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 |
|---|---|---|---|---|---|
A certain portion of a curve is given by the equation y = x^2 + 1, where x ranges from 0 to 2. Find the arc length of the curve in that interval. | tiger_lab_math | [
0.010256635025143623,
0.013705049641430378,
-0.044967152178287506,
-0.047095391899347305,
0.022226890549063683,
-0.028837010264396667,
-0.029949761927127838,
0.06435950100421906,
0.04439136013388634,
-0.027728550136089325,
0.0123654929921031,
0.019808443263173103,
-0.03952081874012947,
-0.... | 0.836273 | 0.830418 | 0.005855 |
Sure! Here's a numerical problem for a math student to solve:
A company produces and sells widgets at a cost of $5 per widget. The demand function for the widgets is given by q(p) = 3000 - 100p, where p is the price per widget and q(p) is the quantity demanded at that price. The company wants to maximize its profits b... | tiger_lab_math | [
0.028823386877775192,
0.01221196074038744,
-0.04891790449619293,
-0.028744295239448547,
0.026398828253149986,
-0.031607698649168015,
-0.03851580619812012,
0.04335935786366463,
0.023236490786075592,
-0.005898585543036461,
0.02817104570567608,
0.03806159645318985,
-0.03755749389529228,
-0.04... | 0.823482 | 0.813695 | 0.009787 |
Let X be a complex projective algebraic variety of dimension 2 and let Y be a Zariski closed subset contained in X. Suppose that the structure sheaves of X and Y generate their derived categories. Then, what is the decomposition of the derived category of X into the Lefschetz decomposition induced by the action of a ge... | tiger_lab_math | [
0.019286390393972397,
0.018763920292258263,
-0.04817274212837219,
-0.029726676642894745,
0.021384984254837036,
-0.009387259371578693,
-0.0037298230454325676,
0.05651632696390152,
0.04823136329650879,
-0.007673575542867184,
0.005094733089208603,
0.04155266284942627,
-0.03928980231285095,
-0... | 0.802201 | 0.789343 | 0.012857 |
Let A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, find the symmetric difference of (A ∆ B) ∪ A and (A ∪ A) ∆ (B ∪ B). | tiger_lab_math | [
0.043025705963373184,
0.019842160865664482,
-0.02574784867465496,
-0.03194529935717583,
0.031570810824632645,
-0.01792404055595398,
-0.03538113832473755,
0.05604426562786102,
0.07429370284080505,
-0.013550886884331703,
0.004727935418486595,
0.026104331016540527,
-0.03408246859908104,
-0.01... | 0.836279 | 0.831274 | 0.005005 |
Let $n$ equal the number of sides in a regular polygon. For $3\leq n < 10$, how many values of $n$ result in a regular polygon where the common degree measure of the interior angles is not an integer? | tiger_lab_math | [
0.010101883672177792,
0.011679714545607567,
-0.05180003121495247,
-0.04274819791316986,
0.01084099430590868,
-0.002322903135791421,
-0.019922008737921715,
0.03546590358018875,
0.020993854850530624,
-0.028531759977340698,
0.017056258395314217,
0.04710232838988304,
-0.03462909907102585,
-0.0... | 0.85153 | 0.836671 | 0.014859 |
Sure, a precise numerical problem for a math student to practice finding the critical points of a function could be:
Find the critical points of the function f(x) = 2x^3 - 9x^2 + 12x. | tiger_lab_math | [
0.03894944116473198,
0.013651235029101372,
-0.04163360595703125,
-0.04052504152059555,
0.03736601769924164,
-0.02598857693374157,
-0.021212654188275337,
0.05805375054478645,
0.021774163469672203,
-0.004409280605614185,
0.015491203404963017,
0.03139190003275871,
-0.03545975312590599,
-0.032... | 0.818301 | 0.81717 | 0.001131 |
The equation of the ellipse shown below can be written as
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.\]Find $h + k + a + b.$
[asy]
unitsize(0.3 cm);
int i, n = 10;
for (i = -n; i <= n; ++i) {
draw((i,-n)--(i,n),gray(0.7));
draw((-n,i)--(n,i),gray(0.7));
}
draw((0,-n)--(0,n));
draw((-n,0)--(n,0));
draw... | tiger_lab_math | [
-0.000883535947650671,
0.057406239211559296,
-0.02243844047188759,
-0.04248921945691109,
0.05102210119366646,
-0.03532954677939415,
-0.022672077640891075,
0.07401273399591446,
0.06118853762745857,
-0.02031818777322769,
0.030450336635112762,
0.03331519663333893,
-0.004881922155618668,
-0.02... | 0.82185 | 0.816878 | 0.004972 |
Let M be a smooth manifold with a coordinate chart (U, x) where U is an open subset of M and x:U→R^n is a diffeomorphism. Suppose v = x_1 ∂/∂x_1 + x_2 ∂/∂x_2 + 2x_3 ∂/∂x_3 is a vector field on M with respect to the coordinate chart (U, x). Determine the Lie derivative of v with respect to the vector field w = ∂/∂x_2 in... | tiger_lab_math | [
0.015922371298074722,
0.031550925225019455,
-0.04889770224690437,
-0.049818217754364014,
0.04226893559098244,
-0.016994213685393333,
-0.02354923076927662,
0.07137841731309891,
0.03606515750288963,
-0.008746753446757793,
0.000569304742384702,
0.03373875096440315,
-0.05715150386095047,
-0.01... | 0.818128 | 0.809785 | 0.008343 |
Suppose two players, A and B, are playing the centipede game with a pot of $100. Player A makes the first move and can either take $1 from the pot or pass the turn to player B. If player A passes, player B can either take $1 from the pot or pass the turn back to player A. This continues for a total of 10 turns (so ther... | tiger_lab_math | [
0.01509266160428524,
0.006975920870900154,
-0.035230450332164764,
-0.017543142661452293,
0.014676127582788467,
-0.04199216514825821,
-0.025557948276400566,
0.044538646936416626,
0.046734485775232315,
-0.039937712252140045,
0.03694165125489235,
0.023488324135541916,
-0.04244118928909302,
-0... | 0.814103 | 0.801964 | 0.01214 |
Find the least positive integer $n$ such that $$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$$ | tiger_lab_math | [
-0.006027217488735914,
0.04698965698480606,
-0.020302537828683853,
-0.04960649833083153,
0.029276492074131966,
-0.003866095095872879,
-0.018306078389286995,
0.050727054476737976,
0.03199775144457817,
0.0004998814547434449,
0.029360834509134293,
0.009385185316205025,
-0.0357750840485096,
-0... | 0.854675 | 0.845562 | 0.009113 |
Let's say we have a 3-dimensional sphere $S^3 \subset \mathbb{R}^4$ defined by $x^2 + y^2 + z^2 + w^2 = 1$. Determine the Hopf map $h:S^3 \to S^2$, where $h(z,w) = (2zw,2z^2-1,2w^2-1)$, for all $(z,w) \in S^3$. | tiger_lab_math | [
0.022876499220728874,
0.03310944885015488,
-0.02064286172389984,
-0.047893352806568146,
0.030802985653281212,
-0.026220185682177544,
0.0002981108264066279,
0.04257509484887123,
0.021061312407255173,
-0.013718136586248875,
0.026958057656884193,
0.048661962151527405,
-0.022446325048804283,
-... | 0.825914 | 0.804173 | 0.021742 |
Compute the smallest positive integer $n$ such that $n + i,$ $(n + i)^2,$ and $(n + i)^3$ are the vertices of a triangle in the complex plane whose area is greater than 2015. Let's write a Python program to solve it. | tiger_lab_math | [
0.008033941499888897,
0.03346395492553711,
-0.028866946697235107,
-0.034300386905670166,
0.03571202978491783,
-0.007719994988292456,
-0.01940026879310608,
0.043561577796936035,
0.04190700128674507,
0.003084989031776786,
0.04057132825255394,
0.03737039119005203,
-0.03183026239275932,
-0.027... | 0.841619 | 0.822608 | 0.019011 |
In pentagon $ABCDE$, $BC=CD=DE=2$ units, $\angle E$ is a right angle and $m \angle B = m \angle C = m \angle D = 135^\circ$. The length of segment $AE$ can be expressed in simplest radical form as $a+2\sqrt{b}$ units. What is the value of $a+b$? | tiger_lab_math | [
0.022545650601387024,
0.019622139632701874,
