instruction_seed stringlengths 56 2.52k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.77 0.86 | mean_negative_score float64 0.76 0.85 | difference_score float64 0 0.03 |
|---|---|---|---|---|---|
"statement 1": Tim and Stacy need to select seats in a row of chairs that are numbered in order from 1 to 5, "statement 2" :Tim and Stacy need to select seats in a row of chairs that are numbered in order from less than 4 to 5, "options: " Entailment or contradiction or neutral? Let's write a Python program to solve i... | tiger_lab_math | [
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0.0... | 0.806766 | 0.805248 | 0.001518 |
A high school has decided to form groups of students for a special project. The school requires that each group must have exactly 3 students and each student must belong to exactly one group. If there are 21 students in the school, how many different groups can be formed? (Hint: This problem involves finding the number... | tiger_lab_math | [
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A delivery company needs to deliver products to 5 different cities such that the total distance traveled by the delivery truck is minimized. The distance between each pair of cities is given below:
City A to City B: 120 km
City A to City C: 90 km
City A to City D: 200 km
City A to City E: 150 km
City B to City C: 100 ... | tiger_lab_math | [
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Consider an auction mechanism for selling an item where there are three bidders: A, B, and C. The bidders have the following valuations for the item: A values the item at $1000, B values the item at $800, and C values the item at $1200.
The auction mechanism works as follows: each bidder submits a sealed bid for the ... | tiger_lab_math | [
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Find
\[\sum_{n = 1}^\infty \frac{n^2 + n - 1}{(n + 2)!}.\] Let's write a program. | tiger_lab_math | [
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In a Centipede Game, there are two players, player A and player B. The game starts with a pile of money $10. Player A can either take the entire pile, or pass it to player B. If player B receives the pile, they can either take the entire pile or pass it back to player A, and so on. Each time a player passes, the pile i... | tiger_lab_math | [
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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... | tiger_lab_math | [
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Sharon wants to get kitchen supplies. She admired Angela's kitchen supplies which consist of 20 pots, 6 more than three times as many plates as the pots, and half as many cutlery as the plates. Sharon wants to buy half as many pots as Angela, 20 less than three times as many plates as Angela, and twice as much cutlery ... | tiger_lab_math | [
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Let A, B, and C be matrices, and let A,B: M -> N and B,C: N -> P be matrix transformations such that the transformation B*C : M -> P is invertible. Which of the following must be true? Answer Choices: (A) A,B is invertible. (B) B,C is invertible. (C) Both A,B and B,C are invertible. (D) Neither A,B nor B,C need to be ... | tiger_lab_math | [
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If $a>0$ and $b>0,$ a new operation $\nabla$ is defined as follows: $$a \nabla b = \frac{a + b}{1 + ab}.$$For example, $$3 \nabla 6 = \frac{3 + 6}{1 + 3 \times 6} = \frac{9}{19}.$$Calculate $(1 \nabla 2) \nabla 3.$ Please write a program to solve it | tiger_lab_math | [
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Two lines are perpendicular. One line has a direction vector of $\begin{pmatrix} 3 \\ -7 \end{pmatrix}.$ The other line has a direction vector of $\begin{pmatrix} a \\ 2 \end{pmatrix}.$ Find $a.$ | tiger_lab_math | [
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To raise funds for her local soup kitchen, Didi enlisted the help of her family, friends, and neighbors. They donated 10 same-size cakes that she sliced into 8 slices per cake and started selling a slice for $1. A local business owner was so impressed by Didi's efforts that she offered to donate 50 cents for each slice... | tiger_lab_math | [
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Consider a triangle with vertices at coordinates (0,0), (3,0) and (0,4). If we subdivide this triangle into four smaller triangles using a centroidal subdivision, what are the coordinates of the centroid of the top-right small triangle? | tiger_lab_math | [
0.02659752406179905,
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Sorry, I apologize for my previous response.
Here's a numerical problem related to paraconsistent logic:
A paraconsistent logic system contains three statements: P, Q, and R. The truth values of these statements are as follows: P is true, Q is false, and R is true. Determine the truth value of the following statemen... | tiger_lab_math | [
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0.06735023856163025,
0.02048819698393345,
-0.03152252733707428,
0.031703487038612366,
0.03055463172495365,
-0.031890660524368286,
-0.0... | 0.839534 | 0.834176 | 0.005358 |
Let $g = dx^2 + 2xy dy^2$ be a Riemannian metric on $\mathbb{R}^2$. Find the Levi-Civita connection of the given metric. | tiger_lab_math | [
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0.03392326831817627,
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-0.... | 0.825936 | 0.815416 | 0.01052 |
A manufacturing company produces two types of products: Product A and Product B. The profit per unit of A is $10 and of B is $8. The production process involves a machine that has a 90% reliability rating, meaning that there is a 10% chance that the machine will break down during production. If the machine breaks down ... | tiger_lab_math | [
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Consider the following game in which two players, A and B, must simultaneously choose between two options, X and Y. If both players choose X, then player A gets a payoff of 3 and player B gets a payoff of 2. If both players choose Y, then player A gets a payoff of 1 and player B gets a payoff of 4. If one player choose... | tiger_lab_math | [
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Let $\theta$ be the angle between the line
\[\frac{x + 1}{2} = \frac{y}{3} = \frac{z - 3}{6}\]and the plane $-10x - 2y + 11z = 3.$ Find $\sin \theta.$
[asy]
import three;
size(150);
currentprojection = perspective(6,3,2);
triple I = (1,0,0), J = (0,1,0), K = (0,0,1), O = (0,0,0);
draw(surface((2*I + 2*J)--(2*I - 2... | tiger_lab_math | [
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0.002416072878986597,
0.045900747179985046,
0.02891985885798931,
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-0.0098... | 0.824899 | 0.815085 | 0.009815 |
Statement 1| The set of all integers under addition is an abelian group. Statement 2 | The intersection of two subgroups of a group G is also a subgroup of G. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True. | tiger_lab_math | [
0.04182789847254753,
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0.006529017351567745,
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-0.0... | 0.808551 | 0.801233 | 0.007317 |
A surface S is given by the equation 2x^2 + 3y^2 - z^2 = 6. Determine the principal curvatures of S at the point (1, 1, 1). | tiger_lab_math | [
0.004970706533640623,
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0.029700759798288345,
0.039887841790914536,
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-... | 0.829011 | 0.820912 | 0.008099 |
A company wants to minimize the cost of producing a cylindrical can that can hold 500 mL of soda. The can must have a top and a bottom that are twice as thick as the sides. Find the dimensions of the can that will minimize the cost of materials. Use calculus to solve this optimization problem. | tiger_lab_math | [
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0.02939743548631668,
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Let G be a cyclic group of order 10 generated by a. Find all the subgroups of G and determine whether each of them is cyclic or not. | tiger_lab_math | [
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To win a brand new Bible at Tom Sawyer's Sunday school, a pupil has to win 10 yellow tickets; each yellow ticket is obtained by trading in 10 red tickets; each red ticket is obtained by trading in 10 blue tickets; and blue tickets are earned by memorizing two Bible verses. But rather than go to all that trouble, Tom Sa... | tiger_lab_math | [
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-0.01296... | 0.814357 | 0.808223 | 0.006134 |
If the function f(x) is continuous on the interval [a, b] and differentiable on (a, b), such that f(a) = f(b), then according to Rolle's Theorem, there exists a number c in the interval (a, b) such that Answer Choices: (A) f(c) = 0 (B) f'(c) = 0 (C) f''(c) = 0 (D) f(a) = f(b) = f(c) | tiger_lab_math | [
0.04007377848029137,
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0.038131531327962875,
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0.019354917109012604,
0.028805797919631004,
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-0... | 0.817017 | 0.812977 | 0.00404 |
In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. So the middle number in Row 2 is $2$ because $1+1=2.$ What is the sum of the numbers in Row 8 of Pascal's Triangle?
