inclusionAI/Ring-lite-rl-data huggingface dataset:
A cube with edge length 1 can freely flip inside a regular tetrahedron with edge length $a$. Find the minimum value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
The arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \) has the following properties:

- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224.

What is the sum of the whole sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{608}
A staircase-shaped brick consists of 12 unit cubes, has a width of 2, and has 3 steps. Find all positive integers \( n \) such that several of these bricks can form a cube with side length \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{394}
Let \( p(x) = 2x^3 - 3x^2 + 1 \). How many squares of integers are there among the numbers \( p(1), p(2), \ldots, p(2016) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $ ABCDE $ be a convex pentagon with $ AB || CE, BC || AD, AC || DE, \angle ABC=120^\circ, AB=3, BC=5, $ and $DE = 15. $ Given that the ratio between the area of triangle $ ABC $ and the area of triangle $ EBD $ is $ m/n, $ where $ m $ and $ n $ are relatively prime positive integers, find $ m+n. $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{484}
The perimeter of triangle \(ABC\) is 1. Circle \(\omega\) is tangent to side \(BC\), the extension of side \(AB\) at point \(P\), and the extension of side \(AC\) at point \(Q\). A line passing through the midpoints of \(AB\) and \(AC\) intersects the circumcircle of triangle \(APQ\) at points \(X\) and \(Y\). Find the length of segment \(XY\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
A certain rectangle had its dimensions expressed in whole numbers of decimeters. Then, it changed its dimensions three times. First, one of its dimensions was doubled and the other was adjusted so that the area remained the same. Then, one dimension was increased by $1 \mathrm{dm}$ and the other decreased by $4 \mathrm{dm}$, keeping the area the same. Finally, its shorter dimension was reduced by $1 \mathrm{dm}$, while the longer dimension remained unchanged.

Determine the ratio of the lengths of the sides of the final rectangle.

(E. Novotná)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:1}
Given a triangle \( ABC \). A circle with a radius \( R \) is tangent to lines \( AB \) and \( BC \) at points \( A \) and \( C \), and intersects the median \( BD \) at point \( L \), such that \( BL = \frac{5}{9} BD \).

Find the area of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27R^2}{100}}
A ball invites 2018 couples, each assigned to areas numbered $1, 2, \cdots, 2018$. The organizer specifies that at the $i$-th minute of the ball, the couple in area $s_i$ (if any) moves to area $r_i$, and the couple originally in area $r_i$ (if any) exits the ball. The relationship is given by:
$$
s_i \equiv i \pmod{2018}, \quad r_i \equiv 2i \pmod{2018},
$$
with $1 \leq s_i, r_i \leq 2018$. According to this rule, how many couples will still be dancing after $2018^2$ minutes? (Note: If $s_i = r_i$, the couple in area $s_i$ remains in the same area and does not exit the ball).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
设 $a_{1}, a_{2}, \cdots, a_{18}$ 都是非零实数。试求下式的最小可能值:  
$$  
S=\sum_{i=1}^{18} \sum_{j=1}^{18} \frac{\left(a_{i}+a_{j}\right)^{2}\left(1+a_{i}^{2} a_{j}^{2}\right)}{a_{i}^{2} a_{j}^{2}}  
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1296}
How many ternary relations are possible from a set with 4 elements to a set with 3 elements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{36}}
A function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x) + f'(x) < 1$ and $f(0) = 3$. Find the solution set for the inequality $f(x) > \frac{2}{e^{x}} + 1$, where $e$ is the base of the natural logarithm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0)}
Find the last two digits of \(\left[(\sqrt{29}+\sqrt{21})^{1984}\right]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{71}
A man can row his boat with the stream at some speed and against the stream at 12 km/h. The man's rate is 2 km/h. What is the man's speed when he rows with the stream?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A museum has 7 adjacent exhibition rooms. A guard starts walking to the first room, then to the second room... after reaching the seventh room, he turns back and goes to the sixth room, then to the fifth room... Every time he enters a room, he switches the light in that room on or off. If all 7 rooms are illuminated at the beginning, how many rooms will still be illuminated after he has walked through 100 rooms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
若存在正整数 $k<n\left(n \geqslant 3, n \in \mathrm{~N}^{*}\right)$, 使得 $\mathrm{C}_{n}^{k-1}, \mathrm{C}_{n}^{k}, \mathrm{C}_{n}^{k+1}$ 构成等差数列，其中组合数 $\mathrm{C}_{n}^{k}=$ $\frac{n!}{k!(n-k)!}(n!=1 \cdot 2 \cdot 3 \cdot \cdots \cdot n$, 约定 $0!=1)$ ，则称 $n$ 为理想数。不超过 2020 的理想数的个数是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
A right triangle \(ABC\) \((\angle A = 90^\circ)\) has two squares \(BEFC\) and \(AMNC\) positioned such that points \(E\) and \(A\) lie on opposite sides of line \(BC\), and points \(M\) and \(B\) lie on opposite sides of line \(AC\). Find the distance between the centers of the squares if \(AB = a\), \(AC = b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{a^2}{2} + ab + b^2}}
What is the annual interest rate when the continuous compound interest on a principal of Rs. 600 amounts to Rs. 160 in 4 years, given the amount is compounded continuously?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.91\%}
In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.6 times the probability of earning a B. The probability of earning a D is 0.3 times the probability of earning a B. Assuming that all grades are A, B, C, or D, how many B's will there be in a mathematics class of 50 students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let the sequence $\{ a_n \}_{n = -2}^\infty$ satisfy the following initial conditions and recursive formula:
- $a_{-1} = a_{-2} = 0$
- $a_0 = 1$

