inclusionAI/Ring-lite-rl-data huggingface dataset:
In a right triangle $DEF$ where leg $DE = 30$ and leg $EF = 40$, determine the number of line segments with integer length that can be drawn from vertex $E$ to a point on hypotenuse $\overline{DF}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
A number is called *6-composite* if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is *composite* if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)

*Ray Li.*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{441}
The sequence \( a_{0}, a_{1}, \dots \) is defined as follows:

\[ a_{0} = 1995, \]
\[ a_{n} = n a_{n-1} + 1, \quad \text{for } n \geq 1. \]

What is the remainder when \( a_{2000} \) is divided by 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
已知直线 $L: x+y-9=0$ 和圆 $M: 2 x^{2}+2 y^{2}-8 x-8 y-1=0$, 点 $A$ 在直线 $L$ 上, $B, C$ 为圆 $M$ 上两点, 在 $\triangle A B C$ 中, $\angle B A C=45^{\circ}, A B$ 过圆心 $M$,则点 $A$ 横坐标范围为 $\qquad$. 请给出该范围的两个端点之和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
Given proposition p: The function y = kx + 1 is an increasing function on R, and proposition q: The function y = x^2 - 2x + 3 has a maximum value of 3 and a minimum value of 2 on the interval [0, k]. If p ∧ q is a false proposition, and p ∨ q is a true proposition, find the range of values for k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \in (0, 1) \cup (2, +\infty)}
The power function $f(x)=(m^{2}+2m-2)x^{m}$ is a decreasing function on $(0,+\infty)$. Find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
In parallelogram $A B C D$ with angle $A$ equal to $60^{\circ}$, the bisector of angle $B$ intersects side $C D$ at point $E$. A circle with radius $R$ is inscribed in triangle $E C B$. Another circle is inscribed in trapezoid $A B E D$. Find the distance between the centers of these circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R\sqrt{7}}
Given the angles of a triangle and side $c$, express the height corresponding to side $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{c \sin \alpha \sin \beta}{\sin \gamma}}
one day , raviraj left home and cycled 20 km southwards , turned right and cycled 10 km and turned right and cycled 20 km and turned left and cycled 20 km . how many kilometres will he have to cycle to reach his home straight ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Consider the sum $\overline{a b} + \overline{ c d e}$, where each of the letters is a distinct digit between $1$ and $5$. How many values are possible for this sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let $z$ be a complex number satisfying $(z+\tfrac{1}{z})(z+\tfrac{1}{z}+1)=1$. Evaluate $(3z^{100}+\tfrac{2}{z^{100}}+1)(z^{100}+\tfrac{2}{z^{100}}+3)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
已知椭圆$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$，过椭圆上一点M作圆$x ^ { 2 } + y ^ { 2 } = b ^ { 2 }$的两条切线，切点为P，Q，设直线P与x轴、y轴分别交于点E、F，求$\triangle EOF$面积的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { b ^ { 3 } } { a }$}
On a very long narrow highway where overtaking is impossible, $n$ cars are driving in random order, each at its preferred speed. If a fast car catches up to a slow car, the fast car has to slow down and drive at the same speed as the slow car. As a result, the cars form groups. Find the expected number of "lonely" cars, i.e., groups consisting of a single car.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the number of ordered quadruples of positive integers \((a,b,c,d)\) such that \(a,b,c,\) and \(d\) are all (not necessarily distinct) factors of \(30\) and \(abcd > 900\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1940}
Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from $-10$ to $10$ in both the $x$ and $y$ coordinates. Consequently, when he illustrates the graph of a quadratic, he likes to use a quadratic with the following properties:

1. The quadratic sends integers to integers.
2. The quadratic has distinct integer roots both between $-10$ and $10$, inclusive.
3. The vertex of the quadratic has integer $x$ and $y$ coordinates both between $-10$ and $10$, inclusive.