\begin{tabular}{rccccccccccc}
Row 0:& & & & & & 1\\\noalign{\smalls... | tiger_lab_math | [
0.054220303893089294,
0.01793539710342884,
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0.030001286417245865,
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0.056308913975954056,
0.020383570343255997,
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0.012310431338846684,
0.03012032061815262,
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-0.... | 0.831932 | 0.817109 | 0.014823 |
Let $a$, $b$, and $c$ be the roots of $x^3 - 20x^2 + 18x - 7 = 0$. Compute \[(a+b)^2 + (b+c)^2 + (c+a)^2.\] | tiger_lab_math | [
0.046681396663188934,
0.04676993563771248,
-0.059309620410203934,
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0.03354806825518608,
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0.0012753184419125319,
0.063129723072052,
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0.002478964626789093,
0.02727784775197506,
-0.0513710118830204,
-0.046950... | 0.843041 | 0.839816 | 0.003225 |
Suppose there are three bidders in an auction for an item with a reserve price of $1000. Each bidder values the item differently and submits a bid as follows: Bidder 1 offers $1200, Bidder 2 offers $1300, and Bidder 3 offers $1100.
Using the second-price sealed-bid auction (Vickrey auction) method, determine which bi... | tiger_lab_math | [
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0.0019717877730727196,
-0.0348917581140995,
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0.021081876009702682,
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0.0562068335711956,
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0.02485230565071106,
0.021867522969841957,
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-0.01... | 0.794325 | 0.791487 | 0.002838 |
Two players, Alice and Bob, are trying to decide what movie to watch. Alice wants to watch "The Notebook" and Bob wants to watch "Avengers: Endgame". They both prefer to watch the same movie over watching different movies, but Alice would rather watch "The Notebook" alone than watch "Avengers: Endgame" with Bob, and vi... | tiger_lab_math | [
0.027973156422376633,
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0.016935257241129875,
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-0.022484205663204193,
0.06072525680065155,
0.023293329402804375,
-0.03156111761927605,
0.02544577792286873,
-0.013033738359808922,
-0.04071330651640892,
-... | 0.793222 | 0.774078 | 0.019145 |
Find the Laurent series expansion of the function f(z) = (z^2 - 1) / (z - 2i) centered at z = 2i, and determine the radius of convergence of the series. | tiger_lab_math | [
0.01623612456023693,
0.03763153403997421,
-0.04378627613186836,
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0.03560575470328331,
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0.03253409266471863,
0.024717755615711212,
-0.012176846154034138,
0.03811923414468765,
-0.0007803160697221756,
-0.030888423323631287,
-0.... | 0.819655 | 0.817441 | 0.002214 |
Here's a precise numerical problem for a math student to solve in the subtopic of solving optimization problems using whale optimization algorithm:
Suppose you are a biologist studying the growth of a certain species of plant. You have collected data from the field and want to use whale optimization algorithm to find ... | tiger_lab_math | [
0.007427634205669165,
0.013174373656511307,
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0.02767847664654255,
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0.04970648139715195,
0.027799487113952637,
-0.014987983740866184,
-0.000012998926649743225,
0.025762241333723068,
-0.02442350424826145,
... | 0.821931 | 0.808768 | 0.013163 |
Let $r$ be a complex number such that $r^5 = 1$ and $r \neq 1.$ Compute
\[(r - 1)(r^2 - 1)(r^3 - 1)(r^4 - 1).\] | tiger_lab_math | [
0.03341501206159592,
0.05862664058804512,
-0.01896941103041172,
-0.04330406337976456,
0.045842818915843964,
-0.022619403898715973,
-0.01027501467615366,
0.057631492614746094,
0.04289451986551285,
-0.013153580948710442,
0.02344375103712082,
0.03305500000715256,
-0.03923847898840904,
-0.0199... | 0.850876 | 0.839059 | 0.011817 |
Consider a reaction-diffusion equation for pattern formation, given as:
$$
u_t = D_u\nabla^2 u + f(u, v) \\
v_t = D_v\nabla^2 v + g(u, v)
$$
where $u(x, y)$ and $v(x, y)$ are concentrations of two chemicals, $D_u$ and $D_v$ are diffusion coefficients, and $f(u, v)$ and $g(u, v)$ are reaction terms.
Suppose we have a 2... | tiger_lab_math | [
0.005311912391334772,
0.03124552220106125,
-0.04075712710618973,
-0.03327875956892967,
0.032639000564813614,
-0.04948297142982483,
-0.050691865384578705,
0.10317553579807281,
0.022280409932136536,
-0.005147277843207121,
0.015533234924077988,
0.02399962954223156,
-0.025316337123513222,
-0.0... | 0.817626 | 0.806552 | 0.011074 |
Consider the topological space X = {a, b, c, d} with topology T = {{}, {a, b}, {c, d}, {a, b, c, d}}.
Find out whether the subset A = {a, b} is open or not in the given topology. | tiger_lab_math | [
0.03627573698759079,
0.005653469357639551,
-0.05293278768658638,
-0.011970334686338902,
0.03608664870262146,
-0.0189529936760664,
-0.00800742395222187,
0.0665108785033226,
0.03446192666888237,
-0.015361670404672623,
0.013088159263134003,
0.005226602777838707,
-0.02630753070116043,
-0.02342... | 0.821639 | 0.80502 | 0.01662 |
Find a harmonic function $u(x,y)$ on the square $0 \leq x \leq 1$, $0 \leq y \leq 1$ that satisfies the following conditions:
\begin{align*}
u(0,y) &= y^2, \\
u(1,y) &= 1+y^2, \\
u(x,0) &= x^3, \\
u(x,1) &= 1+x^3.