For all non-negative integers $n$, the sequence satisfies:
$$n^2 = \sum_{k = 0}^n a_{n - k}a_{k - 1} + \sum_{k = 0}^n a_{n - k}a_{k - 2}$$

Given that $a_{2018}$ is a rational number, find the maximum integer $m$ such that $2^m$ divides the denominator of the reduced form of $a_{2018}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4029}
In the 100-digit number 12345678901234...7890, all digits in odd positions were crossed out; in the resulting 50-digit number, all digits in odd positions were crossed out again, and so on. The crossing out continued as long as there were digits to cross out. Which digit was crossed out last?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns $16$ dollars per week for his paper route. He will use all of his birthday money and all of the money he earns from his paper route. In how many weeks will he be able to buy the mountain bike?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$25$}
Given the function $y=\cos(2x+\frac{\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\frac{\pi}{6}$ units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11\pi}{24}}
In a coastal village, 7 people go fishing every day, 8 people go fishing every other day, 3 people go fishing every three days, and the rest do not fish at all. Yesterday, 12 people went fishing, today 10 people are fishing. How many people will go fishing tomorrow?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the greatest possible value of \(\sum_{i=1}^{10} \cos(3x_i)\) for real numbers $x_1,x_2,\dots,x_{10}$ satisfying \(\sum_{i=1}^{10} \cos(x_i) = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{480}{49}}
Find the sum of the two smallest possible values of $x^\circ$ (in degrees) that satisfy the following equation if $x$ is greater than $2017^\circ$:  
\[ 
\cos^5(9x) + \cos^5(x) = 32\cos^5(5x)\cos^5(4x) + 5\cos^2(9x)\cos^2(x)(\cos(9x) + \cos(x)). 
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4064}
What is the smallest number of points that need to be marked inside a convex \( n \)-gon so that every triangle with vertices at the vertices of the \( n \)-gon contains at least one marked point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-2}
There are $2n$ cards, each numbered from 1 to $n$ (each number appearing exactly on two cards). The cards are placed face down on the table. A set of $n$ cards is called good if each number appears exactly once in the set. Baron Munchausen claims that he can point to 80 sets of $n$ cards, at least one of which is definitely good. For what maximum value of $n$ can the baron's words be true?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Peyton puts 30 L of oil and 15 L of vinegar into a large empty can. He then adds 15 L of oil to create a new mixture. What percentage of the new mixture is oil?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75\%}
Dima has written the numbers $\frac{1}{80!}, \frac{1}{81!}, \ldots, \frac{1}{99!}$ on 20 infinite pieces of paper as decimal fractions. For example, the following is written on the last piece: \( \frac{1}{99!} = 0.\underbrace{00\ldots00}_{155\text{ zeros}}10715\ldots \). Sasha wants to cut a fragment of $N$ consecutive digits from one of these pieces without the comma. What is the maximal $N$ for which Sasha can do this so that Dima cannot guess from which piece Sasha has cut his fragment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
John has saved up $5555_8$ dollars for a new laptop. A laptop he desires costs $1500_{10}$ dollars. How many dollars will he have left after purchasing the laptop if all his money is in base eight and the laptop price is in base ten?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1425}
Consider the set $\Omega$ of all bijections $\phi: \mathbb{N} \cup \{0\} \rightarrow \mathbb{Q}$. We can visualize each $\phi \in \Omega$ as a "staircase" in the real plane, where for $x \in \mathbb{R}^+$, we have $x \mapsto \phi(\lfloor x \rfloor)$. The largest step in this staircase is given by $\sup_{n \geq 0} |\phi(n+1)-\phi(n)|$. What is the smallest possible value of this largest step among all bijections in $\Omega$? That is, what is \[\inf_{\phi \in \Omega} \sup_{n \geq 0} |\phi(n+1)-\phi(n)|\ \ ?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In the tetrahedron \( OABC \), \(\angle AOB = 45^\circ\), \(\angle AOC = \angle BOC = 30^\circ\). Find the cosine value of the dihedral angle \(\alpha\) between the planes \( AOC \) and \( BOC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2} - 3}
A person spends 1/3 of the money he has on clothes, 1/5 of the remaining on food, and 1/4 of the remaining on travel. He had Rs 999.9999999999998 with him in the beginning. How much money is he left with after spending on clothes, food, and travel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Let $x_1<x_2<x_3$ be the three real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If it takes a machine some time to produce one item, the machine will produce 150 items in 2 hours. How long does it take for the machine to produce one item?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8}
In a town with a population of 10,000, the number of children under the age of 18 is 3,500, the number of adults from the age of 18 to 64 is 5,000, and the number of adults from the age of 65 and above is 1,500. The population of children increases annually at the rate of 10%, the population of working-age adults increases at the rate of 15%, and the population of adults above 65 increases at the rate of 5%. What will be the population of the town in two years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12,501}
In $\triangle ABC$, $\overrightarrow{AB} \cdot \overrightarrow{AC} = 2\sqrt{3}$ and $\angle BAC = 30^\circ$. Let $M$ be a point inside $\triangle ABC$. Define $f(M) = (x, y, z)$, where $x$, $y$, and $z$ represent the areas of $\triangle MBC$, $\triangle MAC$, and $\triangle MAB$, respectively. If $f(M) = \left(x, y, \frac{1}{2}\right)$, find the minimum value of $\frac{1}{x} + \frac{4}{y}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval [0, 1]. This process is repeated two independent times, obtaining two real numbers x and y. Calculate the probability that |x - y| > 1/2. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
A cuckoo clock chimes "cuckoo" as many times as the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock at 9:05 and started turning the minute hand until the clock advanced by 7 hours. How many times did the clock chime "cuckoo" during this period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Let  $k$  be a positive integer. Marco and Vera play a game on an infinite grid of square cells. At the beginning, only one cell is black and the rest are white. 
	A turn in this game consists of the following. Marco moves first, and for every move he must choose a cell which is black and which has more than two white neighbors. (Two cells are neighbors if they share an edge, so every cell has exactly four neighbors.) His move consists of making the chosen black cell white and turning all of its neighbors black if they are not already. Vera then performs the following action exactly  $k$  times: she chooses two cells that are neighbors to each other and swaps their colors (she is allowed to swap the colors of two white or of two black cells, though doing so has no effect). This, in totality, is a single turn. If Vera leaves the board so that Marco cannot choose a cell that is black and has more than two white neighbors, then Vera wins; otherwise, another turn occurs.
	Let  $m$  be the minimal  $k$  value such that Vera can guarantee that she wins no matter what Marco does. For  $k=m$ , let  $t$  be the smallest positive integer such that Vera can guarantee, no matter what Marco does, that she wins after at most  $t$  turns. Compute  $100m + t$ .