-0.049786586314439774,
-0.055575281381607056,
0.01635570265352726,
-0.03684721514582634,
-0.018226295709609985,
0.04967570677399635,
0.043464962393045425,
-0.013707549311220646,
0.019508741796016693,
0.02944522723555565,
-0.03939599543809891,
-0.... | 0.844807 | 0.832538 | 0.012268 |
We have a triangle $\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$? Let's write a program. | tiger_lab_math | [
0.02932954952120781,
-0.004634200129657984,
-0.04589371383190155,
-0.0345628522336483,
0.02828262932598591,
-0.02329031564295292,
-0.032337486743927,
0.04534370079636574,
0.047457486391067505,
-0.02942686900496483,
0.018899448215961456,
0.03326535224914551,
-0.03366205096244812,
-0.0218328... | 0.835425 | 0.834053 | 0.001372 |
Consider the following optimization problem:
Maximize f(x) = x^2 + 3x + 5
subject to the constraint g(x) = cos(x) ≤ 0
Use ant colony optimization to find the maximum value of f(x) subject to the given constraint g(x). Assume that the search space for x is [0, 2π] and use 10 ants to explore the space. Use a pherom... | tiger_lab_math | [
0.006559635046869516,
0.01614132523536682,
-0.03526832535862923,
-0.023883003741502762,
0.025468969717621803,
-0.037555381655693054,
-0.04287903010845184,
0.0504094734787941,
0.029377326369285583,
-0.013761469163000584,
0.016641901805996895,
0.01702689938247204,
-0.03233690559864044,
-0.01... | 0.814953 | 0.798467 | 0.016486 |
A math student is analyzing the relationship between the scores of 4 different subjects (English, Math, Science, and History) of 20 students in a class. The student wants to know if there is a significant correlation between the subjects. The following table shows the scores of the students:
| Student | English | Math... | tiger_lab_math | [
0.021645424887537956,
0.023283835500478745,
-0.03771532326936722,
-0.04631603881716728,
0.034624237567186356,
-0.014648565091192722,
-0.04356261342763901,
0.04713694751262665,
0.036952171474695206,
-0.005584826692938805,
0.006441773381084204,
0.02953224815428257,
-0.02983318269252777,
0.00... | 0.795004 | 0.792318 | 0.002686 |
Suppose there are two players A and B playing the matching pennies game. If player A chooses "heads" and player B chooses "tails", A wins and B loses $2. If player A chooses "tails" and player B chooses "heads", A loses $2 and B wins. If both players choose the same side, there is no payoff. If player A chooses "heads"... | tiger_lab_math | [
0.012544147670269012,
0.020007744431495667,
-0.0576234832406044,
-0.01619022898375988,
0.020876286551356316,
-0.00632500322535634,
-0.005879588890820742,
0.03624085336923599,
0.031207570806145668,
-0.029518362134695053,
0.02572746016085148,
0.03641987591981888,
-0.016384266316890717,
-0.00... | 0.828784 | 0.821416 | 0.007369 |
A manufacturing company produces two types of products, A and B. The production of product A requires 2 hours of machine time and 3 hours of labor, while the production of product B requires 1 hour of machine time and 4 hours of labor. The company has a total of 10,000 hours of machine time and 18,000 hours of labor av... | tiger_lab_math | [
0.031149541959166527,
0.006722093094140291,
-0.058626439422369,
0.0006391091155819595,
0.023545607924461365,
-0.006159639451652765,
-0.05486728996038437,
0.09201865643262863,
0.028979944065213203,
-0.005996345542371273,
0.005304498132318258,
0.03209212049841881,
-0.007006951607763767,
-0.0... | 0.811227 | 0.796802 | 0.014424 |
Find the number of ordered pairs $(a,b)$ of complex numbers such that
\[a^3 b^5 = a^7 b^2 = 1.\] | tiger_lab_math | [
0.0242999866604805,
0.025872262194752693,
-0.056753501296043396,
-0.048600465059280396,
0.028645768761634827,
-0.011095422320067883,
-0.005535947624593973,
0.03494204208254814,
0.033928465098142624,
-0.01607534848153591,
0.017583640292286873,
0.01541389711201191,
-0.03853462636470795,
-0.0... | 0.849165 | 0.848083 | 0.001082 |
Consider a graph with 6 vertices, labeled $A$, $B$, $C$, $D$, $E$, and $F$, and 9 edges. Determine if it has a perfect matching, and if so, find one. | tiger_lab_math | [
0.02490946650505066,
0.0028584639076143503,
-0.040225204080343246,
-0.03098132833838463,
0.029916979372501373,
-0.038604896515607834,
-0.016977151855826378,
0.02267966978251934,
0.037694212049245834,
-0.036312226206064224,
0.02315216325223446,
0.01422584243118763,
-0.011395961977541447,
-0... | 0.839258 | 0.826954 | 0.012303 |
Statement 1 | Every finite integral domain is a field. Statement 2 | The intersection of two subgroups is always a subgroup. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.027637509629130363,
0.02347869798541069,
-0.02291899546980858,
-0.04576242342591286,
0.04686528816819191,
-0.03623928502202034,
-0.004053452517837286,
0.035598024725914,
0.05511349439620972,
-0.025802817195653915,
0.01815829984843731,
0.02775576524436474,
-0.013661317527294159,
-0.025371... | 0.814991 | 0.808974 | 0.006017 |
Given a group G with a normal subgroup N and an element g in G, statement 1 asserts that the order of the coset gN in the quotient group G/N is equal to the order of g in G. Statement 2 asserts that for every pair of subgroups H, K of G, their set product HK is also a subgroup of G. Answer Choices: (A) True, True (B) F... | tiger_lab_math | [
0.03662317246198654,
0.020444367080926895,
-0.02554522454738617,
-0.04547255486249924,
0.03918066993355751,
-0.033427730202674866,
-0.0002758670598268509,
0.07287408411502838,
0.03295312449336052,
-0.027987515553832054,
0.008758177980780602,
0.020169081166386604,
-0.023224027827382088,
-0.... | 0.813574 | 0.805852 | 0.007722 |
A simple example of a chaotic system is the logistic map: $$x_{n+1} = rx_n(1-x_n)$$
Suppose we want to analyze the behavior of this system for a parameter value of $r=3.6$. Using the Poincaré map, determine the attractor period of the system. | tiger_lab_math | [
0.029903007671236992,
0.004949300549924374,
-0.03140866011381149,
-0.04542478919029236,
0.023329563438892365,
-0.03785109892487526,
-0.02744685485959053,
0.04855538532137871,
-0.0009296670905314386,
-0.009383169002830982,
0.0035419901832938194,
0.020871827378869057,
-0.04107894003391266,
-... | 0.798571 | 0.794059 | 0.004512 |
Let $A$ and $B$ be noncommutative polynomial rings over a field $\mathbb{F}$, such that $A$ has a single variable, $x$, and $B$ has two variables, $x$ and $y$. Suppose $\phi : A \rightarrow B$ is a noncommutative algebra homomorphism that sends $x$ to $xy$. What is the image of the element $f(x) = x^2 + x + 1$ under th... | tiger_lab_math | [
0.02643701061606407,
0.0077319215051829815,
-0.03670303896069527,
-0.04973827674984932,
0.035545893013477325,
-0.039026740938425064,
0.00995688233524561,
0.07282049208879471,
0.05604545772075653,
-0.019888194277882576,
0.016454491764307022,
0.023618903011083603,
-0.028776682913303375,
-0.0... | 0.824653 | 0.808858 | 0.015794 |
Consider the complex function f(z) = z^4 - 4z^3 + 2z^2 + 9z - 5. Determine the number of roots (zeros) and poles of f(z) that lie inside the unit circle |z| < 1. | tiger_lab_math | [
0.027676193043589592,
0.0360993854701519,
-0.04714638367295265,
-0.03722892701625824,
0.04349004477262497,
-0.026590140536427498,
-0.004328891634941101,
0.03180675208568573,
0.02400491200387478,
0.007980920374393463,
0.024145448580384254,
0.02060692198574543,
-0.034937985241413116,
-0.0341... | 0.830011 | 0.827966 | 0.002045 |
Consider the function $f(x,y)=x^3+3xy^2-3x^2-3y^2+2x$.