How many quadratics are there with these properties?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{478}
Tim has a bag of some orange-flavoured sweets and Peter has a bag of 44 grape-flavoured sweets. They have to divide up the sweets into small trays with equal number of sweets; each tray containing either orange-flavoured or grape-flavoured sweets only. If the largest possible number of sweets in each tray without any remainder is 4, how many orange-flavoured sweets does Tim have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given condition $p: x^2 + 2x > 3$, and condition $q: x > a$, and $\neg p$ is a sufficient but not necessary condition for $\neg q$, determine the range of possible values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \geq 1}
On Fedor's bookshelf, there are volumes of works by various authors. When a friend borrowed two volumes of Pushkin, Fedor noticed that he had fully read at least half of the remaining books. After the friend returned the two volumes of Pushkin and borrowed two volumes of Lermontov, Fedor realized that he had fully read less than one-third of the remaining volumes. What is the maximum number of volumes that could have been on Fedor's shelf?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Evaluate the integral: $$\int_{0}^{2\pi} \frac{\cos^2(mx)}{\cos^2(x)} \mathrm dx$$ where $m$ is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2m\pi}
A flagpole is 12 feet tall. It breaks, folding over at a certain point, such that what was the tip of the flagpole is now dangling two feet above the ground. The flagpole broke 7 feet from the base. At what point did the flagpole break in relation to its height?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7/12}
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$37:1$}
Points \( K \) and \( M \) are the midpoints of the edges \( AB \) and \( AC \) of the triangular pyramid \( ABCD \) with a base area \( p \). Find the area of face \( BCD \), given that the cross-section \( DKM \) has an area \( q \), and the base of the pyramid's height falls on the intersection of the medians of the base triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\sqrt{4q^2 + \frac{p^2}{12}}}
The Knave of Hearts tells only the truth on Mondays, Tuesdays, Wednesdays, and Thursdays. He tells only lies on all the other days. The Knave of Diamonds tells only the truth on Fridays, Saturdays, Sundays, and Mondays. He tells only lies on all the other days. On one day last week, they both said, "Yesterday I told lies." On which day of the week did they make this statement? Provide your answer as the full day of the week.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Friday}
A mathematical demonstration showed that there were distinct positive integers such that $97^4 + 84^4 + 27^4 + 3^4 = m^4$. Calculate the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Given that $\alpha$ and $\beta$ are both acute angles, and $\cos(\alpha+\beta)=\sin\alpha\sin\beta$, then $\tan \alpha=$\_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the maximum number of sides of a convex polygon that has no two adjacent obtuse angles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( D \) be a point inside \( \triangle ABC \) such that \( \angle BAD = \angle BCD \) and \( \angle BDC = 90^\circ \). If \( AB = 5 \), \( BC = 6 \), and \( M \) is the midpoint of \( AC \), find the length of \( DM \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{11}}{2}}
On a computer keyboard, the key for the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will actually print.

Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers satisfy this condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
On an 8x8 board, chips are placed in such a way that each chip lies in a row or column that contains only one chip. What is the maximum possible number of chips on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Triangle \(ABC\) is given with \(AB = 13\), \(BC = 14\), and \(CA = 15\). Let \(E\) and \(F\) be the feet of the altitudes from \(B\) and \(C\), respectively. Let \(G\) be the foot of the altitude from \(A\) in triangle \(AFE\). Find \(AG\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{396}{65}}
A right triangle with legs each measuring 7 cells was drawn on graph paper. Then all the lines of the grid inside the triangle were outlined. What is the maximum number of triangles that can be found in this drawing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Given a bivariate standard Gaussian distribution $(A, B) \sim \mathbb{N}(0, I_2)$ on $\mathbb{R}^2$, where $I_2$ is the $2 \times 2$ identity matrix, find the expected value of the maximum between $2.5A + B$ and $A + 2.5B$:

$$\mathbb{E}\left[\max(2.5A + B, A + 2.5B)\right]$$

The initial attempt to express this as a conditional expectation did not simplify the calculation:

