inclusionAI/Ring-lite-rl-data huggingface dataset:
Count the number of sequences $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ of integers such that $a_{i} \leq 1$ for all $i$ and all partial sums $\left(a_{1}, a_{1}+a_{2}\right.$, etc.) are non-negative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Let $a$ and $b$ be prime numbers such that $a + b = 10000$. Find the sum of the smallest possible value of $a$ and the largest possible value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
Lee mows one lawn and charges $33. Last week he mowed 16 lawns and some customers each gave him a certain amount as a tip. Lee earned $558 mowing lawns last week. How much did each of those customers tip him?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$30}
Triangle $ABC$ is scalene. Points $P$ and $Q$ are on segment $BC$ with $P$ between $B$ and $Q$ such that $BP=21$, $PQ=35$, and $QC=100$. If $AP$ and $AQ$ trisect $\angle A$, then $\frac{AB}{AC}$ can be written uniquely as $\frac{p\sqrt{q}}{r}$, where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. Determine $p+q+r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
已知点 $A(0,2)$ 和抛物线 $y^{2}=x+4$ 上两点 $B, C$, 使得 $A B \perp B C$, 求点 $C$ 的纵坐标的取值范围。请提供该范围的除无穷以外的端点之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}{r}$, where $m$, $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0$. Find $m+n+r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
8. For any real number $x$, $[x]$ represents the greatest integer not exceeding $x$. When $0 \leqslant x \leqslant 100$, the range of $f(x)=[3 x]+[4 x]+[5 x]+[6 x]$ contains $\qquad$ different integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1201}
Let $g(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $s_1,$ $s_2,$ $\dots,$ $s_{2010}.$ There are exactly 1010 distinct values among
\[|s_1|, |s_2|, \dots, |s_{2010}|.\] What is the minimum number of real roots that $g(x)$ can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
12 sparrows, 5 pigeons, 9 crows, and 8 storks were sitting on a fence. Later, 15 more storks and 4 more pigeons joined them. How many more storks than birds (sparrows, pigeons, and crows combined) are sitting on the fence now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that  
  
$P(1) = P(3) = P(5) = P(7) = a,$  and  
  
$P(2) = P(4) = P(6) = P(8) = -a.$ 
  
What is the smallest possible value of $a$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{315}
Points $E, F, G, H$ are chosen on segments $A B, B C, C D, D A$, respectively, of square $A B C D$. Given that segment $E G$ has length 7 , segment $F H$ has length 8 , and that $E G$ and $F H$ intersect inside $A B C D$ at an acute angle of $30^{\circ}$, then compute the area of square $A B C D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{784}{19}}
When researching the relationship between two variables, the effectiveness of model fitting can be judged by the residuals $\hat{e}_1$, $\hat{e}_2$, …, $\hat{e}_n$. This process can also help identify any suspicious data within the original dataset. The analysis work in this area is called ________ analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Residual}}
Let $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $\triangle ABC$. If $XA = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Define $f(x) = \frac{3}{27^x + 3}.$ Calculate the sum
\[ f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + f\left(\frac{3}{2001}\right) + \dots + f\left(\frac{2000}{2001}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Pipe B can fill a tank in an hour and Pipe A can fill the tank in half an hour. They are opened simultaneously when the tank is empty. Pipe A is shut 15 minutes before the tank overflows. When will the tank overflow?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Numbers 1 through 9 are written in the cells of a $3 \times 3$ table. All possible sums of the numbers in adjacent (by side) cells are then listed in a notebook. What is the minimum number of different sums that could be listed in the notebook?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a deck of cards, $J, Q, K$ are counted as $11, 12, 13$ points respectively. From the 13 cards with points from 1 to 13, what is the maximum number of cards that can be selected such that no two pairs of cards have the same sum of points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given $p$: "For all $k \in \mathbb{R}$, the line $y=kx+1$ has two distinct common points with the ellipse $\frac{x^2}{1} + \frac{y^2}{a} = 1$"; $q$: "There exists $x_0 \in \mathbb{R}$, such that the inequality $4^{x_0} - 2^{x_0} - a \leqslant 0$ holds"; If "$p$ and $q$" is a false statement, and "$p$ or $q$" is a true statement, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \frac{1}{4} \leqslant a \leqslant 1}
Let U = {1,2,3,4,5,6}, and let A and B be subsets of U. An "ideal matching set" is defined as a pair (A,B) where A ∩ B = {1,3,5}. Calculate the total number of ideal matching sets. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Determine the number of subsets \( S \) of \(\{1,2,3, \ldots, 10\}\) with the following property: there exist integers \(a < b < c\) with \(a \in S, b \notin S, c \in S\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{968}
In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$.  Al initially starts at city $C_0$. 