1. Determine the critical points of the function.
2. Use the Morse theory to compute the number of critical points for the function. | tiger_lab_math | [
0.022884665057063103,
0.028744416311383247,
-0.04723216965794563,
-0.02282816357910633,
0.02848832868039608,
-0.024099625647068024,
-0.03567438945174217,
0.052124958485364914,
0.01832173578441143,
0.0011555413948372006,
0.005777027457952499,
0.02556820958852768,
-0.033152468502521515,
-0.0... | 0.832888 | 0.824141 | 0.008747 |
A company needs to transport goods from its warehouse to four different retail stores located in different parts of the city. There are five potential delivery routes to choose from, each with a different distance and time requirement. The goal is to determine the most efficient delivery route that allows the company t... | tiger_lab_math | [
0.026621004566550255,
0.021984748542308807,
-0.02374875359237194,
-0.03136478364467621,
0.03679760545492172,
-0.0055758352391421795,
-0.06670530885457993,
0.03490768000483513,
0.046647220849990845,
-0.009450211189687252,
0.026333007961511612,
0.04392823949456215,
-0.008079344406723976,
-0.... | 0.791616 | 0.7907 | 0.000915 |
Consider the following statements about groups and subgroups in abstract algebra. Statement 1: If G is a group and H is a normal subgroup of G, then the factor group G/H is isomorphic to H. Statement 2: If G is a group and H and K are subgroups of G, then the intersection of H and K, denoted by H ∩ K, is also a subgr... | tiger_lab_math | [
0.04158255457878113,
0.037242937833070755,
-0.024500180035829544,
-0.03618503734469414,
0.05480474978685379,
-0.043095335364341736,
-0.004111913498491049,
0.06088535487651825,
0.045396625995635986,
-0.014086311683058739,
0.009731348603963852,
0.013752019964158535,
-0.017819968983530998,
-0... | 0.811766 | 0.800544 | 0.011222 |
A company has 3 options to invest their money. Option 1 has a 50% chance of generating a profit of $10,000, a 30% chance of generating a profit of $5,000, and a 20% chance of generating no profit. Option 2 has a 40% chance of generating a profit of $12,000, a 40% chance of generating a profit of $4,000, and a 20% chanc... | tiger_lab_math | [
-0.002855616854503751,
-0.009845472872257233,
-0.037359561771154404,
-0.02260456047952175,
0.03178904205560684,
-0.020079685375094414,
-0.01964271254837513,
0.08140812069177628,
0.029742179438471794,
-0.012942727655172348,
0.027207856997847557,
0.031066546216607094,
-0.03198250010609627,
-... | 0.80143 | 0.79956 | 0.00187 |
Find the Fourier series of the periodic function f(x) with period 2π defined as:
f(x) = 2, 0 ≤ x ≤ π
f(x) = −2, π ≤ x ≤ 2π | tiger_lab_math | [
0.018308551982045174,
0.028720399364829063,
-0.05143548548221588,
-0.038372334092855453,
0.017278337851166725,
-0.0282316692173481,
-0.021031871438026428,
0.05971645191311836,
0.02574472688138485,
0.018350958824157715,
0.02878570929169655,
0.03951346501708031,
-0.031448788940906525,
-0.014... | 0.821658 | 0.820672 | 0.000986 |
In a building, there are a hundred ladies on the first-floor studying. There are three times that many girls at a party being held on the second floor of the building. How many ladies are on the two floors in total? Let's write a program. | tiger_lab_math | [
0.012024147436022758,
0.004526943899691105,
-0.05974779278039932,
-0.026744067668914795,
0.0468427874147892,
-0.016783440485596657,
-0.02522609382867813,
0.04758177325129509,
0.045728832483291626,
-0.014917071908712387,
0.0103970468044281,
0.04218960925936699,
-0.037643350660800934,
-0.021... | 0.827061 | 0.826253 | 0.000808 |
A beverage company wants to test two different flavors of drink to determine which one is more preferred by their customers. In total, 100 customers were randomly chosen to participate in the taste test. 50 customers were given drink A and 50 customers were given drink B. Out of the 50 customers who tried drink A, 25 r... | tiger_lab_math | [
0.006294477265328169,
0.005854187998920679,
-0.04402397572994232,
-0.03952420875430107,
0.031249871477484703,
-0.036505915224552155,
-0.01688268594443798,
0.06005512550473213,
0.04735582694411278,
-0.000777978973928839,
0.012888336554169655,
0.014896495267748833,
-0.02937084436416626,
0.00... | 0.80233 | 0.797237 | 0.005093 |
Let X be the unit circle in the complex plane and consider a complex line bundle E over X with Chern class c1(E) = 2. Find the rank and determinant of E. | tiger_lab_math | [
-0.003994929604232311,
0.027616428211331367,
-0.018603427335619926,
-0.04556621238589287,
0.03700670227408409,
-0.023508982732892036,
-0.0116499662399292,
0.028768250718712807,
0.036176200956106186,
-0.002964099869132042,
0.014992396347224712,
0.037311844527721405,
-0.047558560967445374,
-... | 0.822944 | 0.812246 | 0.010699 |
A manufacturing company produces two types of products, X and Y with a profit margin of $15 and $10 per unit respectively. The company's monthly production capacity is 8000 units of the two products combined. The company wants to maximize its monthly profit while meeting the following constraints:
- The demand for pro... | tiger_lab_math | [
0.037406861782073975,
0.016191689297556877,
-0.06603128463029861,
-0.004859824199229479,
0.01667969860136509,
-0.014700657688081264,
-0.03901773318648338,
0.06892026960849762,
0.017861563712358475,
-0.015292797237634659,
0.005158411804586649,
0.02460075356066227,
-0.016649279743433,
-0.018... | 0.820854 | 0.807323 | 0.013531 |
The first three stages of a pattern are shown below, in which each line segment represents a toothpick. If the pattern continues such that at each successive stage, three toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangement for the 250th stage? [asy]
size(150);
d... | tiger_lab_math | [
0.018257049843668938,
0.025051547214388847,
-0.0386362299323082,
-0.056759726256132126,
0.03688579797744751,
-0.007641837932169437,
-0.03303471580147743,
0.08119960874319077,
0.03258378058671951,
-0.010785851627588272,
0.015740258619189262,
0.029269305989146233,
-0.027517154812812805,
-0.0... | 0.824804 | 0.821098 | 0.003706 |
Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. For example, $(\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each... | tiger_lab_math | [
0.038578540086746216,
0.011572953313589096,
-0.033773407340049744,
-0.011255920864641666,
0.04165918007493019,
-0.006447912659496069,
-0.03037291206419468,
0.052319977432489395,
0.03739332780241966,
-0.01422371156513691,
0.02964196167886257,
0.007195405662059784,
-0.02057730033993721,
-0.0... | 0.845291 | 0.814994 | 0.030296 |
Let $f(z) = u(x,y) + iv(x,y)$ be a harmonic function in the rectangular region $\{z \in \mathbb{C} : 0 < x < 1, 0 < y < \frac{\pi}{2}\}$ such that $f(z)$ satisfies the boundary conditions:
$$
f(z) = 0, \quad \text{when }x = 0,\\
u(x,\frac{\pi}{2}) = 1+x, \quad u(1,y) - u(0,y) = \cos(y), \quad \text{when }0 \leq y \leq... | tiger_lab_math | [
0.015155207365751266,
0.05258258059620857,
-0.042050302028656006,
-0.01726265624165535,
0.03677808493375778,
0.013989399187266827,
-0.013624724932014942,
0.06442209333181381,
0.03810939937829971,
0.026844171807169914,
0.023153958842158318,
0.019286900758743286,
-0.016995802521705627,
-0.00... | 0.828633 | 0.801006 | 0.027627 |
A company wants to construct a rectangular storage container with a volume of 500 cubic meters. The cost of materials for the bottom of the container is twice the cost of the materials for the sides, and the cost of the materials for the top is equal to the cost of the materials for the bottom. If the cost of materials... | tiger_lab_math | [
0.018629329279065132,
0.027870001271367073,
-0.05870871990919113,
-0.016917603090405464,
0.032089293003082275,
-0.015327147208154202,
-0.049514979124069214,
0.07097720354795456,
0.039857394993305206,
-0.0009349200990982354,
0.00854045245796442,
0.024936789646744728,
-0.032797325402498245,
... | 0.810524 | 0.803202 | 0.007321 |
Let G be a cyclic group generated by 8. Determine if the subgroup H = {1, 8, 64} is cyclic. If not, find a subset of H that is cyclic. | tiger_lab_math | [
0.010811501182615757,
0.010752794332802296,
-0.02071920782327652,
-0.030440369620919228,
0.04356718435883522,
-0.05132410675287247,
-0.013931645080447197,
0.05548854544758797,
0.03133251518011093,
-0.02269805036485195,
0.023767923936247826,
0.008579705841839314,
-0.03841511905193329,
-0.01... | 0.821771 | 0.819908 | 0.001863 |
Teresa is collecting pencils. She has 14 colored pencils and 35 black pencils. Her three younger siblings need pencils for class and their dad asks her to share all her pencils, giving each an equal number of pencils, regardless of color. He tells her she can keep 10 of them for herself. How many pencils does each sibl... | tiger_lab_math | [
0.010903150774538517,
0.02930237166583538,
-0.037406694144010544,
-0.037374623119831085,
0.032609790563583374,
-0.0307888463139534,
-0.024328453466296196,
0.05153566598892212,
0.027846645563840866,
-0.03473016619682312,
0.03361307829618454,
0.02381265163421631,
-0.03447119891643524,
-0.024... | 0.820739 | 0.817815 | 0.002924 |
Suppose that a country's population is 10 million, and on the first day of a pandemic, 100 people become infected with a contagious disease. Assume that each infected person spreads the disease to two other people per day, and also assume that a total of 10% of the population is already vaccinated and cannot become inf... | tiger_lab_math | [
0.02568204700946808,
0.0197101142257452,
-0.04175649955868721,
-0.034009240567684174,
0.03403965011239052,
-0.04721266031265259,
-0.028099635615944862,
0.07380002737045288,
0.020014552399516106,
-0.01518319919705391,
0.006210203282535076,
0.03705612197518349,
-0.019812799990177155,
-0.0016... | 0.822214 | 0.810442 | 0.011773 |
A positive real number $x$ is such that \[
\sqrt[3]{1-x^3} + \sqrt[3]{1+x^3} = 1.