\end{align*}
Then find the value of $\frac{\partial^2 u}{\partial x^2}(1/2,1/2)$. | tiger_lab_math | [
0.011914122849702835,
0.04738066717982292,
-0.05529283359646797,
-0.034453026950359344,
0.04282598942518234,
0.0010358726140111685,
-0.012986935675144196,
0.06796577572822571,
0.03637195751070976,
-0.0027809888124465942,
0.028067314997315407,
0.028046254068613052,
-0.02868984267115593,
-0.... | 0.826574 | 0.80202 | 0.024554 |
Kendra wants enough shirts that she only has to do laundry once every two weeks. She wears one shirt to school for each of the five weekdays. Three days a week, she changes into a different shirt for an after-school club. On Saturday, she wears one shirt all day. On Sunday, she wears a different shirt to church than sh... | tiger_lab_math | [
0.0032539304811507463,
0.01796208694577217,
-0.05909242480993271,
-0.024763166904449463,
0.023903004825115204,
-0.025257844477891922,
-0.0587710440158844,
0.07143743336200714,
0.01663197949528694,
-0.0021241658832877874,
-0.0006925686611793935,
0.034410059452056885,
-0.040721360594034195,
... | 0.814371 | 0.810683 | 0.003688 |
Ming and Catherine walked to the store together to buy some pencils. Ming bought $40$ pencils and Catherine bought $24$. If each package of pencils sold at the store contain the same number of pencils, what is the largest possible number of pencils in a package? Please respond by writing a program in Python. | tiger_lab_math | [
0.021777769550681114,
0.03298648074269295,
-0.041825734078884125,
-0.04186556115746498,
0.04757678508758545,
-0.03337929770350456,
-0.03018350899219513,
0.03501110523939133,
0.028309527784585953,
-0.028461430221796036,
0.03354872390627861,
0.023227278143167496,
-0.030637167394161224,
-0.01... | 0.829399 | 0.826225 | 0.003174 |
The Schrödinger equation of a particle in a 1-dimensional box is given by:
$\frac{-\hbar^2}{2m}\frac{\partial^2\Psi(x)}{\partial x^2} + V(x)\Psi(x) = E\Psi(x)$
where $\hbar$ is the reduced Planck constant, $m$ is the mass of the particle, $V(x)$ is the potential energy function, $\Psi(x)$ is the wave function of the ... | tiger_lab_math | [
0.022890755906701088,
0.045117683708667755,
-0.021106740459799767,
-0.04732762649655342,
0.03939232602715492,
-0.05118950083851814,
-0.028524404391646385,
0.08072914928197861,
0.035254400223493576,
0.0021094712428748608,
0.023350829258561134,
0.03328912705183029,
-0.036113619804382324,
-0.... | 0.805567 | 0.794404 | 0.011162 |
A company wants to segment its customer base into 3 clusters based on their purchase behavior. The company collected data on the number of items purchased by each customer over the past year. Using cluster analysis, determine the center and radius of the clusters. The data is as follows:
Customer #1: bought 10 items
C... | tiger_lab_math | [
0.02713806740939617,
0.028752578422427177,
-0.06512697786092758,
-0.042645376175642014,
0.0263350997120142,
-0.006667060777544975,
-0.049688246101140976,
0.02348831295967102,
0.018734006211161613,
0.01482576783746481,
0.01464002300053835,
0.01594756916165352,
-0.06754373759031296,
-0.01047... | 0.789887 | 0.779624 | 0.010263 |
Let u(x,y) be a harmonic function in the region R: 0<x<1, 0<y<1, such that u(x,0) = 1, u(0,y) = 0, and u(1,y) = ln(y+1), for 0<y<1. Find u(x,y). | tiger_lab_math | [
0.012002105824649334,
0.03178177773952484,
-0.047074731439352036,
-0.037544362246990204,
0.03458157181739807,
0.001701541244983673,
-0.02220812626183033,
0.06336963176727295,
0.04127024859189987,
-0.011831207200884819,
0.03028915263712406,
0.027029670774936676,
-0.03339787945151329,
-0.015... | 0.821 | 0.812009 | 0.008991 |
A company produces two products, A and B, and has contract obligations to produce at least 200 units of product A and 150 units of product B each month. It takes 2 hours to make one unit of product A and 3 hours to make one unit of product B. The company has 600 hours of labor available per month. Product A sells for $... | tiger_lab_math | [
0.020663155242800713,
0.002053994918242097,
-0.06307314336299896,
-0.011108528822660446,
0.031082067638635635,
0.0016074551967903972,
-0.036141324788331985,
0.07784117758274078,
0.038962699472904205,
-0.002101803896948695,
0.01679886132478714,
0.027288617566227913,
-0.010518093593418598,
-... | 0.80978 | 0.798471 | 0.011309 |
The set of vectors $\mathbf{v}$ such that
\[\operatorname{proj}_{\begin{pmatrix} 5 \\ 2 \end{pmatrix}} \mathbf{v} = \begin{pmatrix} -\frac{5}{2} \\ -1 \end{pmatrix}\]lie on a line. Enter the equation of this line in the form "$y = mx + b$". | tiger_lab_math | [
0.030120184645056725,
0.014412332326173782,
-0.06214756518602371,
-0.04336557537317276,
0.06653667986392975,
-0.0034088066313415766,
-0.009277197532355785,
0.08941003680229187,
0.05359392240643501,
-0.0055669923312962055,
0.005685228854417801,
0.04031866416335106,
-0.01922954060137272,
-0.... | 0.820276 | 0.813664 | 0.006613 |
A manufacturing company produces two types of products, A and B. The cost to make each unit of product A is $3 and the cost to make each unit of product B is $5. The selling price for each unit of product A is $8 and for B it is $10. The company wants to maximize their profit. How many units of each product should they... | tiger_lab_math | [
0.022189060226082802,
0.0014535817317664623,
-0.06782953441143036,
-0.002930005080997944,
0.027193646878004074,
-0.012624016031622887,
-0.047404877841472626,
0.07463439553976059,
0.037573251873254776,
-0.013910518027842045,
0.003557389136403799,
0.03607102110981941,
-0.024210236966609955,
... | 0.817708 | 0.814291 | 0.003417 |
Suppose you have a set of integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Using fuzzy logic, assign a grade for each integer ranging from 0 to 1 based on how "big" the integer is. For example, you might assign a grade of 0.9 to the integer 10 because it is the largest integer in the set. Once you have assigned grades to each... | tiger_lab_math | [
0.005467097740620375,
0.010309563018381596,
-0.02126825414597988,
-0.0238201804459095,
0.04244512319564819,
-0.046153198927640915,
-0.02056739293038845,
0.02749732695519924,
0.023321442306041718,
-0.01679595746099949,
0.017730893567204475,
0.03350723162293434,
-0.049895841628313065,
-0.000... | 0.814078 | 0.800255 | 0.013823 |
the vertex of a rectangle are ( 1 , 0 ) , ( 9 , 0 ) , ( 1 , 2 ) and ( 9 , 2 ) respectively . if line l passes through the origin and divided the rectangle into two identical quadrilaterals , what is the slope of line l ? Please write a program to solve it | tiger_lab_math | [
-0.0053666881285607815,
0.02354339137673378,
-0.04721206799149513,
-0.04290672764182091,
0.02490491420030594,
-0.027508871629834175,
-0.022971082478761673,
0.06794978678226471,
0.03279319033026695,
-0.017509721219539642,
0.027445891872048378,
0.02459292858839035,
-0.029304245486855507,
-0.... | 0.83254 | 0.827933 | 0.004607 |
Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2008,0),(2008,2009),$ and $(0,2009)$. What is the probability that $x > 2y$? Express your answer as a common fraction. | tiger_lab_math | [
0.0015267700655385852,
0.015733543783426285,
-0.04960281029343605,
-0.028012951835989952,
0.012009861879050732,
-0.012388442642986774,
-0.026050813496112823,
0.06678513437509537,
0.04176192358136177,
0.013374424539506435,
0.012495371513068676,
0.03601433336734772,
-0.0325547531247139,
-0.0... | 0.83511 | 0.825123 | 0.009987 |
If x and y are positive integers such that x < y and z=x/y, which of the following must be true?
