inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that the first term of an arithmetic sequence is 24 and it starts to become negative from the 10th term, find the range of the common difference d. Express your answer as an inequality using standard notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3 \leq d < -\frac{8}{3}}
Solve the equation \(a b c d e f = a + b + c + d + e + f\) in natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1, 1, 1, 2, 6) in any order.}
Calculate the flux of the vector field $\bar{V} = z\bar{i} + x\bar{j} + y\bar{k}$ through the surface defined by $x^2 + y^2 + z^2 = 1, x \ge 0, y \ge 0, z \ge 0$ using Gauss's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $p$ be the largest prime less than $2013$ for which \[ N = 20 + p^{p^{p+1}-13} \] is also prime. Find the remainder when $N$ is divided by $10^4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1781}
In a 200-meter race, Sonic is 16 meters ahead of Dash when Sonic finishes the race. The next time they race, Sonic starts 2.5 times this lead distance behind Dash, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Determine the distance Sonic is ahead of Dash when Sonic finishes the second race.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.2}
Given a set of 2023 positive numbers, some possibly zero but all being less than 1, with the sum of the elements equal to 1, determine the maximum number of ordered pairs \((j,k)\) such that \(a_k + \frac{a_j^2}{4} \ge \frac{1}{2022}\) and \(k < j\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2023 \times 2022}{2}}
Determine the number of integers \(1 \le a \le 2015\) that are primitive residues modulo 2015. An integer \(a\) is a primitive residue modulo \(n\) if it is exactly one of the following types of residues: quadratic, cubic, or quintic, modulo \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{464}
Find the smallest positive integer $n$ that has at least $7$ positive divisors $1 = d_1 < d_2 < \ldots < d_k = n$, $k \geq 7$, and for which the following equalities hold:

$$d_7 = 2d_5 + 1 \text{ and } d_7 = 3d_4 - 1$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
Find a vector field \( Z = a\partial_x + b\partial_y + c\partial_z \) in \( \mathbb{R}^3 \) such that \( \alpha(Z) = 1 \) and \( d\alpha(Z,.) = 0 \), where \( \alpha = dz - ydx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\partial_z}
Rachel has two indistinguishable tokens, and places them on the first and second squares of a \(1 \times 6\) grid of squares. She can move the pieces in two ways:
- If a token has a free square in front of it, then she can move this token one square to the right.
- If the square immediately to the right of a token is occupied by the other token, then she can "leapfrog" the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token.

If a token reaches the 6th square, then it cannot move forward anymore, and Rachel must move the other one until it reaches the 5th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the 5th square and the 6th square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Given the polynomial \[ f(x)=x^{2020} + \sum_{i=0}^{2019} c_{i} x^{i} \quad (c_{i} \in\{-1,0,1\}), \] determine the maximum number of positive integer roots of \( f(x) \), including their multiplicities, given that the equation \( f(x) = 0 \) has no negative integer roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the minimum value of $$B=\frac{x}{y}+4\sqrt{1+\frac{y}{z}}+6\sqrt[3]{1+\frac{z}{x}}$$ given that $x,y,z>0$ and $x\geq max${$y,z$}.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+4\sqrt{2}+6\sqrt[3]{2}}
Two identical cylindrical vessels are connected at the bottom by a small-diameter pipe with a valve. While the valve was closed, water was poured into the first vessel and oil into the second in such a way that the levels of the liquids were equal and reached $h = 40$ cm. At what level will the water be in the first vessel if the valve is opened? The density of water is 1000 kg/m³ and the density of oil is 700 kg/m³. The volume of the connecting pipe can be neglected. Provide the answer in centimeters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34 \, \text{cm}}
Evaluate the definite integral \( \int_0^a \frac{2a-x}{a+x} \sqrt{\frac{a-x}{a+x}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a(7 - 2\pi)}
Jacqueline has 2 liters of soda. Liliane has 60% more soda than Jacqueline, and Alice has 40% more soda than Jacqueline. Calculate the percentage difference between the amount of soda Liliane has compared to Alice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.29\%}
A hollow box, shaped like a rectangular prism and with negligible thickness, has a volume of $108$ cubic units. The top of the box is removed, exposing the faces on the inside of the box. What is the minimum possible value for the sum of the areas of the faces on the outside and inside of the box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{216}
Given that $X_1, \dots, X_7$ and $Y_1, \dots, Y_8$ are independently and identically distributed (i.i.d.) according to a $N(\mu, \sigma^2)$ distribution, find the probability $P(X_1<Y_{(8)})$, where $Y_{(8)}$ denotes the maximum of $Y_1, \dots, Y_8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{8}{9}\)}
John wants to lose weight. He eats 1800 calories a day and burns a certain amount of calories a day. If he needs to burn 4000 calories to lose 1 pound, it will take him 80 days to lose 10 pounds. How many calories does John burn a day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2300}
Derek is deciding between two different-sized pizzas at his favorite restaurant. The menu lists a 14-inch pizza and an 18-inch pizza. Calculate the percent increase in area if Derek chooses the 18-inch pizza over the 14-inch pizza.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65.31\%}
Find the remainder when\[\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}\]is divided by $1000$.
 ~ pi_is_3.14
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A, B, C, and D enter into a partnership. A subscribes 1/3 of the capital, B 1/4, C subscribes some fraction, and D the rest. A gets a share of Rs. 805 in a profit of Rs. 2415. What fraction of the capital did C subscribe?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5/12}
Evaluate the integral \( \int_{\{ p \}} f(x) \, \text{dvol}_{\{ p \}}(x) \), where \( f: S^2 \rightarrow \mathbb{R} \) is a continuous function on the standard sphere, \( p \in S^2 \) is a fixed point, and \( \text{dvol}_{\{ p \}}(x) \) is the Riemannian volume element of the manifold \( \{ p \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(p)}
Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
There are 7 balls of each of the three colors: red, blue, and yellow. When randomly selecting 3 balls with different numbers, determine the total number of ways to pick such that the 3 balls are of different colors and their numbers are not consecutive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
## Task A-1.6.