\]Find $x^6.$ | tiger_lab_math | [
0.027445059269666672,
0.035800766199827194,
-0.03947696089744568,
-0.030639737844467163,
0.04202142357826233,
-0.023326022550463676,
-0.01792788691818714,
0.05878068134188652,
0.03687925264239311,
-0.010838834568858147,
0.014209054410457611,
0.033855240792036057,
-0.039125483483076096,
-0.... | 0.844032 | 0.82594 | 0.018092 |
Let M be a compact 3-manifold with the fundamental group isomorphic to the alternating group on 4 letters, A4. Determine the Euler characteristic of M. | tiger_lab_math | [
0.022131865844130516,
0.019014613702893257,
-0.016417156904935837,
-0.04943601042032242,
0.04870672523975372,
-0.015150044113397598,
-0.01798110641539097,
0.05293775722384453,
0.03530610725283623,
0.0024027538020163774,
0.0070215510204434395,
0.03221570700407028,
-0.03994748368859291,
-0.0... | 0.831134 | 0.822298 | 0.008835 |
A delivery truck has to deliver packages to six locations. The road distance (in miles) between each pair of locations is given in the table below. Find the shortest possible distance that the truck must travel to complete all six deliveries, starting and ending at the same location.
| |Location A|Location B|Location... | tiger_lab_math | [
0.015129195526242256,
0.02137383073568344,
-0.022036463022232056,
-0.03147304803133011,
0.030210625380277634,
-0.025742214173078537,
-0.043292086571455,
0.03377852588891983,
0.03047574684023857,
-0.01602540910243988,
0.014503924176096916,
0.03612757846713066,
-0.013782910071313381,
-0.0122... | 0.821189 | 0.817459 | 0.00373 |
Consider the noncommutative algebraic variety defined by the following system of equations:
$$
\begin{cases}
x^2 + xy - yx - y^2 = 0 \\
xy - yx = 2
\end{cases}
$$
Find all possible solutions in $\mathbb{C}$ for the variables $x$ and $y$. | tiger_lab_math | [
0.03260383382439613,
0.02414567768573761,
-0.04107322916388512,
-0.028552813455462456,
0.03292808681726456,
-0.02662089094519615,
-0.0150009049102664,
0.07262091338634491,
0.06786581128835678,
-0.03257032856345177,
0.014757577329874039,
0.029121197760105133,
-0.02608051896095276,
-0.027860... | 0.837479 | 0.817902 | 0.019577 |
Suppose that $a^2$ varies inversely with $b^3$. If $a=7$ when $b=3$, find the value of $a^2$ when $b=6$. | tiger_lab_math | [
0.02809421718120575,
0.050000086426734924,
-0.03518680855631828,
-0.025243481621146202,
0.03517048433423042,
-0.02561437524855137,
-0.0043812403455376625,
0.061152804642915726,
0.05288751795887947,
-0.022165575996041298,
0.019031770527362823,
0.03653717041015625,
-0.05242697522044182,
-0.0... | 0.84707 | 0.842935 | 0.004134 |
Find the polynomial $p(x),$ with real coefficients, such that
\[p(x^3) - p(x^3 - 2) = [p(x)]^2 + 12\]for all real numbers $x.$ | tiger_lab_math | [
0.02782580256462097,
0.027199218049645424,
-0.05142402648925781,
-0.027656514197587967,
0.015463998541235924,
-0.03764791786670685,
0.004241766408085823,
0.06851823627948761,
0.03447930887341499,
-0.014962190762162209,
0.0038311341777443886,
0.04264147952198982,
-0.03903157263994217,
-0.02... | 0.844321 | 0.830852 | 0.013469 |
A ball is thrown straight up with an initial velocity of 20 m/s. Use calculus to find the maximum height the ball reaches and the time it takes to get there. (Assume air resistance is negligible) | tiger_lab_math | [
0.011568688787519932,
0.03437579795718193,
-0.038202498108148575,
-0.0204542875289917,
0.051940515637397766,
-0.02313878946006298,
-0.038792964071035385,
0.06355984508991241,
0.04589606076478958,
-0.025734612718224525,
0.011735321022570133,
0.04397230222821236,
-0.03040088340640068,
-0.015... | 0.817141 | 0.815829 | 0.001312 |
Statement 1 | The product of two cyclic groups is always a cyclic group. Statement 2 | A group is Abelian if it is isomorphic to its own commutator subgroup. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.035032790154218674,
0.029271498322486877,
-0.03280022740364075,
-0.032001908868551254,
0.03597448021173477,
-0.04754659906029701,
0.007290938403457403,
0.07772418856620789,
0.03307007625699043,
-0.023675810545682907,
-0.001148088718764484,
0.020619243383407593,
-0.014279401861131191,
-0.... | 0.805504 | 0.799936 | 0.005568 |
Let G_n(s) be the probability generating function of the size Z_n of the n-th generation of a branching process, where Z_0=1 and var(Z_1)>0. Let H_n be the inverse function of the function G_n, viewed as a function on the interval [0, 1]. Is M_n= {H_n(s)}^{Z_n} defines a martingale with respect to the sequence Z? Retu... | tiger_lab_math | [
0.00865551084280014,
0.02998809516429901,
-0.03133830428123474,
-0.022315241396427155,
0.044695015996694565,
-0.02897661179304123,
-0.002727799117565155,
0.07728663086891174,
0.024610331282019615,
0.0034448641818016768,
0.003793529001995921,
0.020982254296541214,
-0.03313680365681648,
0.00... | 0.821788 | 0.810012 | 0.011776 |
is the following function $f(t, y) = \frac{t^3+t^2y+ty+y^3}{t^3 + ty^2}$ scale invariant function Let's write a Python program to solve it. | tiger_lab_math | [
0.013278823345899582,
0.040633175522089005,
-0.05056115239858627,
-0.030077246949076653,
0.045602649450302124,
-0.02552616223692894,
-0.032398391515016556,
0.07491251081228256,
0.036896590143442154,
0.005939553026109934,
0.02637389488518238,
0.026864539831876755,
-0.03352952003479004,
-0.0... | 0.828464 | 0.827434 | 0.00103 |
Consider the 2-dimensional Euclidean space, i.e. $\mathbb{R}^2$ with flat metric $g=dx^2+dy^2$. Let $\gamma(t)=(\cos t,\sin t)$ be a curve in $\mathbb{R}^2$. Find the Riemann curvature tensor $R_{ijkl}$ at the point $\gamma(0)$. | tiger_lab_math | [
-0.005386605858802795,
0.04048449546098709,
-0.023430339992046356,
-0.04934464767575264,
0.02895895577967167,
-0.00372010818682611,
-0.015778565779328346,
0.056374356150627136,
0.028517577797174454,
-0.02268068678677082,
-0.00015787790471222252,
0.017330894246697426,
-0.04158530384302139,
... | 0.821816 | 0.796768 | 0.025049 |
two pipes a and b can separately fill a cistern in 60 minutes and 120 minutes respectively . there is a third pipe in the bottom of the cistern to empty it . if all the three pipes are simultaneously opened , then the cistern is full in 60 minutes . in how much time , the third pipe alone can empty the cistern ? Let's ... | tiger_lab_math | [
0.011014873161911964,
0.02687520533800125,
-0.023403141647577286,
-0.01236923597753048,
0.035539787262678146,
-0.04527663439512253,
-0.030942251905798912,
0.05635594204068184,
0.0256815068423748,
-0.004144213628023863,
0.029260635375976562,
0.046149417757987976,
-0.027739478275179863,
-0.0... | 0.814256 | 0.811568 | 0.002688 |
Using the Fourier transform, analyze the stability of the system described by the equation:
$\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$
where $k$ is a constant and $u(x, t)$ is a function of position $x$ and time $t$.