Each minute for ten minutes, Al flips a fair coin.  If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$.  If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$.

[i]Proposed by Ray Li[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{147}
Given an obtuse triangle \(ABC\) with obtuse angle \(C\). Points \(P\) and \(Q\) are marked on its sides \(AB\) and \(BC\) respectively, such that \(\angle ACP = CPQ = 90^\circ\). Find the length of segment \(PQ\) if it is known that \(AC = 25\), \(CP = 20\), and \(\angle APC = \angle A + \angle B\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given that \(a, b, c\) are integers such that \(abc = 60\), and that the complex number \(\omega \neq 1\) satisfies \(\omega^3 = 1\), find the minimum possible value of \(\left| a + b\omega + c\omega^2 \right|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Given that \( a, b, \) and \( c \) are all positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If both \( A \) and \( B \) are less than 1 unit away from the origin, what is the minimum value of \( a + b + c \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given \(\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}\), representing a cube root of unity, specifically \(\omega = e^{2\pi i / 3}\). Let \(T\) denote all points in the complex plane of the form \(a + b\omega + c\omega^2\), where \(0 \leq a \leq 2\), \(0 \leq b \leq 1\), and \(0 \leq c \leq 1\). Determine the area of \(T\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
The parabola $y=x^{2}-\left(2m-1\right)x-6m$ intersects the $x$-axis at points $(x_{1}$,$0)$ and $(x_{2}$,$0)$, where $x_{1}x_{2}=x_{1}+x_{2}+49$. To make this parabola pass through the origin, it should be shifted to the right by ____ units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere, where \( A B C \) is the base and \( A A_{1}, B B_{1}, C C_{1} \) are the lateral edges. The segment \( C D \) is the diameter of this sphere, and point \( K \) is the midpoint of the edge \( A A_{1} \). Find the volume of the prism if \( C K = 2 \sqrt{3} \) and \( D K = 2 \sqrt{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\sqrt{2}}
Let  $ABC$  be a triangle with  $AB = 9$ ,  $BC = 10$ , and  $CA = 17$ . Let  $B'$  be the reflection of the point  $B$  over the line  $CA$ . Let  $G$  be the centroid of triangle  $ABC$ , and let  $G'$  be the centroid of triangle  $AB'C$ . Determine the length of segment  $GG'$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{48}{17}}
The graph of $y = ax^2 + bx + c$ has a maximum value of 75, and passes through the points $(-3,0)$ and $(3,0)$. Find the value of $a + b + c$ at $x = 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{125}{3}}
200 people stand in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist standing next to two conformists always tells the truth. A conformist standing next to at least one liar can either tell the truth or lie. 100 of the standing people said: "I am a liar," the other 100 said: "I am a conformist." Find the maximum possible number of conformists among these 200 people.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Two circles with radii \( R \) and \( r \) are externally tangent to each other. An external common tangent is drawn to these circles, and a circle is inscribed in the curvilinear triangle formed by these elements. Find the area of the inscribed circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi R^2 r^2}{(\sqrt{R} + \sqrt{r})^4}}
Given the hyperbola $$\frac {x^{2}}{4}-y^{2}=1$$, find the distance from a point on the hyperbola to an asymptote.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{5}}{5}}
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given real numbers $x$ and $y$ satisfying $e^{x}=xy\left(2\ln x+\ln y\right)$, the range of $xy$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[e, +\infty)}
A man disposes of sufficiently many metal bars of length  $2$  and wants to construct a grill of the shape of an  $n \times n$  unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n+1)}
A bug starts at a vertex of a grid made of equilateral triangles of side length $1$. At each step the bug moves in one of the $6$ possible directions along the grid lines randomly and independently with equal probability. What is the probability that after $5$ moves the bug never will have been more than $1$ unit away from the starting position?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{13}{108}$}
On Black Friday, a popular electronics store sells 327 televisions. This year, they observed a 20% increase in sales compared to the previous Black Friday. If this sales trend continues with a consistent 20% increase in sales each year for the next three years, how many televisions will they sell on Black Friday three years from now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{565}
A subset of $\{1, 2, 3, \ldots, 2017, 2018\}$ has the property that none of its members are 5 times another. What is the maximum number of elements that such a subset could have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1682}
A square \(ABCD\) with side length 4 is folded along \(BD\) to form a dihedral angle of \(60^{\circ}\). What is the distance between the midpoint of \(BC\) and point \(A\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
A bird is building a nest from twigs. The bird has put twelve twigs together already to make a circle. For each twig it placed into the circle, it wants to weave in six more twigs to fill out the nest. It knows a tree that has dropped a certain fraction of the twigs it needs. The bird still needs to find 48 twigs to finish its nest. What fraction of the needed twigs has the tree dropped?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
On September 10, 2005, the following numbers were drawn in the five-number lottery: 4, 16, 22, 48, 88. All five numbers are even, exactly four of them are divisible by 4, three by 8, and two by 16. In how many ways can five different numbers with these properties be selected from the integers ranging from 1 to 90?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15180}
Evaluate the following integral:
$$ \int_0^\infty \frac{x^2 \ln(x)}{(1+x^2)^3} {\rm d}x $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For a geometric sequence $\{a_{n}\}$ where all terms are positive, the sum of the first $n$ terms is $S_{n}$. If $a_{2}a_{6}=4$, $a_{3}=1$, then the minimum value of $\frac{{{{({S_n}+\frac{9}{4})}^2}}}{{2{a_n}}}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given \(n\) distinct positive integers \(a_{1}, a_{2}, \cdots, a_{n}\) whose sum is 2000, let 
\[ A = \max \{a_{1}, a_{2}, \cdots, a_{n}\}. \]