I. z>(x − 1)/(y − 2)
II. z<(x − 1)/(y − 2)
III. z>(x + 1)/(y + 2)
Answer Choices: (A) I only (B) I and II (C) II and III (D) II only (E) I and III | tiger_lab_math | [
0.03833010420203209,
0.02400818280875683,
-0.035005684942007065,
-0.018629226833581924,
0.03043876774609089,
-0.008810684084892273,
-0.011784432455897331,
0.05481205880641937,
0.032367315143346786,
-0.03715016692876816,
0.017155751585960388,
0.020201897248625755,
-0.03368200734257698,
-0.0... | 0.830336 | 0.829117 | 0.001219 |
a gambler bought $ 3,000 worth of chips at a casino in denominations of $ 20 and $ 100 . that evening , the gambler lost 16 chips , and then cashed in the remainder . if the number of $ 20 chips lost was 2 more or 2 less than the number of $ 100 chips lost , what is the largest amount of money that the gambler could ha... | tiger_lab_math | [
0.021271120756864548,
0.02769608423113823,
-0.04853105545043945,
-0.030305761843919754,
0.03595849499106407,
-0.02643236704170704,
-0.03495674580335617,
0.04754427447915077,
0.01164326723664999,
-0.010306905023753643,
0.031125517562031746,
0.04732022061944008,
-0.044596485793590546,
-0.009... | 0.816958 | 0.815371 | 0.001586 |
Let $M$ be a 3-dimensional manifold equipped with the following smooth vector field:
$$X = x^2\frac{\partial}{\partial x} + 2y\frac{\partial}{\partial y} + z\frac{\partial}{\partial z}.$$
Determine the Lie derivative of $X$ with respect to the vector field
$$Y = y\frac{\partial}{\partial x} + x\frac{\partial}{\par... | tiger_lab_math | [
0.024423226714134216,
0.0469512939453125,
-0.041072044521570206,
-0.06186290457844734,
0.035501088947057724,
0.002970315283164382,
-0.01894925720989704,
0.06435974687337875,
0.025965042412281036,
-0.01130057591944933,
0.012323927134275436,
0.034425023943185806,
-0.05180621147155762,
-0.015... | 0.819262 | 0.803072 | 0.016189 |
In a conference room, 40 chairs with a capacity of 2 people each were arranged in rows in preparation for the board meeting of a company, whose number of members was the same as the chairs' capacity. If 2/5 of the chairs were not occupied, and the rest each had two people, calculate the number of board members who did ... | tiger_lab_math | [
0.021704034879803658,
0.02961697429418564,
-0.04849154129624367,
-0.0225636288523674,
0.032937031239271164,
-0.025259267538785934,
-0.028010109439492226,
0.04950368404388428,
0.037931010127067566,
-0.014761283993721008,
0.01572353206574917,
0.02364414371550083,
-0.03178063780069351,
-0.014... | 0.831341 | 0.825433 | 0.005908 |
Suppose there are five boxes labeled A, B, C, D, and E. You know that Box A contains 3 apples, Box C contains 7 apples, and Box E contains 8 apples. You do not know how many apples Boxes B and D contain. You make the assumption that Boxes B and D each contain at least 5 apples. However, you then find out that Box B act... | tiger_lab_math | [
0.021740803495049477,
0.017236636951565742,
-0.03073735348880291,
-0.020260455086827278,
0.029222413897514343,
-0.03214621543884277,
-0.02291891910135746,
0.06827875971794128,
0.028052736073732376,
-0.02963961660861969,
0.0021568879019469023,
0.02369554154574871,
-0.024065544828772545,
-0.... | 0.824081 | 0.808625 | 0.015455 |
As a bicycle salesperson, Norman earns a fixed salary of $20 per week plus $6 per bicycle for the first six bicycles he sells, $12 per bicycle for the next six bicycles he sells, and $18 per bicycle for every bicycle sold after the first 12. This week, Norman earned more than twice as much as he did last week. If he so... | tiger_lab_math | [
0.016161369159817696,
0.020407376810908318,
-0.040629997849464417,
-0.022055014967918396,
0.038778554648160934,
-0.03703263774514198,
-0.021979650482535362,
0.0754082202911377,
0.03725292161107063,
-0.011748725548386574,
0.03026287443935871,
0.0369093082845211,
-0.03506414592266083,
-0.008... | 0.819214 | 0.817475 | 0.00174 |
What is the curvature of a hyperbolic triangle with angles measuring 60 degrees, 70 degrees, and 50 degrees?
Hint: The curvature of a hyperbolic triangle is given by K = α + β + γ - π, where α, β, and γ represent the interior angles in radians and π is the constant Pi. | tiger_lab_math | [
-0.009074782021343708,
0.019522447139024734,
-0.015829244628548622,
-0.030527843162417412,
0.02409970760345459,
-0.010883155278861523,
-0.016950413584709167,
0.04203536733984947,
0.029338574036955833,
-0.031076712533831596,
0.018949002027511597,
0.030458752065896988,
-0.026297887787222862,
... | 0.796326 | 0.792302 | 0.004023 |
Given that $\theta$ and $\phi$ are acute angles such that $\tan \theta = \frac{1}{7}$ and $\sin \phi = \frac{1}{\sqrt{10}},$ find $\theta + 2 \phi,$ measured in radians. Please respond by writing a program in Python. | tiger_lab_math | [
0.026388483121991158,
0.04731128364801407,
-0.040869832038879395,
-0.05248463898897171,
0.03518124297261238,
-0.016696510836482048,
-0.02988208830356598,
0.04958558827638626,
0.035816341638565063,
-0.010012449696660042,
0.019098734483122826,
0.013427266851067543,
-0.04359786584973335,
-0.0... | 0.827562 | 0.816885 | 0.010677 |
Suppose there is a used car market in which the distribution of types of cars is divided into two types - high-quality and low-quality. A seller knows the true quality of their car but the buyer does not. The probability of the car being high-quality is 0.4 and low-quality is 0.6. The seller knows the price they paid f... | tiger_lab_math | [
-0.005246289074420929,
0.007943147793412209,
-0.03967617452144623,
-0.006176878698170185,
0.02996777929365635,
0.0015494157560169697,
-0.03307291492819786,
0.050680458545684814,
0.01670251414179802,
-0.024449996650218964,
0.01757730357348919,
0.040131520479917526,
-0.016069136559963226,
-0... | 0.817427 | 0.806911 | 0.010516 |
Consider a curve in 3D space given by the parameterization:
$$C(t) = (2\cos t, 2\sin t, 3t), \ \ \ \ 0\leq t \leq \pi/2$$
Determine the parallel transport of the vector $\vec{v} = (1,0,0)$ along the curve $C(t)$. | tiger_lab_math | [
0.021402373909950256,
0.046328265219926834,
-0.03909766301512718,
-0.007499735336750746,
0.0419967882335186,
-0.003967486321926117,
-0.04030013456940651,
0.07155002653598785,
0.02704109624028206,
-0.00596466613933444,
0.01605251058936119,
0.04769369587302208,
-0.02163190208375454,
0.007153... | 0.832399 | 0.815477 | 0.016923 |
Consider the graph G shown below. Determine whether G is planar or not using the Kuratowski's theorem.
