inclusionAI/Ring-lite-rl-data huggingface dataset:
A circle is inscribed in a square, then a square is inscribed in this circle. Following this, a regular hexagon is inscribed in the smaller circle and finally, a circle is inscribed in this hexagon. What is the ratio of the area of the smallest circle to the area of the original largest square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{32}}
Find the work done by the vector field $F = (x^2 + y^2)\mathbf{i} + (x^2 - y^2)\mathbf{j}$ along the closed curve formed by the line segments from (0, 0) to (2, 0), from (2, 0) to (2, 2), and from (2, 2) to (0, 0).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Two objects start from points $A$ and $B$, and move towards each other. If the total distance between $A$ and $B$ is $m$ and the product of the individual distances traveled before meeting is $n^2$, determine the polynomial whose roots are the distances traveled by each object from $A$ and $B$ before meeting.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - mx + n^2}
A cyclist rode 96 km 2 hours faster than expected. At the same time, he covered 1 km more per hour than he expected to cover in 1 hour 15 minutes. What was his speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a table tennis match between players A and B, they follow the rules that: the winner of each game gets 1 point and the loser gets 0 points. The match stops when one player has 2 more points than the other or after a maximum of 6 games. Let the probability that A wins each game be $\frac{3}{4}$ and the probability that B wins each game be $\frac{1}{4}$, and assume the outcomes of each game are independent. Find the expected number of games $\xi$ played when the match stops, denoted as $\mathrm{E} \xi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{97}{32}}
Given the inequality $e^{2x} - e^{-2x} - 4x - ae^{x} + ae^{-x} + 2ax \geq 0$ holds true for any positive real number $x$, determine the maximum value of the real number $a$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider all $1000$-element subsets of the set $\{1, 2, 3, ... , 2015\}$ . From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{431}
A collection of coins includes only pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents). Using the coins in this collection, it is possible to create any amount of money less than one dollar (100 cents). What is the smallest possible number of coins in the collection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $P=\{1,2,\ldots,6\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
Inside a non-isosceles acute triangle \(ABC\) with \(\angle ABC = 60^\circ\), point \(T\) is marked such that \(\angle ATB = \angle BTC = \angle ATC = 120^\circ\). The medians of the triangle intersect at point \(M\). The line \(TM\) intersects the circumcircle of triangle \(ATC\) at point \(K\) for the second time. Find \( \frac{TM}{MK} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
Two circles with centers \( O_1 \) and \( O_2 \) intersect at points \( A \) and \( B \). The first circle passes through the center of the second circle and its chord \( BD \) intersects the second circle at point \( C \), dividing the arc \( ACB \) in the ratio \( AC:CB = n \). In what ratio does point \( D \) divide the arc \( ADB \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{n+2}}
A 100-litre mixture of milk and water contains 36 litres of milk. Some amount of this mixture is removed and replaced with an equal quantum of water. If the process is repeated once, then the concentration of the milk stands reduced at 9%. What is the volume of the mixture that was removed and replaced with water?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
A polynomial P satisfies the following criterion: Its coefficients are integers. For all real $(a, b, c, d)$ we have $(P(a) + P(b))(P(c) + P(d)) = P(ac - bd) + P(ad + bc)$. Determine all possible values of $P(2014)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4056196}
What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Calculate the flow of the vector field $F(x,y,z)=(xz,yz,-z^2)$ out of the surface $S=\{(x,y,z) \in \mathbb{R}^3:z^2=1+x^2+y^2, 2 \leq z \leq 3\}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-60π}
In a box, there are 200 balls numbered consecutively from 101 to 300. Two people, A and B, each take balls from the box until they each have 100 balls. If person A takes ball number 102 and person B takes ball number 280, calculate the maximum difference between the sum of the numbers on the balls taken by A and the sum of the numbers on the balls taken by B. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9644}
Given the function f(x) = x^5 / (5x^4 - 10x^3 + 10x^2 - 5x + 1), calculate the sum of f(i/2009) from i=1 to 2009. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
Define
\[ S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1). \]
Then $S$ can be written as $\frac{m \pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4045}