$$\mathbb{E}(2.5A + B | A > B) \cdot \mathbb{P}(A > B) + \mathbb{E}(A + 2.5B | A < B) \cdot \mathbb{P}(A < B)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{3}{2\sqrt{\pi}}\)}
The circle inscribed in the isosceles triangle \( ABC \) with base \( AB \) touches the side \( BC \) at point \( M \). Segment \( AM \) intersects the circle at point \( N \). Find \(\frac{MN}{AN}\) if \(\frac{AB}{BC} = k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(2 - k)}
Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing in a circle around a Christmas tree. Afterwards, each of them was asked if her neighbor to the right was in a blue dress. It turned out that only those girls who stood between girls in dresses of the same color answered correctly. How many girls could have answered affirmatively?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Find the maximum real number \( k \) such that for any positive numbers \( a \) and \( b \), the following inequality holds:
$$
(a+b)(ab+1)(b+1) \geqslant k \, ab^2.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27}{4}}
Seven dwarfs lined up by height, starting with the tallest. The first (i.e., the tallest) said: "My height is 60 cm." The second said: "My height is 61 cm." Then in order: "My height is 62 cm," "My height is 63 cm," "My height is 64 cm," "My height is 65 cm," and finally the shortest said: "My height is 66 cm." What is the largest number of dwarfs that could be telling the truth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the acute triangle \(ABC\), the altitudes \(AA_1\) and \(BB_1\) are drawn. The bisector of the exterior angle at vertex \(C\) intersects lines \(AB\) and \(A_1B_1\) at points \(L\) and \(K\) respectively. It turns out that \(CL = 2CK\). Find the angle \(C\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
In triangle \(ABC\), a median \(AM\) is drawn. Circle \(\alpha\) passes through point \(A\), touches line \(BC\) at point \(M\), and intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. On arc \(AD\) that does not contain point \(E\), a point \(F\) is chosen such that \(\angle BFE = 72^\circ\). It is found that \(\angle DEF = \angle ABC\). Find \(\angle CME\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36^\circ}
Calculate the limit of the function:
$$\lim _{x \rightarrow 0}(\cos \pi x)^{\frac{1}{x \cdot \sin \pi x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-\frac{\pi}{2}}}
We call a number antitriangular if it can be expressed in the form \(\frac{2}{n(n+1)}\) for some natural number \(n\). For how many numbers \(k\) (where \(1000 \leq k \leq 2000\)) can the number 1 be expressed as the sum of \(k\) antitriangular numbers (not necessarily distinct)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
设复数 \(z\) 满足 \(|3z-7i|=3\)，求 \(\left|\frac{z^{2}-2z+2}{z-1+i}\right|\) 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}
Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^{3} + a x^{2} + b x \). If there exist three real numbers \( x_{1}, x_{2}, x_{3} \) such that \( x_{1} + 1 \leqslant x_{2} \leqslant x_{3} - 1 \), and \( f(x_{1}) = f(x_{2}) = f(x_{3}) \), find the minimum value of \( |a| + 2|b| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
设函数 $f(x)=a \ln x+\frac{2 a}{x}-\frac{e^{x}}{x^{2}}$.
若 $a \leq e$, 求 $f(x)$ 的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$a \ln 2+a-\frac{e^{2}}{4}$}
For a numerical sequence $\left\{x_{n}\right\}$, in which all terms are distinct starting from $n \geq 2$, the relationship $x_{n} = \frac{x_{n-1} + 98 x_{n} + x_{n+1}}{100}$ holds. Find $\sqrt{\frac{x_{2023} - x_{1}}{2022} \cdot \frac{2021}{x_{2023} - x_{2}}} + 2021$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2022}
Determine the monotonically increasing interval of the function $y = \ln(4-x) + \ln(2+x)$. Express your answer as an interval in standard interval notation (e.g., (a, b)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 1)}
From the 4040 integers ranging from -2020 to 2019, three numbers are randomly chosen and multiplied together. Let the smallest possible product be $m$ and the largest possible product be $n$. What is the value of $\frac{m}{n}$? Provide the answer in simplest fraction form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{2018}{2019}}
How to calculate the value of the sum $$\sum_{k = 0}^{\infty} \frac{(-1)^k}{k} \sin(2k)$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
There are six piles of apples, and the number of apples in each pile forms an arithmetic sequence. Junjun selects one of the piles and takes out 150 apples, then distributes these to the remaining 5 piles with 10, 20, 30, 40, and 50 apples respectively. After the distribution, Junjun magically finds that the number of apples in these 5 piles correspond to the 2, 3, 4, 5, and 6 times the number of apples in the pile he selected. How many apples are there in total in the six piles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{735}
Let $p(x)=x^{2}-x+1$. Let $\alpha$ be a root of $p(p(p(p(x))))$. Find the value of $(p(\alpha)-1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given the sequence \(\left\{a_{n}\right\}\) satisfying \(a_{1}=a > 2\), \(a_{2017} = 2017\), and for any positive integer \(n\), \(a_{n+1} = a_{n}^{2} - 2\). Compute \(\left\lfloor \frac{\sqrt{a-2}}{10^{6}} a_{1} a_{2} \cdots a_{2017} \right\rfloor\), where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding the real number \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( \triangle ABC \) be a triangle with \( \angle A = 18^{\circ} \) and \( \angle B = 36^{\circ} \). Let \( M \) be the midpoint of \( AB \), \( D \) a point on ray \( CM \) such that \( AB = AD \), \( E \) a point on ray \( BC \) such that \( AB = BE \), and \( F \) a point on ray \( AC \) such that \( AB = AF \). Find \( \angle FDE \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
已知点集 $A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leqslant\left(\frac{5}{2}\right)^{2}\right.\right\}$,
$B=\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\}$. 则点集 $A \cap B$ 中的整点（即横、纵坐标均为整数的点）的个数是 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A right circular cone is sliced into five pieces of equal height by planes parallel to its base. Determine the ratio of the volume of the second-largest piece to the volume of the largest piece.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{37}{61}}
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers  ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$th number said by Tadd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5979}
Bob's Rice ID number has six digits, each a number from 1 to 9, and any digit can be used any number of times. The ID number satisfies the following property: the first two digits is a number divisible by 2, the first three digits is a number divisible by 3, etc. so that the ID number itself is divisible by 6. One ID number that satisfies this condition is 123252. How many different possibilities are there for Bob's ID number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324}
How many rectangles can be formed where each vertex is a point on a 4x4 grid of equally spaced points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Evaluate the product $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{6561} \cdot \frac{19683}{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Given the ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. A line passing through $F_{1}$ intersects the ellipse at points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$. If the radius of the incircle of $\triangle ABF_{2}$ is $1$, $|F_{1}F_{2}|=2$, and $|y_{1}-y_{2}|=3$, then the eccentricity of the ellipse is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {2}{3}}
Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=92^\circ$ and $\measuredangle ADC=68^\circ$, find $\measuredangle EBC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$68^\circ$}
Given a line $l$ passes through point $P(0, -2)$, and it always intersects with the line segment $AB$ where $A(1, -1)$ and $B(2, -4)$ are endpoints, find the range of values for the inclination angle of line $l$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \frac{\pi}{4}] \cup [\frac{3\pi}{4}, \pi)}
There are 84 positive integer solutions for an equation involving x, y, z, and t. The equation is x + y + z + t = 10. What is the value of t in one of the solutions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( f \) be a function taking positive integers to positive integers, such that:

(i) \( f \) is increasing (\( f(n + 1) > f(n) \) for all positive integers \( n \))
(ii) \( f(mn) = f(m) f(n) \) for all positive integers \( m \) and \( n \)
(iii) if \( m \neq n \) and \( m^n = n^m \), then \( f(m) = n \) or \( f(n) = m \)

Find all possible values of \( f(60) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3600}
What is the value of \(b\) such that the graph of the equation \[ 3x^2 + 9y^2 - 12x + 27y = b\] represents a non-degenerate ellipse?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{129}{4}}
There are six lines in the plane. No two of them are parallel, and no point lies on more than three lines. What is the minimum possible number of points that lie on at least two lines?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The vending machine fails to drop a snack at a certain rate. One in ten times, it will accidentally drop two snacks. The rest of the time, it drops the purchased snack. If thirty people have used the vending machine once each to purchase a snack, and the vending machine has dropped 28 snacks, what is the rate at which the vending machine fails to drop a snack?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/6,}
A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is squarefree. Find $m + n + p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
1. For all positive real numbers $a, b, c, d$ we have $\left(\frac{a^{3}}{a^{3}+15 b c d}\right)^{\frac{1}{2}} \geq \frac{a^{x}}{a^{x}+b^{x}+c^{x}+d^{x}}$, then $x=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{8}}
A sum was put at simple interest at a certain rate for 10 years. Had it been put at a 5% higher rate, it would have fetched Rs. 200 more. What was the sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}