inclusionAI/Ring-lite-rl-data huggingface dataset:
From the sequence of natural numbers $1, 2, 3, 4, \ldots$, erase every multiple of 3 and 4, but keep every multiple of 5 (for example, 15 and 20 are not erased). After removing the specified numbers, write the remaining numbers in a sequence: $A_{1}=1, A_{2}=2, A_{3}=5, A_{4}=7, \ldots$. Find the value of $A_{1988}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3314}
Each of the nine letters in "STATISTICS" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "TEST"? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
5. Let two regular tetrahedra $P-A B C$ and $Q-A B C$ be inscribed in the same sphere. If the dihedral angle between a lateral face and the base of the regular tetrahedron $P-A B C$ is $45^{\circ}$, then the tangent value of the dihedral angle between a lateral face and the base of the regular tetrahedron $Q-A B C$ is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Stuart has drawn a pair of concentric circles. He draws chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the smaller circle. If $m\angle ABC=60^\circ$, then how many segments will he draw before returning to his starting point at $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
GM Bisain's IQ is so high that he can move around in $10$-dimensional space. He starts at the origin and moves in a straight line away from the origin, stopping after $3$ units. How many lattice points can he land on? A lattice point is one with all integer coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88660}
Given that the even function $y=f\left(x+1\right)$ is monotonically decreasing on the interval $\left[0,+\infty \right)$, the interval of monotonically increasing for the function $y=f\left(x-1\right)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 2]}
Michael jogs daily around a track consisting of long straight lengths connected by a full circle at each end. The track has a width of 4 meters, and the length of one straight portion is 100 meters. The inner radius of each circle is 20 meters. It takes Michael 48 seconds longer to jog around the outer edge of the track than around the inner edge. Calculate Michael's speed in meters per second.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
There are $360$ permutations of the letters in $MMATHS$. When ordered alphabetically, starting from $AHMMST$, $MMATHS$ is in the $n$th permutation. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{173}
Cyclic quadrilateral \(ABCD\) has side lengths \(AB = 1\), \(BC = 2\), \(CD = 3\), and \(DA = 4\). Points \(P\) and \(Q\) are the midpoints of \(\overline{BC}\) and \(\overline{DA}\). Compute \(PQ^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{116}{35}}
2000 students take a paper of five multiple-choice questions each with four possible answers. Each student chooses one answer per question. What is the smallest \( n \) such that the students could have answered the questions in such a way that given any \( n \) students we can find four every two of whom answered at least two questions differently?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Three. (50 points) Let $M=\{1,2, \cdots, 65\}, A \subseteq M$ be a subset. If $|A|=33$, and there exist $x, y \in A, x<y$, $x \mid y$, then $A$ is called a "good set". Find the largest $a \in M$ such that any 33-element subset containing $a$ is a good set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Find the largest integer less than 2012 all of whose divisors have at most two 1's in their binary representations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1536}
设$S = \{ A = ( a _ { 1 } , \cdots , a _ { 8 } ) \vert a _ { i } = 0$或中的两个元素$A = ( a _ { 1 } , \cdots , a _ { 8 } )$和$B = ( b _ { 1 } , \cdots , b _ { 8 } )$，记$d ( A , B ) = \sum _ { i = 1 } ^ { 8 } \vert a _ { i } - b _ { i } \vert$并称其为A和B之间的距离．问S中最多能取出多少元素，它们之中任何两个的距离≥5？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A piece of music consisiting of three movements lasts for 60 minutes. No single movement is longer than the other two movements combined. The duration difference between any two movements is at least 3 minutes. What are the possible range limits for the duration of the shortest movement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \le a \le 17}
Suppose that $a$, $b$, $c$, and $d$ are real numbers such that $a+b+c+d=8$. Compute the minimum possible value of 
\[20(a^2+b^2+c^2+d^2)-\sum_{\text{sym}}a^3b,\]
where the sum is over all $12$ symmetric terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
The CMU Kiltie Band is attempting to crash a helicopter via grappling hook.  The helicopter starts parallel (angle  $0$  degrees) to the ground.  Each time the band members pull the hook, they tilt the helicopter forward by either  $x$  or  $x+1$  degrees, with equal probability, if the helicopter is currently at an angle  $x$  degrees with the ground.  Causing the helicopter to tilt to  $90$  degrees or beyond will crash the helicopter.  Find the expected number of times the band must pull the hook in order to crash the helicopter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{269}{32}}
27.4. (GDR, 78). A circle is circumscribed around a regular $2 n$-gon. A triplet of its distinct vertices is called one-sided if there exists a semicircle on which these vertices lie (the ends of the semicircle belong to it). What is the probability that a randomly chosen triplet of vertices will be one-sided?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3n}{2(2n-1)}}
Find the least real number  $k$  with the following property: if the real numbers  $x$ ,  $y$ , and  $z$  are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{16}{9} }
Find the smallest root of the equation

$$
\sin (\pi x)+\operatorname{tg} \chi=x+x^{3}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given an arithmetic sequence {a_n}, the sequence {b_n} is defined by: b_1 = a_1, b_2 = a_2 + a_3, b_3 = a_4 + a_5 + a_6, etc. If the limit of b_n/n^3 as n approaches infinity is 2, find the common difference d of the sequence {a_n}. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the sum of   $$ \frac{\sigma(n) \cdot d(n)}{ \phi (n)} $$   over all positive  $n$  that divide  $ 60$ .