Image: https://i.imgur.com/DYo7tYw.png | tiger_lab_math | [
-0.005799662787467241,
0.038584113121032715,
-0.040534790605306625,
-0.05418822914361954,
0.024605903774499893,
-0.027262413874268532,
-0.007363101001828909,
0.051632389426231384,
0.02131744660437107,
-0.027743231505155563,
0.013248568400740623,
0.02159455232322216,
-0.03326696529984474,
0... | 0.815009 | 0.800716 | 0.014293 |
Here's a possible numerical optimization problem for the student:
A company is trying to schedule the production of three different products (A, B, and C) using two machines (X and Y). Each product takes a certain amount of time to produce on each machine, and each machine can only work on one product at a time. The c... | tiger_lab_math | [
0.0336054302752018,
0.006385486107319593,
-0.04189622029662132,
-0.02481851726770401,
0.03195633366703987,
-0.01711932010948658,
-0.03784111142158508,
0.09493908286094666,
0.02412346377968788,
0.004222397692501545,
0.009485209360718727,
0.02787412516772747,
-0.018438205122947693,
-0.013883... | 0.816542 | 0.793612 | 0.02293 |
Suppose a company has only $10,000 to invest in two different projects. Project A will return $2,000 for every unit invested and project B will return $3,000 for every unit invested. However, project A requires a minimum investment of 5 units and project B requires a minimum investment of 3 units. The company wants to ... | tiger_lab_math | [
0.011267527006566525,
0.013269812799990177,
-0.06274634599685669,
-0.005698281340301037,
0.023078450933098793,
-0.004093477968126535,
-0.025847505778074265,
0.05851593613624573,
0.024081578478217125,
-0.009134300984442234,
0.008079206570982933,
0.03845994174480438,
-0.018876971676945686,
-... | 0.811553 | 0.806449 | 0.005105 |
A company wants to test the effectiveness of two different cleaning agents for removing grease stains. They randomly selected 15 grease-stained shirts and divided them into two groups of 8 and 7. The first group was cleaned with cleaning agent A and the second group was cleaned with cleaning agent B. The time (in secon... | tiger_lab_math | [
0.008234378881752491,
0.004374350421130657,
-0.04345516860485077,
-0.06204866245388985,
0.029871545732021332,
-0.021860845386981964,
-0.001580303767696023,
0.08710731565952301,
0.039648864418268204,
0.011198379099369049,
-0.0030564910266548395,
0.04118828475475311,
-0.028045015409588814,
-... | 0.800423 | 0.796893 | 0.003529 |
A common problem in topological optimization is to find the optimal shape of a structure given certain conditions. Here's an example:
A design engineer wants to create a beam with a length of 10 meters that can support a load of at least 20 kN with a maximum allowable deformation of 1 centimeter. The beam is made of a... | tiger_lab_math | [
0.022325143218040466,
0.03700071945786476,
-0.05702384188771248,
-0.04267469421029091,
0.031031401827931404,
-0.020461013540625572,
-0.03904564678668976,
0.06274495273828506,
0.013559763319790363,
-0.007724510971456766,
0.008597216568887234,
0.01648244634270668,
-0.03064841218292713,
0.011... | 0.781252 | 0.758697 | 0.022555 |
Suppose a company produces two types of products, A and B. The company can produce a maximum of 500 units of product A and 300 units of product B per day. The cost of producing one unit of product A is $6 and the cost of producing one unit of product B is $8. The selling price of product A is $15 per unit and the selli... | tiger_lab_math | [
0.02179226279258728,
0.003257346572354436,
-0.06772816926240921,
-0.0019106602994725108,
0.02390526793897152,
-0.025717489421367645,
-0.04740610346198082,
0.08473338931798935,
0.038919560611248016,
-0.021535221487283707,
0.0033104177564382553,
0.03335719183087349,
-0.013482300564646721,
-0... | 0.814135 | 0.800497 | 0.013638 |
Let X be the projective variety defined by the equation
xy^2 - x^2y + z^3 = 0 in P^2.
1. Compute the dimension of X.
2. Compute the intersection cohomology groups of X, i.e., IH^k(X), for all k.
3. Determine the number of rational curves of degree 4 in X which pass through a given point P in the projective plane. | tiger_lab_math | [
0.013729654252529144,
0.034116230905056,
-0.029481392353773117,
-0.034434009343385696,
0.027996327728033066,
-0.020464450120925903,
-0.004953023046255112,
0.03453584387898445,
0.04680684953927994,
-0.00805283896625042,
0.006050651427358389,
0.030527887865900993,
-0.018907202407717705,
-0.0... | 0.826658 | 0.805617 | 0.021041 |
Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{3}$. The diagonals have length $10\sqrt {21}$, and point $E$ is at distances $10\sqrt {7}$ and $30\sqrt {7}$ from vertices $A$ and $D$, respectively. Let $F$ be the foot of the altitud... | tiger_lab_math | [
0.03212575614452362,
0.022615045309066772,
-0.03236071392893791,
-0.04608701914548874,
0.020173076540231705,
-0.019963808357715607,
-0.01614248938858509,
0.055926647037267685,
0.03176640719175339,
-0.013345722109079361,
0.036667536944150925,
0.040397681295871735,
-0.04315171018242836,
-0.0... | 0.853009 | 0.825212 | 0.027797 |
Determine the number of distinct groups of order 30 that are isomorphic to either the cyclic group C30, the dihedral group D15 or the non-abelian group of order 30. | tiger_lab_math | [
-0.00796028133481741,
0.01753923110663891,
-0.043425917625427246,
-0.031762827187776566,
0.02000880427658558,
-0.035005200654268265,
-0.009360257536172867,
0.044447578489780426,
0.0328427292406559,
-0.03228286653757095,
0.020690998062491417,
0.006514688953757286,
-0.04231807962059975,
-0.0... | 0.813787 | 0.812536 | 0.001251 |
at a restaurant , glasses are stored in two different - sized boxes . one box contains 12 glasses , and the other contains 16 glasses . if the average number of glasses per box is 15 , and there are 16 more of the larger boxes , what is the total number of glasses w at the restaurant ? ( assume that all boxes are fille... | tiger_lab_math | [
0.023142371326684952,
0.03145185112953186,
-0.05173633247613907,
-0.03328503295779228,
0.05469530448317528,
-0.017285069450736046,
-0.0343480221927166,
0.07022669166326523,
0.041314829140901566,
-0.022772125899791718,
0.034908100962638855,
0.020321251824498177,
-0.03866146504878998,
-0.013... | 0.825902 | 0.824573 | 0.001328 |
In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respectively, after a $180^\circ$ rotation about $G.$ What is the area of the union of the two regions enclosed by the triangles $ABC$ and $A'B'C'?$
| tiger_lab_math | [
0.026548810303211212,
0.019674459472298622,
-0.045329686254262924,
-0.04648658633232117,
0.026574674993753433,
-0.0013674009824171662,
-0.023501668125391006,
0.044533759355545044,
0.03574509546160698,
-0.0038182795979082584,
0.014547630213201046,
0.025506267324090004,
-0.006289349868893623,
... | 0.826915 | 0.819259 | 0.007656 |
Consider a surface given by the equation $z = x^2 + y^2$, with metric $ds^2 = dx^2 + dy^2 + 4(z^2 + 1)dz^2$. Determine the components of the Levi-Civita connection for this metric. | tiger_lab_math | [
0.007227548398077488,
0.03186185285449028,
-0.03713390231132507,
-0.04640467092394829,
0.03709216043353081,
-0.01567421481013298,