Find the range of values of $k$ for which the system is stable, and the r... | tiger_lab_math | [
0.0331253707408905,
0.034868210554122925,
-0.04558638855814934,
-0.017847705632448196,
0.048438556492328644,
-0.035287804901599884,
-0.011537090875208378,
0.08164295554161072,
0.052117034792900085,
-0.011957532726228237,
0.030886555090546608,
0.0299377478659153,
-0.023158850148320198,
-0.0... | 0.814653 | 0.806873 | 0.00778 |
Consider the electromagnetic field given by the Maxwell's equations in a region with no charges or currents, i.e.,
$\nabla \cdot {\bf E} = 0$,
$\nabla \cdot {\bf B} = 0$,
$\nabla \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}$,
$\nabla \times {\bf B} = \frac{1}{c^2}\frac{\partial {\bf E}}{\partial t}$,
wh... | tiger_lab_math | [
0.04410359263420105,
0.04696711525321007,
-0.0425693653523922,
-0.04435959830880165,
0.02554183639585972,
-0.03464507684111595,
-0.03493192046880722,
0.06812505424022675,
0.037181101739406586,
-0.01819433830678463,
0.005894427187740803,
0.007730874698609114,
-0.015646956861019135,
-0.04031... | 0.796976 | 0.790743 | 0.006233 |
Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 8 unit squares. The second ring contains 16 unit squares. If we continue this process, then what is the number of unit squares in the $100^{th}$ ring?
[asy]
fill((2,2)--(7,2)--(7,7... | tiger_lab_math | [
0.017129547894001007,
0.035714585334062576,
-0.01911727339029312,
-0.03467254340648651,
0.029150603339076042,
-0.026952309533953667,
-0.025469623506069183,
0.07074248045682907,
0.025661960244178772,
-0.012995890341699123,
0.03311982378363609,
0.014076766557991505,
-0.024608928710222244,
-0... | 0.840682 | 0.831951 | 0.008731 |
In triangle $PQR,$ $\angle Q = 30^\circ,$ $\angle R = 105^\circ,$ and $PR = 4 \sqrt{2}.$ Find $QR.$ | tiger_lab_math | [
0.001952589489519596,
0.03367331624031067,
-0.032216381281614304,
-0.05237970128655434,
0.023525036871433258,
-0.0033887161407619715,
-0.0224543996155262,
0.049161769449710846,
0.03971309959888458,
-0.019458845257759094,
0.03233896940946579,
0.03299286216497421,
-0.02780437469482422,
-0.03... | 0.838012 | 0.83033 | 0.007682 |
A factory produces 600 units of a product each day. There are two distribution centers, A and B, that the factory can send the products to. Center A can receive up to 400 units per day, while center B can receive up to 300 units per day. How many units should the factory send to each center to maximize the total amount... | tiger_lab_math | [
0.020526740700006485,
-0.0023659749422222376,
-0.05366060882806778,
0.00016039246111176908,
0.015068911015987396,
-0.01867053657770157,
-0.04944716766476631,
0.05762142688035965,
0.034228287637233734,
-0.019270163029432297,
0.008995170705020428,
0.04137633740901947,
-0.02719578891992569,
-... | 0.798057 | 0.797102 | 0.000954 |
Let's say a fluid flow inside a pipe is represented by a periodic function f(t) of period 2π, where t is time. The function is defined as follows:
f(t) = 5sin(t) - 3cos(2t) + 2sin(3t)
Using the Fourier transform, find the frequency content of the fluid flow and determine the amplitude and phase angle for each frequency... | tiger_lab_math | [
0.022104308009147644,
0.03347073495388031,
-0.04322914406657219,
-0.03321260213851929,
0.03246810659766197,
-0.045281924307346344,
-0.02845843695104122,
0.06749235838651657,
0.014587926678359509,
0.03164750710129738,
0.03933194279670715,
0.02888949029147625,
-0.027966467663645744,
-0.02367... | 0.79593 | 0.793315 | 0.002614 |
Let X be an algebraic variety over the field of complex numbers given by the equation x^2 + y^2 = 1, and let D(X) be the derived category of X. Find the Euler characteristic of D(X). | tiger_lab_math | [
0.001940157962962985,
0.01786671206355095,
-0.03142162039875984,
-0.05076061189174652,
0.035662323236465454,
-0.02694692648947239,
-0.00826739240437746,
0.06381013244390488,
0.05301922559738159,
-0.007326124701648951,
0.0032745548523962498,
0.05516200140118599,
-0.038382355123758316,
-0.00... | 0.828738 | 0.813452 | 0.015286 |
One example of a specific problem related to solving polynomial equations in Algebra is:
Find all the roots (real or complex) of the polynomial equation 3x^3 + 7x^2 - 5x - 9 = 0. | tiger_lab_math | [
0.06453980505466461,
0.015058604069054127,
-0.05023719370365143,
-0.0563235841691494,
0.01348080113530159,
-0.037606097757816315,
0.010157736018300056,
0.03627333045005798,
0.06079152226448059,
-0.01368948258459568,
0.022504176944494247,
0.006082823965698481,
-0.031639985740184784,
-0.0196... | 0.819682 | 0.813957 | 0.005725 |
A triangle has two equal sides of length 8 cm and a base of length 10 cm. Find the perimeter of the triangle using trigonometry. | tiger_lab_math | [
0.02616679109632969,
0.020810244604945183,
-0.02711578831076622,
-0.04423139989376068,
0.027485137805342674,
-0.03022553212940693,
-0.03337029367685318,
0.04807209596037865,
0.028774144127964973,
-0.020138124004006386,
0.013857576996088028,
0.041683051735162735,
-0.038917552679777145,
-0.0... | 0.801736 | 0.798268 | 0.003468 |
Two hunters, A and B, are trying to hunt a stag. Each hunter can either hunt a rabbit alone, in which case they get 1 unit of food or hunt a stag together, in which case they get 3 units of food. If only one hunter decides to hunt a stag alone, they get nothing. If both hunters decide to hunt a rabbit alone, they each ... | tiger_lab_math | [
0.027807021513581276,
0.0035081859678030014,
-0.04232734441757202,
-0.012432309798896313,
0.003462406573817134,
-0.013595819473266602,
-0.019512705504894257,
0.06712375581264496,
0.026615548878908157,
-0.04131750762462616,
0.01743876375257969,
0.014510424807667732,
-0.05207572877407074,
-0... | 0.815947 | 0.806277 | 0.009669 |
miller street begins at baker street and runs directly east for 4.5 kilometers until it ends when it meets turner street . miller street is intersected every 250 meters by a perpendicular street , and each of those streets other than baker street and turner street is given a number beginning at 1 st street ( one block ... | tiger_lab_math | [
0.03148965165019035,
0.0171990767121315,
-0.013211164623498917,
-0.0024895595852285624,
0.023986687883734703,
0.004928655922412872,
-0.033086709678173065,
0.027303548529744148,
0.0609903484582901,
0.0011467479635030031,
0.026777850463986397,
0.03762643411755562,
-0.020674824714660645,
-0.0... | 0.791282 | 0.789538 | 0.001744 |
Sure, here's a problem for the math student to solve:
Bob and Alice are using the Diffie-Hellman key exchange to establish a secret key. They agree to use a prime number p=17 and a primitive root g=3. Bob chooses a secret integer b=5, and computes g^b mod p. Alice chooses a secret integer a=3, and computes g^a mod p. ... | tiger_lab_math | [
0.04891965165734291,
0.04508410766720772,
-0.04734541475772858,
-0.0629696249961853,
0.021753182634711266,
-0.03037659265100956,
