inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the area of the region in the first quadrant \(x>0, y>0\) bounded above by the graph of \(y=\arcsin(x)\) and below by the graph of \(y=\arccos(x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - \sqrt{2}}
In the diagram, there is a map consisting of 8 cities connected by 12 roads. How many ways are there to close 5 roads simultaneously for repairs such that it is still possible to travel from any city to any other city?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{384}
Given the circle \( M: 2x^{2} + 2y^{2} - 8x - 8y - 1 = 0 \), and the line \( l: x + y - 9 = 0 \), let \( A \) be a point on \( l \). Construct \(\triangle ABC\) such that \(\angle BAC = \frac{\pi}{4}\), with side \( AB \) passing through the center of the circle \( M \) and points \( B \) and \( C \) lying on the circle \( M \). Determine the range of the y-coordinate of point \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[3, 6]}
Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
(x, y, p) = (2, 2, 3)

}
Let $a, b, c$ be orthonormal vectors in $\mathbb{R}^n$. Determine whether the sum $\sum_{k=1}^n a_{k}b_{k}c_{k}$ is equal to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
 \( \frac{49}{216} \) 
}
Let  $ABCD$  be a unit square. For any interior points  $M,N$  such that the line  $MN$  does not contain a vertex of the square, we denote by  $s(M,N)$  the least area of the triangles having their vertices in the set of points  $\{ A,B,C,D,M,N\}$ . Find the least number  $k$  such that  $s(M,N)\le k$ , for all points  $M,N$ .

*Dinu Șerbănescu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
Triangle $\vartriangle ABC$ has sidelengths $AB = 10$, $AC = 14$, and, $BC = 16$. Circle $\omega_1$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $B$. Circle $\omega_2$ is tangent to rays $\overrightarrow{AB}$, $\overrightarrow{AC}$ and passes through $C$. Let $\omega_1$, $\omega_2$ intersect at points $X, Y$ . The square of the perimeter of triangle $\vartriangle AXY$ is equal to $\frac{a+b\sqrt{c}}{d}$ , where $a, b, c$, and, $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6272}
Given the group \(G=\mathbb{Z}_6 \times \mathbb{Z}_3\), the element \(a=(4,1)\) of \(G\), the subgroup \(H=\langle a\rangle\) of \(G\), and the group \(G_1=G/H\), find the order \(k\) of the class of \((3,1)\) in \(G_1\) and the order \(\ell\) of the class of \((2,2)\) in \(G_1\). What is \((k+\ell) \mod{5}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If sequences $x_n$ and $y_n$ converge almost surely to constants $a$ and $b$ respectively, does the sequence $x_n + y_n$ converge almost surely to $a + b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + b}
A light ray falls at an angle $\alpha=60^{\circ}$ on the front surface of a plane-parallel glass plate. The refractive index of the glass is $n=1.6$. By what angle from the direction of the incident ray is the ray, reflected from the back surface of the plate and exiting back through the front surface, deviated?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
At a table tennis team meeting, each member sits on a stool. There are two types of stools: square stools (with four legs) or round stools (with three legs). A child walks into the meeting room and counts a total of 33 legs (not including the child's own legs). How many team members are attending the meeting?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
At a race track, a maximum of 6 horses can race together at a time. There is a total of 30 horses. Races not only depend on speed, but also rely on track conditions and unpredictable horse behavior during each race. Track conditions vary with each race. What is the minimum number w of races we need to conduct to confidently determine the top 3 fastest horses under different conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A centaur, a chess piece that can move one square up, left, or diagonally up-right, stands on the square marked with a star on a $1000 \times 1000$ board. Four rectangles $2 \times 994$ have been removed from the board (as shown in the diagram). Two players take turns moving the centaur. The player who cannot make a move loses. Who wins with optimal play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Second player}
Problem 6.3. A country has the shape of a square and is divided into 25 identical square counties. In each county, either a knight-count, who always tells the truth, or a liar-count, who always lies, rules. One day, each count said: "Among my neighbors, there are an equal number of knights and liars." What is the maximum number of knights that could have been? (Counts are neighbors if their counties share a common side.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A circle touches the sides of an angle at points $A$ and $B$. The distance from a point $C$ on the circle to the line $A B$ is 6. Find the sum of the distances from point $C$ to the sides of the angle, given that one of these distances is 5 more than the other.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A list of six positive integers \( p, q, r, s, t, u \) satisfies \( p < q < r < s < t < u \). The sums of the 15 pairs of numbers that can be formed by choosing two different numbers from this list are:

\[ 25, 30, 38, 41, 49, 52, 54, 63, 68, 76, 79, 90, 95, 103, 117 \]