-0.010488206520676613,
0.048496462404727936,
0.032895028591156006,
-0.0033558167051523924,
0.014389975927770138,
0.04351485148072243,
-0.0293585155159235,
-0.00... | 0.823448 | 0.810436 | 0.013012 |
a rectangular wall is covered entirely with two kinds of decorative tiles : regular and jumbo . 1 / 3 of the tiles are jumbo tiles , which have a length three times that of regular tiles and have the same ratio of length to width as the regular tiles . if regular tiles cover 90 square feet of the wall , and no tiles ov... | tiger_lab_math | [
0.00380786694586277,
0.04370943084359169,
-0.046986110508441925,
-0.03303362801671028,
0.022571824491024017,
-0.03311626985669136,
-0.025427505373954773,
0.05377110093832016,
0.01777815818786621,
-0.010534689761698246,
0.043410524725914,
0.03964647650718689,
-0.04909072816371918,
-0.012034... | 0.81592 | 0.810392 | 0.005528 |
Statement 1 | The center of a group is a subgroup of that group. Statement 2 | The order of the dihedral group D_n is n. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.03763119503855705,
0.02720710262656212,
-0.026863953098654747,
-0.041475601494312286,
0.03181185573339462,
-0.02935873717069626,
-0.010172180831432343,
0.054915621876716614,
0.03474333509802818,
-0.013309318572282791,
0.01617462746798992,
0.027343764901161194,
-0.04449641704559326,
-0.01... | 0.809912 | 0.808626 | 0.001286 |
An oil refinery has two production units, A and B, with a total production capacity of 500 barrels of oil per day. The revenues from unit A and B are $10 and $15 per barrel, respectively. However, due to uncertain weather conditions, production rates are not constant and vary randomly. On an average day, unit A produce... | tiger_lab_math | [
0.0003958052839152515,
0.015474467538297176,
-0.06559392809867859,
-0.00692713214084506,
0.016684457659721375,
-0.03601120412349701,
-0.04420776292681694,
0.0833398625254631,
0.0392875149846077,
-0.005239098332822323,
0.01313318032771349,
0.003822013735771179,
-0.011221032589673996,
-0.003... | 0.788405 | 0.779478 | 0.008926 |
Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true?
Answer Choices: (A) The minimum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 – 2m + 1 (B) The maximum possible value of a^2 + b^2 + c^2 + d^2 is 4m^2 – 2m + 1. (C) ... | tiger_lab_math | [
0.036641597747802734,
0.029333289712667465,
-0.032902054488658905,
-0.03492261469364166,
0.05682535097002983,
-0.031664103269577026,
-0.01853298582136631,
0.055065713822841644,
0.04828977212309837,
-0.031077124178409576,
0.012647289782762527,
0.03614593297243118,
-0.017605066299438477,
-0.... | 0.844644 | 0.837086 | 0.007558 |
Consider the following 2D metric:
$ds^2 = 4x(dx)^2 + dy^2$
Calculate the Christoffel symbols $\Gamma^i_{jk}$ for this metric. | tiger_lab_math | [
0.01258812379091978,
0.03082902729511261,
-0.034757357090711594,
-0.0564623661339283,
0.04568353667855263,
-0.03376106172800064,
-0.02291155606508255,
0.05862666293978691,
0.05496848374605179,
-0.008519061841070652,
0.01041433122009039,
0.0314452238380909,
-0.028068270534276962,
-0.0242407... | 0.831698 | 0.821004 | 0.010694 |
Let X be a Calabi-Yau variety defined by the equation x^3 + y^3+ z^3 = 0 in projective three-space P^3, and let L be the line defined by the equations x = y = 0. Find the intersection of the divisor [L] and the canonical divisor of X. | tiger_lab_math | [
0.028088483959436417,
0.027541453018784523,
-0.04893086478114128,
-0.03630547970533371,
0.03952081501483917,
-0.008201353251934052,
-0.016886433586478233,
0.05670928955078125,
0.04259566590189934,
-0.010896963067352772,
0.022911474108695984,
0.0301816388964653,
-0.033804915845394135,
-0.01... | 0.833969 | 0.821179 | 0.012789 |
Suppose you have $100,000 to invest in two stocks, A and B. The expected annual return for stock A is 12% and the standard deviation is 15%, while for stock B the expected return is 10% and the standard deviation is 10%. The correlation coefficient between the two stocks is 0.3. Determine the optimal portfolio allocati... | tiger_lab_math | [
0.012460228987038136,
-0.0037222381215542555,
-0.06121731922030449,
-0.033468134701251984,
0.026972290128469467,
-0.02088434435427189,
-0.03242838382720947,
0.06136946380138397,
0.04908788204193115,
0.003017992479726672,
0.020066488534212112,
0.04414120316505432,
-0.03010934591293335,
-0.0... | 0.80667 | 0.800626 | 0.006045 |
Suppose C[0,1] denotes the space of all the continuous functions on the interval [0,1]. Is (C[0,1],\|\cdot\|_1 ) a Banach space? Here $\|f(x)\|_1=\int_0^1 |f(t)|dt$ with $f\in C[0,1]$. Answer 1 for yes and 0 for no. | tiger_lab_math | [
0.023958954960107803,
0.034913137555122375,
-0.04761496186256409,
-0.021856607869267464,
0.039771668612957,
-0.0025217458605766296,
-0.01308163907378912,
0.06802698224782944,
0.04218701273202896,
-0.006878129672259092,
0.016242342069745064,
0.023214280605316162,
-0.013965965248644352,
-0.0... | 0.820239 | 0.805706 | 0.014533 |
Let $S$ be a region in the plane with area 4. When we apply the matrix
\[\begin{pmatrix} 2 & -1 \\ 7 & 2 \end{pmatrix}\]to $S,$ we obtain the region $S'.$ Find the area of $S'.$ Please write a program to solve it | tiger_lab_math | [
0.04346900433301926,
0.04074343293905258,
-0.06729618459939957,
-0.05072616785764694,
0.049923524260520935,
-0.008855754509568214,
-0.016289573162794113,
0.05649612471461296,
0.037381310015916824,
-0.004988320637494326,
0.018186308443546295,
0.044458985328674316,
-0.027271337807178497,
-0.... | 0.823855 | 0.822015 | 0.00184 |
A company has three factories producing the same type of product and three warehouses in different cities. The demand for the product in each city is 2300, 1600, and 2100 units, respectively. The transportation cost per unit from each factory to each warehouse is given as follows:
- Factory 1: $5, $7, $3
- Factory 2: ... | tiger_lab_math | [
0.017768818885087967,
0.0074646975845098495,
-0.06722423434257507,
-0.012068627402186394,
0.022934315726161003,
-0.0005607089842669666,
-0.042585402727127075,
0.052781667560338974,
0.02812386304140091,
-0.014077120460569859,
0.015179461799561977,
0.040104158222675323,
-0.03187379613518715,
... | 0.809743 | 0.800769 | 0.008974 |
Suppose a car manufacturing company produces two types of cars: Sedans and SUVs. The company has a budget of $600,000 to spend on advertising. The cost of advertising per sedan is $2000, and the cost of advertising per SUV is $3000. The profit per sedan is $4000, and the profit per SUV is $5000. Using the firefly algor... | tiger_lab_math | [
-0.012896494008600712,
0.014769481495022774,
-0.0374409444630146,
-0.020369969308376312,
0.0212539192289114,
-0.02098197489976883,
-0.027583254501223564,
0.08374696224927902,
0.034082040190696716,
-0.020631752908229828,
0.01936614327132702,
0.017747940495610237,
-0.01315270084887743,
-0.03... | 0.809139 | 0.80148 | 0.007658 |
Let $M$ be a 2-dimensional manifold equipped with a metric tensor $g$ given by $g = dx^2 + 2dy^2$. Consider a tensor field $T$ on $M$ given by $T = xy^2\frac{\partial}{\partial x}\otimes(dx) + x^2y\frac{\partial}{\partial y}\otimes(dy)$.