*Proposed by Ashwin Sah*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{203}
For which \( k \) do there exist \( k \) pairwise distinct primes \( p_{1}, p_{2}, \ldots, p_{k} \) such that

\[ p_{1}^{2} + p_{2}^{2} + \cdots + p_{k}^{2} = 2010? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
There is a unique angle $\theta$ between $0^{\circ}$ and $90^{\circ}$ such that for nonnegative integers $n$, the value of $\tan{\left(2^{n}\theta\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\theta$ is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{547}
In a tetrahedron $ABCD$ where all edges have a length of $1$, $E$ is the midpoint of $AD$, and $P$ is a moving point on the line $CE$. Find the minimum value of $|BP|+|DP|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{1 + \frac{\sqrt{6}}{3}}}
## 2. Megacube

From identical small cubes, one large cube is assembled. The length of the edge of the large cube is a hundred times larger than the length of the edge of each small cube. The large cube is placed on a table, and all its sides, except the one that touches the table, are painted red.

How many small cubes have exactly two red sides?

Result:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{788}
Rebecca runs a hair salon. She charges $30 for haircuts, $40 for perms, $60 for dye jobs, and $80 for hair extensions. In addition, she needs to buy supplies for each service, which costs $5 for every haircut, $10 for a box of hair dye for every dye job, and $25 for hair extension materials per client. She offers a 10% discount to students and a 15% discount to seniors. 

