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 |
|---|---|---|---|---|---|
Here's a numerical problem related to non-monotonic logic:
Suppose a reasoning system uses non-monotonic logic to make inferences based on a set of rules. The system has the following rules:
1. If it is raining, the grass will be wet.
2. If someone is mowing the lawn, the grass will be dry.
3. If the grass is wet, th... | tiger_lab_math | [
0.029899243265390396,
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0.02248293161392212,
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-0.03220915421843529,
0.05860919877886772,
0.03932317718863487,
-0.027089659124612808,
0.0022468282841145992,
0.020743003115057945,
-0.025262922048568726,
... | 0.820087 | 0.807218 | 0.012869 |
Consider two firms A and B in a Cournot duopoly setting with the demand function given by P = 120 - Q, where Q is the total quantity supplied by both firms. The marginal cost of production for firm A is 10 and for firm B is 20. What is the Nash equilibrium quantity supplied by each firm and the resulting market price? | tiger_lab_math | [
0.025288302451372147,
0.004497602581977844,
-0.04617513343691826,
-0.03215842694044113,
0.010614026337862015,
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-0.02191789261996746,
0.06557672470808029,
0.03470322862267494,
-0.029644984751939774,
0.014638382941484451,
0.046529144048690796,
-0.042398933321237564,
-0.... | 0.809177 | 0.804915 | 0.004263 |
A factory can produce a maximum of 1200 units of a product in a day. The cost of producing 'x' number of units of the product can be given as:
C(x) = 1000 + 4x + 0.002x^2
If the product can be sold for $10 per unit, find the number of units the factory should produce in a day to maximize the profit. | tiger_lab_math | [
0.01892981305718422,
0.015620534308254719,
-0.0595218762755394,
0.0144943967461586,
0.014592724852263927,
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0.0833810493350029,
0.03381484001874924,
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0.002177471062168479,
0.036458954215049744,
-0.015522094443440437,
-0.03377... | 0.805632 | 0.804442 | 0.00119 |
A math student is given the equation of a curve for a two-dimensional space:
y^2 = x^3 + 3x^2
The student is asked to use Mori theory to determine the number of rational curves that can be drawn through 9 general points on the above curve. How many rational curves pass through the 9 general points? | tiger_lab_math | [
0.020673567429184914,
0.011056466959416866,
-0.03909385949373245,
-0.046091414988040924,
0.016915971413254738,
-0.03097396157681942,
-0.0257324930280447,
0.05126272886991501,
0.03620798885822296,
-0.025243598967790604,
-0.004830182995647192,
0.025400493294000626,
-0.029155081138014793,
-0.... | 0.836384 | 0.822328 | 0.014056 |
huey ' s hip pizza sells two sizes of square pizzas : a small pizza that measures 9 inches on a side and costs $ 10 , and a large pizza that measures 14 inches on a side and costs $ 20 . if two friends go to huey ' s with $ 30 apiece , how many more square inches of pizza can they buy if they pool their money than if t... | tiger_lab_math | [
0.00006725026469212025,
0.02905871532857418,
-0.04599248617887497,
-0.027747537940740585,
0.02711041085422039,
-0.02208041027188301,
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0.03317314758896828,
0.036527201533317566,
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0.04532866179943085,
0.03895789012312889,
-0.04723726585507393,
-0.03... | 0.815873 | 0.814925 | 0.000948 |
Suppose $E$ is an elliptic curve over $\mathbb{C}$ with equation $y^2=x^3+ax+b$ where $a,b$ are integers. Let $P=(x_0,y_0)$ be a point on $E$. Define the set $E[n]=\{P\in E(\overline{\mathbb{C}}) : nP=\mathcal{O}\}$, where $nP=\underbrace{P+P+\cdots+P}_{\text{$n$ times}}$.
Now, let $E$ and $P$ be given as above with $... | tiger_lab_math | [
0.00525725819170475,
0.0425439216196537,
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0.03442761301994324,
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0.0008616393315605819,
0.030185556039214134,
-0.038327332586050034,
-0.020... | 0.848847 | 0.824552 | 0.024295 |
Suppose a farmer has a rectangular plot of land with a fixed perimeter of 400 meters. The farmer wants to maximize the area of the plot that can be used for farming and also wants to minimize the length of the shorter side of the rectangle.
What are the dimensions of the rectangular plot that will satisfy the farmer's... | tiger_lab_math | [
0.009769505821168423,
0.028086833655834198,
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-0.04579314962029457,
0.015774164348840714,
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-0.029080988839268684,
0.04062259569764137,
0.04178859293460846,
0.004285281524062157,
-0.009849119931459427,
0.04731615260243416,
-0.031114697456359863,
-0.0... | 0.802031 | 0.798357 | 0.003673 |
Suppose we have a genus 2 curve defined by the equation $y^2 = x^5 + x^3 + x + 1$ over the field $\mathbb{F}_5$. Find the number of rational points on the curve (including the point at infinity). | tiger_lab_math | [
0.0009645798127166927,
0.023613592609763145,
-0.04287442937493324,
-0.05607331544160843,
0.030277907848358154,
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0.04504503309726715,
0.04039512947201729,
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0.0323236808180809,
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-0.... | 0.844464 | 0.831672 | 0.012792 |
Let M be a two-dimensional manifold with a single chart x,y where x,y are Cartesian coordinates. Given the vector field V(x,y) = (x^2 + y^2, xy), find the Lie derivative of V(x,y) with respect to the vector field W(x,y) = (2x, y). Evaluate the Lie derivative at the point (1,1). | tiger_lab_math | [
0.013601010665297508,
0.02453078329563141,
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0.03208044543862343,
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-0.029794683679938316,
0.07535705715417862,
0.03986101970076561,
-0.005190840922296047,
0.0025932167191058397,
0.034784335643053055,
-0.041070375591516495,
-0.... | 0.81246 | 0.807181 | 0.005279 |
Monica is trying to solve the following equation by completing the square: $$x^2-8x+8 = 0.$$She successfully rewrites the above equation in the following form: $$(x + b)^2 = c,$$where $b$ and $c$ are integers. What is the value of $b + c$? | tiger_lab_math | [
0.031071726232767105,
0.028326300904154778,
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0.03711531683802605,
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0.004128390923142433,
0.09133201837539673,
0.0572391003370285,
-0.035663411021232605,
0.005540915299206972,
0.025361193343997,
-0.03229636326432228,
-0.05196... | 0.829794 | 0.825069 | 0.004726 |
A manufacturing plant has two machines, A and B, and needs to produce 1,000 units of product P. Machine A can produce one unit of P in 4 minutes, and Machine B can produce one unit of P in 6 minutes. The product goes through two stages of processing, with each stage taking 3 minutes of Machine A and 2 minutes of Machin... | tiger_lab_math | [
0.030057940632104874,
0.014285224489867687,
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0.020827816799283028,
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0.08417040854692459,
0.03738930821418762,
0.0034340161364525557,
0.005564560182392597,
0.016911325976252556,
-0.0015762464608997107,
-... | 0.804973 | 0.800998 | 0.003974 |
Let \[f(x) = \left\{
\begin{array}{cl}
\sqrt{x} &\text{ if }x>4, \\
x^2 &\text{ if }x \le 4.