Calculate the sum \( r+s \) using the given information. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
圆周上 \(A_{1}, A_{2}, \cdots, A_{7}\) 七个点两两相连。求任选两条线段，这两条线段无公共点的概率。答案的形式为\(\frac{k}{m}\)，请计算k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compare the following numbers: $2\sqrt{2}-3\_\_\_\_\_\_0.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{<}
What is the maximum number of rooks that can be placed on an 8x8 chessboard such that each rook attacks at most one other? A rook attacks all squares on its file and rank.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Viola has been practising the long jump. At one point, the average distance she had jumped was $3.80 \mathrm{~m}$. Her next jump was $3.99 \mathrm{~m}$, and that increased her average to $3.81 \mathrm{~m}$. After the following jump, her average had become $3.82 \mathrm{~m}$. Calculate the distance of her final jump. Express your answer in meters, rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.01}
The number $K$ is divisible by 7 and its binary representation is given by:

$$
K = 10110101010101xyz110
$$

Determine the digits $x$, $y$, and $z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, 1, 0}
To make nine circular cardboard discs for Carnival, square sheets with sides of 10 cm are used, as indicated in the figure. Calculate the area (in cm²) of the unused paper after cutting out the circular discs. Express your answer as a decimal number, rounded to one decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21.5}
Determine all real numbers $a > 0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region \[ R = \{ (x,y) ; 0 \le x \le a, 0 \le y \le f(x) \} \] has perimeter $k$ units and area $k$ square units for some real number $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{All real numbers $a > 2$.}
If $a+b=1$, find the supremum of $$- \frac {1}{2a}- \frac {2}{b}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \frac {9}{2}}
设复数 $z$ 满足 $|3 z-7 \mathrm{i}|=3$, 求 $\left|\frac{z^{2}-2 z+2}{z-1+\mathrm{i}}\right|$ 的最大值。答案的形式为 $\frac{k}{m}$，请给出 $k + m$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given the inequality $x^{3}-3x+3- \frac{x}{e^{x}}-a\leqslant 0$ with respect to $x$, where $x\geqslant -2$, find the minimum value of the real number $a$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1- \frac{1}{e}}
Suppose that $A$ is an abelian group with order $p^4$ where $p$ is a prime number. And there exists a subgroup $N$ such that $A/N\simeq \mathbb{Z}/p^3\mathbb{Z}$. How many abelian groups $A$ satisfy the condition above?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{184147}
Find the numbers \( x \) between 0 and 30 for which the sine of \( x \) degrees equals the sine of \( x \) radians. How many such numbers exist between 30 and 90?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Tom, Dick, and Harry started out on a $100$-mile journey. Tom and Harry traveled by automobile at a speed of $25$ mph, while Dick walked at a speed of $5$ mph. After a certain distance, Harry got off and continued walking at $5$ mph, while Tom went back for Dick and brought him to the destination at the same time that Harry arrived. Find the number of hours required for the trip.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the last 5 digits of the number $M = 5^{2010}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65625}
Points \( K, L, \) and \( M \) are the midpoints of sides \( AB, BC \), and \( CD \) of parallelogram \( ABCD \). It turns out that quadrilaterals \( KBLM \) and \( BCDK \) are inscribed. Find the ratio \( AC:AD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Example 9 In $\triangle A B C$, $O$ is the intersection of $A A^{\prime}$ and $B B^{\prime}$, $A^{\prime}$ is on $B C$, $B^{\prime}$ is on $A C$, and $S_{\triangle A O R^{\prime}}=3, S_{\triangle O A B}=2, S_{\triangle O A^{\prime} B}=1$. Find $S_{\triangle C A^{\prime} B^{\prime}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{45}{2}}
Calculate the enthalpy change (ΔH) for the reaction when 1 mole of CH3CH2OH reacts with 1.5 moles of O2 to form 1 mole of H2O and 1 mole of CH3COOH using the following bond dissociation energies:

CH3-CH2 (C-C): 347 kJ/mol
CH3-O (C-O): 358 kJ/mol
CH2-OH (O-H): 463 kJ/mol
C=O (COOH): 745 kJ/mol
O-H (COOH): 467 kJ/mol
O=O (O2): 498 kJ/mol
O-H (H2O): 467 kJ/mol
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{236}
A deck of playing cards without jokers has 4 suits with a total of 52 cards, each suit has 13 cards numbered from 1 to 13. Feifei draws 2 hearts, 3 spades, 4 diamonds, and 5 clubs. If the sum of the face values of these 4 drawn cards is exactly 34, how many of them are 2s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
设取整 $\left[\frac{10^{2022}}{10^{100}-9}\right]$ 被 $10^{n}$ 整除, 则非负整数 $n$ 的最大可能值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Quadrilateral $ABCD$ satisfies the following conditions: 
- $AB = 8$, 
- $BC = 5$, 
- $CD = 17$, 
- $DA = 10$. 

Let $E$ be the intersection of diagonals $AC$ and $BD$. It is given that the ratio $BE : ED = 1 : 2$. 

Find the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
A box contains $900$ cards, labeled from $100$ to $999$. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Find the smallest positive integer $m$ such that the coefficient of $x^2$ in the expansion of the sum $(1+x)^2 + (1+x)^3 + \ldots + (1+x)^{49} + (1+mx)^{50}$ is equal to $(3n+1)\binom{51}{3}$ for some positive integer $n$. Determine the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The positive integer \( n \) can be divided exactly by 1990, and \( n \) has exactly 12 positive divisors (including 1 and \( n \)). Find \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3980}
Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{4}{15}$}
Given a triangle $ \triangle ABC $ with sides that are consecutive integers, and the largest angle $ \angle B $ is twice the smallest angle $ \angle A $, find the lengths of the sides of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5, 4, 6}
Let the functions  $f(\alpha,x)$  and  $g(\alpha)$  be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then  $g(\alpha)$  is a polynomial is  $\alpha$ .  Find the leading coefficient of  $g(\alpha)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/16}
Perpendiculars \( AP \) and \( AK \) are dropped from vertex \( A \) of triangle \( ABC \) onto the angle bisectors of the external angles at \( B \) and \( C \) respectively. Find the length of segment \( PK \), given that the perimeter of triangle \( ABC \) is \( P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{P}{2} }
On Qingming Festival, students take a bus to the Martyrs' Cemetery. If the bus increases its speed by one-fifth after traveling for 1 hour at its original speed, it can arrive 20 minutes earlier than the scheduled time. If the bus first travels 72 km at its original speed and then increases its speed by one-third, it can arrive 30 minutes earlier than the scheduled time. How many kilometers is it from the school to the Martyrs' Cemetery?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216 \text{ km}}
The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Let $a = 4 + 3i,$ $b = 1 -2i,$ and $c = 8 - 5i.$  The complex number $d$ is such that $a,$ $b,$ $c,$ and $d$ form the vertices of a parallelogram, when plotted in the complex plane.  Enter all possible values of $d,$ separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3 + 6i}
Figure 0 consists of a square with side length 18. For each integer \(n \geq 0\), Figure \(n+1\) consists of Figure \(n\) with the addition of two new squares constructed on each of the squares that were added in Figure \(n\). The side length of the squares added in Figure \(n+1\) is \(\frac{2}{3}\) of the side length of the smallest square(s) in Figure \(n\). Define \(A_{n}\) to be the area of Figure \(n\) for each integer \(n \geq 0\). What is the smallest positive integer \(M\) with the property that \(A_{n}<M\) for all integers \(n \geq 0\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2916}
由数字0，1，2，3，4，5，6，7组成没有重复数字的四位数，则能被15整除且0不在个位的四位数共有________个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Given $x_1=4$, $x_2=5$, $x_3=6$, calculate the standard deviation of this sample. Express your answer as a fraction with a square root, in the form $\boxed{\frac{\sqrt{a}}{b}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{3}}
Toner Drum and Celery Hilton are both running for president. A total of  $2015$  people cast their vote, giving  $60\%$  to Toner Drum. Let  $N$  be the number of "representative'' sets of the  $2015$  voters that could have been polled to correctly predict the winner of the election (i.e. more people in the set voted for Drum than Hilton). Compute the remainder when  $N$  is divided by  $2017$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{605}
Tom weighs 150 kg.  He manages to hold 1.5 times his weight in each hand while wearing a weight vest weighing half his weight.  How much total weight was he moving with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{525}