In a pond, there are 6 ducks and they all eat different amounts of the total pieces of bread thrown into the water. The first duck eats 3/7 of all the pieces of bread, the second duck eats 1/5 of the remaining pieces after the first duck, the third duck eats 2/9 of the remaining pieces after the second duck, the fourth duck eats 1/7 of the remaining pieces after the third duck, the fifth duck eats 15 pieces of the bread, and the sixth duck eats 9 pieces of the bread. There are 50 pieces of bread left in the water after all the ducks have eaten. How many total pieces of bread were thrown in the pond?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
The numbers  $1, 2\ldots11$  are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{33}}
In a positive geometric sequence {a_n}, it is known that a_1a_2a_3 = 4, a_4a_5a_6 = 12, and a_{n-1}a_na_{n+1} = 324. Find the value of n. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
11.5. In a convex pentagon $P Q R S T$, angle $P R T$ is half the size of angle $Q R S$, and all sides are equal. Find angle $P R T$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
If a number is selected from the whole numbers 1 to \( m \), and if each number has an equal chance of being selected, the probability that the number is a factor of \( m \) is \(\frac{p}{39}\). Find \( p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A psychologist believes one in every four people has a disorder he calls ZYX syndrome. The clinic where he works has recently doubled its previous number of 26 patients. The distribution of the disorder is not even, as the clinic found that 20% of females and 30% of males have this syndrome. Out of the total patients, 60% are female. The psychologist's diagnostic accuracy is 80% for males and 70% for females to correctly identify patients with ZYX syndrome. On the other hand, the probability of false negatives (incorrectly classifying patients without the syndrome as having it) is 15% for males and 10% for females. Determine how many patients the psychologist is likely to diagnose with ZYX syndrome considering the varying diagnostic accuracies and probability of false negatives.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In how many ways can the positive integers from 1 to 100 be arranged in a circle such that the sum of every two integers placed opposite each other is the same? (Arrangements that are rotations of each other count as the same.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49!\cdot 2^{49}}
To make the statement $(6\ ?\ 3) + 4 - (2 - 1) = 5$ true, replace the question mark with the correct arithmetic operation. Convert the correct arithmetic operation to its ASCII code value and provide this value as your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Evaluate the integral $$\int_0^{\pi/2}(\cos^{10}x )(\sin 12x )dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The Bell Zoo and the Carlton Zoo have various animals. The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. It also has three more elephants than the Carlton Zoo has lions and the same number of elephants as the Carlton Zoo has rhinoceroses. Additionally, the Carlton Zoo has two more elephants than rhinoceroses and twice as many monkeys as it has rhinoceroses, elephants, and lions combined. It also has two more penguins than monkeys. The Bell Zoo has two-thirds as many monkeys as the Carlton Zoo has penguins, two more penguins than monkeys, and only half as many lions as penguins. The total number of rhinoceroses, elephants, lions, monkeys, and penguins in the Bell Zoo is $48$. Find the total number of rhinoceroses, elephants, lions, monkeys, and penguins in the Carlton Zoo.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
Given a sample of $40$ data points, which is divided into six groups, with the frequencies of the first four groups being $10$, $5$, $7$, and $6$ respectively, and the frequency of the fifth group being $0.10$, what is the frequency of the sixth group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $\theta$ be the smallest acute angle for which $\sin \theta,$ $\sin 2 \theta,$ $\sin 3 \theta$ form an arithmetic progression, in some order.  Find $\cos \theta.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4}}
Given the function f(x) = sin(2x) + sqrt(3)cos(2x), find the value of varphi that makes the graph of y = f(x + varphi) symmetric about the line x = 0. Express your answer in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{12}}
Mike had 20 video games but 11 of them weren't working. If he wanted to sell the working games for $8 each and there is a 12% sales tax applied to the selling price, how much money could he earn after considering the sales tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$72}
If several students participate in three competitions where the champion earns 5 points, the runner-up earns 3 points, and the third-place finisher earns 1 point, and there are no ties, what is the minimum score a student must achieve to definitely have a higher score than any other student? 