\end{array}
\right.\]Find $f(f(f(2)))$. Please respond by writing a program in Python. | tiger_lab_math | [
0.014373935759067535,
0.04686189070343971,
-0.03803842514753342,
-0.04126724973320961,
0.03268413618206978,
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0.06790617108345032,
0.03990449383854866,
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0.03233576565980911,
0.019767288118600845,
-0.033738233149051666,
-0.0... | 0.816196 | 0.814223 | 0.001973 |
At Central Middle School the $108$ students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of $15$ cookies, lists these items:
$\bullet$ $1\frac{1}{2}$ cups of flou... | tiger_lab_math | [
0.01386711560189724,
0.04062148928642273,
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0.023351777344942093,
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0.06646361947059631,
0.02851034328341484,
-0.006499671842902899,
0.013544909656047821,
0.02862820029258728,
-0.03052045963704586,
-0.0279... | 0.818956 | 0.813473 | 0.005483 |
At the gym, every day, there are 300 towels laid out for guest use. Each guest is given 1 towel during their visit. The gym is open from 10:00 a.m. to 2:00 p.m. In the first hour, 50 guests enter the gym. In the second hour, 20% more guests enter than during the first hour. In the third hour, 25% more guests enter ... | tiger_lab_math | [
0.021128619089722633,
0.02556494250893593,
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0.025152046233415604,
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0.010420872829854488,
0.01865435391664505,
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-0.03... | 0.813352 | 0.812161 | 0.001191 |
Suppose $X=V(y-x^2)$ is a projective algebraic curve in $\mathbb{P}^2$ over the complex numbers. Compute the cohomology groups $H^i(X,\mathbb{C})$ for all $i$. | tiger_lab_math | [
0.013456578366458416,
0.032602451741695404,
-0.030691849067807198,
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0.020554978400468826,
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-0.002909016329795122,
0.06003756821155548,
0.058578550815582275,
-0.022097809240221977,
0.00948150921612978,
0.0322406105697155,
-0.019971594214439392,
-0.0... | 0.826212 | 0.807303 | 0.018909 |
Triangle $ABC$ has sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 43, 13, and 48, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$ and let $D$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{AC}$ that is not on the same side of $\overline{... | tiger_lab_math | [
0.030894679948687553,
0.03076089359819889,
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0.02856074832379818,
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0.05231824517250061,
0.03949658200144768,
0.007773690391331911,
0.03798319771885872,
0.029276661574840546,
-0.03782925754785538,
-0.01... | 0.854705 | 0.825659 | 0.029046 |
Let M be a smooth manifold with dimension 3, and let g be a Riemannian metric on M. Let $\nabla$ be the Levi-Civita connection of g. Suppose that the torsion tensor of $\nabla$ vanishes, and that $\nabla R=0$, where R is the Riemann curvature tensor of g. Calculate the Cartan connection of M. | tiger_lab_math | [
0.01891494169831276,
0.0277637280523777,
-0.03452591225504875,
-0.037927381694316864,
0.04337389022111893,
0.002266855211928487,
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0.06029808893799782,
0.020916426554322243,
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0.025138232856988907,
-0.051059357821941376,
-0.0... | 0.814647 | 0.79631 | 0.018338 |
Squares of side length 1 are arranged to form the figure shown. What is the perimeter of the figure? [asy]
size(6cm);
path sqtop = (0, 0)--(0, 1)--(1, 1)--(1, 0);
path sqright = (0, 1)--(1, 1)--(1, 0)--(0, 0);
path horiz = (0, 0)--(1, 0); path vert = (0, 0)--(0, 1);
picture pic;
draw(pic, shift(-4, -2) * unitsqua... | tiger_lab_math | [
0.011326612904667854,
0.03770154342055321,
-0.02809675596654415,
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0.04244118556380272,
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0.05554112419486046,
0.037762757390737534,
-0.0042505753226578236,
0.025029318407177925,
0.04712516441941261,
-0.016704481095075607,
-0.... | 0.825997 | 0.819684 | 0.006313 |
Sure, here's a numerical problem for a student to solve:
A company's profit function is given by:
P(x,y) = 500x - 10x^2 + 50xy - 2y^2 - 100y
where x is the number of units of product X produced and y is the number of units of product Y produced.
Using the gradient descent method, determine the optimal values of x... | tiger_lab_math | [
0.008282962255179882,
0.005875885020941496,
-0.040802694857120514,
-0.03336581587791443,
0.022929102182388306,
-0.027211418375372887,
-0.034264836460351944,
0.06130285933613777,
0.036120470613241196,
-0.006334180943667889,
0.01846962422132492,
0.029237158596515656,
-0.02434132806956768,
-0... | 0.827755 | 0.823265 | 0.00449 |
Two suspects, A and B, are arrested by the police. The police have enough evidence to convict both on a lesser charge, but need confessions to convict both on a more serious charge. The police have interrogated the suspects separately, and neither knows what the other has said. They offer each prisoner the same deal: i... | tiger_lab_math | [
0.017271269112825394,
0.008354565128684044,
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0.024167384952306747,
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0.07505863904953003,
0.05451887473464012,
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0.006314926780760288,
0.033768896013498306,
-0.025285007432103157,
-... | 0.818858 | 0.802699 | 0.016159 |
Let A = {1, 2, 3} and B = {2, 3, 4}. Find (A Δ B) ∪ B, where Δ denotes the symmetric difference of two sets. | tiger_lab_math | [
0.03893487527966499,
0.011289093643426895,
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0.03233713656663895,
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0.059126004576683044,
0.0691121369600296,
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0.013097218237817287,
0.03133617341518402,
-0.03422955423593521,
-0.004... | 0.837694 | 0.829094 | 0.0086 |
Statement 1 | Let G be a group and let a and b be elements in G. If ab=ba, then G is an abelian group. Statement 2 | Let H be a subgroup of a group G. If the order of G is finite and prime, then H must be either the trivial group or G itself. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, T... | tiger_lab_math | [
0.03861082345247269,
0.029912356287240982,
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0.038826096802949905,
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0.0035298725124448538,
0.06389361619949341,
0.02893144264817238,
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0.01693466491997242,
-0.026859764009714127,
-0... | 0.810121 | 0.803006 | 0.007115 |
Let X be the cubic curve defined by the equation $y^2 = x^3 - 2x$ over the field $\mathbb{Q}$. Compute the first cohomology group $H^1(X,\mathbb{Q})$. | tiger_lab_math | [
0.0037289222236722708,
0.02284158021211624,
-0.03827866539359093,
-0.039545975625514984,
0.02924596145749092,
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0.06389924883842468,
0.04848960041999817,
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0.013617102988064289,
0.021090831607580185,
-0.026006100699305534,
-0... | 0.830864 | 0.812257 | 0.018607 |
A math student is analyzing a mirror symmetry problem that involves a pair of Calabi-Yau manifolds. The student has already determined that the degrees of the two manifolds are (3,3) and (1,1), respectively. The student now needs to calculate the Hodge numbers of each of the both manifolds.
What are the Hodge numbers ... | tiger_lab_math | [
0.027477627620100975,
0.0198009442538023,
-0.04373025894165039,
-0.06420060992240906,
0.04314883053302765,
0.004377876408398151,
-0.02206464856863022,
0.05639500916004181,
0.03614978492259979,
-0.01826815865933895,
-0.0033380440436303616,
0.022972015663981438,
-0.027477433905005455,
-0.022... | 0.818091 | 0.807594 | 0.010497 |
A company is planning to produce two types of products P1 and P2. Each product requires two types of raw materials R1 and R2. Given the limited availability of raw materials, the company cannot produce more than 800 units of P1 and 700 units of P2. Also, each unit of P1 requires 6 units of R1 and 8 units of R2, and eac... | tiger_lab_math | [
0.03815102204680443,
0.02716408483684063,
-0.048791833221912384,
-0.007172320503741503,
0.008451836183667183,
-0.022461483255028725,
-0.041185472160577774,
0.0902649462223053,
0.012610948644578457,
-0.013814538717269897,
0.002825292060151696,
0.03778904303908348,
-0.009626890532672405,
-0.... | 0.809004 | 0.794714 | 0.01429 |
One possible numerical problem related to Abductive Logic is:
A magician has 7 rabbits, 5 hats, and 3 tricks. If he selects one rabbit, one hat, and one trick randomly, what is the probability that he will perform a disappearing act using that rabbit, that hat, and that trick?
