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TIGER-Lab

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TheoremQA

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As scotch whiskey ages, its value increases. One dollar of scotch at year 0 is worth $V(t) = exp{2\sqrt{t} - 0.15t}$ dollars at time t. If the interest rate is 5 percent, after how many years should a person sell scotch in order to maximize the PDV of this sale?

25

integer

How many ways are there to divide a set of 6 elements into 3 non-empty ordered subsets?

1200

integer

Given an image $$ \begin{array}{llllllll} 6 & 5 & 6 & 7 & 7 & 7 & 7 & 7 \\ 7 & 7 & 7 & 7 & 6 & 7 & 7 & 7 \\ 8 & 8 & 8 & 6 & 5 & 5 & 6 & 7 \\ 8 & 8 & 8 & 6 & 4 & 3 & 5 & 7 \\ 7 & 8 & 8 & 6 & 3 & 3 & 4 & 6 \\ 7 & 8 & 8 & 6 & 4 & 3 & 4 & 6 \\ 8 & 8 & 8 & 7 & 5 & 5 & 5 & 5 \\ 8 & 9 & 9 & 8 & 7 & 6 & 6 & 4 \end{array} $$ . Find an appropriate threshold for thresholding the following image into 2 regions using the histogram.

6.25

float

A function f(x) is given by f(0)=3, f(2)=7, f(4)=11, f(6)=9, f(8)=3. Approximate the area under the curve y=f(x) between x=0 and x=8 using Trapezoidal rule with n=4 subintervals.

60.0

float

For the signal f(t)=3sin(200πt)+ 6sin(400πt) + sin(500πt), determine the minimum sampling requency (in πHz) satisfying the Nyquist criterion.

500

integer

Is the differential equation $2tyy' + 2t + ty^2 = 0$ the total derivative of the potential function $\phi(t, y) = t^2 + ty^2$?

False

bool

Given image \begin{tabular}{|llll|} \hline 7 & 1 & 6 & 0 \\ 3 & 3 & 7 & 6 \\ 6 & 6 & 5 & 7 \\ \hline \end{tabular} , and the bit-depth of the image is 4. Is the contrast of the image is poor? Judge it based on the histogram of the image.

True

bool

Approximate the area under the curve y=2^{x} between x=-1 and x=3 using the Trapezoidal rule with n=4 subintervals.

11.25

float

How many trees are there on 5 unlabeled vertices?

3

integer

For (10236, 244), use the Euclidean algorithm to find their gcd.

4

integer

For matrix A = [[2, 4, 3], [3, 3, 1], [42, 20, 51]], what is its determinant?

-376

integer

In a group of 1000 people, at least how many people have to share the same birthday?

3

integer

A pizza parlor offers 8 different toppings. In how many ways can a customer order a pizza with 3 toppings?

56

integer

In a CSMA/CD network with a data rate of 10 Mbps, the minimum frame size is found to be 512 bits for the correct operation of the collision detection process. What should be the minimum frame size (in bits) if we increase the data rate to 1 Gbps?

51200

integer

An image has the gray level PDF $p_r(r)$ shown in Fig. Q1a. One wants to do histogram specification SO that the processed image will have the specified $p_z(z)$ shown in Fig. Q1b. Can we use intensity mapping function $T: z=1-r$ to achieve the goal?

False

bool

the monotone function f on [0,1] is differentiable almost everywhere. This can be proved by: (a) Fubini Theorem; (b) Tonelli Theorem; (c) Vitali Cover Theorem; (d) None of the above. Which option is correct?

(c)

option

Suppose a fair coin is tossed 50 times. The bound on the probability that the number of heads will be greater than 35 or less than 15 can be found using Chebyshev's Inequality. What is the upper bound of the probability?

0.125

float

Place the little house mouse into a maze for animal learning experiments, as shown in the figure ./mingyin/maze.png. In the seventh grid of the maze, there is a delicious food, while in the eighth grid, there is an electric shock mouse trap. Assuming that when the mouse is in a certain grid, there are k exits that it can leave from, it always randomly chooses one with a probability of 1/k. Also, assume that the mouse can only run to adjacent grids each time. Let the process $X_n$ denote the grid number where the mouse is located at time n. Calculate the probability that the mouse can find food before being shocked if: the mouse start from 0, $X_0=0$; the mouse start from 4, $X_0=4$? Return the two answers as a list.