-0.013389660976827145,
0.0381910465657711,
0.04888143017888069,
-0.02582433819770813,
0.02995811589062214,
0.011470669880509377,
-0.0385851226747036,
-0.03645405... | 0.833925 | 0.827813 | 0.006112 |
Rationalize the denominator of $\frac{3}{2\sqrt[3]{5}}$. The answer can be written in the form of $\frac{A\sqrt[3]{B}}{C}$, where $A$, $B$, and $C$ are integers, $C$ is positive, and $B$ is not divisible by the cube of any prime. Find $A+B+C$. | tiger_lab_math | [
0.03574224188923836,
0.04777791351079941,
-0.04470822215080261,
-0.019030893221497536,
0.043165676295757294,
-0.02030627615749836,
0.0030753384344279766,
0.060513585805892944,
0.04114873707294464,
-0.018542684614658356,
0.015344180166721344,
0.029644891619682312,
-0.01908954419195652,
-0.0... | 0.837472 | 0.826064 | 0.011409 |
What is the area of the largest circular disk on the complex plane that does not contain any element of the Julia set related to the quadratic polynomial f(z) = z^2 + c, where c is a point on the Mandelbrot set? | tiger_lab_math | [
-0.0023565569426864386,
0.048922885209321976,
-0.05507368594408035,
-0.054793767631053925,
0.014684480614960194,
0.005783209577202797,
-0.017115864902734756,
0.03416477516293526,
0.044130243360996246,
-0.012555637396872044,
0.017217138782143593,
0.033217329531908035,
-0.02486857958137989,
... | 0.820043 | 0.817075 | 0.002968 |
Two prisoners, A and B, are held in separate cells and cannot communicate with each other. They are both offered a plea deal:
- If both prisoners confess, they each get a sentence of 5 years in prison.
- If neither prisoner confesses, they each get a sentence of 2 years in prison.
- If one prisoner confesses and the ... | tiger_lab_math | [
0.012820092961192131,
0.008116001263260841,
-0.05077101290225983,
-0.026766952127218246,
0.012214962393045425,
-0.028567194938659668,
-0.010294406674802303,
0.06901226937770844,
0.046708643436431885,
-0.03340177237987518,
0.01756855472922325,
0.03134724870324135,
-0.0353226400911808,
-0.00... | 0.8188 | 0.815148 | 0.003652 |
Let $X$ be the unit circle in the complex plane $\mathbb{C}$. Consider the covering space of $X$ given by the map $p: \mathbb{C} -\{0\} \rightarrow X$ defined by $p(z) = z/|z|$. Calculate the fundamental group of $X$ with respect to this covering space. | tiger_lab_math | [
0.0018184843938797712,
0.04193864390254021,
-0.04527031257748604,
-0.04137713462114334,
0.036877844482660294,
-0.023935237899422646,
0.010318916290998459,
0.04778618738055229,
0.034656547009944916,
-0.003971739672124386,
0.020650912076234818,
0.0186031274497509,
-0.033789731562137604,
-0.0... | 0.83074 | 0.807756 | 0.022985 |
Consider the system of linear equations:
\begin{equation}
\begin{aligned}
x + 2y + z &= 6 \\
2x + 4y + 2z &= 12 \\
x + 3y + 3z &= 9 + d
\end{aligned}
\end{equation}
where $d$ is a small perturbation.
Determine the value of $d$ such that the system becomes ill-conditioned. Then, find the solution to the system using... | tiger_lab_math | [
0.011827688664197922,
0.020059728994965553,
-0.04748895764350891,
-0.04888441786170006,
0.0461617074906826,
-0.02872185781598091,
0.005208719056099653,
0.07048135995864868,
0.022941093891859055,
-0.004824822302907705,
0.00791673269122839,
0.03283632546663284,
-0.027422171086072922,
-0.0321... | 0.834071 | 0.820694 | 0.013377 |
Here's a precise numerical problem related to Paraconsistent logic:
Suppose you have a set of statements:
A) "The number 7 is even."
B) "The number 7 is odd."
C) "The number 7 is neither odd nor even."
Using Paraconsistent logic, determine whether it is possible for more than one of these statements to be true. If... | tiger_lab_math | [
0.04367053508758545,
0.011489166878163815,
-0.0078049288131296635,
-0.03361472859978676,
0.0278281569480896,
-0.012785703875124454,
-0.016686053946614265,
0.0641159638762474,
0.03418371453881264,
-0.01894216425716877,
0.011514831334352493,
0.02041585184633732,
-0.035587128251791,
-0.022220... | 0.830121 | 0.814362 | 0.015759 |
A furniture company produces two types of chairs: wooden and plastic chairs. Each wooden chair requires 3 hours of labor and 2 kilograms of wood, while each plastic chair requires 2 hours of labor and 3 kilograms of plastic. If the company has 120 hours of labor and 180 kilograms of wood and plastic available, and they... | tiger_lab_math | [
0.01619105413556099,
0.006371599622070789,
-0.05928434059023857,
-0.016367711126804352,
0.019152866676449776,
-0.015006817877292633,
-0.03226839005947113,
0.06910835206508636,
0.0315617173910141,
-0.00615428714081645,
0.025926917791366577,
0.041967056691646576,
-0.028131039813160896,
-0.02... | 0.814171 | 0.810926 | 0.003245 |
Here's a numerical problem for a math student to solve:
Suppose there are 10 sellers in a market with a 50% chance of having a high-quality product (valued at $10) and a 50% chance of having a low-quality product (valued at $5). Each seller knows the true quality of their product, but buyers do not. Assume that buyers... | tiger_lab_math | [
0.012774360366165638,
-0.00016096049512270838,
-0.05511805787682533,
-0.015735235065221786,
0.037183113396167755,
0.011322205886244774,
-0.03587109595537186,
0.05891508236527443,
0.03340348228812218,
-0.005516650155186653,
0.013305197469890118,
0.03393544256687164,
-0.04373767599463463,
-0... | 0.835722 | 0.814195 | 0.021528 |
A rectangular piece of cardboard measuring 10 inches by 12 inches will be made into a box by cutting out equal squares from each corner and folding up the sides. What is the maximum volume of the box if the box must have a capacity of at least 150 cubic inches? (Hint: Use calculus to find the maximum volume of the box ... | tiger_lab_math | [
0.04725353792309761,
0.0402313731610775,
-0.036294467747211456,
-0.006452701520174742,
0.030312543734908104,
-0.03042014129459858,
-0.02869170531630516,
0.05232374742627144,
0.009868307039141655,
-0.016240840777754784,
0.020135868340730667,
0.036130934953689575,
-0.02904641069471836,
-0.01... | 0.791482 | 0.7881 | 0.003382 |
Grady distributed $x$ pieces of candy evenly among nine Halloween bags such that every bag received the greatest possible number of whole pieces of candy, but some candy was left over. What is the greatest possible number of pieces that could have been left over? | tiger_lab_math | [
0.013110051862895489,
0.00816216692328453,
-0.018711306154727936,
-0.04144097864627838,
0.02236996963620186,
-0.034181997179985046,
-0.027137598022818565,
0.05141520872712135,
0.027801068499684334,
-0.031890012323856354,
0.03549174964427948,
0.026884010061621666,
-0.04921884089708328,
-0.0... | 0.837449 | 0.832492 | 0.004957 |
A particle is moving in a one-dimensional box with a length of 2 nm. The wave function of the particle is given as:
Ψ(x, t) = A sin(πx/L)e^((-iE/ħ)t)
where A = 1.5 nm^(-1/2), E = 3 eV, L = 2 nm, t = 0, and x is the position of the particle.