Assume that the magician can only use on... | tiger_lab_math | [
0.02404608018696308,
0.017754511907696724,
-0.03500116989016533,
-0.021659819409251213,
0.03228891268372536,
-0.016738170757889748,
-0.02979085221886635,
0.04128098860383034,
0.027240129187703133,
-0.00728190690279007,
-0.0039052378851920366,
0.021180137991905212,
-0.029315141960978508,
-0... | 0.828678 | 0.823617 | 0.005061 |
A charity organization's goal is to distribute 400 boxes of food to a community that was recently destroyed by a flood. Each box contains $80 worth of food and $165 worth of additional supplies such as toiletries and fresh linens. Then, an anonymous donor gives the organization 4 times the amount of money they had alre... | tiger_lab_math | [
0.01887124963104725,
0.015730157494544983,
-0.03319840878248215,
-0.027353515848517418,
0.02530721202492714,
-0.02230992168188095,
-0.028584878891706467,
0.04777761176228523,
0.03472255542874336,
-0.017760712653398514,
0.015575114637613297,
0.020184580236673355,
-0.04629702866077423,
-0.02... | 0.822922 | 0.816006 | 0.006916 |
What is the area of the main cardioid of the Mandelbrot set, which is defined by the equation:
|z(n+1)| = |z(n)|^2 + c
where c is a complex number and z(0) = 0.
Hint: The main cardioid is the largest bulb in the Mandelbrot set that is centered at the origin and is completely enclosed by the circle of radius 1/4. | tiger_lab_math | [
0.03724391385912895,
0.039404261857271194,
-0.03361165151000023,
-0.04553359001874924,
0.028714898973703384,
-0.01451123133301735,
-0.019684795290231705,
0.05232875794172287,
0.034708112478256226,
-0.009626204147934914,
0.02511054091155529,
0.010044392198324203,
-0.03462648019194603,
-0.02... | 0.810667 | 0.801612 | 0.009055 |
Let $\mathbf{A}$ be a $2 \times 2$ matrix, with real entries, such that $\mathbf{A}^3 = \mathbf{0}.$ Find the number of different possible matrices that $\mathbf{A}^2$ can be. If you think the answer is infinite, then enter "infinite". | tiger_lab_math | [
0.044242776930332184,
0.015337662771344185,
-0.0417935885488987,
-0.02340948022902012,
0.02846529521048069,
-0.014978060498833656,
-0.021846799179911613,
0.06909392029047012,
0.04549586772918701,
-0.018281949684023857,
0.005862722173333168,
0.01760263182222843,
-0.03956867754459381,
-0.028... | 0.846413 | 0.828974 | 0.017439 |
Let A, B, C be subspaces of a vector space V. Consider the set D = A ∩ (B + C), where B + C is the smallest subspace containing B and C. Which of the following is true? Answer Choices: (A) D is a subspace of A. (B) D is a subspace of B. (C) D is a subspace of C. (D) D is a subspace of B + C. | tiger_lab_math | [
0.012578343041241169,
0.02250610664486885,
-0.019361846148967743,
-0.022297058254480362,
0.04720379039645195,
-0.009764224290847778,
0.0012441998114809394,
0.0747302994132042,
0.06463190913200378,
-0.001986986957490444,
0.01543208584189415,
0.023323701694607735,
-0.024575792253017426,
-0.0... | 0.814585 | 0.806161 | 0.008424 |
Let $a,$ $b,$ $c$ be complex numbers such that
\[a + b + c = ab + ac + bc = abc = 1.\]Enter the values $a,$ $b,$ $c,$ separated by commas, in any order. | tiger_lab_math | [
0.049004167318344116,
0.026418937370181084,
-0.03604944050312042,
-0.034319378435611725,
0.04285188391804695,
-0.02889329381287098,
-0.01216091401875019,
0.049703530967235565,
0.041546791791915894,
-0.022648939862847328,
0.02107030712068081,
0.04245183244347572,
-0.048683248460292816,
-0.0... | 0.839578 | 0.827286 | 0.012292 |
Let $M$ be a smooth manifold with a Riemannian metric $g$ and let $f: M \rightarrow \mathbb{R}$ be a smooth function. Consider the vector field $X=\frac{\partial}{\partial x}$ on $M$. Find the covariant derivative $\nabla_X(df)$ of the differential $df$ of $f$ with respect to $X$. | tiger_lab_math | [
0.00211923336610198,
0.029027486220002174,
-0.05366320163011551,
-0.039119552820920944,
0.03458112105727196,
0.0020893155597150326,
-0.01883886195719242,
0.07656393200159073,
0.03966331109404564,
-0.0065954518504440784,
-0.0038468532729893923,
0.02056974358856678,
-0.040864478796720505,
-0... | 0.807335 | 0.7947 | 0.012635 |
A manufacturing company has to allocate its resources, i.e., labor and raw materials, to produce two products, A and B. It takes 20 hours to produce one unit of A and 30 hours to produce one unit of B. Similarly, it takes 5 kilograms of raw material to produce one unit of A and 10 kilograms of raw material to produce o... | tiger_lab_math | [
0.02524939738214016,
0.024508625268936157,
-0.04258665069937706,
-0.01333280373364687,
0.01375321950763464,
-0.008842178620398045,
-0.07787813991308212,
0.08697009086608887,
0.01829960010945797,
-0.022928938269615173,
-0.0025294863153249025,
0.03833997622132301,
-0.01630244590342045,
-0.03... | 0.804538 | 0.801159 | 0.003379 |
A factory has 3 different machines (machine 1, machine 2, and machine 3) and 4 different tasks (task A, task B, task C, and task D) that they need to complete. Each machine can only work on one task at a time, and each task can only be worked on by one machine at a time. The time (in hours) that each machine takes to c... | tiger_lab_math | [
0.0278684813529253,
0.02055376209318638,
-0.05760977789759636,
-0.014510062523186207,
0.01936591975390911,
-0.013804532587528229,
-0.05249502509832382,
0.05714792013168335,
0.03603947535157204,
-0.0011079215910285711,
0.015786537900567055,
0.014730983413755894,
-0.013715576380491257,
-0.01... | 0.811081 | 0.808851 | 0.00223 |
A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the ... | tiger_lab_math | [
0.01835220493376255,
0.04185028746724129,
-0.05359024181962013,
-0.04401302710175514,
0.028716260567307472,
-0.03377627953886986,
-0.02240998111665249,
0.06312629580497742,
0.02015163190662861,
-0.008993866853415966,
0.02836286649107933,
0.026809288188815117,
-0.016005301848053932,
-0.0222... | 0.829572 | 0.820571 | 0.009001 |
Suppose a Dictator Game is played between two persons. The first person, the dictator, is given 100 tokens and must decide how many tokens to give to the second person, the recipient. If the recipient accepts the offer, they each keep their allocated tokens. However, if the recipient rejects the offer, neither player r... | tiger_lab_math | [
0.010580639354884624,
-0.0023597097024321556,
-0.03771044313907623,
-0.031027985736727715,
0.01631024293601513,
-0.007411790546029806,
-0.017250265926122665,
0.03332343325018883,
0.050760768353939056,
-0.050523702055215836,
0.005571114365011454,
0.027635054662823677,
-0.03680393099784851,
... | 0.824693 | 0.816738 | 0.007955 |
A right circular cone has a base with radius $600$ and height $200\sqrt{7}.$ A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is $125$, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}.$... | tiger_lab_math | [
0.0048857321962714195,
0.045476034283638,
-0.0483841598033905,
-0.03833742067217827,
0.01924656517803669,
-0.013246475718915462,
-0.027607455849647522,
0.06355449557304382,
0.04233430698513985,
-0.008483047597110271,
0.015941735357046127,
0.03978124260902405,
-0.023473626002669334,
-0.0417... | 0.841303 | 0.827857 | 0.013446 |
In $\triangle PQR,$ the sum of $\angle P$ and $\angle Q$ is $60^\circ.$ What is the measure of $\angle R?$ | tiger_lab_math | [
0.0034802681766450405,
0.023028457537293434,
-0.038584135472774506,
-0.04551133140921593,
0.019125763326883316,
0.00481864158064127,
-0.019860755652189255,
0.03982732072472572,
0.03165898099541664,
-0.029219331219792366,
0.023407746106386185,
0.02449965849518776,
-0.03421812132000923,
-0.0... | 0.833116 | 0.83157 | 0.001545 |
Circle $C$ has radius 6 cm. How many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle $C$? Let's write a Python program. | tiger_lab_math | [
0.006981811951845884,
0.04853185638785362,
-0.04488848149776459,
-0.027097035199403763,
0.04697248339653015,
-0.015211240388453007,
-0.033674370497465134,