Compute the covariant derivative of $T$ with respect to the Levi-Civita connectio... | tiger_lab_math | [
0.00015522770991083235,
0.03540651127696037,
-0.038175567984580994,
-0.0385214239358902,
0.023922178894281387,
-0.0047748195938766,
-0.017651652917265892,
0.07463086396455765,
0.034038204699754715,
0.00805920921266079,
-0.004609068389981985,
0.0390876829624176,
-0.03494073823094368,
-0.007... | 0.799182 | 0.788101 | 0.011081 |
In a Traveler's Dilemma game between two players A and B, the payoff matrix is given below:
| | Player B's Offer |
|-----------|-----------------|
| | $2$ | $3$ |
| Player A's Offer | $2$ | $x_A, x_B$ | $y_A, y_B$ |
| | $3$ | $z_A, z_B$ | $w_A, w_B$ |
Assume that both players choose their str... | tiger_lab_math | [
0.027956048026680946,
0.020755615085363388,
-0.04883589968085289,
-0.04146410897374153,
0.012564457952976227,
-0.007123053073883057,
-0.020222987979650497,
0.06843031942844391,
0.023273343220353127,
-0.029269453138113022,
0.03511141985654831,
0.030572040006518364,
-0.03300347551703453,
-0.... | 0.827305 | 0.810455 | 0.01685 |
A certain collection of numbered indexed cards includes one card with a 1 written on it, two cards with a 2, and so forth up to $n$ cards showing an $n,$ for some positive integer $n$. Determine $n,$ if the average value of a card in this collection is 2017. | tiger_lab_math | [
0.014134366065263748,
0.006950655952095985,
-0.05144885182380676,
-0.028276095166802406,
0.031721703708171844,
-0.029207440093159676,
-0.04334165155887604,
0.037225041538476944,
0.033424459397792816,
-0.0151179488748312,
0.021501148119568825,
0.017340293154120445,
-0.052054263651371,
0.010... | 0.841158 | 0.826154 | 0.015003 |
Let X be a path-connected topological space such that Π1(X, x₀) is isomorphic to ℤ₂, where x₀ is a fixed basepoint. Suppose Y is a covering space of X such that Y has two sheets. Determine the fundamental group of Y. | tiger_lab_math | [
0.021925756707787514,
0.010244888253509998,
-0.03212306275963783,
-0.03342337906360626,
0.0240767989307642,
-0.025922589004039764,
0.006125557702034712,
0.060733016580343246,
0.034845396876335144,
-0.010537483729422092,
0.015779098495841026,
0.0268505048006773,
-0.03687267377972603,
-0.007... | 0.827814 | 0.806064 | 0.02175 |
A small square is entirely contained in a larger square, as shown. The side length of the small square is 3 units and the side length of the larger square is 7 units. What is the number of square units in the area of the black region?
[asy]
fill((0,0)--(21,0)--(21,21)--(0,21)--cycle,black);
fill((9,4)--(9,13)--(18,13)... | tiger_lab_math | [
0.022304005920886993,
0.03516925126314163,
-0.04542699083685875,
-0.049755245447158813,
0.039980825036764145,
-0.006619863212108612,
-0.02414967492222786,
0.05615250766277313,
0.03709815442562103,
-0.005658825393766165,
0.024239659309387207,
0.040985096246004105,
-0.036580681800842285,
-0.... | 0.83299 | 0.829119 | 0.003871 |
Suppose you are the owner of a small bakery that produces two types of cakes: chocolate and vanilla. On a given day, you have 9 pounds of flour, 7 pounds of sugar, and 5 pounds of cocoa powder at your disposal. A chocolate cake requires 2 pounds of flour, 2 pounds of sugar, and 1 pound of cocoa powder, while a vanilla ... | tiger_lab_math | [
0.020365195348858833,
0.03198112174868584,
-0.048473846167325974,
-0.027550695464015007,
0.01883208006620407,
-0.006522743031382561,
-0.037851057946681976,
0.0650988221168518,
0.028072994202375412,
-0.02200026996433735,
0.004416192881762981,
0.02797640487551689,
-0.02707367204129696,
-0.03... | 0.830862 | 0.815836 | 0.015026 |
Here is a numerical problem for a math student to solve related to the subtopic of Applying the Fourier transform to signal processing:
A continuous-time signal in the time domain is defined as x(t) = 2cos(2π(10t + 3)) + 4cos(2π(25t - 1)).
a) What is the period of the signal?
b) Compute the Fourier transform of the s... | tiger_lab_math | [
0.026400715112686157,
0.03041156753897667,
-0.06624233722686768,
-0.042803384363651276,
0.05091289058327675,
-0.011556107550859451,
-0.010679291561245918,
0.06558497250080109,
0.028156334534287453,
0.01265082135796547,
0.0323164202272892,
0.02402627281844616,
-0.03346952795982361,
-0.01221... | 0.814185 | 0.811122 | 0.003063 |
In right triangle $ABC$, shown below, $\cos{C}=\frac{9\sqrt{130}}{130}$. Find $AC$.
