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TheoremQA_standardized

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What's the maximum number of edges in a simple planar graph with 30 vertices?

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84

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Find the minimum of $f(x,y)=2x - 5y$, subject to the constraint $x^2+y^2=144$.

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-64.62

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Find the smallest positive integer that leaves a remainder of 1 when divided by 4, a remainder of 2 when divided by 3, and a remainder of 5 when divided by 7.

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17

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Consider a 900 Kbytes file stored in a web server. Client A sends a request to the server to retrieve the file from a remote location. There are 3 links (2 intermediate nodes) between server and client and each has a transmission rate of 10Mbps. Given that the segment size is 15 Kbytes, the round trip time (RTT) between the server and client is 30ms, the initial slow-start threshold is 8 and the client's buffer has a storage space of 150 K bytes. Assume that TCP Reno is used, there is no loss during transmission and the headers of protocols are ignored. It is noted that the segments do experience a store-and-forward delay in intermediate routers. Determine how many ms client A takes to receive the whole file from the server after sending a request.

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918

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Let a undirected graph G with edges E = {<2,6>,<2,8>,<2,5>,<6,5>,<5,8>,<6,10>,<10,8>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 2 to node 10? Represent the path as a list.

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[2, 8, 10]

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For any poitie integer $n$, let $\langle n\rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^{\infty} \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$.

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3.0

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Let $x_1$ and $x_2$ be the roots of the equation $x^2 + 3x + 1 =0$. Compute $(x_1/(x_2 + 1))^2 + (x_2 / (x_1 + 1))^2$.

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18.0

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Use divergence therem to evaluate $\iint_S \vec{F} \cdot d \vec{S}$ where $\vec{F} = xy \vec{i} - \frac{1}{2}y^2\vec{j} + z\vec{k}$ and the surface $S$ consists of the three surfaces, $z=4 - 3*x^2 - 3y^2, 1 \le z \le 1$ on the sides and $z=0$ on the bottom.

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7.853

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You want to move a 500-N crate across a level floor. To start thecrate moving, you have to pull with a 230-N horizontal force.Once the crate breaks loose and starts to move, you can keep itmoving at constant velocity with only 200 N. What are the coefficients of static and kinetic friction?

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0.4

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What is \lim_{x \to 0} (x \lfloor 1/x floor)?

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1

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Photoelectrons may be emitted from sodium (phi = 2.36 eV) even for light intensities as low as 10^-8 W/m^2. Calculate classically how much time (in seconds) the light must shine to produce a photoelectron of kinetic energy 1.00 eV. Return the numeric value.

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463000000.0

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A disadvantage of the contention approach for LANs, such as CSMA/CD, is the capacity wasted due to multiple stations attempting to access the channel at the same time. Suppose that time is divided into discrete slots, with each of 5 stations attempting to transmit with probability 0.35 during each slot. What fraction of slots is wasted due to multiple simultaneous transmission attempts?

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0.572

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What is the determinant of the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

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-3

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What is the coefficient of $x^2y^5$ for the formula $(x + 2y)^7$?

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672

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What's phi(29791) where phi is Euler's Totient Function?

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28830

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Let {X_n: n \geq 1} be independent, identically distributed random variables taking integer values {1,-1}. Let S_0=0, S_n=\sum_{i=1}^n X_i. Let P(X_i=1)=0.8 and P(X_i=-1)=0.2. The range R_n of S_0,S_1,...,S_n is the number of distinct values taken by the sequence. Then what is the limit of n^{-1}E[R_n] as n goes to infinity? Here E[R_n] is the expectation over the random variable R_n.

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0.6

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matrix $A=(\begin{array}{rrrr} -2 & -1 & -1 & -1 \ 2 & 1 & 3 & 2 \ 1 & 1 & 0 & 1 \ -1 & -1 & -2 & -2 \end{array})$. Suppose f is the minimal polynomial of A. What is f(99)? Return the numeric without explanation.

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990000.0

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A state issues a 15 year $1000 bond that pays $25 every six months. If the current market interest rate is 4%, what is the fair market value of the bond?

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1111.97

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A positive-definite kernel function satisfies the Cauchy-Schwartz inequality. True or false?

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True

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V is a vector space over the real field R. It is known that the vector group u_1, u_2, u_3 in V are linearly independent. Finding the rank of vector group ${u_1-\lambda u_2, u_2-\lambda u_3, u_3-\lambda u_1}$ for $\lambda=\sqrt{5}$ and $\lambda=1$ separately. Return the answer as a list.

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[3, 2]

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suppose $-\pi<x<\pi$. what is the value of $(\sum_{n=1}^{\infty}(-1)^{n-1} \frac{cos(nx)}{n})/log(2cos(x/2))$? Rounding it to the hundredths place and return the value.

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1.0

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Find the smallest positive integer that leaves a remainder of 3 when divided by 5, a remainder of 4 when divided by 7, and a remainder of 2 when divided by 9.