Today, she has five haircuts, three perms, two dye jobs, and one hair extension appointment scheduled. Among these clients, two haircuts are for students, one perm is for a senior, and one dye job is for a first-time visitor who receives a $5 discount. She also makes $75 in total tips. After paying her daily expenses of $45, how much money will she have in dollars at the end of the day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$413}
In triangle $ABC$, point $D$ is on side $AB$, $CD \perp BC$, $AC = 5\sqrt{3}$, $CD = 5$, $BD = 2AD$. Find the length of $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The function $f(x) = x^2 + 2(a-1) + 3$ is increasing on the interval $[1, +\infty)$. Determine the range of real numbers for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, +\infty)}
Find the smallest positive integer $n$ such that $\underbrace{2^{2^{\cdots^{2}}}}_{n} > 3^{3^{3^3}}$. 

(The notation $\underbrace{2^{2^{\cdots^{2}}}}_{n}$ is used to denote a power tower with $n$ $2$’s. For example, $\underbrace{2^{2^{\cdots^{2}}}}_{4}$ would equal $2^{2^{2^2}}$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In $\triangle ABC$, let $\vec{m} = \vec{CB} - 3\vec{AC}$ and $\vec{n} = \vec{CB}$. If $\vec{m} \perp \vec{n}$, find the maximum value of $\sin A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{5}}
6. In space, there are 2017 points. The midpoints of the line segments connecting each pair of points are colored red. The minimum number of red points is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4031}
9.1. At a round table, 2015 people are sitting, each of them is either a knight or a liar. Knights always tell the truth, and liars always lie. Each person was given a card with a number on it; all the numbers on the cards are different. After looking at their neighbors' cards, each person said: "My number is greater than the number of each of my two neighbors." After this, $k$ of the people said: "My number is less than the number of each of my two neighbors." For what largest $k$ could this have happened?

(O. Podlipsky)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2013}
Five students $A, B, C, D$, and $E$ have participated in a competition. Before the competition, two people, $X$ and $Y$, guessed the rankings. $X$ thought the ranking would be $A, B, C, D, E$. $Y$ thought the ranking would be $D, A, E, C, B$. At the end, it was revealed that $X$ didn't guess any rankings of the participants correctly and didn't guess any orderings of pairs of consecutive participants correctly. $Y$ guessed the correct rankings of two participants and the correct ordering of two pairs of consecutive participants. Determine the rankings of the competition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E,D,A,C,B}
Five lanterns of the same size but different colors (red, orange, yellow, green, purple) are connected from top to bottom and hung together end to end. It is required that red and purple are adjacent, orange and green are not adjacent, and orange cannot be at the top. The number of different ways to hang the lanterns is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Problem 10. Points $M, N$, and $K$ are located on the lateral edges $A A_{1}, B B_{1}$, and $C C_{1}$ of the triangular prism $A B C A_{1} B_{1} C_{1}$ such that $A M: A A_{1}=3: 7, B N: B B_{1}=2: 5, C K: C C_{1}=4: 9$. Point $P$ belongs to the prism. Find the maximum possible value of the volume of the pyramid $M N K P$, if the volume of the prism is 40.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a park there are 10,000 trees planted in a square grid pattern (100 rows and 100 trees per row). What is the maximum number of trees that can be cut down such that if you stand on any stump, no other stump is visible? (You can assume the trees are sufficiently thin.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5000}
Define the sequences \( a_n \), \( b_n \), \( c_n \) as follows: \( a_0 = k \), \( b_0 = 4 \), \( c_0 = 1 \). If \( a_n \) is even, then \( a_{n+1} = \frac{a_n}{2} \), \( b_{n+1} = 2b_n \), \( c_{n+1} = c_n \). If \( a_n \) is odd, then \( a_{n+1} = a_n - \frac{b_n}{2} - c_n \), \( b_{n+1} = b_n \), \( c_{n+1} = b_n + c_n \). Find the number of positive integers \( k < 1995 \) such that some \( a_n = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
$\triangle GHI$ is inscribed inside $\triangle XYZ$ such that $G, H, I$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle GZC, \triangle HYD, \triangle IXF$ have centers $O_1, O_2, O_3$, respectively. Also, $XY = 26, YZ = 28, XZ = 27$, and $\stackrel{\frown}{YI} = \stackrel{\frown}{GZ},\ \stackrel{\frown}{XI} = \stackrel{\frown}{HZ},\ \stackrel{\frown}{XH} = \stackrel{\frown}{GY}$. The length of $GY$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}