0.02948344126343727,
0.03849700093269348,
-0.030509669333696365,
0.02984064444899559,
0.052595749497413635,
-0.011672419495880604,
-0.0... | 0.827267 | 0.823434 | 0.003833 |
Suppose $E$ and $F$ are vector bundles over a compact topological space $X$. Given that both $E$ and $F$ are of rank $n$, what is the rank of the vector bundle $E \oplus F$? | tiger_lab_math | [
0.01206550095230341,
0.012230928055942059,
-0.03543495014309883,
-0.038146644830703735,
0.030399765819311142,
-0.007942045107483864,
-0.011864875443279743,
0.04717281088232994,
0.04641123488545418,
-0.009706741198897362,
0.007352238986641169,
0.02291950024664402,
-0.058599647134542465,
-0.... | 0.819321 | 0.817163 | 0.002158 |
An entrepreneur is crowdfunding a new business effort. He has three different dollar amount levels of financial backing options and each level is ten times as high as the previous one. He needs to raise $12000 to get his business off the ground. He succeeded after getting two backers at the highest level of financial b... | tiger_lab_math | [
0.018392736092209816,
-0.00026173883816227317,
-0.035329584032297134,
-0.0193686094135046,
0.025607245042920113,
0.0007639958639629185,
-0.02313045971095562,
0.0674431174993515,
0.031131034716963768,
-0.0174388624727726,
0.015496902167797089,
0.03703206405043602,
-0.040325649082660675,
-0.... | 0.811824 | 0.80964 | 0.002185 |
In the figure below, $ABCD$ is a square piece of paper 6 cm on each side. Corner $C$ is folded over so that it coincides with $E$, the midpoint of $\overline{AD}$. If $\overline{GF}$ represents the crease created by the fold such that $F$ is on $CD,$ what is the length of $\overline{FD}$? Express your answer as a commo... | tiger_lab_math | [
0.022943338379263878,
0.017224328592419624,
-0.04078546166419983,
-0.06305466592311859,
0.029836999252438545,
-0.02610289305448532,
-0.007380313705652952,
0.04957403987646103,
0.03306002542376518,
-0.021022098138928413,
0.015294522047042847,
0.02519533969461918,
-0.006743318401277065,
-0.0... | 0.848727 | 0.830133 | 0.018595 |
Equilateral triangle $ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \sqrt{11}$. Find $\sum_{k=1}^4(CE_k)^2$. | tiger_lab_math | [
0.031821057200431824,
0.005011747591197491,
-0.023371269926428795,
-0.059143148362636566,
0.029829304665327072,
-0.02560766413807869,
-0.023372292518615723,
0.053493060171604156,
0.04194643720984459,
0.021294787526130676,
0.02238556742668152,
0.038146015256643295,
-0.04525831341743469,
-0.... | 0.850049 | 0.825522 | 0.024527 |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | tiger_lab_math | [
-0.008159425109624863,
0.005835161544382572,
-0.0451524555683136,
-0.0463729165494442,
0.022166868671774864,
-0.024207893759012222,
-0.020452963188290596,
0.04843360185623169,
0.04273068904876709,
0.0037530940026044846,
0.002502798568457365,
0.03557306155562401,
-0.023721782490611076,
-0.0... | 0.830113 | 0.825345 | 0.004768 |
Let S = {1, 2, 3}. Determine the size of the power set of S, and compare this to the size of S itself. Does this support the statement that the power set of a set is always larger than the set itself? | tiger_lab_math | [
0.03234762325882912,
0.03664007782936096,
-0.03324142470955849,
-0.04121983051300049,
0.038793615996837616,
-0.05355946719646454,
-0.001920592156238854,
0.0534394346177578,
0.040690433233976364,
-0.038824256509542465,
0.02006901241838932,
0.019667621701955795,
-0.06121226027607918,
-0.0229... | 0.818855 | 0.808147 | 0.010708 |
A manufacturing company produces two products: product A and product B. The production of each product requires a specific amount of three different chemicals: x, y, and z. The company has limited resources of these chemicals available each week: 240 units of x, 320 units of y, and 400 units of z. Each unit of product ... | tiger_lab_math | [
0.010904579423367977,
0.01379935909062624,
-0.03552845120429993,
-0.022225720807909966,
0.02065384015440941,
-0.03213487192988396,
-0.0657062977552414,
0.0859096571803093,
0.01566425897181034,
-0.012019678950309753,
0.009110010229051113,
0.03032584674656391,
-0.022508099675178528,
-0.01762... | 0.8004 | 0.785755 | 0.014645 |
Here's a numerical problem that a math student can solve related to using the Fourier transform to solve heat transfer problems:
The temperature distribution in a one-dimensional rod of length 10 cm is given by the function f(x) = x(10 - x). Apply the Fourier transform to determine the temperature distribution in the ... | tiger_lab_math | [
0.0332273505628109,
0.023360108956694603,
-0.04087292402982712,
-0.03337406367063522,
0.031392768025398254,
-0.03501700982451439,
-0.02704557217657566,
0.05469726771116257,
0.03448617085814476,
0.003999690990895033,
0.018270816653966904,
0.02762724459171295,
-0.028152862563729286,
-0.01748... | 0.803859 | 0.802377 | 0.001482 |
The figure shows three squares with non-overlapping interiors. The area of the shaded square is 1 square inch. What is the area of rectangle $ABCD$, in square inches?
[asy]size(100);
pair A = (0,0), D = (3,0),C = (3,2),B = (0,2);
draw(A--B--C--D--cycle);
draw(A--(1,0)--(1,1)--(0,1)--cycle);
filldraw(B--(1,2)--(1,1)--(... | tiger_lab_math | [
0.02125491574406624,
0.025030869990587234,
-0.05284776911139488,
-0.04495680332183838,
0.036979470402002335,
-0.014311231672763824,
-0.003037496469914913,
0.05651159957051277,
0.03630078211426735,
-0.015505065210163593,
0.029129154980182648,
0.032978642731904984,
-0.03306989371776581,
-0.0... | 0.830087 | 0.821989 | 0.008099 |
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 |
Four prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction. Let's write a program. | tiger_lab_math | [
0.01844176836311817,
0.01621880754828453,
-0.026439033448696136,
-0.03421874716877937,
0.04445381462574005,
-0.01898251660168171,
-0.012453577481210232,
0.04052465409040451,
0.04143470525741577,
-0.01388708595186472,
0.0037276321090757847,
0.00948275811970234,
-0.02772342786192894,
-0.0225... | 0.830355 | 0.828528 | 0.001828 |
A logistic map is given by the recursive equation: Xn+1 = r*Xn*(1 - Xn), where X0 = 0.25. Find the Feigenbaum constant δ for this logistic map to determine if the system exhibits universality. | tiger_lab_math | [
-0.004790134262293577,
0.007939361967146397,
-0.03693237155675888,
-0.05642754212021828,
0.04119884595274925,
-0.02707979455590248,
-0.020054113119840622,
0.06744394451379776,
0.0218917578458786,
-0.015375182032585144,
0.016362255439162254,
0.029495086520910263,
-0.020329346880316734,
-0.0... | 0.821576 | 0.818451 | 0.003125 |
Find $q(x)$ if the graph of $\frac{x^3-2x^2-5x+3}{q(x)}$ has vertical asymptotes at $2$ and $-2$, no horizontal asymptote, and $q(3) = 15$. | tiger_lab_math | [
-0.0012808067258447409,
0.015282548032701015,
-0.0451081208884716,
-0.032837510108947754,
0.023692825809121132,
-0.030949609354138374,
-0.020416326820850372,
0.07741833478212357,
0.015328658744692802,
-0.022827379405498505,
0.02864428050816059,
0.009395451284945011,
-0.025723930448293686,
... | 0.829788 | 0.826672 | 0.003116 |
A company has a manufacturing plant in Boston that needs to ship goods to three different cities: New York, Philadelphia, and Washington D.C. The cost of shipping one unit of goods from Boston to each of these cities is given in the following table:
| Destination | Cost per unit |
| ----------- | ------------- |
| New... | tiger_lab_math | [
0.014781761914491653,
0.007257706020027399,
-0.05686098709702492,
-0.008518008515238762,
0.0334787480533123,
-0.020109044387936592,
-0.03939031809568405,
0.05422164872288704,
0.044895075261592865,
-0.0165390744805336,
0.01083891000598669,
0.04910719022154808,
-0.035992223769426346,
-0.0267... | 0.803435 | 0.795043 | 0.008393 |
The function $f$ is defined on positive integers as follows:
\[f(n) = \left\{
\begin{array}{cl}
n + 10 & \text{if $n < 10$}, \\
f(n - 5) & \text{if $n \ge 10$}.