Note: The function  $d(i)$  outputs the number of divisors of  $i$ ,   $\sigma (i)$  outputs the sum of the factors of  $i$ , and  $\phi  (i)$  outputs the number of positive integers less than or equal to  $i$  that are relatively prime to  $i$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{350}
For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes, respectively. A positive integer $n$ is called fair if $s(n) = c(n) > 1$. Find the number of fair integers less than $100$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A bus is traveling uphill along a road with varying slope gradients. The bus accelerates at a rate of 2 m/s^2 on a 5% slope, 3 m/s^2 on a 10% slope, and 4 m/s^2 on a 15% slope. If the bus starts from a stop (initial velocity of 0 m/s) and travels 50 meters on the 5% slope, followed by 80 meters on the 10% slope, and finally 70 meters on the 15% slope, what is the final velocity of the bus at the end of these three sections?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35.21}
Baker Ram is storing cake boxes in cartons that measure 25 inches by 42 inches by some inches. If the measurement of each cake box is 8 inches by 7 inches by 5 inches, and the maximum number of cake boxes that can be placed in each carton is 210, what is the height of the carton?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
A hexagonal prism has a height of \(165 \text{ cm}\). Its two hexagonal faces are regular hexagons with sides of length \(30 \text{ cm}\). Its other six faces are rectangles. A fly and an ant start at point \(X\) on the bottom face and travel to point \(Y\) on the top face. The fly flies directly along the shortest route through the prism. The ant crawls around the outside of the prism along a path of constant slope so that it winds around the prism exactly \(n+\frac{1}{2}\) times, for some positive integer \(n\). The distance crawled by the ant is more than 20 times the distance flown by the fly. What is the smallest possible value of \(n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Let $P(x)$ be a polynomial with positive integer coefficients and $P(0) > 1$. The product of all the coefficients is $47$ and the remainder when $P(x)$ is divided by $(x - 3)$ is $9887$. What is the degree of $P(x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In the complex plane, the vertices of the right-angled triangle $\triangle ABC$ correspond to the complex numbers $z+1$, $2z+1$, and $(z+1)^2$, with $A$ being the right-angled vertex, and $|z|=2$. Define the set $M=\left\{m \mid z^{m} \in \mathbf{R}, m \in \mathbf{N}_{+}\right\}$ and set $P=\left\{x \mid x=\frac{1}{2^{m}}, m \in M\right\}$. Find the sum of all elements in set $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/7}
In measuring the sides of a rectangle, one side is taken in excess, and the other 5% in deficit. The error percent in the area calculated from these measurements is 0.7%. What is the percentage in excess for one side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\%}
The base of a right prism is a right triangle with a hypotenuse of length \( c \) and an acute angle of \( 30^\circ \). A plane is drawn through the hypotenuse of the lower base and the vertex of the right angle of the upper base, forming an angle of \( 45^\circ \) with the base's plane. Determine the volume of the triangular pyramid cut off from the prism by this plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{c^3}{32}}
Compute
\[
\log_2 \left( \prod_{a=1}^{2015} \prod_{b=1}^{2015} (1+e^{2\pi i a b/2015}) \right)
\]
Here $i$ is the imaginary unit (that is, $i^2=-1$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13725}
Given the expressions x = 0.82^0.5, y = sin(1), and z = log_3(sqrt(7)), determine the correct order of magnitudes of x, y, and z. Express your answer in the format x < y < z or x > y > z, etc.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y < z < x}
In the game of *Winners Make Zeros*, a pair of positive integers  $(m,n)$  is written on a sheet of paper.  Then the game begins, as the players make the following legal moves:


- If  $m\geq n$ , the player choose a positive integer  $c$  such that  $m-cn\geq 0$ , and replaces  $(m,n)$  with  $(m-cn,n)$ .