Find the normalized probability density function of the particle at x = 1 ... | tiger_lab_math | [
0.00774207292124629,
0.03847723826766014,
-0.045123837888240814,
-0.02712717466056347,
0.03791417181491852,
-0.02223780006170273,
-0.049383968114852905,
0.05820416659116745,
0.02535797655582428,
0.012020484544336796,
0.022608201950788498,
0.01472192257642746,
-0.009028032422065735,
-0.0083... | 0.80803 | 0.802511 | 0.005519 |
Calculate the fundamental group of the space obtained by attaching a 2-dimensional disk to a circle via a continuous map that wraps the circle around the perimeter of the disk exactly twice. | tiger_lab_math | [
0.021020054817199707,
0.03655707463622093,
-0.0218651182949543,
-0.04102326184511185,
0.026195133104920387,
-0.015380315482616425,
-0.020691776648163795,
0.0408448725938797,
0.030862437561154366,
-0.009139155969023705,
0.014082830399274826,
0.03451712802052498,
-0.025993390008807182,
-0.01... | 0.825078 | 0.806892 | 0.018186 |
A rectangular pool is 10 meters long, 5 meters wide, and 2 meters deep. Water is flowing into the pool at a rate of 5 cubic meters per minute. Find the rate at which the water level is rising when the water in the pool is 1 meter deep using the Fourier transform to solve the fluid dynamics problem. | tiger_lab_math | [
0.029584573581814766,
0.03067968413233757,
-0.04323021322488785,
-0.027533048763871193,
0.023615935817360878,
-0.04965044930577278,
-0.0409078523516655,
0.06357812136411667,
0.034586939960718155,
0.00810743123292923,
0.021228428930044174,
0.018365241587162018,
-0.039200861006975174,
-0.009... | 0.805697 | 0.804758 | 0.000939 |
Monica tosses a fair 6-sided die. If the roll is a prime number, then she wins that amount of dollars (so that, for example, if she rolls 3, then she wins 3 dollars). If the roll is composite, she wins nothing. Otherwise, she loses 3 dollars. What is the expected value of her winnings on one die toss? Express your an... | tiger_lab_math | [
0.0264616496860981,
0.013225273229181767,
-0.034361425787210464,
-0.02952297404408455,
0.027988145127892494,
-0.031498752534389496,
-0.01661253720521927,
0.0758838951587677,
0.047822292894124985,
-0.01641840860247612,
0.040003515779972076,
0.04125973954796791,
-0.026211844757199287,
-0.044... | 0.836228 | 0.829492 | 0.006736 |
The integers $G$ and $H$ are chosen such that
\[\frac{G}{x+5}+\frac{H}{x^2-4x}=\frac{x^2-2x+10}{x^3+x^2-20x}\]for all real values of $x$ except $-5$, $0$, and $4$. Find $H/G$. | tiger_lab_math | [
0.018065255135297775,
0.02717798575758934,
-0.04267070069909096,
-0.030061103403568268,
0.04988477751612663,
-0.05335477739572525,
-0.011226748116314411,
0.06807860732078552,
0.051843203604221344,
-0.02135959081351757,
0.0019439694005995989,
0.02383720502257347,
-0.009828800335526466,
-0.0... | 0.835301 | 0.831234 | 0.004068 |
Statement 1 | The zero element in a vector space is unique. Statement 2 | In a group, every element must have an inverse. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True, | tiger_lab_math | [
0.03167077153921127,
0.03563879057765007,
-0.003428833559155464,
-0.03639017790555954,
0.05225874111056328,
-0.031200289726257324,
-0.006165684666484594,
0.08177728950977325,
0.041594259440898895,
-0.011466925963759422,
0.010930587537586689,
0.0023672685492783785,
-0.015317974612116814,
-0... | 0.808795 | 0.804548 | 0.004247 |
Consider a compact and connected 2-dimensional surface without boundary $M$ and a closed 1-form $\omega\in\Omega^1(M)$. Let $[\alpha]$ denote the cohomology class of a closed 1-form $\alpha\in\Omega^1(M)$. Suppose that $[\omega]=3[\alpha]$. If $H^2(M;\mathbb{Z})\cong\mathbb{Z}$, determine the value of $[\omega]\cdot[\o... | tiger_lab_math | [
0.03699185699224472,
0.04026295244693756,
-0.04346000403165817,
-0.0424964539706707,
0.03444446623325348,
-0.000025315141101600602,
-0.008682682178914547,
0.06743142008781433,
0.04187272861599922,
-0.009919929318130016,
0.010110598057508469,
0.03839513659477234,
-0.01868480071425438,
-0.01... | 0.827447 | 0.79502 | 0.032428 |
Let f(z) = z^2 + 2iz - 1, find the maximum and minimum values of |f(z)| on the unit circle |z| = 1. | tiger_lab_math | [
0.024529779329895973,
0.03253693878650665,
-0.03262653946876526,
-0.044167064130306244,
0.05037588253617287,
-0.009777586907148361,
-0.008113845251500607,
0.04202025383710861,
0.0215911827981472,
-0.0003952017577830702,
0.02904599905014038,
0.02163299359381199,
-0.012068168260157108,
-0.01... | 0.828551 | 0.818137 | 0.010414 |
A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$. Find $|x-y|$.
$\text{(A) } 12\quad \text{(B) } 13\qua... | tiger_lab_math | [
0.014998553320765495,
0.02561253495514393,
-0.03769341856241226,
-0.034697964787483215,
0.022539321333169937,
-0.018021129071712494,
-0.019015038385987282,
0.06045039743185043,
0.029166728258132935,
-0.01786612905561924,
0.02314935438334942,
0.04898571968078613,
-0.03269268199801445,
0.000... | 0.843299 | 0.825069 | 0.018229 |
Solve the following boundary value problem using complex analysis techniques:
(𝑥^2 − 4)𝑢″(𝑥) − 2𝑥𝑢′(𝑥) + 2𝑢(𝑥) = 𝑥^2, 𝑢(−2) = 1, 𝑢(2) = 2.
(Note: 𝑢′ is the first derivative of 𝑢 with respect to 𝑥, and 𝑢″ is the second derivative of 𝑢 with respect to 𝑥.) | tiger_lab_math | [
0.009671754203736782,
0.031111285090446472,
-0.036287713795900345,
-0.038648612797260284,
0.046510010957717896,
-0.030801493674516678,
0.005485824774950743,
0.06112821027636528,
0.05332290008664131,
0.003535047871991992,
0.016736624762415886,
0.031672462821006775,
-0.022920943796634674,
-0... | 0.816545 | 0.813809 | 0.002736 |
An integer $n$ is said to be square-free if the only perfect square that divides $n$ is $1^2$. How many positive odd integers greater than 1 and less than $100$ are square-free? Please respond by writing a program in Python. | tiger_lab_math | [
0.018572505563497543,
0.0269508957862854,
-0.04633054509758949,
-0.04930437356233597,
0.03503141924738884,
-0.028343195095658302,
-0.019202902913093567,
0.04542131721973419,
0.047542113810777664,
-0.02236419916152954,
0.018466422334313393,
0.009883861057460308,
-0.03472810238599777,
-0.029... | 0.826554 | 0.819665 | 0.006889 |
Here's a precise numerical problem for a math student to solve in the subtopic Nonlinear dynamics and chaos theory of Differential Equations:
Suppose you have a nonlinear system described by the Lorenz equations:
dx/dt = σ(y - x)
dy/dt = x(ρ - z) - y
dz/dt = xy - βz
Where σ = 10, ρ = 28, β = 8/3.