Determine the smallest natural number $n$ such that half of $n$ is the square of some natural number, a third of $n$ is the cube of some natural number, and a fifth of $n$ is the fifth power of some natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{15}\cdot3^{10}\cdot5^{6}}
Calculate the variance of the difference between the maximum and minimum values of a set of n independent and identically distributed (i.i.d.) uniform variables, $X_i$, where each $X_i$ is uniformly distributed on the interval (0,1). Specifically, find $variance[\max(\{X_i\}) - \min(\{X_i\})]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2n-2}{(n+1)^2(n+2)}\)}
How can you measure 15 minutes using a 7-minute hourglass and an 11-minute hourglass?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given that F<sub>1</sub> and F<sub>2</sub> are the left and right foci of the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$ respectively, if there exists a point P on the left branch of the hyperbola such that |PF<sub>1</sub>|=|F<sub>1</sub>F<sub>2</sub>|, and the distance from F<sub>1</sub> to the line PF<sub>2</sub> is $$\sqrt {7}a$$, then the eccentricity of the hyperbola e is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the interval for the parameter \( c \) such that the expression \( \frac{x^2-x+c}{x^2+x+2c} \) can take all real values for any real number \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-6, 0)}
Four objects of different masses are available. What is the minimum number of weighings on a balance scale without weights needed to determine the heaviest and the lightest of these objects?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the volume of a tetrahedron whose vertices are located at points with coordinates \(\left(F_{n}, F_{n+1}, F_{n+2}\right), \quad\left(F_{n+3}, F_{n+4}, F_{n+5}\right), \quad\left(F_{n+6}, F_{n+7}, F_{n+8}\right),\) and \(\left(F_{n+9}, F_{n+10}, F_{n+11}\right)\), where \(F_{i}\) is the \(i\)-th term of the Fibonacci sequence: \(1, 1, 2, 3, 5, 8 \ldots\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose that  $a_1 = 1$ , and that for all  $n \ge 2$ ,  $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$  Suppose furthermore that  $b_n = a_1 + a_2 + \ldots + a_n$  for all  $n$ . If  $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$  for some  $k$ , find  $k$ .

*Proposed by Andrew Wu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k = 2022}
Determine the convergence of the series:

$$ \sum_{k=1}^{\infty} (\sqrt k - 2\sqrt {k + 1} + \sqrt {k + 2}) $$

Assume that methods like the Root Test or comparing the series directly are not applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \sqrt{2}}
Let $f(x) = g(x) \cdot |x-1| \cdot |x-2| \cdot |x-3|^2 \cdot |x-4|^3$, where $g(x) = x^3 + ax^2 + bx + c$. If $f(x)$ is differentiable for all $x$ and $f'(3) + f'''(4) = 0$, find the third factor of $g(x)$ given that $x-1$ and $x-2$ are factors and $f'(3) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x - 4}
Find the length of a curve of constant width \( h \), composed of an odd number of arcs of circles with radius \( h \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi h}
Five cards labeled 1, 3, 5, 7, 9 are laid in a row in that order, forming the five-digit number 13579 when read from left to right. A swap consists of picking two distinct cards and then swapping them. After three swaps, the cards form a new five-digit number \( n \) when read from left to right. Compute the expected value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50308}
由曲线 $x^{2}=4 y, x^{2}=-4 y, x=4, x=-4$ 围成的图形绕 $y$ 轴旋转一周所得的旋转体的体积为 $V_{1}$; 满足 $x^{2}+y^{2} \leq 16, x^{2}+(y-2)^{2} \geqslant 4, x^{2}+(y+2)^{2} \geqslant 4$ 的点绕 $y$ 轴旋转一周所得的旋转体的体积为 $V_{2}$。设$V_{1}=aV_{2}+b$，请给出a+b的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the number of elements of order $p^n$ in the group ${\bf Z}_{p^n}\oplus{\bf Z}_p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^n(p-1)}
Determine if the limit \( L = \lim_{n \rightarrow \infty} \int_{0}^{n} \frac{f(x/n)}{(1+x)^{4}} \, dx \) exists for a continuous function \( f : [0, \infty) \rightarrow \mathbb{R} \) such that \( |f(x)| \leq 1 + x^{2} \) for all \( x \geq 0 \), and find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{f(0)}{3}}
Determine the Galois group of the polynomial $x^5 - 5$ over the field of rational numbers $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{20}}
Virginia had 13 Sweettarts. She gave equal numbers of Sweettarts to some friends and then she ate what was left. Each person ate 3 Sweettarts. How many friends did Virginia give Sweettarts to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the function $y(x)$ that makes the integral \( \int_{0}^{1} (x^{2} + y^{'2}) \, dx \) stationary, subject to the constraints \( \int_{0}^{1} y^2 \, dx = 2 \), \( y(0) = 0 \), and \( y(1) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sin(\pi x)}
in a fraction , if numerator is added by 2 and denominator by 1 it results 1 , and if numerator added again by 4 and denominator by 2 it results 1 / 2 . find the fraction ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{4}}
Calculate the probability that a Brownian motion $W$ with $T_1 = 1$ year and $T_2 = 2$ years satisfies $W_{T_1} > 0$ and $W_{T_2} < 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
Akash's birthday cake is a $4 \times 4 \times 4$ cube. The top and four sides of the cake are frosted, but the bottom is not. If the cake is cut into 64 smaller $1 \times 1 \times 1$ cubes, how many of the smaller cubes have frosting on exactly two faces?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
$O$ 是凸四边形 $A B C D$ 对角线 $A C$ 和 $B D$ 的交点。已知三角形 $A O B, B O C, C O D, D O A$ 的周长相同，且三角形 $A O B, B O C, C O D$ 的内切圆半径分别为 $3, 4, 6$。求三角形 $D O A$ 的内切圆半径。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}
The faces of a cube are inscribed with six natural numbers (possibly repeating) such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the possible degree of the polynomial \(P(x)\) with real coefficients, given that \(P(a_1) = 0\) and \(P(a_{i+1}) = a_i\) for \(i = 1, 2, \ldots\), where \(\{a_i\}_{i=1,2,\ldots}\) is an infinite sequence of distinct natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Solve the exponential equation \(2^x + 3^x = 10000\) for \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.354}
The triangular pyramid \(ABCD\) intersects with the plane \(P\) along the quadrilateral \(EFGH\) such that the vertices \(E\) and \(F\) lie on the edges \(AB\) and \(AC\), respectively. It is known that the plane \(P\) is parallel to the edges \(AD\) and \(BC\), the ratio of segment \(EA\) to segment \(EB\) is 2, and the edges \(AD\) and \(BC\) are equal. Find the ratio \(EF:EH\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In $\triangle ABC$, point $D$ is on side $AB$, and $\overrightarrow{BD}=\dfrac{1}{2}\overrightarrow{DA}$. Let $\overrightarrow{CB}=a$ and $\overrightarrow{CA}=b$. Calculate $\overrightarrow{CD}$ in terms of $a$ and $b$, expressing your answer as a linear combination of $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}a+ \dfrac{1}{3}b}
A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{10}}
Determine the radius of convergence of the product of two power series, each with radii of convergence \( R \) and \( S \), respectively, where the functions are analytic in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min\{R, S\}}
Determine the number of solutions to the equation \(x^3 = (1\ 2\ 3\ 4)\) in the symmetric group \(S_7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a cube \(ABCD-A_1B_1C_1D_1\), construct a line \(l\) passing through vertex \(A_1\) such that the angles between \(l\) and the lines \(AC\) and \(BC_1\) are both \(60^\circ\). How many such lines \(l\) are there? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If \(n\) is a multiple of \(4\), find the sum \(s=1+2i+3i^2+ \ldots +(n+1)i^{n}\), where \(i=\sqrt{-1}\). The original answer is in the form \(\frac{k}{m}(n+a-b\cdot i)\). Please calculate the value of \(k + m + a + b\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A child spends their time drawing pictures of Native Americans (referred to as "Indians") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo. 