(The 7th American Junior High School Mathematics Examination, 1991)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
设x,y,z∈R* ，满足x+y+z=xyz，则函数

$f ( x , y , z ) = x ^ { 2 } ( y z - 1 ) + y ^ { 2 } ( z x - 1 ) + z ^ { 2 } ( x y - 1 )$

的最小值是
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A sum of money is to be distributed among a, b, c, d. If c gets $500 more than d, and d's share is $1500, what is the proportion of their shares?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:3}
设 $f(x)=x^{3}+3\left(x^{2}+x+\sin \pi x\right)$, 则 $f(1-\pi)+f(\pi-3)=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
将两个相同的白球、三个相同的红球、四个相同的黑球全部放入三个不同的袋子中. 则没有空袋的放法数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{723}
The sum
\[
\frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!}
\]
can be expressed as a rational number \(N\). Find the last 3 digits of \(2021! \cdot N\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{977}
In $\Delta ABC$, $a+c=6$, and $(3-\cos A)\tan \frac{B}{2}=\sin A$, then the maximum area of $\Delta ABC$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
2. The numbers from 1 to 8 are arranged in a circle. A number is called large if it is greater than its neighbors, and small if it is less than its neighbors. Each number in the arrangement is either large or small. What is the greatest possible sum of the small numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given \( AB \) as the diameter of the smallest radius circle centered at \( C(0,1) \) that intersects the graph of \( y = \frac{1}{|x|-1} \), where \( O \) is the origin. Find the value of \( \overrightarrow{OA} \cdot \overrightarrow{OB} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
If the vector $\vec{a} + 3\vec{b}$ is perpendicular to the vector $7\vec{a} - 5\vec{b}$, and the vector $\vec{a} - 4\vec{b}$ is perpendicular to the vector $7\vec{a} - 2\vec{b}$, then the angle between the vectors $\vec{a}$ and $\vec{b}$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
In triangle ABC, the sides opposite angles A, B, and C are denoted as a, b, and c, respectively. Given that $10a\cos(B) = 3b\cos(A)$ and $\cos(A) = \frac{5\sqrt{26}}{26}$, find the value of angle C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\pi}{4}}
Vasya wrote a note on a piece of paper, folded it in four, and wrote the inscription "MAME" on top. Then he unfolded the note, wrote something else, folded it again along the crease lines at random (not necessarily in the same way as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" is still on top.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
To hit at least 90% of his targets in 60 rounds, John successfully hits 54 targets. After 40 rounds, he has 20 targets remaining. Calculate the maximum number of rounds he can afford to miss.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let's consider all possible natural seven-digit numbers, in the decimal notation of which the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995^{th}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3651427}
The integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1,2,3,4,$ and $5,$ then the expected column number of the entry $23$ can be written as $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given a triangular pyramid with areas of faces S_1, S_2, S_3, S_4 and distances from a point within the pyramid to the faces H_1, H_2, H_3, H_4, and a constant k such that S_1/1 = S_2/2 = S_3/3 = S_4/4 = k, calculate the value of 1H_1 + 2H_2 + 3H_3 + 4H_4 in terms of the volume V of the pyramid and k. Express your answer as a fraction involving V and k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3V}{k}}
Given the circle \( O: x^{2}+y^{2}=4 \) and the curve \( C: y=3|x-t| \), and points \( A(m, n) \) and \( B(s, p) \) \((m, n, s, p \in \mathbb{N}^*) \) on the curve \( C \), such that the ratio of the distance from any point on the circle \( O \) to point \( A \) and to point \( B \) is a constant \( k (k>1) \), find the value of \( t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
There are six thin sticks with lengths of \(3, 2 \sqrt{2}, 2, 2, 2, 2\). Using them to form a triangular prism, find the cosine of the angle formed by the lines containing the two longer edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13 \sqrt{2}}{24}}
Given the point \( P \) on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \left(a>b>0, c=\sqrt{a^{2}-b^{2}}\right)\), and the equation of the line \( l \) is \(x=-\frac{a^{2}}{c}\), and the coordinate of the point \( F \) is \((-c, 0)\). Draw \( PQ \perp l \) at point \( Q \). If the points \( P \), \( Q \), and \( F \) form an isosceles triangle, what is the eccentricity of the ellipse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
Given two concentric circles with radii 1 and 3 and common center $O$, a third circle touches both of them. Find the angle between the tangents to the third circle drawn from point $O$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
Evaluate the limit $\lim \limits_{n\to \infty} \sum \limits_ {k=0}^n \left(1-\frac{k^2}{2n}\right)^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2\pi}}{2}}
Calculate the real solutions $x\in\mathbb{R}$ to $$ \tag1\sqrt{4^x-6^x+9^x}+\sqrt{9^x-3^x+1}+\sqrt{4^x-2^x+1} = 2^x+3^x+1 $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the set M = {1,2,3,4,5,6,7} and the proposition p: ∀n∈M, n > 1, what is the negation of p? Express your answer as a logical proposition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∃n∈M, n ≤ 1}
Let $X=\{(x,y):x^2+y^2<5\}$ and $K=\{(x,y):1\leq x^2+y^2\leq 2 or 3\leq x^2+y^2\leq 4\}$. Then: Find the number of relatively compact connected components of $X\setminus K$ in $X$.
