inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the remainder when \(35^{32} + 51^{24}\) is divided by 1785. Use the fact that 1785 factors as \(3 \times 5 \times 7 \times 17\), and that 35 is congruent to 0 modulo 5 and 7, while 51 is congruent to 0 modulo 3 and 17. Apply the Chinese Remainder Theorem to solve the problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the value of \(a > 0\) such that the integral \(I = \int_1^\infty \int_1^\infty (x+y)^{-a} \, dx \, dy\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that f(x) = log(-ax) is an odd function, find the value of the real number a that satisfies the condition f(x) + f(-x) = 0. Express your answer as a numerical value, including the ± symbol if applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
For $0 \leq d \leq 9$, we define the numbers 
\[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]
Find the last digit of the number 
\[S_{0}+S_{1}+\cdots+S_{9}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
In a parallelogram $ABCD$, a perpendicular is dropped from vertex $C$ to line $CD$, intersecting a perpendicular from vertex $A$ to diagonal $BD$ at point $F$. Another perpendicular is dropped from point $B$ to line $AB$, intersecting the perpendicular bisector of segment $AC$ at point $E$. Determine the ratio in which segment $EF$ divides side $BC$. Provide your answer as a ratio of two integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
In the game of projective set, each card contains some nonempty subset of six distinguishable dots. A projective set deck consists of one card for each of the 63 possible nonempty subsets of dots. How many collections of five cards have an even number of each dot? The order in which the cards appear does not matter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{109368}
Given the function \( f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \) for \( x \in [-1, 1] \), find the minimum value of the maximum of \( |f(x)| \) as \( b \) and \( c \) vary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - 2\sqrt{2}}
Given the quadratic equation \(x^2 - 4x + 7 = 0\) with roots \(x_1\) and \(x_2\), calculate the value of the expression:
\[
\frac{x_1^4 - 2x_1^3 + 3}{x_1^3 + x_1^2} + \frac{x_2^4 - 2x_2^3 + 3}{x_2^3 + x_2^2}
\]
using Vieta's formulas without manually calculating the roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{29}{49}}
Determine the dimension of the 2 by 2 orthogonal group, \( O(2) = \{ A \in M(2,2,\mathbb{R}): AA^T = I \} \), where \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). Given the equations \( a^2 + b^2 = 1 \), \( c^2 + d^2 = 1 \), and \( ac + bd = 0 \), find the dimension of this group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Right triangle  $ABC$  has its right angle at  $A$ . A semicircle with center  $O$  is inscribed inside triangle  $ABC$  with the diameter along  $AB$ . Let  $D$  be the point where the semicircle is tangent to  $BC$ . If  $AD=4$  and  $CO=5$ , find  $\cos\angle{ABC}$ .

[asy]
import olympiad;