\end{array}
\right.\]Find the maximum value of the function. | tiger_lab_math | [
0.04789011552929878,
0.030321238562464714,
-0.0422920361161232,
-0.0238942988216877,
0.0387265607714653,
-0.029794232919812202,
-0.0368429534137249,
0.05654033273458481,
0.016658905893564224,
0.0026986293960362673,
0.028492504730820656,
0.043982114642858505,
-0.04387488588690758,
-0.006188... | 0.821774 | 0.817597 | 0.004177 |
Consider the matrix A = ([7, 5],[5, 9]) and use power iteration method to find the largest (in absolute value) eigenvalue and its corresponding eigenvector with an initial guess of [1,1]^T. Also, compute the error between the actual eigenvalue and the estimated eigenvalue after 5 iterations. | tiger_lab_math | [
0.03445141762495041,
0.028204383328557014,
-0.024813244119286537,
-0.0462864525616169,
0.04395205155014992,
-0.02745763212442398,
-0.007164014037698507,
0.06804683804512024,
0.03757927194237709,
-0.01901407167315483,
0.022276340052485466,
0.024139689281582832,
-0.043301697820425034,
0.0037... | 0.818323 | 0.810236 | 0.008087 |
Find a quadratic with rational coefficients and quadratic term $x^2$ that has $\sqrt{3}-2$ as a root. | tiger_lab_math | [
0.01622622460126877,
0.036146312952041626,
-0.05428483337163925,
-0.04351034760475159,
0.02600356563925743,
-0.030319225043058395,
-0.0047298637218773365,
0.062310636043548584,
0.020524881780147552,
-0.0353846475481987,
0.027541909366846085,
0.027919214218854904,
-0.03226077929139137,
-0.0... | 0.839065 | 0.836564 | 0.002501 |
Let X be the space obtained by identifying the boundary circle of a Mobius strip with a point. Find the Morse homology groups H_k(X) for k = 0, 1, 2. | tiger_lab_math | [
0.009984214790165424,
0.017382733523845673,
-0.03177175670862198,
-0.04482216760516167,
0.02709774114191532,
-0.031254060566425323,
-0.03701860457658768,
0.04587537795305252,
0.04544372484087944,
-0.001960001653060317,
0.003259885823354125,
0.041413407772779465,
-0.019029658287763596,
-0.0... | 0.820225 | 0.804356 | 0.015869 |
Let $f(x,y,z) = x^3y^2z$. Find the covariant derivative of the tensor field $T$ on the manifold $\mathbb{R}^3$, where $T_{ij} = \frac{\partial f}{\partial x^i \partial x^j}$ for $i,j \in \{1,2,3\}.$ | tiger_lab_math | [
0.017790453508496284,
0.039501842111349106,
-0.05361340194940567,
-0.045304276049137115,
0.01911162957549095,
0.0002782315423246473,
-0.011602111160755157,
0.06190907955169678,
0.023527169600129128,
-0.005864116363227367,
0.014156038872897625,
0.024343954399228096,
-0.04783220961689949,
-0... | 0.822164 | 0.80847 | 0.013695 |
There are two concentric circles such that the area of the outer circle is four times the area of the inner circle. Let A, B and C be three distinct points on the perimeter of the outer circle such that AB and AC are tangents to the inner circle. If the area of the outer circle is 12 square centimetres then the area (i... | tiger_lab_math | [
0.029707886278629303,
0.03658987581729889,
-0.04676904156804085,
-0.03250933066010475,
0.035794906318187714,
-0.01073203794658184,
-0.017760159447789192,
0.0406096875667572,
0.045440562069416046,
0.0023337474558502436,
0.0034196986816823483,
0.0340900681912899,
-0.024115143343806267,
-0.03... | 0.829447 | 0.813963 | 0.015484 |
The $52$ cards in a deck are numbered $1, 2, \cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $... | tiger_lab_math | [
0.010619907639920712,
0.018414469435811043,
-0.05441662296652794,
-0.04157857224345207,
0.024653567001223564,
-0.02218959853053093,
-0.018142087385058403,
0.05346980690956116,
0.020384449511766434,
-0.022595852613449097,
0.02022792026400566,
0.02221810631453991,
-0.02228354848921299,
0.018... | 0.83687 | 0.817482 | 0.019388 |
A fluid is flowing through a pipe with a circular cross-section. The fluid's velocity at the center of the pipe is 0.5 m/s. Given that the pipe has a diameter of 0.1 m and a length of 1 m, find the velocity profile of the fluid using the Navier-Stokes equations for turbulence modeling. Assume that the fluid has a visco... | tiger_lab_math | [
-0.018333110958337784,
0.04617922753095627,
-0.025158075615763664,
-0.011541462503373623,
0.042747654020786285,
-0.027273986488580704,
-0.05771256610751152,
0.07241982221603394,
0.015233900398015976,
0.011316451244056225,
0.02058437094092369,
-0.011646255850791931,
-0.007225652225315571,
-... | 0.778155 | 0.77521 | 0.002944 |
Suppose we have a non-monotonic logic framework in which there are 20 premises, labeled 1 through 20. A conclusion C is derived from these premises. However, if premise 4 is removed, the conclusion C is no longer true. If premise 17 is also removed, then C becomes true again. How many different combinations of premises... | tiger_lab_math | [
0.01692911796271801,
0.010024975053966045,
-0.048348601907491684,
-0.020111417397856712,
0.02290104329586029,
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0.05083583667874336,
0.05031321570277214,
-0.025325721129775047,
-0.00935952365398407,
0.02354160137474537,
-0.04027076065540314,
-0.006... | 0.829346 | 0.821179 | 0.008167 |
A toy manufacturing company has 8 pounds of plastic and 6 pounds of metal to make toys. Each plastic toy requires 2 pounds of plastic and 1 pound of metal, while each metal toy requires 1 pound of plastic and 3 pounds of metal. The company wants to maximize the profit by selling the toys. The profit for each plastic to... | tiger_lab_math | [
0.022874431684613228,
0.011329792439937592,
-0.05140875652432442,
-0.03416541591286659,
0.0254763662815094,
-0.030168429017066956,
-0.04224028438329697,
0.05597909912467003,
0.02321191504597664,
-0.00785305816680193,
0.005725621245801449,
0.0449833869934082,
-0.02862471342086792,
-0.030855... | 0.811912 | 0.808028 | 0.003885 |
Sure, here's a numerical problem related to the Nonlinear Schrödinger equation for soliton waves:
Consider a one-dimensional wave governed by the Nonlinear Schrödinger equation:
$$i \frac{\partial \psi}{\partial t} + \frac{1}{2} \frac{\partial^2 \psi}{\partial x^2} + |\psi|^2 \psi = 0$$
with the initial condition
... | tiger_lab_math | [
0.020063860341906548,
0.04304604232311249,
-0.03496547415852547,
-0.029773710295557976,
0.025743674486875534,
-0.043671779334545135,
-0.043585993349552155,
0.07779575139284134,
0.04553251340985298,
-0.007438503671437502,
0.006422154139727354,
0.03075336292386055,
-0.035561759024858475,
-0.... | 0.80789 | 0.801397 | 0.006493 |