[0.5, 0.66667]

list of float

Obtain the number of real roots between 0 and 3 of the equation P(x) = x^4 -4x^3 + 3x^2 + 4x - 4 = 0 using Sturm's sequence.

2

integer

What is the limit of the sequence a_n = n/(\sqrt{n^2 + 1})?

1

integer

Let a undirected graph G with edges E = {<0,4>,<4,1>,<0,3>,<3,4>,<3,2>,<1,3>}, which <A,B> represent Node A is connected to Node B. What is the minimum vertex cover of G? Represent the vertex cover in a list of ascending order.

[3, 4]

list of integer

What is \lim_{x \to 1} ((x - 1) sin((\pi)/(x - 1))?

0

integer

Use Green's Theorem to evaluate $\oiint_{s} y^3 dx + x^3dy$ where $C$ is the positively oriented circle of radius 2 centered at origin.

-75.396

float

A hospital has a 3.0 x 10^14 Bq Co-60 source for cancer therapy. The rate of gamma rays incident on a patient of area 0.30 m^2 located 4.0 m from the source is $X*10^11$ Bq, what is X? Co-60 emits a 1.1- and a 1.3-MeV gamma ray for each disintegration.

8.95

float

We are interested in the capacity of photographic film. The film consists of silver iodide crystals, Poisson distributed, with a density of 100 particles per unit area. The film is illuminated without knowledge of the position of the silver iodide particles. It is then developed and the receiver sees only the silver iodide particles that have been illuminated. It is assumed that light incident on a cell exposes the grain if it is there and otherwise results in a blank response. Silver iodide particles that are not illuminated and vacant portions of the film remain blank. We make the following assumptions: We grid the film very finely into cells of area $dA$. It is assumed that there is at most one silver iodide particle per cell and that no silver iodide particle is intersected by the cell boundaries. Thus, the film can be considered to be a large number of parallel binary asymmetric channels with crossover probability $1 - 100dA$. What is the capacity of a 0.1 unit area film?

10.0

float

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

18.0

float

Let N be a spatial Poisson process with constant intensity $11$ in R^d, where d\geq2. Let S be the ball of radius $r$ centered at zero. Denote |S| to be the volume of the ball. What is N(S)/|S| as $r\rightarrow\infty$?

11.0

float

Perform 2 iterations with the Müller method for the following equation: x^3 - 1/2 = 0, x_0 = 0, x_1 = 1, x_2 = 1/2. What's the decimal value of x_3?

0.7929

float

A bungee cord is 30.0 m long and, when stretched a distance x, it exerts a restoring force of magnitude kx. Your father-in-law (mass 95.0 kg) stands on a platform 45.0 m above the ground, and one end of the cord is tied securely to his ankle and the other end to the platform. You have promised him that when he steps off the platform he will fall a maximum distance of only 41.0 m before the cord stops him. You had several bungee cords to select from, and you tested them by stretching them out, tying one end to a tree, and pulling on the other end with a force of 380.0 N. When you do this, what distance (in m) will the bungee cord that you should select have stretched?

0.602

float

Universal Fur is located in Clyde, Baffin Island, and sells high-quality fur bow ties throughout the world at a price of $5 each. The production function for fur bow ties (q) is given by q = 240x - 2x^2, where x is the quantity of pelts used each week. Pelts are supplied only by Dan's Trading Post, which obtains them by hiring Eskimo trappers at a rate of $10 per day. Dan's weekly production function for pelts is given by x = \sqrt{l}, where l represents the number of days of Eskimo time used each week. For a quasi-competitive case in which both Universal Fur and Dan's Trading Post act as price-takers for pelts, what will be the equilibrium price (p_x) for pelt?

600

integer

In a certain nuclear reaction initiated by 5.5-MeV alpha particles, the outgoing particles are measured to have kinetic energies of 1.1 MeV and 8.4 MeV. What is the Q value of the reaction in MeV?

4.0

float

Company A is currently trading at $150 per share, and earnings per share are calculated as $10. What is the P/E ratio?