- If  $m<n$ , the player choose a positive integer  $c$  such that  $n-cm\geq 0$ , and replaces  $(m,n)$  with  $(m,n-cm)$ .

When  $m$  or  $n$  becomes  $0$ , the game ends, and the last player to have moved is declared the winner.  If  $m$  and  $n$  are originally  $2007777$  and  $2007$ , find the largest choice the first player can make for  $c$  (on his first move) such that the first player has a winning strategy after that first move.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $\gcd(a, b, c, d) = 77$ and $\operatorname{lcm}(a, b, c, d) = n$. What is the smallest possible value for $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27,720}
On the radius \( AO \) of a circle with center \( O \), a point \( M \) is selected. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 12 and \( \cos \alpha = \frac{1}{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. Find the number of distinct lines that pass through at least two members of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
In a party with $1982$ people, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ 1979 \]}
Find all positive integers $n$ such that $n$ has exactly 6 positive divisors $1 < d_{1} < d_{2} < d_{3} < d_{4} < n$ and $1 + n = 5(d_{1} + d_{2} + d_{3} + d_{4})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1519}
Mary is baking a cake. The recipe calls for 11 cups of sugar, 6 cups of flour, and 9 cups of salt. She already put in some cups of flour. She needs to add 2 more cups of sugar than cups of salt now. How many cups of flour has she already put in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
The six-digit number $20210A$ is prime for only one digit $A$. What is the value of $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A TV show costs $100,000 per episode for the first season and twice that much for every other season. The first season had a certain number of episodes, and every season after that had 50% more episodes except the last season, which had 24 episodes. The show had 5 seasons. It cost $16,800,000 to produce all the episodes. How many episodes were in the first season?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Aaron and his brother Carson each saved up $150 for a weekend outing. They first go to a fancy restaurant where the bill amounts to 3/4 of their total money inclusive of a 15% service charge. Later, they decide to treat themselves with some expensive gourmet ice cream. Each scoop of ice cream costs $4. If they both get the same number of ice cream scoops and they are both left with $4 at the end of the day, how many scoops of ice cream did they each buy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
8. Consider each permutation of the nine digits $1,2, \cdots, 9$ as a nine-digit number. Among them, the number of those that are divisible by 11 is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31680}
Find the smallest positive integer $n$, such that $3^k + n^k + (3n)^k + 2014^k$ is a perfect square for all natural numbers $k$, but not a perfect cube for all natural numbers $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the triangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is $m$ times the area of the square. The ratio of the area of the other small right triangle to the area of the square is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{4m}$}
It is given that the number $4^{11} + 1$ is divisible by some prime greater than $1000$. Determine this prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2113}
A rigid board with a mass \( m \) and a length \( l = 20 \) meters partially lies on the edge of a horizontal surface, overhanging it by three quarters of its length. To prevent the board from falling, a stone with a mass of \( 2m \) was placed at its very edge. How far from the stone can a person with a mass of \( m / 2 \) walk on the board? Neglect the sizes of the stone and the person compared to the size of the board.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15 \, \text{m}}
A $12 \times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non-attacking if they are not in the same column or same row. Find the least number $N$ such that if $N$ rooks are placed on the unit squares, one rook per square, we can always find $7$ rooks such that no two are attacking each other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
给定正整数 $n$, 求最大的正整数 $k$, 使得如下命题成立: 对每个 $1 \leq i \leq 2 n$,设 $A_{i}$ 是若干个相邻的整数构成的集合（即每个 $A_{i}$ 都形如 $\{a+1, a+2, \cdots, a+r\}$ ，其中 $a$ 是整数, $r$ 是正整数），如果对任何 $1 \leq i \leq n, n+1 \leq j \leq 2 n$ 都有 $A_{i} \cap A_{j} \neq \emptyset$ ，则存在整数 $x$ ，使得集合 $\left\{1 \leq i \leq 2 n \mid x \in A_{i}\right\}$ 包含至少 $k$ 个不同的元素. 请给出当n=5时的k值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Suppose that global mean temperature is normally distributed $N(\mu, \sigma)$ and $N$ samples have been drawn from this distribution: $T_1, \cdots, T_N$. Let $T_{maxN}$ be the maximum of those first $N$ samples. What is the probability that the next three samples drawn are successively greater than the maximum of the first $N$ samples, i.e., $$P(T_{N+3} > T_{N+2} > T_{N+1} > T_{maxN})$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{(N+1)(N+2)(N+3)}}