Let A = {1, 2, 3} and B = {a, b}. Find (AxB) ∪ (AxC) where C = {x, y, z}. | tiger_lab_math | [
0.04734906926751137,
0.02749823033809662,
-0.03838410601019859,
-0.021301593631505966,
0.028060702607035637,
-0.01269275601953268,
-0.007264229469001293,
0.0379851758480072,
0.04606969282031059,
-0.029513219371438026,
0.018552854657173157,
0.030384361743927002,
-0.0321478508412838,
-0.0144... | 0.845627 | 0.83434 | 0.011287 |
A manufacturing company has an assembly line that produces 3 types of products, A, B and C. The assembly line can produce a maximum of 700 units per day. The profit per unit of A, B, and C is $10, $15, and $25 respectively. However, due to limited resources, the maximum number of units for A, B and C that can be produc... | tiger_lab_math | [
0.02003082074224949,
-0.0061442372389137745,
-0.0700567290186882,
-0.0019329636124894023,
0.02549966238439083,
-0.030145075172185898,
-0.05033442750573158,
0.07844880223274231,
0.021655011922121048,
-0.01482404489070177,
0.0014310297556221485,
0.02752874791622162,
-0.011605003848671913,
-0... | 0.812222 | 0.797473 | 0.014749 |
In a game of Stag Hunt, Alice and Bob can choose to either hunt a stag (worth 5 points) or a hare (worth 3 points). If they both choose to hunt the stag, they each earn 5 points. If one hunts a stag and the other hunts a hare, the one who hunted the stag earns 1 point and the one who hunted the hare earns 3 points. If ... | tiger_lab_math | [
0.0051994542591273785,
0.0047379229217767715,
-0.046831145882606506,
-0.031417395919561386,
0.017046773806214333,
-0.02833322063088417,
-0.019468583166599274,
0.04476979002356529,
0.035357844084501266,
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0.00888285506516695,
0.021318353712558746,
-0.03713168948888779,
... | 0.799704 | 0.789645 | 0.010059 |
Let f(x) = e^(6x-3) be a function defined on the range [0,π]. Obtain the f'(x) and f''(x). Using the Fourier Transform of the function, find its Fourier series F(w) and determine its convergence. | tiger_lab_math | [
0.025249946862459183,
0.04498940333724022,
-0.041192151606082916,
-0.02811860665678978,
0.031190983951091766,
-0.0302395261824131,
-0.014928855001926422,
0.05167458951473236,
0.037975385785102844,
0.004194741602987051,
0.026465989649295807,
0.030400488525629044,
-0.034995824098587036,
-0.0... | 0.815155 | 0.814212 | 0.000943 |
For positive integers $n$, define $S_n$ to be the minimum value of the sum
\[\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},\]where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $n$ for which $S_n$ is also an integer. | tiger_lab_math | [
0.019704025238752365,
0.028647582978010178,
-0.047126419842243195,
-0.04492420330643654,
0.03498482331633568,
-0.02050430327653885,
-0.026752158999443054,
0.055542584508657455,
0.041639190167188644,
-0.011169984936714172,
0.023697473108768463,
0.028328044340014458,
-0.023133056238293648,
-... | 0.861928 | 0.831645 | 0.030283 |
A surface S is represented by the equation z = 2x^3 + 3y^3, find the sectional curvature of S at the point (1, 1, 5). | tiger_lab_math | [
0.005851455964148045,
0.0205568578094244,
-0.05911065265536308,
-0.052697475999593735,
0.0437704473733902,
-0.007700818125158548,
-0.01837575063109398,
0.06479847431182861,
0.012193598784506321,
-0.0041184318251907825,
0.0317302905023098,
0.041025131940841675,
-0.03267578035593033,
-0.0040... | 0.827229 | 0.822801 | 0.004427 |
A math student is studying birational geometry and wants to evaluate their understanding of the subject. They are given the following problem:
Let X be an algebraic curve defined by the equation X: y^2 = x^3 + 2x^2 - 4x. Find the equation of the birational transformation from X to the affine line. | tiger_lab_math | [
0.03500164672732353,
0.020747492089867592,
-0.05038990080356598,
-0.049348074942827225,
0.029856329783797264,
-0.02673567831516266,
-0.0058403885923326015,
0.06728526949882507,
0.06156405434012413,
-0.014816471375524998,
-0.008779083378612995,
0.020385531708598137,
-0.04593626409769058,
-0... | 0.832204 | 0.814826 | 0.017378 |
Find the largest constant $m,$ so that for any positive real numbers $a,$ $b,$ $c,$ and $d,$
\[\sqrt{\frac{a}{b + c + d}} + \sqrt{\frac{b}{a + c + d}} + \sqrt{\frac{c}{a + b + d}} + \sqrt{\frac{d}{a + b + c}} > m.\] | tiger_lab_math | [
0.026827296242117882,
0.04351692274212837,
-0.03391806781291962,
-0.0425235740840435,
0.034410081803798676,
-0.009723548777401447,
-0.028853541240096092,
0.057672854512929916,
0.04852474853396416,
-0.016500966623425484,
-0.007001824211329222,
0.034411776810884476,
-0.03719855844974518,
-0.... | 0.854892 | 0.83566 | 0.019232 |
Let X be the algebraic variety defined by the equation:
X = {(x, y) ∈ ℂ^2 | x^2 + y^2 - 1 = 0}.
Calculate the second cohomology group of X with coefficients in ℤ. | tiger_lab_math | [
0.02859223447740078,
0.02398923970758915,
-0.04353014752268791,
-0.029699034988880157,
0.02964390628039837,
-0.025250891223549843,
-0.0009620726923458278,
0.07198036462068558,
0.05744083970785141,
-0.022329164668917656,
0.0018958606524392962,
0.030856352299451828,
-0.024268200621008873,
-0... | 0.823247 | 0.808799 | 0.014447 |
At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers? Please write a program to solve it | tiger_lab_math | [
0.03022831492125988,
0.0073545752093195915,
-0.0548860989511013,
-0.048066116869449615,
0.04931192472577095,
-0.045768339186906815,
-0.042639415711164474,
0.03507963940501213,
0.034446340054273605,
-0.01351488009095192,
0.020263493061065674,
0.0325469896197319,
-0.020551996305584908,
-0.01... | 0.83173 | 0.828157 | 0.003573 |
Imagine two travelers lose their luggage in an airport, each traveler has a suitcase with undisclosed valuable content including five $100 bills. Airline employees offer to compensate them for the lost baggage, but they need to disclose the value of the lost items. The compensation for each traveler would be the highes... | tiger_lab_math | [
0.02110803872346878,
0.0055376142263412476,
-0.047891609370708466,
-0.01582290418446064,
0.018468504771590233,
-0.006579729728400707,
-0.022202976047992706,
0.04203593730926514,
0.04676557704806328,
-0.036864060908555984,
0.02131866291165352,
0.0245134849101305,
-0.027061333879828453,
-0.0... | 0.811574 | 0.795337 | 0.016236 |
A student is given the following problem to solve:
Consider a rational curve defined by the equation Y^2Z = X^3 - XZ^2.