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263

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Suppose V is a finite-dimensional vector space on F. $M1={a_1,a_2,a_3}$ is a basis of V, $M2={b_1,b_2,b_3}$ is another basis of V. Suppose the coordinates of b_1,b_2,b_3 under M1 are $c_1=(1,1,-1),c_2=(1,-1,1),c_3=(-1,1,1)$. Suppose the coordinate of $d\in V$ under M1 is (1,3,5). What is the coordinate of d under M2? Return the three coordinate values as a list.

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[2, 3, 4]

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Let a undirected graph G with edges E = {<1,2>,<2,4>,<5,4>,<5,6>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 1 to node 6? Represent the path as a list.

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[1, 2, 4, 5, 6]

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Given $V_s$ = 5V, $R_1$ = 1kΩ, $R_2$ = 2.2kΩ, $R_3$ = 2.2kΩ, $R_4$ = 1.5kΩ, and $R_L$ = 4.7kΩ. Determine the voltage and current across $R_L$. Answer in unit of V (3 sig.fig.).

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1.06

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Use the Birge-Vieta method to find a real root correct to three decimals of the following equation: x^5 - x + 1 = 0, p=-1.5.

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-1

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The atomic mass of the 4He atom is 4.002603 u. Find the binding energy of the 4He nucleus in MeV.

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28.3

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Compute the real integral $I=\int_{-\infty}^{\infty} 1/(x^2 + 1)^2 dx$.

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1.57

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X rays scattered from rock salt (NaCl) are observed to have an intense maximum at an angle of 20° from the incident direction. Assuming n = 1 (from the intensity), what must be the Wavelength of the incident radiation in nm?

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0.098

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Suppose that f is analytic on the closed unit disk, f(0) = 0, and $|Rf(z)| \leq |e^z|$ for |z| < 1. What's the maximum value of f((1 + i)/2)?

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17.95

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Given the following equation: x^4 - x - 10 = 0. determine the initial approximations for finding the smallest positive root. Use these to find the root correct to three decimal places with Secant method.

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1.856

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A box contains 4 red, 3 green, and 2 blue balls. Balls are distinct even with the same color. In how many ways can we choose 4 balls, if at least 2 are red?

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81

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The cross section for a 2.0-MeV neutron (a typical energy for a neutron released in fission) being absorbed by a U-238 nucleus and producing fission is 0.68 barn. For a pure U-238 sample of thickness 3.2 cm, what is the probability of a 2.0-MeV neutron producing fission?

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0.1

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Consider the infinitely long chain of resistors shown below. What is the resistance between terminals a and b if R=1?

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0.73

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The image produced by a concave mirror is at -16.2m, and the magnification is 1.79. What is the object distance in terms of meter?

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9.05

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Find the last 3 digits of 2003^(2002^2001).

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241

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In the process of searching circles in an image, object O is detected. The contour of the object O is represented with the Fourier Descriptors (35,129,0,1,0,0,-1,0). Given that the Fourier Descriptors of a circle are (0,40,0,0,0,0,0,0). Is the object O a circle-like polygon in the image? Bear in mind that there is some high frequency noise in the image. You should take this into account when you make your judgment.

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True

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A random variable $X$ takes on $m$ values and has entropy $H(X)$. An instantaneous ternary code is found for this source, with an average length $L=H_3(X)$ that achieves the entropy bound. Then $m$ must be odd. True or False?

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True

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Is there an eigenbasis for the identity matrix I_n?

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True

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Point charges q1=50μC and q2=−25μC are placed 1.0 m apart. What is the force on a third charge q3=20μC placed midway between q1 and q2?

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53.94

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suppose sequence x_n satisfies x_n*x_{n+1}=n for all n>=1, and $\lim_{n\rightarrow\infty}\frac{x_n}{x_{n+1}}=1$. What's the value of $\pi*x_1^2$?

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2.0

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what is the limit of $2/\sqrt{\pi}*\sqrt{n}\int_0^1(1-x^2)^n dx$ as n goes to infinity?

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1.0

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ABC is a right triangle. AM is perpendicular to BC. The size of angle ABC is equal to 55 degrees. Find the size of angle MAC.

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55

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In a set of 20 positive integers, at least how many pairs of numbers have a difference that is a multiple of 10?

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10

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If p is a prime number and a is an integer, what is (a^p - a) mod p?

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0

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Fig 1(a) and 1(b) show the situation of a reference frame and a current block for block matching motion estimation. The size of searching window is 14x2 while the block size is 2x2. The numbers within the squares are the pixel values. Determine the optimum motion vector.

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[-4, 0]

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Find the sum of all positive integers less than 196 and relatively prime to 98.

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8232

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suppose $lim_{n \rightarrow \infty}a_n=1$, what is the limit of (a_1+2a_2+...+na_n)/n^2?

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0.5

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Mr. Jackson bought his house in 1995, and financed the loan for 30 years at an interest rate of 7.8%. His monthly payment was $1260. In 2015, Mr. Jackson decides to pay off the loan. Find the balance of the loan he still owes.

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104761.48

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the matrix in ./mingyin/mc.png represents a markov chain. What is the period of state 0? What is the period of state 1? Return the two answers as a list.

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[2, 2]

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