Find the minimum value of \(A + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110}
1. Let $n$ be the sum of all ten-digit numbers that have each of the digits $0,1, \ldots, 9$ in their decimal representation. Determine the remainder when $n$ is divided by seventy-seven.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Let \( a, b, c, d \) be integers such that \( a > b > c > d \geq -2021 \) and
\[ \frac{a+b}{b+c} = \frac{c+d}{d+a} \]
(and \( b+c \neq 0 \neq d+a \)). What is the maximum possible value of \( a \cdot c \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510050}
3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. If
$$
\begin{array}{l}
a=\frac{\sqrt{6+\sqrt{6+\cdots+\sqrt{6}}}}{2016 \text { nested radicals }}, \\
b=\frac{\sqrt[3]{6+\sqrt[3]{6+\cdots+\sqrt[3]{6}}}}{2017 \text { nested radicals }},
\end{array}
$$

then $[a+b]=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(f_{0}(x)=\frac{1}{c-x}\) and \(f_{n}(x)=f_{0}\left(f_{n-1}(x)\right)\), \(n=1,2,3, \ldots\). If \(f_{2000}(2000)=d\), find the value of \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Let $n$ be a given positive integer. Find the smallest positive integer $u_{n}$ such that for every positive integer $d$, any $u_{n}$ consecutive odd positive integers contain at least as many numbers divisible by $d$ as there are among $1, 3, 5, \cdots,$ and $2n-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n-1}
On the side \( AB \) of the parallelogram \( ABCD \), a point \( F \) is chosen, and on the extension of the side \( BC \) beyond the vertex \( B \), a point \( H \) is taken such that \( \frac{AB}{BF} = \frac{BC}{BH} = 5 \). The point \( G \) is chosen such that \( BFGH \) forms a parallelogram. \( GD \) intersects \( AC \) at point \( X \). Find \( AX \), if \( AC = 100 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
James decides to buy school supplies for his class from an international online store. He buys 2 notebooks with 50 pages each, a box of 10 pens, and a pack of 15 folders. The price of each item is as follows:

- Notebooks: $12 each.
- Box of 10 pens: $9.
- Pack of 15 folders: $19.

Additionally, he is charged $5 for local shipping and $10 for international shipping. There are also different discounts applied to these items:

- Notebooks: Buy one, get the second one with a 50% discount.
- Box of 10 pens: 20% off.
- Pack of 15 folders: No discount.