Find the coordinates of all points on the curve of the form (u^2, u^3, u) where u is a nonzero rational number. | tiger_lab_math | [
0.014429655857384205,
0.030375489965081215,
-0.046741239726543427,
-0.05672518163919449,
0.023565376177430153,
-0.017474360764026642,
0.0012312084436416626,
0.03842094913125038,
0.04975440353155136,
-0.022783881053328514,
0.005619140341877937,
0.023109275847673416,
-0.030230721458792686,
-... | 0.836629 | 0.817674 | 0.018955 |
Consider the projective variety $V$ in $\mathbb{P}^2$ defined by the homogeneous polynomial $F(x,y,z) = x^2y + y^2z + z^2x$. Find the intersection of $V$ with the line $L$ given by the equations $x=z$ and $y=1$. | tiger_lab_math | [
0.015631167218089104,
0.028775207698345184,
-0.03432294726371765,
-0.051179610192775726,
0.033008404076099396,
-0.02380242757499218,
-0.0066695003770291805,
0.04268839955329895,
0.03314274176955223,
-0.017580663785338402,
0.02231687679886818,
0.04146108403801918,
-0.011173180304467678,
-0.... | 0.84161 | 0.813909 | 0.027701 |
Let $f(x) = \sin{x} + 2\cos{x} + 3\tan{x}$, using radian measure for the variable $x$. Let $r$ be the smallest positive value of $x$ for which $f(x) = 0$. Find $\lfloor r \rfloor.$ | tiger_lab_math | [
0.012172091752290726,
0.041163068264722824,
-0.0580013208091259,
-0.030598729848861694,
0.027179032564163208,
-0.012468368746340275,
-0.02221846953034401,
0.05516507104039192,
0.026876820251345634,
0.006496209651231766,
0.021315382793545723,
0.015728505328297615,
-0.034948043525218964,
-0.... | 0.841229 | 0.829946 | 0.011283 |
Five points $A$, $B$, $C$, $D$, and $O$ lie on a flat field. $A$ is directly north of $O$, $B$ is directly west of $O$, $C$ is directly south of $O$, and $D$ is directly east of $O$. The distance between $C$ and $D$ is 140 m. A hot-air balloon is positioned in the air at $H$ directly above $O$. The balloon is held i... | tiger_lab_math | [
0.0193770881742239,
0.01626366190612316,
-0.04236674681305885,
-0.03574089705944061,
0.04203660413622856,
-0.03966003656387329,
-0.03608288615942001,
0.050035957247018814,
0.0412864051759243,
-0.00284448079764843,
0.0031075349543243647,
0.03682815656065941,
-0.008773809298872948,
-0.024200... | 0.835115 | 0.830337 | 0.004778 |
We define two sequences of vectors $(\mathbf{v}_n)$ and $(\mathbf{w}_n)$ as follows: First, $\mathbf{v}_0 = \begin{pmatrix} 1 \\ 3 \end{pmatrix},$ $\mathbf{w}_0 = \begin{pmatrix} 4 \\ 0 \end{pmatrix}.$ Then for all $n \ge 1,$ $\mathbf{v}_n$ is the projection of $\mathbf{w}_{n - 1}$ onto $\mathbf{v}_0,$ and $\mathbf{w}... | tiger_lab_math | [
0.03951982408761978,
0.041813239455223083,
-0.03558368608355522,
-0.03979787975549698,
0.048522938042879105,
-0.00573388347402215,
-0.01716356724500656,
0.07802489399909973,
0.04537627846002579,
-0.01089541893452406,
0.012945937924087048,
0.02422172576189041,
-0.026284053921699524,
-0.0106... | 0.838646 | 0.818102 | 0.020543 |
two equal triangles constitute rhombus a . if the diagonals of rhombus are 15 cm and 20 cm long . what is the area of each triangle ? Let's write a Python program. | tiger_lab_math | [
0.027293335646390915,
0.01829828880727291,
-0.05127347260713577,
-0.04153236001729965,
0.026429077610373497,
-0.02387891709804535,
-0.01681976206600666,
0.05404886230826378,
0.036176662892103195,
-0.02605769969522953,
0.02552308514714241,
0.04307938739657402,
-0.029102837666869164,
-0.0154... | 0.821803 | 0.819951 | 0.001852 |
A manufacturer produces three types of smartphones - A, B, and C. A sample of 20 smartphones was selected, where 7 were type A, 6 were type B, and 7 were type C. The battery life (in hours) of each smartphone was recorded. Conduct an ANOVA test at a 5% level of significance to determine if there is a significant differ... | tiger_lab_math | [
0.01227219495922327,
0.01941761001944542,
-0.0721428319811821,
-0.04757227376103401,
0.02210322581231594,
-0.04538092017173767,
-0.0038595593068748713,
0.10554134100675583,
0.0034595245961099863,
-0.028309429064393044,
0.009098676033318043,
0.041706331074237823,
-0.040970150381326675,
0.00... | 0.792119 | 0.786201 | 0.005918 |
A furniture manufacturer produces two types of chairs: a standard chair and a deluxe chair. Each standard chair can be sold for $100 and each deluxe chair can be sold for $150. The manufacturer has 50 hours of carpentry time and 30 hours of painting time each week. The standard chair requires 2 hours of carpentry time ... | tiger_lab_math | [
0.0017193392850458622,
0.005748273339122534,
-0.061236683279275894,
-0.00941593199968338,
0.025986289605498314,
-0.017321711406111717,
-0.044698283076286316,
0.0763816311955452,
0.04583444446325302,
-0.014160210266709328,
0.036623623222112656,
0.035183560103178024,
-0.029744019731879234,
-... | 0.810128 | 0.80045 | 0.009678 |
Compute the smallest positive integer $n$ such that
\[\sum_{k = 0}^n \log_2 \left( 1 + \frac{1}{2^{2^k}} \right) \ge 1 + \log_2 \frac{2014}{2015}.\] Let's write a program. | tiger_lab_math | [
-0.003969983663409948,
0.048522260040044785,
-0.029167400673031807,
-0.024472007527947426,
0.05042339116334915,
-0.006659408565610647,
-0.02692391723394394,
0.05293714627623558,
0.05108688399195671,
-0.012927787378430367,
0.020343204960227013,
0.019085608422756195,
-0.018414145335555077,
-... | 0.834271 | 0.830036 | 0.004235 |
Suppose M is a smooth manifold of dimension n, and let $f:M \rightarrow \mathbb{R}$ be a smooth function. Compute the de Rham cohomology classes of the differential form $d(f \cdot \omega)$, where $\omega$ is a closed $k$-form on $M$ and $0 \leq k \leq n$. | tiger_lab_math | [
0.026425933465361595,
0.035088784992694855,
-0.04092792421579361,
-0.04604382440447807,
0.023495929315686226,
-0.005373781081289053,
-0.021174566820263863,
0.06637254357337952,
0.03988092020153999,
0.0015022759325802326,
-0.0035768598318099976,
0.03950430452823639,
-0.03882337361574173,
-0... | 0.805114 | 0.784542 | 0.020572 |
What is the number of critical points of the following function on the sphere $\mathbb{S}^2$:
$$f(x,y,z) = x^3 + y^3 + z^3 - 3xyz$$ | tiger_lab_math | [
-0.002576500875875354,
0.03336181864142418,
-0.0654948279261589,
-0.04390412196516991,
0.01626407541334629,
-0.0036552271340042353,
-0.02523490972816944,
0.04184271767735481,
0.02685338445007801,
-0.0044478545896708965,
0.021013598889112473,
0.03440200909972191,
-0.03452150151133537,
-0.03... | 0.840885 | 0.83657 | 0.004315 |
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 12$ and $\angle O_{1}PO_{2} = 120^{\circ}$, then $AP = \sqrt{a} + \sqrt{b}$, where $a$ and $b$ are positive integers. Find $a + b$.