Suppose the initia... | tiger_lab_math | [
0.02189154177904129,
0.0276314876973629,
-0.04608678072690964,
-0.03885428607463837,
0.033793482929468155,
0.011525745503604412,
-0.026304826140403748,
0.059675611555576324,
0.03292764350771904,
-0.011952285654842854,
0.021947288885712624,
0.020190062001347542,
-0.03963305801153183,
-0.016... | 0.825126 | 0.811564 | 0.013562 |
For all integers $n$ greater than 1, define $a_n =
\dfrac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c=
a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find $b - c.$ | tiger_lab_math | [
0.031736597418785095,
0.053649477660655975,
-0.053760550916194916,
-0.02938304841518402,
0.024405861273407936,
-0.017339453101158142,
-0.025243906304240227,
0.06395742297172546,
0.035196639597415924,
-0.018427111208438873,
0.014270560815930367,
0.031875573098659515,
-0.047223545610904694,
... | 0.856945 | 0.845111 | 0.011834 |
Here is a precise numerical problem for a student to solve in Solving optimization problems using gradient descent method:
A manufacturing company has found that the cost (in thousands of dollars) of producing x units of a particular product can be modeled by the function C(x) = 0.1x^2 + 25x + 30,000. What is the mini... | tiger_lab_math | [
0.00040893140248954296,
0.014533868990838528,
-0.03886914625763893,
-0.024113832041621208,
0.035109829157590866,
-0.023322369903326035,
-0.035159122198820114,
0.07284335047006607,
0.0346028171479702,
-0.011710859835147858,
0.013775309547781944,
0.018985578790307045,
-0.016659142449498177,
... | 0.82524 | 0.821167 | 0.004073 |
A group of 6 volunteers is discussing whether or not to organize a charity event. If at least 4 of them agree, they will organize the event, otherwise, they won't. Each of the volunteers has a personal value of attending the event, which ranges between 0 and 10 with increments of 1 (0,1,2,...,10). The value represents ... | tiger_lab_math | [
0.003941682167351246,
0.01115576084703207,
-0.04082532227039337,
-0.01860726997256279,
0.036905352026224136,
-0.03768492490053177,
-0.01544228009879589,
0.03580845519900322,
0.046500351279973984,
-0.01842832937836647,
0.027033787220716476,
0.028590740635991096,
-0.03456141799688339,
-0.012... | 0.817305 | 0.811485 | 0.00582 |
Consider the Fano variety of lines on a smooth cubic surface $S$ in $\mathbb{P}^3$. Let $C$ be a curve on $S$ which is the complete intersection of $S$ with a hyperplane section $H$ of $S$. Find the degree of $C$ in terms of the degree of $S$. | tiger_lab_math | [
0.024556832388043404,
0.0304720439016819,
-0.05613497644662857,
-0.03337836638092995,
0.02466210350394249,
-0.023839745670557022,
-0.01837719976902008,
0.0502190925180912,
0.02348720282316208,
-0.01575813814997673,
0.022406000643968582,
0.044296715408563614,
-0.031543247401714325,
-0.00175... | 0.836417 | 0.810811 | 0.025606 |
Camille goes to the Ice Cream Palace with her husband and two children. Each one orders a different ice cream. Camille orders a banana split, her husband orders a waffle bowl, her daughter orders a single cone, and her son orders a double cone. The cone has 1 scoop of ice cream, the banana split has 3 times as many sco... | tiger_lab_math | [
0.00326071516610682,
0.03430933505296707,
-0.018248800188302994,
-0.03805183246731758,
0.03641999512910843,
-0.006292680744081736,
-0.036557987332344055,
0.05633766949176788,
0.04033788666129112,
-0.029263125732541084,
0.031979117542505264,
0.033804040402173996,
-0.03392661735415459,
-0.02... | 0.832797 | 0.829658 | 0.003138 |
A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he increases the number of members in each row by... | tiger_lab_math | [
0.021019335836172104,
0.0290207676589489,
-0.05449546128511429,
-0.032245934009552,
0.021667838096618652,
-0.016193386167287827,
-0.019233815371990204,
0.053461626172065735,
0.027991345152258873,
-0.00887436792254448,
-0.004016976337879896,
0.033649418503046036,
-0.04922780022025108,
-0.00... | 0.84436 | 0.825342 | 0.019018 |
Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6)$ of $(1,2,3,4,5,6)$ that satisfy
\[\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{2} \cdot \frac{a_3 + 3}{2} \cdot \frac{a_4 + 4}{2} \cdot \frac{a_5 + 5}{2} \cdot \frac{a_6 + 6}{2} > 6!.\] | tiger_lab_math | [
0.022344456985592842,
0.038666922599077225,
-0.02259628288447857,
-0.02456943318247795,
0.04480498656630516,
-0.019715748727321625,
-0.023512329906225204,
0.05583295226097107,
0.021990733221173286,
-0.02513156831264496,
0.011525087989866734,
0.021694259718060493,
-0.029599417001008987,
-0.... | 0.84625 | 0.837912 | 0.008339 |
in triangle pqr , the angle q = 90 degree , pq = 2 cm , qr = 8 cm . x is a variable point on pq . the line through x parallel to qr , intersects pr at y and the line through y , parallel to pq , intersects qr at z . find the least possible length of xz ? Let's write a program. | tiger_lab_math | [
0.021166956052184105,
0.01744401827454567,
-0.0380985289812088,
-0.03940458595752716,
0.030311603099107742,
-0.001563992933370173,
-0.03347936272621155,
0.051397308707237244,
0.04534991458058357,
-0.017235562205314636,
0.024281801655888557,
0.03920343518257141,
-0.017411377280950546,
-0.01... | 0.830648 | 0.821989 | 0.008659 |
Let A, B, and C be matrices. If the product of the matrices A and B equals C, i.e., AB=C, and the determinant of C is 6, which of the following must be true? Answer Choices: (A) The determinant of A is 6. (B) The determinant of B is 6. (C) The product of the determinants of A and B is 6. (D) None of the above. | tiger_lab_math | [
0.04470840096473694,
0.02566719986498356,
-0.041095633059740067,
-0.013386677950620651,
0.06210571900010109,
-0.030301999300718307,
-0.014313873834908009,
0.07068027555942535,
0.031201479956507683,
-0.04348999261856079,
-0.008525612764060497,
0.025268621742725372,
-0.02632911317050457,
-0.... | 0.814276 | 0.810517 | 0.003759 |
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and K... | tiger_lab_math | [
0.01574084907770157,
0.054212603718042374,
-0.0568220429122448,
-0.03431538864970207,
0.03442593291401863,
0.0010862298076972365,
-0.03814685717225075,
0.07270514965057373,
0.039605893194675446,
-0.01289542205631733,
0.006460236385464668,
0.01871989294886589,
-0.03040466643869877,
-0.00962... | 0.845978 | 0.841924 | 0.004054 |
Consider a population of 1000 people in a town where a new infectious disease is spreading rapidly. It is estimated that the rate at which people are getting infected is proportional to the product of the number of infected people and the number of susceptible people, with a constant of proportionality of 0.0004. Suppo... | tiger_lab_math | [
0.01970166712999344,
0.0269294623285532,
-0.047070153057575226,
-0.026622433215379715,
0.03815297409892082,
-0.03932686522603035,
-0.025650935247540474,
0.07307035475969315,
0.03114640899002552,
-0.010003700852394104,
0.005849155597388744,
0.04765891283750534,
-0.022693656384944916,
-0.012... | 0.823082 | 0.813992 | 0.009091 |
A Sierpinski tetrahedron starts with one equilateral triangle. In each iteration, each equilateral triangle is divided into four smaller equilateral triangles and the middle one is removed. If the length of one side of the starting equilateral triangle is 6 cm, what is the area of the Sierpinski tetrahedron after 5 ite... | tiger_lab_math | [
0.007900492288172245,
0.014979701489210129,
-0.03816768899559975,
-0.046990156173706055,
0.03616391867399216,
-0.014267596416175365,
-0.025753997266292572,
0.06439539045095444,
0.035927362740039825,
-0.018821468576788902,
0.010048190131783485,
0.02368779107928276,
-0.035625167191028595,
-0... | 0.826356 | 0.821934 | 0.004421 |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\le x\le 4$ and $1\le y\le 4$?
$\text{(A) } 496\quad \text{(B) } 500\quad \text{(C) } 512\quad \text{(D) } 516\quad \text{(E) } 560$
Let's program in Python in the response. | tiger_lab_math | [
0.003914977423846722,
0.016985636204481125,
-0.051669564098119736,
-0.023604799062013626,
0.0339680053293705,
-0.02317390777170658,
-0.02799239754676819,
0.03161270543932915,
0.05017073079943657,
-0.0018451851792633533,
0.010867035947740078,
0.05558006465435028,
-0.015650516375899315,
-0.0... | 0.840478 | 0.828961 | 0.011518 |
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