At a chamber music festival, six musicians have gathered. In each concert, some of the musicians perform while the rest listen from the audience. What is the minimum number of concerts required so that each of the six musicians can listen to all the others from the audience?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \text{ concerts}}
In the geometric sequence $\{a_{n}\}$, $a_{5}a_{8}=6$, $a_{3}+a_{10}=5$, then $\frac{a_{20}}{a_{13}}=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
$5.$ Let x be a real number with  $0<x<1$  and let  $0.c_1c_2c_3...$  be the decimal expansion of x.Denote by  $B(x)$  the set of all subsequences of  $c_1c_2c_3$  that consist of 6 consecutive digits.
For instance ,  $B(\frac{1}{22})={045454,454545,545454}$ Find the minimum number of elements of  $B(x)$  as  $x$  varies among all irrational numbers with  $0<x<1$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the diagram, line segments meet at 90 degrees as shown. If the short line segments are each 3 cm long, what is the area of the shape? Express your answer in square centimeters (cm^2) as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
In the diagram, $COB$ is a sector of a circle with $\angle COB=45^\circ.$ $OZ$ is drawn perpendicular to $CB$ and intersects $CB$ at $W.$ What is the length of $WZ$? Assume the radius of the circle is 10 units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 - 5 \sqrt{2+\sqrt{2}}}
Question 233: Let the difference between any two adjacent interior angles of a convex $\mathrm{n}$-sided polygon be $20^{\circ}$, find the maximum possible value of $\mathrm{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Given the general term of the sequence $\{a_n\}$ is $a_n=n^2\left(\cos^2 \frac{n\pi}{3}-\sin^2 \frac{n\pi}{3}\right)$, calculate the value of $S_{30}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{470}
Given \( a \) is a given real number, the number of subsets for the set \( M = \{ x \in \mathbf{R} \mid |x|x^2 - 3x - a^2 + 2 = 0 \} \) is what?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon. Find the area of this hexagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Misha drew a triangle with a perimeter of 11 and cut it into parts with three straight cuts parallel to the sides, as shown in the figure. The perimeters of the three shaded figures (trapezoids) were found to be 5, 7, and 9. Find the perimeter of the small triangle that resulted from the cutting.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a three-dimensional coordinate system with origin $O,$ points $A,$ $B,$ and $C$ are on the $x$-, $y$-, and $z$-axes, respectively.   If the areas of triangles $OAB,$ $OAC,$ and $OBC$ are 4, 6, and 12, respectively, compute the area of triangle $ABC.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Given the function $y= \sqrt{3}\tan \frac{x}{2}$, with $x\neq (2k+1)\pi (k\in \mathbb{Z})$, find the expression for $x$ when $y=1$. Express your answer as an equation for $x$ in terms of $k$ and $\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k\pi + \frac{\pi}{3}}
For any positive integer $n, S_{n}$ be the set of all permutations of \{1,2,3, \ldots, n\}. For each permutation $\pi \in S_{n}$, let $f(\pi)$ be the number of ordered pairs $(j, k)$ for which $\pi(j)>\pi(k)$ and $1 \leq j<k \leq n$. Further define $g(\pi)$ to be the number of positive integers $k \leq n$ such that $\pi(k) \equiv k \pm 1(\bmod n)$. Compute $$\sum_{\pi \in S_{999}}(-1)^{f(\pi)+g(\pi)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{995 \times 2^{998}}
Marisa has a collection of  $2^8-1=255$  distinct nonempty subsets of  $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process  $2^8-2=254$  times until there is only one set left in the collection. What is the expected size of this set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1024}{255}}
Given the set \( S = \{1, 2, \cdots, 100\} \), determine the smallest possible value of \( m \) such that in any subset of \( S \) with \( m \) elements, there exists at least one number that is a divisor of the product of the remaining \( m-1 \) numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
A certain shopping mall sells toy $A$ with a purchase price of $60$ yuan per unit. According to the requirement, the profit margin per unit at the time of sale must not exceed $40\%$. Based on market research, when the selling price is set at $80$ yuan, 200 units can be sold per day, and for every $1$ yuan decrease in the selling price, an additional 20 units can be sold per day. Find the selling price at which the shopping mall can make a profit of $2500$ yuan per day selling toy $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Let $f$ be an entire function on $\mathbb{C}$ such that $|f(z)|\le 100 \ln |z|$ for $|z|\ge 2$ and $f(i)=2i$. What is $f(1)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2i}