| tiger_lab_math | [
0.021411577239632607,
0.0248824842274189,
-0.052134815603494644,
-0.043600838631391525,
0.024956198409199715,
-0.00951636303216219,
0.0015883406158536673,
0.06774228811264038,
0.05023461580276489,
0.009103856049478054,
0.012702065519988537,
0.037619806826114655,
-0.033905863761901855,
-0.0... | 0.843728 | 0.818372 | 0.025356 |
Suppose you have a box containing 10 red balls and 5 blue balls. You randomly select a ball from the box, write down its color, and return it to the box. You repeat this process 6 times. What is the probability that exactly 3 red balls and 3 blue balls are chosen? To solve this problem, apply linear logic concepts of c... | tiger_lab_math | [
-0.0020288764499127865,
0.019830530509352684,
-0.049798667430877686,
-0.01518967468291521,
0.03248470649123192,
-0.005276195239275694,
-0.027252310886979103,
0.04443005844950676,
0.030689312145113945,
-0.012665941379964352,
0.01332450844347477,
0.016891030594706535,
-0.01642714999616146,
-... | 0.828736 | 0.817299 | 0.011437 |
Consider a network with 8 nodes and the following adjacency matrix:
$$
\begin{bmatrix}
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\
0... | tiger_lab_math | [
0.03531956300139427,
0.02554858848452568,
-0.040166109800338745,
-0.02569691650569439,
0.03383057564496994,
-0.005117131397128105,
-0.019613662734627724,
0.05955712869763374,
0.03169230371713638,
0.012703957036137581,
0.013678444549441338,
0.020894235000014305,
-0.04041304066777229,
-0.026... | 0.821967 | 0.816725 | 0.005242 |
Let $r$ be a real number, $|r| < 2,$ and let $z$ be a complex number such that
\[z + \frac{1}{z} = r.\]Find $|z|.$ Let's write a program. | tiger_lab_math | [
0.01972305215895176,
0.03633125126361847,
-0.0331907756626606,
-0.03143245354294777,
0.043944720178842545,
-0.002772662090137601,
-0.0062340484000742435,
0.045615553855895996,
0.04048991948366165,
-0.02718040905892849,
0.013112034648656845,
0.031554099172353745,
-0.02907993458211422,
-0.02... | 0.838724 | 0.829428 | 0.009295 |
In the Traveler's Dilemma game, two travelers each declare a whole number between 0 and 100. The traveler with the lower number gets paid the amount he declared, but the traveler with the higher number gets paid $2 less than the amount he declared. If both travelers declare the same number, they each get paid that numb... | tiger_lab_math | [
0.015989212319254875,
0.00413855304941535,
-0.050382182002067566,
-0.016864672303199768,
0.017292547971010208,
-0.010241440497338772,
-0.021384278312325478,
0.04386691376566887,
0.043275121599435806,
-0.033565353602170944,
0.03469300642609596,
0.03666945919394493,
-0.03733135014772415,
-0.... | 0.827865 | 0.825221 | 0.002645 |
Find the analytic continuation of the function f(z) = sqrt(z-1) to the complex plane with a branch cut along the negative real axis, evaluated at the point z = 2+3i. | tiger_lab_math | [
0.018955575302243233,
0.04687612131237984,
-0.030195994302630424,
-0.055469732731580734,
0.02494189329445362,
0.008387373760342598,
-0.008103668689727783,
0.05314350128173828,
0.03397798165678978,
-0.006910704541951418,
0.025940602645277977,
0.018974581733345985,
-0.038447119295597076,
-0.... | 0.827507 | 0.821852 | 0.005655 |
Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order)... | tiger_lab_math | [
0.022433646023273468,
0.006753514986485243,
-0.050921108573675156,
-0.04181426763534546,
0.036047790199518204,
-0.017677564173936844,
-0.0421021394431591,
0.046949103474617004,
0.04134511947631836,
-0.028298258781433105,
0.011686335317790508,
0.017602942883968353,
-0.039798520505428314,
-0... | 0.824115 | 0.819867 | 0.004249 |
The numbers 1, 3, 6, 10, $\ldots$, are called triangular numbers, as shown geometrically here. What is the $20^{\text{th}}$ triangular number?
[asy]
dot((0,0));
label("1",(0,-1.5));
dot((3,0));
dot((4,0));
dot((3,1));
label("3",(3.5,-1.5));
dot((7,0));
dot((8,0));
dot((9,0));
dot((7,1));
dot((7,2));
dot((8,1));
la... | tiger_lab_math | [
0.038067471235990524,
0.05451780930161476,
-0.03939249366521835,
-0.03293608874082565,
0.045130811631679535,
-0.03163453936576843,
-0.036154892295598984,
0.05110527575016022,
0.019101379439234734,
0.004911590833216906,
0.03222448378801346,
0.03384896367788315,
-0.04376834258437157,
-0.0163... | 0.821126 | 0.817652 | 0.003474 |
Consider two firms, A and B, producing identical products with identical costs. The market demand curve is represented by Q = 100 - 2P, where Q is the total quantity demanded and P is the price of the product. Both firms simultaneously choose the quantity they will produce. Using Cournot competition, calculate the quan... | tiger_lab_math | [
0.012163427658379078,
0.013647742569446564,
-0.048457223922014236,
-0.02293127402663231,
0.015772098675370216,
-0.017913369461894035,
-0.027357440441846848,
0.0596543550491333,
0.019703209400177002,
-0.023003006353974342,
-0.0016842155018821359,
0.037494152784347534,
-0.03510589897632599,
... | 0.805842 | 0.791253 | 0.014589 |
Consider two companies, A and B, producing identical products. If both companies produce simultaneously and choose the quantities of their products to sell, they will end up in a Cournot Competition.
Suppose that the demand function for the product is: Q = 240 – P. Where Q is the total quantity demanded at a price of... | tiger_lab_math | [
0.028196390718221664,
0.015552029013633728,
-0.053764425218105316,
-0.02979062683880329,
0.017426826059818268,
-0.0036071306094527245,
-0.03243304416537285,
0.07429219037294388,
0.022713718935847282,
-0.019229399040341377,
0.006993402726948261,
0.04198974370956421,
-0.034091413021087646,
-... | 0.815638 | 0.798978 | 0.01666 |
dexter and prexter are competing with each other in a friendly community competition in a pool of 50 m length and the race is for 1000 m . dexter crosses 50 m in 2 min and prexter in 3 min 15 sec . each time they meet / cross each other , they do handshake ' s . how many such handshake ' s will happen if they start fro... | tiger_lab_math | [
0.021530352532863617,
0.00916899461299181,
-0.04145989567041397,
-0.03588179871439934,
0.029362672939896584,
-0.023877954110503197,
-0.04904229938983917,
0.039239551872015,
0.039708659052848816,
-0.028149643912911415,
0.014959597960114479,
0.022885799407958984,
-0.032648950815200806,
-0.03... | 0.82112 | 0.818613 | 0.002508 |
Suppose you are the CEO of a company and want to hire a new employee who will either work hard or shirk. If the new employee works hard, you get a profit of $50,000; if the new employee shirks, you get a profit of $20,000. The probability of the new employee shirking is 0.5. You can offer the new employee a salary of $... | tiger_lab_math | [
-0.004351628944277763,
0.006435511168092489,
-0.056164804846048355,
-0.018259624019265175,
0.03549019619822502,
-0.0035468931309878826,
-0.03436665236949921,
0.055232759565114975,
0.04507316276431084,
-0.019536392763257027,
0.02266787551343441,
0.011070004664361477,
-0.022911693900823593,
... | 0.815904 | 0.812826 | 0.003079 |
Jack and Jill went up the hill to fetch water to fill a 600 gallon tank. Each bucket is five gallons. Jack is stronger, and is able to carry two buckets of water at a time, but Jill can only manage one. Also, Jack can complete three trips to the well and back in the time it takes Jill to make two trips. How many trips ... | tiger_lab_math | [
0.020443145185709,
0.036484718322753906,
-0.04099700227379799,
-0.03334590047597885,
0.03616910055279732,
-0.05315760150551796,
-0.023187078535556793,
0.0829399898648262,
0.017802942544221878,
-0.01858321949839592,
0.0015841471031308174,
0.032573990523815155,
-0.053465958684682846,
-0.0247... | 0.819569 | 0.817279 | 0.00229 |
Suppose two hunters are hunting in a forest. They both have to decide whether to hunt a stag or a rabbit. If they both hunt a stag, their payoff is 10 units each. If one hunter hunts a stag and the other hunts a rabbit, the hunter who hunts the stag gets 8 units and the hunter who hunts the rabbit gets 0 units. If they... | tiger_lab_math | [
0.003607821185141802,
0.017750808969140053,
-0.05831852927803993,
-0.02949036844074726,
0.031378816813230515,
-0.02230200357735157,
-0.016634559258818626,
0.05133432894945145,
0.03714975342154503,
-0.018023192882537842,
0.017943987622857094,
0.018728172406554222,
-0.02829872816801071,
-0.0... | 0.826888 | 0.821901 | 0.004987 |
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