15.0

float

Is the Taylor Series for $f$ at x=5 where $f(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!} absolutely converging?

1.0

float

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

0.73

float

Let $A=\{n+\sum_{p=1}^{\infty} a_p 2^{-2p}: n \in \mathbf{Z}, a_p=0 or 1 \}$. What is the Lebesgue measure of A?

0.0

float

An IPv4 packet contains the following data (in hexadecimal value) in the IP header: 4500 0034 B612 4000 4006 6F80 0A00 008B 5BC6 AEE0 . Does the header contains error?

False

bool

Is (t-y)y' - 2y +3t + y^2/t = 0 an Euler homogeneous equation?

True

bool

In how many ways can a set of 6 distinct letters be partitioned into 2 non-empty groups if each group must contain at least 2 letters?

25

integer

Compute the mean molecular speed v in the light gas hydrogen (H2) in m/s

1750.0

float

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

59

integer

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

1

integer

What is the vector that spans the kernel of A = [[1, 0, 2, 4], [0, 1, -3, -1], [3, 4, -6, 8], [0, -1, 3, 4]]?

[-2, 3, 1, 0]

list of integer

Every published book has a ten-digit ISBN-10 number that is usually of the form x_1 - x_2 x_3 x_4 - x_5 x_6 x_7 x_8 x_9 - x_{10} (where each x_i is a single digit). The first 9 digits identify the book. The last digit x_{10} is a check digit, it is chosen so that 10 x_1 + 9 x_2 + 8 x_3 + 7 x_4 + 6 x_5 + 5 x_6 + 4 x_7 + 3 x_8 + 2 x_9 + x_{10} = 0 (mod 11). Is 3-540-90518-9 a valid ISBN number?

True

bool

An 8% bond with 18 years to maturity has a yield of 9%. What is the price of this bond?

91.17

float

ABCD is a square. Inscribed Circle center is O. Find the the angle of ∠AMK. Return the numeric value.

130.9

float

A curve with a 120 m radius on a level road is banked at the correct angle for a speed of 20 m/s. If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction needed between tires and road to prevent skidding?

0.34

float

ABCD is a parallelogram such that AB is parallel to DC and DA parallel to CB. The length of side AB is 20 cm. E is a point between A and B such that the length of AE is 3 cm. F is a point between points D and C. Find the length of DF in cm such that the segment EF divide the parallelogram in two regions with equal areas.

17

integer

Let rectangle R = [1, 2.5] * [1, 2]. Calculate the Riemann Sum S_{3,2} for \int \int_{R} xy dA for the integral, using the lower-left vertex of rectangles as sample points.

2.812

float

If the annual earnings per share has mean $8.6 and standard deviation $3.4, what is the chance that an observed EPS less than $5.5?

0.1814

float

Compute the are of that part of the helicoid z = arctan(y/x) which lies in the first octant between the cylinder $x^2+y^2 = 1^2$ and $x^2+y^2 = 2^2$.

2.843

float

A certain underlying state graph is a tree where each node has three successor nodes, indexed $a$, $b$, $c$. There are two assets defined on this tree which pay no dividends except at the terminal time $T$. At a certain period it is known that the prices of the two accets are multiplied by factors, depending on the successor node. These factors are shown in the table below: | | a | b | c security | 1 | 1.2 | 1.0 | 0.8 | 2 | 1.2 | 1.3 | 1.4 Is there a short-tem riskless asset for this period? Answer True or False.

True

bool

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)?

17.95

float

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

-64.62

float

Calculate the Gross Domestic Product using the total expenditure approach: Consumption Expenditures | $500 billion Wages and salaries | $400 billion (Gross Private) Investments Expenditures | $80 billion Government Expenditures | $100 billion Taxes | $70 billion Imports | $50 billion Exports | $30 billion What is the GDP (in billions)?

660

integer

Suppose H is a Banach space. Let A be a linear functional on the space H that maps H to H. Suppose operator A satisfies: for all $x\in H$, $||Ax||\geq a ||x||$ for some a>0. If A is not a compact operator on H, Is the dimension of H finite or infinite? Return 1 for finite dimension and 0 for infinite dimension

0.0

float

What is \lim_{x \to (\pi)/2} (cos(x)cos(tan(x)))?