A psychologist noticed the following:
1. The number of Native Americans drawn is twice the number of Eskimos.
2. The number of Eskimos with teepees is equal to the number of Native Americans with igloos.
3. Each teepee drawn with an Eskimo is matched with three igloos.

Based on this information, determine the proportion of Native Americans among the inhabitants of teepees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{8}}
Mr. Curry is playing basketball. Suppose that, on average, he makes two successful three-point shots out of every five attempts. What is the probability that he will be able to make at least two successful 3-point shots in a row out of four attempts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{44}{125}}
5. $\underbrace{11 \cdots 1}_{100 \uparrow 1}+\underbrace{22 \cdots 2}_{50 \uparrow 2}+\underbrace{44 \cdots 4}_{25 \uparrow 4}+2010$ The sum of the digits of the calculation result is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{303}
There was only enough room for 21 people on a raft if no one wore life jackets. There were 7 fewer people who could fit on the raft if everyone wore life jackets. If a certain number of people on the raft needed life jackets, 17 people could fit on the raft. How many people needed life jackets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
给定正整数 $m, n(2 \leq m \leq n)$, 设 $a_{1}, a_{2}, \cdots, a_{m}$ 是 $1,2, \cdots, n$ 中任取 $m$ 个互不相同的数构成的一个排列，如果存在 $k \in\{1,2, \cdots, m\}$ 使得 $a_{k}+k$ 为奇数，或者存在整数 $k, l(1 \leq k<l \leq m)$ ，使得 $a_{k}>a_{l}$ ，则称 $a_{1}, a_{2}, \cdots, a_{m}$ 是一个 "好排列"，试确定所有好排列的个数。当m=4, n=6时，请给出答案。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{355}
Let \( f(n) \) be the number of 0's in the decimal representation of the positive integer \( n \). For example, \( f(10001123) = 3 \) and \( f(1234567) = 0 \). Find the value of
\[ f(1) + f(2) + f(3) + \ldots + f(99999) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38889}
Evaluate the integral $ \iiint_{R} \frac{dxdydz}{\sqrt{(x-a)^{2}+y^{2}+z^{2}}} $ over the solid unit sphere $ 0 \leq x^{2}+y^{2}+z^{2} \leq 1 $ for $ a > 1 $.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{4\pi}{3a}\)}
What is the cardinality of a single equivalence class of Cauchy sequences in the rational numbers, corresponding to a single real number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Micah bought envelopes to send, and depending on the weight of the envelope he will need more stamps. If an envelope weighs more than a certain weight, he will need 5 stamps. If it weighs less than that, it will only need 2 stamps. He bought 52 stamps with 6 envelopes that weigh less than that weight. Micah needed to buy 14 envelopes in total. What is the weight limit for an envelope to need only 2 stamps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the value of $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $ $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
In a school of 1,500 students, 35% wear blue shirts, 20% wear red shirts, 10% wear yellow shirts, 5% wear green shirts, and the remaining students wear other colors. Among those wearing blue shirts, 40% have stripes, 15% have polka dots, 10% have floral patterns, and the rest have no patterns. Among those wearing red shirts, 30% have stripes, 25% have polka dots, 5% have floral patterns, and the rest have no patterns. Among those wearing yellow shirts, 20% have stripes, 10% have polka dots, and the remaining 70% have no patterns. Among those wearing green shirts, 10% have stripes, 30% have polka dots, 50% have floral patterns, and the remaining 10% have no patterns.

In addition to their shirts, 18% of the students wear glasses, 12% wear a hat, and 10% wear a scarf regardless of their other features.

How many students wear a blue shirt with stripes and glasses, a red or yellow shirt with polka dots and a hat, or a green shirt with floral patterns and a scarf?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
9. Tim has a working analog 12 -hour clock with two hands that run continuously (instead of, say, jumping on the minute). He also has a clock that runs really slow-at half the correct rate, to be exact. At noon one day, both clocks happen to show the exact time. At any given instant, the hands on each clock form an angle between $0^{\circ}$ and $180^{\circ}$ inclusive. At how many times during that day are the angles on the two clocks equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Compute $\int_{\gamma}{\log(z)\over z}dz$ for $\gamma(t)=e^{it}$, $0\le t\le 2\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $Z$ denote the set of points in $\mathbb{R}^n$ whose coordinates are 0 or 1. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^n$.) Given a vector subspace $V$ of $\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that lie in $V$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subseteq \mathbb{R}^n$ of dimension $k$, of the number of points in $V \cap Z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^k}