pair A, B, C, D, O;
A = (1,0);
B = origin;
C = (1,1);
O = incenter(C, B, (1,-1));
draw(A--B--C--cycle);
dot(O);
draw(arc(O, 0.41421356237,0,180));
D = O+0.41421356237*dir(135);
label(" $A$ ",A,SE);
label(" $B$ ",B,SW);
label(" $C$ ",C,NE);
label(" $D$ ",D,NW);
label(" $O$ ",O,S);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{4}{5}$}
The sides of a rectangular park are in the ratio 3 : 2 and its area is 3750 sq m. The cost of fencing it at a certain rate per meter is $200. What is the cost of fencing per meter in ps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Let $n \in \mathbb{N}^*$, $a_n$ be the sum of the coefficients of the expanded form of $(x+4)^n - (x+1)^n$, $c=\frac{3}{4}t-2$, $t \in \mathbb{R}$, and $b_n = \left[\frac{a_1}{5}\right] + \left[\frac{2a_2}{5^2}\right] + ... + \left[\frac{na_n}{5^n}\right]$ (where $[x]$ represents the largest integer not greater than the real number $x$). Find the minimum value of $(n-t)^2 + (b_n + c)^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{25}}
设 $a, b>0$, 满足关于 $x$ 的方程 $\sqrt{|x|}+\sqrt{|x+a|}=b$ 恰有三个不同的实数解 $x_{1}, x_{2}, x_{3}$,且 $x_{1}<x_{2}<x_{3}=b$, 则 $a+b$ 的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
There is a certain amount of milk and water in which milk forms 84%. 26.9 liters of water must be added to this solution to make it a solution in which milk forms 58%. What is the initial volume of the milk and water solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Suppose $\frac ab = \frac35$,$\frac bc=\frac{15}6$, and $\frac cd = 6$. What is the value of $\frac ad$? Express your answer in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a cube with 8 vertices, where any two vertices define a line, calculate the probability that two randomly chosen lines from all possible lines determined by these vertices are skew lines. Express your answer as a simplified fraction in the form \(\boxed{\frac{a}{b}}\), where \(a\) and \(b\) are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{29}{63}}
三棱雉 $P-A B C$ 中, $P A \perp P B, P B \perp P C, P C \perp P A, M$ 为底面三角形内一点, 若 $\sin \angle A P M$ $=\frac{4}{5}, \cos \angle B P M=\frac{\sqrt{3}}{3}$, 则 $\cos \angle C P M=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{69}}{15}$}
Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be defined by $f(x) = cx$ for some constant $c \neq 0$, and let $S$ be a regular surface. Express the Gaussian curvature of $f(S)$ in terms of the Gaussian curvature of $S$. Provide your answer in terms of $c$ and the Gaussian curvature of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{K}{c^2}}
Given the function $y=\sqrt{ax^2+4x+1}$ with a range of $[0,+\infty)$, determine the range of values for $a$ that satisfies this condition. Express your answer using interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,4]}
Find the largest positive integer \( n \) such that there exist \( n \) lattice points on the coordinate plane, and any 3 points form a triangle whose centroid is not a lattice point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the functions $f(x)=1+x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots +\frac{x^{2017}}{2017}$ and $g(x)=1-x+\frac{x^{2}}{2}-\frac{x^{3}}{3}+\frac{x^{4}}{4}+\cdots -\frac{x^{2017}}{2017}$, let $F(x)=f(x+4)\cdot g(x-4)$. If all the zeros of the function $F(x)$ are in the interval $[a-1, a]\cup[b-1,b] (a < b, b\in\mathbb{Z})$, find the value of $a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( f(x)=1+\frac{x^{3}}{3 !}+\frac{x^{6}}{6 !}+\cdots \), \( g(x)=\frac{x}{1 !}+\frac{x^{4}}{4 !}+\frac{x^{7}}{7 !} \cdots \), and \( h(x)=\frac{x^{2}}{2 !}+\frac{x^{5}}{5 !}+\frac{x^{8}}{8 !}+\cdots \). If \( v=f(x)^{2}+g(x)^{2}+h(x)^{2} \), find the value of \( v(\ln 3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{29}{9}}
Lucius owns a small business and spends $10 every day on ingredients. He then makes a portion of French Fries and a portion of Poutine and sells them to the market. The price of French fries is $12 and the price of Poutine is $8. He earns $56 every week after selling all his products and buying all the ingredients. What percentage of his weekly income does he pay as tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0\%}
Find the value of the integral $\int_0^\infty \int_0^\infty \int_0^\infty e^{-(x^2+xy+xz)}\ dx\ dy\ dz$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sqrt{\pi}}{2}\)}
若点$M$，$N$在圆${{C}_{1}}$：${{x}^{2}}+{{y2}}=1$上运动，且$\left| MN \right|=\sqrt{3}$，点$P\left( {{x0}},{{y}_{0}} \right)$是圆${{C2}}$：${{x}^{2}}+{{y2}}-6x-8y+24=0$上一点，则$\left| \overrightarrow{PM}+\overrightarrow{PN} \right|$的取值范围为______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left[ 7,13 \right]$}
Calculate the coefficient of the term \(x^6 y^5 z\) in the expansion of \(\left( 2x + xy - z + \frac{1}{xyz} \right)^{10}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5040}
Find the upper bound for the integral $$\int_0^1 \frac{1}{1+x^2e^{x^2}}\,\mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{4}$}
Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90720}
Let $\zeta = e^{2\pi i/99}$ and $\omega = e^{2\pi i/101}$. The polynomial \[ x^{9999} + a_{9998}x^{9998} + \cdots + a_1x + a_0 \] has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In $\triangle ABC$, given that $AB = 3$, $BC = 6$, and $AC = 4$, let $D$ be a point on side $BC$ such that $AD = 3$. Find the value of $\frac{BD}{DC}$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{29}{7}}
Mary typed a six-digit number, but the two $1$s she typed didn't show. What appeared was $2002.$ How many different six-digit numbers could she have typed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the number of divisors of the number $2^{2} \cdot 3^{3} \cdot 5^{3} \cdot 7^{5}$ that are of the form $4n+1$, where $n$ is a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, respectively, and let $P$ be any point on the ellipse. The coordinates of point $M$ are $(6,4)$. Then, the maximum value of $|PM| + |PF_1|$ is ______________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
What is the length of the shortest sequence that contains all possible 4-digit sequences (from 0000 to 9999) as substrings?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10003}
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1932}
Let $(1+\sqrt{2})^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. Find the greatest common divisor of $b$ and $81$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral \( \int_0^\infty Ai(y) \, dy \), where the Airy function \( Ai(y) \) is given by the integral representation \( Ai(y) = \frac{1}{\pi} \int_0^\infty \cos(\alpha y + \frac{\alpha^3}{3}) \, d\alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
已知实数$x、y$满足：
$x>\max\{-3,y\},(x+3)\left(x^{2}-y^{2}\right)=8$。
则$3x+y-4\sqrt6$的最小值为_______。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
Georgia's school club is selling carnations at various prices and in different bundles for a fundraising event. The pricing tiers are as follows:

Tier 1:
- A single carnation costs $0.60
- A dozen carnations cost $5.00
- A bundle of 25 carnations cost $10.00

Tier 2:
- A single carnation costs $0.50
- A dozen carnations cost $4.50
- A bundle of 50 carnations cost $16.00

Tier 3:
- A single carnation costs $0.40
- A dozen carnations cost $4.00
- A bundle of 100 carnations cost $30.00

Georgia wants to send specific quantities of carnations from different pricing tiers to each of her 8 teachers:

- Teacher 1 receives 2 dozen from Tier 1
- Teacher 2 receives 1 dozen + 5 individual carnations from Tier 2
- Teacher 3 receives 1 bundle of 25 carnations from Tier 1
- Teacher 4 receives 3 dozen + 10 individual carnations from Tier 3
- Teacher 5 receives 2 bundles of 50 carnations from Tier 2
- Teacher 6 receives 1 dozen from Tier 3 + 7 individual carnations from Tier 1
- Teacher 7 receives 1 bundle of 100 carnations from Tier 3
- Teacher 8 receives 3 dozen from Tier 1

She also wants to buy different quantities of carnations from various pricing tiers for each of her 20 friends:

- Friends 1-4 receive 3 carnations each from Tier 2
- Friend 5 receives 1 dozen carnations from Tier 1
- Friends 6-9 receive 5 carnations each from Tier 3
- Friend 10 receives 15 individual carnations from Tier 2
- Friends 11-13 receive 1 dozen + 2 individual carnations each from Tier 1
- Friend 14 receives 25 individual carnations from Tier 3
- Friends 15-18 receive 4 carnations each from Tier 1
- Friend 19 receives 1 bundle of 50 carnations from Tier 2
- Friend 20 receives 2 dozen + 5 individual carnations from Tier 1

How much money would Georgia spend on purchasing all these carnations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$221.90}
Given that \(a, b, c\) are positive integers and the quadratic equation \(a x^{2}+b x+c=0\) has two real roots whose absolute values are both less than \(\frac{1}{3}\), find the minimum value of \(a + b + c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
In the plane Cartesian coordinate system \( xOy \), the set \( S=\left\{(x, y) \mid (|x|+|y|-1)\left(x^{2}+y^{2}-1\right) \leqslant 0\right\} \) represents a plane region. Find the area of this region.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi - 2}
A batch of cookies was divided among 3 tins: 2/3 of all the cookies were placed in either the blue tin or the green tin, and the rest were placed in the red tin. If some fraction of all the cookies were placed in the blue tin and 0.5555555555555556 of the cookies that were placed in the other tins were placed in the green tin, what fraction of all the cookies were placed in the blue tin?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8/27}
Let $f:R\to R$ be a continuous odd function, which vanishes at exactly one point and $f(1)=\frac{1}{2}$. Suppose that $F(x)=\int_{-1}^{x}f(t)dt,\hspace{1cm}\forall x\in[-1,2]$, and $G(x)=\int_{-1}^{x}t|f(f(t))|dt\hspace{1cm} \forall x\in[-1,2]$. If $\lim_{x\to 1}\frac{F(x)}{G(x)}=\frac{1}{14},$ then find the value of $f(\frac{1}{2})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Vasya and Petya set off to go fishing at the lake on foot, while Kolya stayed home with a bicycle. Vasya was the first to head back home, and Kolya set off to meet him on the bicycle. Petya left the lake for home at the moment Vasya and Kolya met. Kolya brought Vasya home on the bicycle, then returned to meet Petya and brought him home as well. The time Petya spent traveling from the lake to home was \( \frac{5}{4} \) of the time Vasya took for the same journey. Calculate how much slower Vasya would have been if he walked the entire way home by himself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A certain product of 936 and w, where w is a positive integer, has some factors. If 3^3 and 12^2 are also factors of the product, and the smallest possible value of w is 144, what is the other factor in the form of 2^x?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^4}
Given \( [x] \) denotes the greatest integer less than or equal to the real number \( x \). The sequence \( \{a_n\} \) is defined by
\[
a_1 = a_2 = \frac{1}{2}, \quad a_{n+1} = 2a_n + a_{n-1} \quad (n \in \mathbf{Z}_{+}).
\]