0

integer

Let $N_1(t)$ and $N_2(t)$ be two independent Posson processes with rate $\lambda_1 = 1$ and $\lambda_2 = 2$, respectively. Let N(t) be the merged process N(t) = N_1(t) + N_2(t). Given that N(1) = 2, Find the probability that N_1(1) = 1.

0.4444

float

The diagonals of rhombus FGHJ intersect at K. If m∠FJH = 82, find m∠KHJ.

49

integer

Titan, the largest moon of Saturn, has a mean orbital radius of 1.22x10^9 m. The orbital period of Titan is 15.95 days. Hyperion, another moon of Saturn, orbits at a mean radius of 1.48x10^9 m. Use Kepler's third law of planetary motion to predict the orbital period of Hyperion in days.

21.3

float

Perform 2 iterations with the Müller method for the following equation: log_{10}(x) - x + 3 = 0, x_0 = 1/4, x_1 = 1/2, x_2 = 1. What's the decimal value of x_3?

3.2

float

Compute covariance of x=(1,2,3,4), y=(2,3,4,5)

1.67

float

Square ABCD. CT: tangent to semicircle. Find the angle ∠CTD. Return the numeric value.

63.4

float

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.

[1, 2, 4, 5, 6]

list of integer

Let a undirected graph G with edges E = {<0,1>,<0,2>,<0,3>,<3,5>,<2,3>,<2,4>,<4,5>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 0 to node 5? Represent the path as a list.

[0, 3, 5]

list of integer

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.

[3, 2]

list of integer

An aluminum cylinder 10 cm long, with a cross-sectional area of 20 $cm^2$ is used as a spacer between two steel walls. At 17.2°C it just slips between the walls. Calculate the stress in the cylinder and the total force it exerts on each wall when it warms to 22.3°C assuming that the walls are perfectly rigid and a constant distance apart. (Unit: 10^4 N)

-1.7

float

Is there an eigenbasis for the identity matrix I_n?

True

bool

Light of wavelength 400 nm is incident upon lithium (phi = 2.93 eV). Calculate the photon energy in eV.

3.1

float

For an American perpetual option within the Black-Scholes framework, you are given: (i) $h_1 + h_2$ = 7/9 (ii) The continuously compounded risk-free interest rate is 5%. (iii) σ = 0.30. What is the value of $h_1$?

1.51

float

Let X_1, X_2 , X_3 be independent random variables taking values in the positive integers and having mass functions given by P(X_i=x)=(1-p_i)*p_i^{x-1} for x=1,2,... and i=1,2,3. Suppose p_1=1/2,p_2=1/4,p_3=1/8, what is the probability of X_1<X_2<X_3 (i.e. P(X_1<X_2<X_3))?

0.00153609831

float

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

241

integer

Let f be an entire function such that |f(z)| $\geq$ 1 for every z in C. Is f is a constant function?

True

bool

what is the value of $\int_{0}^\pi (sin(123*x/2)/sin(x/2))^2dx$? Round the answer to the thousands decimal.

386.4158898

float

Assume that the Black-Scholes framework holds. The price of a nondividened-paying stock is $30. The price of a put option on this stock is $4.00. You are given $(i) $\Delta=-0.28$. (ii) $\Gamma=0.10$ Using the delta-gamma approximation, determine the price of the put option if the stock price changes to $31.50.

3.7

float

How many distinct directed trees can be constructed from a undirected tree with 100 nodes?

100

integer

A TCP entity sends 6 segments across the Internet. The measured round-trip times (RTTM) for the 6 segments are 68ms, 42ms, 65ms, 80ms, 38ms, and 75ms, respectively. Assume that the smooth averaged RTT (RTTs) and Deviation (RTTD) was respectively 70ms and 10ms just before the first of these six samples. According to the Jacobson's algorithm, the retransmission timeout (RTO) is given by one RTTs plus 4 times the value of RTTD. Determine the value of RTO (in ms) after the six segments using the Jacobson's algorithm if the exponential smoothing parameters (a and B) are 0.15 and 0.2 for calculating RTTs and RTTD respectively.