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Answer: \boxed{2}
Given \(\left[\prod_{k=1}^{n} \frac{(2k+1)^{4}+(2k+1)^{2}+1}{(2k)^{4}+(2k)^{2}+1}\right] = 44\), find \(n = \qquad\), where \([x]\) denotes the greatest integer less than or equal to the real number \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
If $\log_{3}2^4\cdot\log_{4}3^5\cdot\log_{5}4^6\cdot....\log_{64} {65}^{66}=y!$, what is the value of $y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
在平面直角坐标系 $x O y$ 中，圆 $C_{1}: x^{2}+y^{2}-a=0$ 关于直线 $l$ 对称的圆为 $C_{2}: x^{2}+y^{2}+2 x-2 a y+3=0$, 则直线 $l$ 的方程为 ax + by + c = 0。请给出 a + |b| + c 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The areas of projections of a triangle onto the coordinate planes $Oxy$ and $Oyz$ are equal to $\sqrt{6}$ and $\sqrt{7}$, respectively, while the area of the projection onto the plane $O yz$ is an integer. Find the area of the triangle if it is also known to be an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Miki has 15 oranges and 15 pears. She extracts 12 ounces of orange juice from 3 oranges and 10 ounces of pear juice from 4 pears. Miki then makes a juice blend using 5 pears and 4 oranges. Calculate the percentage of the blend that is pear juice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43.86\%}
Evan writes a random positive integer on a board: the integer  $k$  has probability  $2^{-k}$  of being written. He keeps writing integers in this way repeatedly until he writes an integer that he had written before. He then takes all the integers he has written besides his last, sorts them in the order he first drew them, and also sorts them in increasing order, forming two sequences. For example, if he wrote  $5,8,2,3,6,10,2$  in that order then his two sequences would be  $5,8,2,3,6,10$  and  $2,3,5,6,8,10.$ Find the probability that for all  $k \in \{ 1,4,34 \},$  that  $k$  was written, and  $k$  appears in the same position in both sequences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{-39}}
Given points A, B, C, and P all on the surface of a sphere O with PC as its diameter, AB is perpendicular to BC, AB=2, BC=4. If the volume of sphere O is $8\sqrt{6}\pi$, find the tangent value of the angle formed by the skew line PB and AC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the distances 1, 2, 4, 8, 16, 32 between points on a plane, what is the smallest number of points required to accommodate all these distances? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let $P(3, y)$ be a point on the terminal side of angle $\alpha$, if $\cos\alpha= \frac {3}{5}$, then $y= \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±4}
Given positive integers \(a\), \(b\) \((a \leq b)\), a sequence \(\{ f_{n} \}\) satisfies:
\[ f_{1} = a, \, f_{2} = b, \, f_{n+2} = f_{n+1} + f_{n} \text{ for } n = 1, 2, \ldots \]

If for any positive integer \(n\), it holds that 
\[ \left( \sum_{k=1}^{n} f_{k} \right)^2 \leq \lambda \cdot f_{n} f_{n+1}, \]
find the minimum value of the real number \(\lambda\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \sqrt{5}}
A group of 101 Dalmathians participate in an election, where they each vote independently on either candidate \(A\) or \(B\) with equal probability. If \(X\) Dalmathians voted for the winning candidate, the expected value of \(X^{2}\) can be expressed as \(\frac{a}{b}\) for positive integers \(a, b\) with \(\operatorname{gcd}(a, b)=1\). Find the unique positive integer \(k \leq 103\) such that \(103 \mid a-bk\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
If the power function $f(x)=(m^{2}-9m+19)x^{m-4}$ is monotonically increasing on $(0,+\infty)$, then the real number $m=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If the function
$$
f(x) = |a \sin x + b \cos x - 1| + |b \sin x - a \cos x| \quad (a, b \in \mathbf{R})
$$
attains a maximum value of 11, then $a^{2} + b^{2} = \, \, \, $ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}