[asy]
draw((0,0)--(7,0)--(0,9)--cycle,black+linewidth(1));
draw(rightanglemark((7,0),(0,0),(0,9),20),black+linewidth(1));
label("$A$",(0,0),W);
label("$B$",(7,0),E);
label("$C$",(0,9),W);
label("$\sqrt{130}$",(7,0)--(0,9),NE);
[/asy] ... | tiger_lab_math | [
0.027866899967193604,
0.03326555714011192,
-0.04073719307780266,
-0.03592658415436745,
0.03613731265068054,
-0.025630751624703407,
-0.01978231780230999,
0.0503108985722065,
0.047099918127059937,
-0.005868109874427319,
0.030649708583950996,
0.03575385734438896,
-0.007152475416660309,
-0.013... | 0.840943 | 0.827585 | 0.013358 |
Candice buys all the bread she and her family needs for the week from a local bakery. She needs 2 loaves of white bread for sandwiches that cost $3.50 each. She also needs a baguette that costs $1.50 and 2 loaves of sourdough bread that cost $4.50 each. She also treats herself to a $2.00 almond croissant each visit.... | tiger_lab_math | [
0.014813442714512348,
0.04361102357506752,
-0.04675838723778725,
-0.0313819944858551,
0.018302248790860176,
-0.019990503787994385,
-0.03251314535737038,
0.04630010575056076,
0.03994666412472725,
-0.008571921847760677,
0.03478396683931351,
0.014212986454367638,
-0.03779439628124237,
-0.0284... | 0.787987 | 0.785165 | 0.002822 |
A manufacturing company produces two types of products, product A and product B, using two machines, machine X and machine Y. The production time for product A is 5 hours on machine X and 4 hours on machine Y, while the production time for product B is 3 hours on machine X and 6 hours on machine Y. The company wants to... | tiger_lab_math | [
0.013572555035352707,
0.0029674081597477198,
-0.07202106714248657,
-0.012882311828434467,
0.026472628116607666,
-0.003399224253371358,
-0.05254032462835312,
0.09279900789260864,
0.04819028452038765,
-0.0007957236957736313,
0.0009685845579952002,
0.01776473969221115,
-0.0009065835038200021,
... | 0.794302 | 0.785355 | 0.008946 |
A used car dealer has three cars he is trying to sell. He knows the quality of each car, but the potential buyers do not. The dealer labels the cars as "good," "fair," and "poor." The dealer knows that the "good" car is worth $12,000, the "fair" car is worth $8,000, and the "poor" car is only worth $4,000. The potentia... | tiger_lab_math | [
0.007244728039950132,
0.007864443585276604,
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0.02570841647684574,
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0.012978521175682545,
0.03761665150523186,
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-0.00... | 0.822487 | 0.803635 | 0.018851 |
Here is a numerical problem on second-order logic for the math student to solve:
Let P be the set of all positive integers, and let Q be the set of all prime numbers. Define a second-order logic formula Φ(x) as follows: Φ(x) = "for every subset S of Q, there exists a prime number p ∈ S such that p + x is also in S." D... | tiger_lab_math | [
0.032283250242471695,
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0.030177179723978043,
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-0.00021566647046711296,
0.0589330680668354,
0.04600802809000015,
-0.015679344534873962,
0.009560066275298595,
0.023881539702415466,
-0.017256468534469604,
... | 0.834448 | 0.807817 | 0.026631 |
Suppose there are two players, A and B, playing a game where both simultaneously choose to either stay silent or confess to a crime. If both players stay silent, they will each receive a sentence of 2 years in jail. If player A stays silent and player B confesses, then player A will receive a sentence of 10 years in ja... | tiger_lab_math | [
0.00918845646083355,
0.0020320825278759003,
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0.027013685554265976,
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0.05829286202788353,
0.0519338920712471,
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0.01661260984838009,
0.02250095270574093,
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-0.00934... | 0.816946 | 0.807441 | 0.009505 |
For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+... | tiger_lab_math | [
0.01954696886241436,
0.020596103742718697,
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0.024253908544778824,
0.01607193797826767,
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0.0041... | 0.819735 | 0.805633 | 0.014102 |
Suppose a farmer has a rectangular field with a fixed perimeter of 500 meters. Find the dimensions of the field that maximize the area of the field. | tiger_lab_math | [
0.013929547742009163,
0.025705980136990547,
-0.04973291978240013,
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0.013300641439855099,
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0.04421984776854515,
0.036459047347307205,
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0.039413753896951675,
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... | 0.81527 | 0.811146 | 0.004124 |
Consider a graph G with 8 vertices and degree sequence {3,3,3,2,2,2,1,1}. Is such a graph possible? Justify your answer. | tiger_lab_math | [
0.017105311155319214,
0.018170274794101715,
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0.030812503769993782,
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0.02115810476243496,
0.03605994954705238,
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-0.02... | 0.823694 | 0.821919 | 0.001775 |
For all values of $x$ for which it is defined, $f(x) = \cot \frac{x}{4} - \cot x$ can be written as
\[f(x) = \frac{\sin kx}{\sin \frac{x}{4} \sin x}.\]Find the value of $k.$ | tiger_lab_math | [
0.013279440812766552,
0.05090624466538429,
-0.04603475704789162,
-0.018407007679343224,
0.034879013895988464,
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0.0006606628303416073,
0.0649632140994072,
0.06077219545841217,
0.005656904075294733,
0.03831911087036133,
0.027170579880475998,
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0.0077... | 0.820803 | 0.81303 | 0.007774 |
A test has 160 questions. Each question has 5 options, but only 1 option is correct. If test-takers mark the correct option, they are awarded 1 point. However, if an answer is incorrectly marked, the test-taker loses 0.25 points. No points are awarded or deducted if a question is not attempted. A certain group of test-... | tiger_lab_math | [
0.021834155544638634,
-0.009053944610059261,
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0.04587103798985481,
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0.03386308625340462,
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-0... | 0.82186 | 0.82034 | 0.001521 |
Let f(z) = 3z^3 + (1-i)z^2 - 4iz + 2 be an analytic function on the open unit disk |z| < 1. Determine the Laurent series expansion of f(z) about the point z = 0 within the annulus 0 < |z| < 1/2. | tiger_lab_math | [
0.013632086105644703,
0.029157526791095734,
-0.0453830324113369,
-0.03622664883732796,
0.04369252547621727,
0.011169693432748318,
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0.05499151721596718,
0.03746691718697548,
0.010182098485529423,
0.032002776861190796,
0.01645975187420845,
-0.029199473559856415,
-0.0159... | 0.824668 | 0.813665 | 0.011003 |
Let A be a set containing 5 elements. Determine whether the empty set ∅ is a subset of set A or not. | tiger_lab_math | [
0.032939791679382324,
0.016544682905077934,
-0.027885213494300842,
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0.058771297335624695,
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0.048421185463666916,
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0.0141145596280694,
0.017245832830667496,
-0.03004022315144539,
-0... | 0.82368 | 0.820003 | 0.003677 |
Given that $\sum_{k=1}^{35}\sin 5k=\tan \frac mn,$ where angles are measured in degrees, and $m$ and $n$ are relatively prime positive integers that satisfy $\frac mn<90,$ find $m+n.$
| tiger_lab_math | [
0.007453571073710918,
0.03304516524076462,
-0.03760768845677376,
-0.05385342612862587,
0.02851642668247223,
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0.05064043402671814,
0.02478008158504963,
0.014260848052799702,
0.03233460709452629,
0.02552291564643383,
-0.04383868724107742,
-0.008098... | 0.83735 | 0.826825 | 0.010525 |
For the opening home game of the baseball season, the Madd Batters minor league baseball team offered the following incentives to its fans:
Every 75th fan who entered the stadium got a coupon for a free hot dog.
Every 30th fan who entered the stadium got a coupon for a free cup of soda.
Every 50th fan who entered th... | tiger_lab_math | [
0.015901150181889534,
0.010393645614385605,
-0.04580104723572731,
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0.03851640224456787,
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0.036438796669244766,
0.04409089311957359,
-0.03430594131350517,
0.0018401853740215302,
0.019755268469452858,
-0.05422767996788025,
-0.... | 0.790575 | 0.788729 | 0.001846 |
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