114.28

float

In the figure,At what rate is thermal energy generated in the $20 \Omega$ resistor? Answer in unit of W (3 sig.fig.).

1.63

float

For how many positive integral values of x ≤ 100 is 3^x − x^2 divisible by 5?

20

integer

Evaluate $\lim _{x \rightarrow 1^{-}} \prod_{n=0}^{\infty}(\frac{1+x^{n+1}}{1+x^n})^{x^n}$?

0.73575888

float

Let g(x) = 1 / (1 + x^{3/2}), what is g'(x) when x = 1?

-0.375

float

For which 2 * 2 matrices A does there exist a nonzero matrix M such that AM = MD, where D = [[2, 0], [0, 3]]? Give your answer in terms of eigenvalues of A.

[2, 3]

list of integer

Arbitrarily place 19 points in a unit square and cover as many of these points as possible with a circle of diameter $\frac{\sqrt 2}{3}$. Question: At least how many points can be guaranteed to be covered?

3

integer

Let (x_n) be a sequence defined by x_1 = 2 and x_{n+1} = 1 + 1/(1 + x_n). If (x_n) converges, what must its limit be in decimals?

1.414

float

Let W(t) be the standard Brownian motion. Find P(W(1) + W(2) > 2).

0.186

float

suppose f is differentiable in [0,+\infty) and f(0)=0. When x>=0, |f'(x)|<=|f(x)| where f' stands for the derivative of f. What is f(2687) and f(35)? answer the two values in a list

[0, 0]

list of integer

Suppose f is an analytic function defined on $\{z \in C : IM(z) > 0\}$, the upper half plane. Given the information that f(f(z)) = z and f'(z) = 1/z^2 for every z. Find the most general possible expression of f(z). What is f(2)?

-0.5

float

How many ways are there to distribute 13 identical balls into 4 distinct boxes if the boxes are distinguishable and no box can be left empty?

220

integer

Consider two 5 year bonds: one has a 9% coupon and sells for 101.00; the other has a 7% coupon and sells for 93.20. What is the price of a 5-year zero-coupon bond.

65.9

float

Based on field experiments, a new variety green gram is expected to given an yield of 12.0 quintals per hectare. The variety was tested on 10 randomly selected farmers fields. The yield ( quintals/hectare) were recorded as 14.3,12.6,13.7,10.9,13.7,12.0,11.4,12.0,12.6,13.1. Do the results conform the expectation with Level of significance being 5%?

True

bool

Calculate the future value of an ordinary annuity of $800 per year for 4 years at 5% rate of return.

3448.1

float

Is the transformation [[-1, 0], [0, -1]] invertible?

True

bool

Suppose $\Omega$ is a bounded open area in $\mathbb{R}^n$. For any $f\in L^2(\Omega)$, the Laplace equation (with respect to a real function $u$), $\Delta u = f$ with boundary condition $u\mid_{\partial \Omega}=0$, has a unique weak solution. This can be proved by: 1. Poincare inequality and Riesz representation theorem; 2. Cauchy-Schwartz inequality and Hahn-Banach theorem. 3. None of the above. Return the answer as a number

1.0

float

Find the measure of angle A in the figure below. Return the numeric value.

87

integer

Use the linear approximation to estimate (3.99)^3 (1.01)^4 (1.98)^{-1}.

33.36

float

For all $n>1$, define $a_n=\sum_{k=1}^{n-1} \frac{\sin (\frac{(2 k-1) \pi}{2 n})}{\cos ^2(\frac{(k-1) \pi}{2n}) \cos ^2 (\frac{k \pi}{2n})}$. What is the limit of $a_n/n^3$ as $n$ goes to infinity?

0.258

float

Find the smallest positive integer that leaves a remainder of 5 when divided by 8, a remainder of 1 when divided by 3, and a remainder of 7 when divided by 11.

205

integer

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.

17

integer

Figure Q8 shows the contour of an object. Represent it with an 4-directional chain code. Represent the answer as a list with each digit as a element.

[0, 0, 3, 3, 3, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 0, 0, 1]

list of integer

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?

0.572

float

How many ways are there to divide a set of 5 elements into 2 non-empty ordered subsets?

240

integer