inclusionAI/Ring-lite-rl-data huggingface dataset:
Given a function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(1)=1$, and for any $x$, $f'(x) < \frac{1}{2}$ always holds, then the solution set of the inequality $f(\lg^2x) < \frac{\lg^2x}{2}+ \frac{1}{2}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac {1}{10})\cup(10,+\infty)}
Given two functions $f(x) = e^{2x-3}$ and $g(x) = \frac{1}{4} + \ln \frac{x}{2}$, if $f(m) = g(n)$ holds, calculate the minimum value of $n-m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} + \ln 2}
Berlov S.L.

A convex 2011-gon is drawn on the board. Petya sequentially draws diagonals in it such that each newly drawn diagonal intersects no more than one of the previously drawn diagonals at an internal point. What is the maximum number of diagonals Petya can draw?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4016}
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the function $f(x)=(ax-1)(x-b)$, if the solution set of the inequality $f(x) > 0$ is $(-1,3)$, then the solution set of the inequality $f(-x) < 0$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,-3)\cup(1,+\infty)}
For any positive integer  $n$  , let  $s(n)$  denote the number of ordered pairs  $(x,y)$  of positive integers for which  $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$  . Determine the set of positive integers for which  $s(n) = 5$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{p^2 \mid p \text{ is a prime number}\}}
In how many ways can 10 people, including 3 sets of clones (2 clones in each set), be arranged in a row such that no clone is adjacent to another clone from the same set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{236880}
Let a, b, c, d be rational numbers with a > 0. If for every integer n >= 0, the number an^3 + bn^2 + cn + d is also an integer, then find the minimal value of a. Express your answer as a rational number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6}}
Evaluate the limit: $$\lim _{x\to \infty }\frac{\left(x!\right)^3\left(3\left(x+1\right)\right)!}{\left(\left(x+1\right)!\right)^3\left(3x\right)!}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
The domain of the function $f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x))))$ is an interval of length $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$271$}
Determine the radius of convergence of the series:

$$\sum_{n=1}^\infty \left[\frac {z(z+n)}{n}\right]^{n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Positive integers  $ a$ ,  $ b$ ,  $ c$ , and  $ d$  satisfy  $ a > b > c > d$ ,  $ a \plus{} b \plus{} c \plus{} d \equal{} 2010$ , and  $ a^2 \minus{} b^2 \plus{} c^2 \minus{} d^2 \equal{} 2010$ . Find the number of possible values of  $ a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501}
A square $S$ with side length $2n$ units is divided into $4n^2$ unit squares by drawing $2n-1$ horizontal and $2n-1$ vertical lines, each one unit apart. A circle with diameter $2n-1$ is inscribed in $S$, with its center at the intersection of the diagonals. Determine the number of unit squares that have at least a portion of their boundary intersected by the circle's circumference.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8n - 4}
All of the square-shaped tiles are lined up to make a larger square. If the number of tiles on the perimeter is 52, find the number of tiles on the outermost of one side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given any $n$ points on a plane $p_{1}, p_{2}, \cdots, p_{n}$, where no three points are collinear, and each point $p_{i}(i=1,2,\cdots,n)$ is colored either red or blue. Let $S$ be a set of some triangles with vertex sets $\left\{p_{1}, p_{2}, \cdots, p_{n}\right\}$, having the property that for any two line segments $p_{i} p_{j}$ and $p_{h} p_{k}$, the number of triangles in $S$ with $p_{i} p_{j}$ as a side is equal to the number of triangles with $p_{h} p_{k}$ as a side. Find the smallest positive integer $n$ such that there are always two triangles in the set $S$ where each triangle's three vertices are the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $a,$ $b,$ and $c$ be positive real numbers such that $a > b$ and $a + b + c = 4.$  Find the minimum value of
\[4a + 3b + \frac{c^3}{(a - b)b}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the smallest possible number of digits in a natural number \( N \) such that its decimal representation consists only of the digits 1 and 2, and by removing digits from this number, any of the 10,000 numbers consisting of 9999 ones and one two can be obtained.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19999}
A  $13$ -foot tall extraterrestrial is standing on a very small spherical planet with radius  $156$  feet. It sees an ant crawling along the horizon. If the ant circles the extraterrestrial once, always staying on the horizon, how far will it travel (in feet)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130\pi}
The negation of the proposition "The cube of every even number is even" is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{there exists an even number whose cube is odd}
A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{\sqrt{k} - 1}{m}\). Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine all positive integers  $a$  for which there exist exactly  $2014$  positive integers  $b$  such that  $\displaystyle2\leq\frac{a}{b}\leq5$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6710, 6712, 6713}
A cylindrical drinking cup is made from a rectangular piece of paper with length \( L \) and width \( W \) by rolling the paper and joining the edges. Determine the ratio \( \frac{L}{W} \) that maximizes the volume of the resulting cup.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine $x^2+y^2+z^2+w^2$ if $\frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1$ $\frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1$ $\frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1$ $\frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$36$}
A fair coin is flipped indefinitely. What is the probability that a run of three consecutive tails occurs before a run of four consecutive heads?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{22}}
将1、2、3、4、5、6、7、8分别放在正方体的八个顶点上，使得每一个面上的任意三个数之和均不小于10．求每一面上四个数之和的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the number of possible values of \( x \) that satisfy the equation \( \frac{x}{100} = \sin(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Given a certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases, calculate the total number of free cytosine deoxyribonucleotide molecules required when this DNA fragment is replicated twice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1320}
A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{27}{2}$}
A best-of-9 series is to be played between two teams; that is, the first team to win 5 games is the winner. The Mathletes have a chance of  $\tfrac{2}{3}$  of winning any given game. What is the probability that exactly 7 games will need to be played to determine a winner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{20}{81}}
Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. How many distinct lines pass through at least two members of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
For a positive real number $r$, let $M_r$ be the regular surface $M_r = \{(x, y, z) | x^2 + y^2 = z < r^2, x > 0, y > 0\}$. Let $K$ denote the Gaussian curvature of $M_r$. Determine $\int_{M_r} K dA$ and $\lim_{r \rightarrow \infty} \int_{M_r} K dA$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{2}\)}
Let  $Q$  be a  $(2n+1) \times (2n+1)$  board. Some of its cells are colored black in such a way that every  $2 \times 2$  board of  $Q$  has at most  $2$  black cells. Find the maximum amount of black cells that the board may have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2n+1)(n+1)}
Calculate:
$$
\left(10^{4}-9^{4}+8^{4}-7^{4}+\cdots+2^{4}-1^{4}\right)+\left(10^{2}+9^{2}+5 \times 8^{2}+5 \times 7^{2}+9 \times 6^{2}+9 \times 5^{2}+13 \times 4^{2}+13 \times 3^{2}\right) =
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7615}
As shown in the figure, the line segment $AB$ of length 9 is divided into nine equal parts. What is the total length of all the line segments in the figure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Suppose that $P(z), Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z)=R(z)$. What is the minimum possible value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many distinct sets of  $5$  distinct positive integers  $A$  satisfy the property that for any positive integer  $x\le 29$ , a subset of  $A$  sums to  $x$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the sum

$$
\sum_{k=1}^{\infty}\left(\prod_{i=1}^{k} \frac{P_{i}-1}{P_{i+1}}\right)=\frac{1}{3}+\frac{1}{3} \cdot \frac{2}{5}+\frac{1}{3} \cdot \frac{2}{5} \cdot \frac{4}{7}+\frac{1}{3} \cdot \frac{2}{5} \cdot \frac{4}{7} \cdot \frac{6}{11}+\ldots
$$

where $P_{n}$ denotes the $n^{\text {th }}$ prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the molecular weight of some moles of N2O3 if the molecular weight of the given moles is 304?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76.02}
Find the sum of all "auspicious numbers" with up to 3 digits, where a positive integer \( n \) with \( n \) digits is called an "auspicious number" if, when appended to the end of any two positive integers, the product of these two new numbers ends in \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1114}
The side lengths  $a,b,c$  of a triangle  $ABC$  are positive integers. Let:

\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]

for any positive integer  $n$ . If  $\frac{T_{2}}{2T_{1}}=2023$  and  $a>b>c$  , determine all possible perimeters of the triangle  $ABC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
To fold a paper airplane, Austin starts with a square paper  $F OLD$  with side length  $2$ . First, he  folds corners  $L$  and  $D$  to the square’s center. Then, he folds corner  $F$  to corner  $O$ . What is the longest distance between two corners of the resulting figure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Let $X_n$ be a sequence of random variables defined as $X_n=\sum_{i=1}^{n}U_i$, where $U_i$ are independent random variables such that $P(U_i=1)=P(U_i=-1)=\frac{1}{2}$. For $m\geq n$, find the variance of $E(U_n|X_m)$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{m}}
A convex pentagon  $ ABCDE$  is inscribed in a circle. The distances of  $ A$  from the lines  $ BC,CD,DE$  are  $ a,b,c,$  respectively. Compute the distance of  $ A$  from the line  $ BE$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a \cdot c}{b}}
The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says:

"Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers."

How many possibilities are there for the set of truth-tellers on the U.S. Senate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
What is the value of \[2^3 - 1^3 + 4^3 - 3^3 + 6^3 - 5^3 + \dots + 18^3 - 17^3?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3159$}
Let  $ABCDEF$  be a regular hexagon with side length 10 inscribed in a circle  $\omega$ .  $X$ ,  $Y$ , and  $Z$  are points on  $\omega$  such that  $X$  is on minor arc  $AB$ ,  $Y$  is on minor arc  $CD$ , and  $Z$  is on minor arc  $EF$ , where  $X$  may coincide with  $A$  or  $B$  (and similarly for  $Y$  and  $Z$ ). Compute the square of the smallest possible area of  $XYZ$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7500}
A trapezoid \(ABCD\) (\(AD \parallel BC\)) and a rectangle \(A_{1}B_{1}C_{1}D_{1}\) are inscribed in a circle \(\Omega\) with a radius of 13 such that \(AC \parallel B_{1}D_{1}\) and \(BD \parallel A_{1}C_{1}\). Find the ratio of the areas of \(ABCD\) and \(A_{1}B_{1}C_{1}D_{1}\), given that \(AD = 24\) and \(BC = 10\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Determine the behavior of the sequence \( S_n = \frac{p_n-1}{\phi(p_n-1)} \) as \( n \) approaches infinity, where \( p_n \) is the \( n \)th prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x\]?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
设A={1,2,3,4,5,6},B={7,8,9,⋯,n}，在A中任取3个数，在B中取2个数，组成含有5个元素的集合$A _ { i } ( i = 1 , 2 , 3 , \cdots , 2 0 )$，使得$\vert A _ { i } \cap A _ { j } \vert \leq 2 , 1 \leq i < j \leq 2 0 .$求n的最小值. （2002年IMO中国国家集训队选拨考试试题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
One country has  $n$  cities and every two of them are linked by a railroad. A railway worker should travel by train exactly once through the entire railroad system (reaching each city exactly once). If it is impossible for worker to travel by train between two cities, he can travel by plane. What is the minimal number of flights that the worker will have to use?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
There is a  $2n\times 2n$  rectangular grid and a chair in each cell of the grid. Now, there are  $2n^2$  pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at  $(3,3)$  and  $(2,5)$  respectively, then the distance between them is  $|3-2|+|3-5|=3$ . Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4n^3}
学校水果店里有苹果、香蕉、石榴、橘子、葡萄、西梅6种水果，西梅数量不多，只够一人购买.甲、乙、丙、丁4位同学前去购买，每人只选择其中一种，这4位同学购买后，恰好买了其中3种水果，则他们购买水果的可能情况有________种.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{600}
Given the transition probability matrix \( P = \begin{bmatrix} 1 & 0.1 & 0.8 \\ 0 & 0.7 & 0.2 \\ 0 & 0.2 & 0 \end{bmatrix} \) and the initial state \( s_0 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \), calculate the expected number of steps needed for absorption.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{13}}
For how many integer values of $a$ does the equation $$x^2 + ax + 12a = 0$$ have integer solutions for $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let $A$ be an annulus defined as $A=\{z:r<|z|<R\}$, where $r<1<R$. If $f \in H(A)$ and its Laurent series is $f(z) = \sum_{n=-\infty}^\infty a_n z^n$, find the condition on the coefficients $a_n$ such that there exists a function $g \in H(A)$ with $f=g'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_{-1} = 0}