A point is randomly thrown on the segment \([6, 11]\) and let \( k \) be the resulting value. Find the probability that the roots of the equation \(\left(k^{2}-2k-24\right)x^{2}+(3k-8)x+2=0\) satisfy the condition \( x_{1} \leq 2x_{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
If there exists an $x \in [-1, 2]$ such that $a \cdot 3^x \geq x - 1$, find the minimum value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
Three positive integers are each greater than $1$, have a product of $1728$, and are pairwise relatively prime. What is their sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
In the interval [0, 2π], how many solutions does the equation tan(2x) = cos(x/2) have? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the function \( f(x) = \frac{x^5}{5x^4 - 10x^3 + 10x^2 - 5x + 1} \), calculate the sum \( \sum_{i=1}^{2009} f\left( \frac{i}{2009} \right) \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
Given a real number \( a \) such that there is only one real number \( x \) satisfying the inequality \( \left|x^{2}+2 a x+3 a\right| \leqslant 2 \), what is the number of real numbers \( a \) that meet this condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given \( a, b \in \mathbf{N}^{*} \) and satisfying \( \frac{1}{a} - \frac{1}{b} = \frac{1}{2021} \), determine the number of positive integer pairs \( (a, b) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
. Let \( n \geqslant 3 \) be an integer. A frog is to jump along the real axis, starting at the point 0 and making \( n \) jumps: one of length 1, one of length 2, \(\ldots\), one of length \( n \). It may perform these \( n \) jumps in any order. If at some point the frog is sitting on a number \( a \leqslant 0 \), its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number \( a > 0 \), its next jump must be to the left (towards the negative numbers). Find the largest positive integer \( k \) for which the frog can perform its jumps in such an order that it never lands on any of the numbers \( 1, 2, \ldots, k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor\frac{n-1}{2}\right\rfloor}
a typist uses a sheet measuring 20 cm by 30 cm lenghtwise . if a margin of 2 cm is left on each side and a 3 cm margin on the top and bottom , then what is the percentage of page used by the typist ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64\%}
Let  $ m, n \geq 1$  be two coprime integers and let also  $ s$  an arbitrary integer. Determine the number of subsets  $ A$  of   $ \{1, 2, ..., m \plus{} n \minus{} 1\}$  such that  $ |A| \equal{} m$  and  $ \sum_{x \in A} x \equiv s \pmod{n}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{n} \binom{m+n-1}{m}}
Three truck drivers went into a roadside cafe. One driver bought four sandwiches, a cup of coffee, and ten donuts for a total of 1 dollar and 69 cents. The second driver bought three sandwiches, a cup of coffee, and seven donuts for 1 dollar and 26 cents. How much did the third driver pay for a sandwich, a cup of coffee, and a donut?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Byan is playing a game called "raven, raven, falcon" with his three friends. His friends sit in a circle, and Byan walks clockwise around them, tapping each friend he passes on the head and saying either "raven" or "falcon," each with a probability of $\frac{1}{2}$. The game ends when Byan has said "falcon" twice. The probability that one of his friends will be called a "falcon" twice can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{107}
Let $x_i \in \mathbb{R}$ such that $|x_i|<1$ for $i=1, 2,..., n$ and $x_1^2+x_2^2+...+x_n^2 = (x_1+x_2+...+x_n)^2 + 2561$. Find the least value of $n$ so that the equation is true.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2562}
A polynomial $P$ of degree 2015 satisfies the equation $P(n)=\frac{1}{n^{2}}$ for $n=1,2, \ldots, 2016$. Find \lfloor 2017 P(2017)\rfloor.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
A man spends 1/5 of his salary on food, 1/10 of his salary on house rent, 3/5 of his salary on clothes, and saves 1/20 of his salary. After accounting for taxes, which take away 1/8 of his initial salary, he still has $18,000 left. Find his original salary before taxes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$411,428.57}
In a regular quadrangular pyramid, the plane angle at the apex is equal to the angle between the lateral edge and the base plane. Determine the dihedral angles between adjacent lateral faces of this pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos (2-\sqrt{5})}
a cone of height 9 cm with diameter of its base 18 cm is carved out from a wooden solid sphere of radius 9 cm . the percentage of the wood wasted is :
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}