximum value of \( \frac{|MN|}{|AB|} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{243}}
If the angle formed by two skew lines is 60°, then this pair of skew lines is called a "Golden Skew Line Pair." Among all the lines connecting the vertices of a cube, there are _______ pairs of "Golden Skew Line Pairs."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Let the integer $n \geq 3$, and let $P = \{1, 2, \ldots, n\}$. $A$ and $B$ are two non-empty subsets of $P$. The number of pairs of sets $(A, B)$ such that the maximum number in $A$ is less than the minimum number in $B$ is: ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-2) \cdot 2^{n-1} + 1}
Natural numbers $1,2, \ldots, 64$ are written in the cells of an $8 \times 8$ table so that for all $k=1,2,3, \ldots, 63$, the numbers $k$ and $k+1$ are in adjacent cells along one side. What is the maximum possible sum of the numbers on the main diagonal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{484}
If a semicircle is used to form the lateral surface of a cone, what is the maximum angle between any two generatrices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given a sphere resting on a flat surface and a 1.5 m tall post, the shadow of the sphere is 15 m and the shadow of the post is 3 m, determine the radius of the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
Peyton has 3 children - John, Samantha, and Heather. John gets 2 juice boxes and Samantha gets 1 juice box in their lunch, 5 days a week. Heather gets 3 juice boxes on Monday and Wednesday and 2 juice boxes on Tuesday and Thursday while on Friday, she gets just 1 juice box. The school year is 18 weeks long, but Heather has a one-week break, making her school year 17 weeks. Samantha attends after-school activities, which shorten her school year to 16 weeks. Additionally, during the last 2 weeks of the school year, there is a special holiday break where none of the children need packed lunches. How many juice boxes will Peyton need for the entire school year for all of her children, considering all these factors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{395}
How many such five-digit Shenma numbers exist, where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1512}
In $\triangle ABC$ with $AC>AB$, let $D$ be the foot of the altitude from $A$ to side $\overline{BC}$, and let $M$ be the midpoint of side $\overline{AC}$. Let lines $AB$ and $DM$ intersect at a point $E$. If $AC=8$, $AE=5$, and $EM=6$, find the square of the area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{252}
Given two circles $O\_1$: $(x-2)^2 + y^2 = 16$ and $O\_2$: $x^2 + y^2 = r^2 (0 < r < 2)$, a moving circle $M$ is tangent to both circles $O\_1$ and $O\_2$. The locus of the center of the moving circle $M$ forms two ellipses, with eccentricities $e\_1$ and $e\_2 (e\_1 > e\_2)$. Find the minimum value of $e\_1 + 2e\_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3+2\sqrt{2}}{4}}
Four pairs of siblings, in each pair one is a boy and the other is a girl. We divide them into three groups so that each group has at least two people, but siblings do not end up in the same group. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
In the plane Cartesian coordinate system $xOy$, there is a circle $C:(x+2)^{2}+(y-m)^{2}=3$. If circle $C$ has a chord $AB$ with a midpoint $G$, and the length of $AB$ is twice the distance from $G$ to the center $O$ of circle $C$, i.e., $|AB|=2|GO|$, then the range of the real number $m$ is ______________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \sqrt{2} \leq m \leq \sqrt{2}}
In a hockey league, there are 15 teams. The number of times each team faces the other teams varies based on their current ranking. The top 5 ranked teams face off against the other top 5 teams 12 times each, and the remaining 10 teams 8 times each. Rank 6 to 10 teams play against each other 10 times each and against the remaining 5 teams 6 times each. Finally, the bottom 5 ranked teams face off against each other 8 times each. How many games are played in the season?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{850}
Find the angle $\angle XYZ$ where $X, Y, Z$ are points in the complex plane such that $$\frac{X-Y}{X-Z}+\frac{\overline{X-Y}}{\overline{X-Z}}=2.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
现有边长为3、4、5的三角形两个，边长分别为4、5、41的三角形四个，边长分别为$\frac { 5 } { 6 }\sqrt{2} $、4、5的三角形六个．用上述三角形为面，可以拼成__个四面体.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
We define $N$ as the set of natural numbers $n < 10^6$ with the following property: There exists an integer exponent $k$ with $1 \leq k \leq 43$, such that $2012 \mid n^k - 1$. Find $|N|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1989}
if 4^x + y = 34, 2^x - y = 20, for integers of x and y, y ^ 2 = ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{196}
For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \leq 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{632}
A fan can create an airflow of 10 liters per second. If the fan works for a certain amount of time each day, the amount of airflow it will create in one week is 42000 liters. How long does the fan work each day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
已知实数 x 和 y 满足 $5 x^{2}+6 x y+5 y^{2}-8 x-8 y=-3$. 记 $x+y-x^{2}-y^{2}$ 的最大值为 I ，则 $[100 I]=$？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
In 1854, geologist W.K. Loftus excavated two clay tablets in Senkereh (a location in ancient Babylon) on which one of the tablets recorded:  
$9^2 = 81 = 60 + 21 = 1•21$  
$10^2 = 100 = 60 + 40 = 1•40$  
$11^2 = 121 = 2 \times 60 + 1 = 2•1$  
$12^2 = 144 = 2 \times 60 + 24 = 2•24$  
...  
Following this pattern, $58^2 = \_\_\_\_\_\_$．(Write in the form of “a•b”)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56•4}
Given a geometric sequence with an even number of terms, where the sum of the even-numbered terms is twice the sum of the odd-numbered terms, the first term is 1, and the sum of the two middle terms is 24, find the total number of terms in the sequence. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
已知三棱锥$P-ABC$的四个顶点均在某球面上，$PC$为该球的直径，$\vartriangle ABC$是边长为4的等边三角形，三棱锥$P-ABC$的体积为$\frac{16}{3}$，则此三棱锥的外接球的表面积为______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{80\pi }{3}$}
A round robin tournament is held with $2016$ participants. Each player plays each other player once, and no games result in ties. We say a pair of players $A$ and $B$ is a dominant pair if all other players either defeat both $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number of dominant pairs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Evaluate the contour integral $$\oint_{|z|=1}\frac{dz}{1+2z+2/z}$$ using Cauchy's residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let triangle $ABC$ have $\angle BAC = 45^{\circ}$ and circumcircle $\Gamma$. Let $M$ be the intersection of the angle bisector of $\angle BAC$ with $\Gamma$. Let $\Omega$ be the circle tangent to segments $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Gamma$ at point $T$. Given that $\angle TMA = 45^{\circ}$ and that $TM = \sqrt{100 - 50 \sqrt{2}}$, the length of $BC$ can be written as $a \sqrt{b}$, where $b$ is not divisible by the square of any prime. Find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that function $f(x)$ is an odd function defined on $\mathbb{R}$ with its derivative ${f}'(x)$, when $x < 0$, $2f(x) + x{f}'(x) < 0$ always holds. Determine the relationship among $f(1)$, $2016f(\sqrt{2016})$, and $2017f(\sqrt{2017})$, expressing your answer using inequality signs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(1) < 2016f(\sqrt{2016}) < 2017f(\sqrt{2017})}
We are given some three-element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ nice if it doesn't include any of the given subsets. If, no matter how the three-element subsets are selected initially, we can add one more element to every 29-element nice subset while keeping it nice, find the minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{436}