If $|x|<1$, then what is the coefficient of $x^3$ in the expansion of $\dfrac{1}{e^x\cdot (1+x)}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{8}{3}}
Find the remainder when the number \( S = \sum_{i=0}^{99} (n+i)^6 + 2^{2^{2558}} + 1 \) is divided by 100, where \( n \) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Let \(ABCD\) be a cyclic quadrilateral. The side lengths of \(ABCD\) are distinct integers less than \(15\) such that \(BC\cdot CD=AB\cdot DA\). Find the largest possible value of \(BD\) in the form \(\sqrt{\frac{x}{y}}\). Please provide the value of \(x + y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{427}
Given that the solution set of the inequality $\frac{x+1}{x-3}\leqslant 0$ is $A$, and the solution set of the inequality ${x^{2}}-2ax-3{{a}^{2}}\leqslant 0$ is $B$, and $B$ is a subset of $A$, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[- \frac{1}{3},1)}
How many different five-colored flags with five vertical stripes of equal width can be made if 11 different colors of fabric are available? (The flag here is considered simply as a piece of cloth, not attached to a pole or anything else.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55440}
5. Find the number of pairs of integers $(x ; y)$ that satisfy the condition $5 x^{2}-6 x y+y^{2}=6^{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["19594"]}
Evaluate the limit: $$\lim_{x\to0}\frac{\sin^2(\sin{x})-x^2\sin{x}}{x^4}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given that line $l\_1$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l\_2$ passes through points $C(1,m)$ and $D(-1,m+1)$, find the values of the real number $m$ when $l\_1$ is parallel to $l\_2$ or $l\_1$ is perpendicular to $l\_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{9}{2}}
If $\triangle PQR$ is right-angled at $P$ with $PR=12$, $SQ=11$, and $SR=13$, what is the perimeter of $\triangle QRS$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Let $x, y, z, w \in [0,1]$. Find the maximum value of $S=x^{2} y+y^{2} z+z^{2} w+w^{2} x-x y^{2}-$ $y z^{2}-z w^{2}-w x^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{27}}
A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3 : 1$. Then $3$ green frogs moved to the sunny side and $5$ yellow frogs moved to the shady side. Now the ratio is $4 : 1$. What is the difference between the number of green frogs and the number of yellow frogs now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
How many roots does the equation \(\sin x - \log_{10} x = 0\) have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1964}
Determine the minimum girth of a $k$-regular graph with $v_G > 2k$ vertices that ensures the graph is not Hamiltonian.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
4.4. Two identical cylindrical vessels are connected at the bottom by a small-bore pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \) cm. At what level will the water in the first vessel stabilize if the valve is opened? The density of water is 1000 kg \(/ \mathrm{m}^{3}\), and the density of oil is 700 kg \(/ \mathrm{m}^{3}\). Neglect the volume of the connecting pipe. Provide the answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$279$}
Jordan has a new hit song on Spotify. Some months are left in the year, and she currently has 60,000 listens. If the number of listens per month doubles, the song will have 900,000 listens by the end of the year. How many months are left in the year?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A chord of a circle is at a distance $h$ from the center. In each of the segments subtended by the chord, a square is inscribed such that two adjacent vertices of the square lie on the arc and the other two on the chord or its extension. What is the difference in the side lengths of these squares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8h}{5}}
In base- $2$  notation, digits are  $0$  and  $1$  only and the places go up in powers of  $-2$ . For example,  $11011$  stands for  $(-2)^4+(-2)^3+(-2)^1+(-2)^0$  and equals number  $7$  in base  $10$ . If the decimal number  $2019$  is expressed in base  $-2$  how many non-zero digits does it contain ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}