Find \(\left[\sum_{k=2}^{2016} \frac{1}{a_{k-1} a_{k+1}}\right]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the fourteen numbers, $1^4, 2^4, \ldots, 14^4$. Find the smallest natural number $n$ such that these numbers leave distinct remainders when divided by $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given the lengths of the sides of a regular hexagon as follows: 5 inches, 8 centimeters, 6 inches, 10 centimeters, 7 inches, and 12 centimeters. Convert all sides to one unit (either inches or centimeters) and then find the perimeter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29.81}
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2005}\left|x_{k}-x_{k+1}\right|=2007 \). Define \( y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right) \) for \( k=1, 2, \cdots, 2007 \). Find the maximum possible value of \( \sum_{k=1}^{2006}\left|y_{k}-y_{k-1}\right| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2006}
Calculate the volume of the solid that lies outside the sphere defined by $r=2$ and inside the surface defined by $r=1+\cos\phi$ using a triple integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the value of \(\lambda\) that satisfies the equation \(\binom{\lambda}{n} = n+1\) for some positive integer \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Find the maximum number of natural numbers  $x_1,x_2, ... , x_m$   satisfying the conditions:
a) No  $x_i - x_j , 1 \le i < j \le m$  is divisible by  $11$ ,  and
b) The sum  $x_2x_3 ...x_m + x_1x_3  ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$  is divisible by  $11$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
设 $f(x)$ 是定义在 $\mathbf{R}$ 上的函数, 对任意实数 $x$ 有 $f(x+3) \cdot f(x-4)=-1$. 又当 $0 \leqslant x<7$ 时, $f(x)=\log _{2}(9-x)$, 则 $f(-100)$ 的值为 \qquad. 原始答案是以-\frac{m}{n}的形式给出的，请提供m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $O$ be a point outside a given circle. Two lines $OAB$, $OCD$ through $O$ meet the circle at $A$, $B$, $C$, $D$, where $A$, $C$ are the midpoints of $OB$, $OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
${\dfrac{3}{4}}$}
A die with faces showing the numbers $0,1,2,3,4,5$ is rolled until the total sum of the rolled numbers exceeds 12. What is the most likely value of this sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the expected number of draws needed to obtain a sum divisible by 3 when drawing numbers from a bag containing \(1, 3, 5, 7, 9\), with replacement after each draw.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $[x]$ be the greatest integer less than or equal to the real number $x$. Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1$ for $n \in N^{*}$, find the value of $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $\sin^2 A - \sin^2 B = 2 \sin B \sin C$ and $c = 3b$, find the value of angle $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
In a kilometer race, A can give B a 100 meters start and C a certain amount of meters start. B can give C a 111.11111111111109 meters start in a kilometer race. How much start can A give C in the race?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
已知 $f(x)=e^{\sin x^{2}}$, 则 $\left[-f^{(8)}(0)\right]=$？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5040}
A driver travels 30 miles per hour for 3 hours and at a certain speed for 4 hours to deliver goods to a town every day from Monday to Saturday. The driver travels 1140 miles in a week. What is the speed of the driver in the last 4 hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine the limit of the series \(\lim_{n\rightarrow \infty}\left(\frac{(n+1)(n+2)\ldots(3n)}{n^{2n}}\right).\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
The product of three consecutive integers is 384. What is their sum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In the coordinate plane, points (x, 8) and (20, y) are on line k. Line k passes through the origin. The product of x and y is 160. What is the slope of line k?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8}
Find the derivative of the expression \(-a^T X^T y\) with respect to the vector \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-X^\top y}
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Suppose  $a_{1} < a_{2}< \cdots < a_{2024}$  is an arithmetic sequence of positive integers, and   $b_{1} <b_{2} < \cdots <b_{2024}$  is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Evaluate the integral \( \iint_D xy\ dxdy \), where \( D \) is the region bounded by the curves \( y=x^2 \), \( y=2x^2 \), \( x=y^2 \), and \( x=3y^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{18}}
One face of a tetrahedron has sides of length $3$, $4$, and $5$. The tetrahedron’s volume is $24$ and surface area is $n$. When $n$ is minimized, it can be expressed in the form $n = a\sqrt{b} + c$, where $a$, $b$, and $c$ are positive integers and b is not divisible by the square of any prime. Evaluate $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{157}