Also, the sales tax is 5%, and the exchange rate is 2.3 units of his local currency to a dollar. How many units of his local currency did each page of the notebooks cost, considering all other expenses and applied discounts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.43}
In triangle \(ABC\) with \(\angle ABC = 60^\circ\), the angle bisector of \(\angle A\) intersects \(BC\) at point \(M\). Point \(K\) is taken on side \(AC\) such that \(\angle AMK = 30^\circ\). Find \(\angle OKC\), where \(O\) is the circumcenter of triangle \(AMC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
对矩阵 P，定义 e^P=I+\ds{m=1}{\infty}\df{1}{m!}P^m. 设 20 阶实矩阵 A 满足 e^A=I_{20}，且特征值的模长不超过 20，则这样的互不复相似的 A 有__________个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{286}
On graph paper, a square measuring $11 \times 11$ cells is drawn. It is required to mark the centers of some cells in such a way that the center of any other cell lies on the segment connecting any two marked points that are on the same vertical or horizontal line. What is the minimum number of cells that need to be marked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Let \(ABCD\) be an isosceles trapezoid having parallel bases \(\overline{AB}\) and \(\overline{CD}\) with \(AB>CD.\) Line segments from a point inside \(ABCD\) to the vertices divide the trapezoid into four triangles whose areas are \(2, 3, 4,\) and \(5\) starting with the triangle with base \(\overline{CD}\) and moving clockwise as shown in the diagram below. The original answer for the ratio \(\frac{AB}{CD}\) is in the form \(k+\sqrt{m}\). Please find the value of \(k+m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Joan went to 4 football games this year, of which 2 were home games and 2 were away games. She also attended the playoffs, but only at home. She went to 9 games last year, with 6 of them being home games and 3 being away games. Last year, Joan also attended the playoffs, both home and away games. The ticket prices vary: home game tickets cost $60, away game tickets cost $75, home playoff tickets cost $120, and away playoff tickets cost $100. 

Joan also brought a friend to each game she attended in the two-year period. For home games, her friend received a discounted ticket price of $45 and for away games, the friend's ticket cost stayed the same at $75. Additionally, Joan spent $25 on transportation for each home game and $50 for each away game.

Calculate the total cost Joan spent on attending football games, her friend's tickets, and transportation over the two-year period.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$2645}
在正四棱雉 $O-A B C D$ 中, 已知 $\angle A O B=45^{\circ}$, 记 $\alpha$ 为面 $O A B$ 和面 $O B C$ 所成的二面角的大小, 且 $\cos \alpha=m+\sqrt{n}$, 其中 $m, n$ 为整数, 则 $m+n=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Interior numbers begin in the third row of Pascal's Triangle. Calculate the sum of the squares of the interior numbers in the eighth row.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3430}
How many ways are there to arrange the numbers  $1$  through  $8$  into a  $2$  by  $4$  grid such that the sum of the numbers in each of the two rows are all multiples of  $6,$  and the sum of the numbers in each of the four columns are all multiples of  $3$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
What is the ratio of the following two integrals? 

$$I_1 = \int_0^1 \frac{dt}{\sqrt{1-t^4}}$$
$$I_2 = \int_0^1 \frac{dt}{\sqrt{1+t^4}}$$

Find $\frac{I_1}{I_2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Jane can arrange some vases of flowers in a day. If there are 248 vases of flowers to be arranged, and she will arrange 8 vases of flowers on the last day, how many vases of flowers can she arrange in a day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let $u$ and $v$ be real numbers satisfying the inequalities $2u + 3v \le 10$ and $4u + v \le 9.$ Find the largest possible value of $u + 2v$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.1}
In a competition there are  $18$  teams and in each round  $18$  teams are divided into  $9$  pairs where the  $9$  matches are played coincidentally. There are  $17$  rounds, so that each pair of teams play each other exactly once. After  $n$  rounds, there always exists  $4$  teams such that there was exactly one match played between these teams in those  $n$  rounds. Find the maximum value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the mass percentage of Cl in Ammonium chloride
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66.26\%}
已知过点 $P(3,0)$ 斜率为 $k$ 的直线 $l$ 交双曲线 $C: x^{2}-\frac{y^{2}}{3}=1$ 右支于 $A, B$ 两点, $F$ 为双曲线 $C$ 的右焦点, 且 $|A F|+|B F|=16$, 求 $k$ 的值。请给出所有满足条件的|k|的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the number of permutations \(a_1, a_2, \ldots, a_{10}\) of the numbers \(1, 2, \ldots, 10\) such that \(a_{i+1}\) is not less than \(a_i - 1\) for \(i = 1, 2, \ldots, 9\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
The equation \(z(z+i)(z+3i)=2002i\) has a zero of the form \(a+bi\), where \(a\) and \(b\) are positive real numbers. The original answer is in the form \(\sqrt{k}\). Please find the value of \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.\]What is $k?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$137$}
Given that $θ$ is a real number, if the complex number $z=\sin 2θ-1+i( \sqrt {2}\cos θ-1)$ is a purely imaginary number, then the imaginary part of $z$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-2"]}