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 |
|---|---|---|---|---|---|
If a rectangular prism has a length of $l$, a width of $w$, and a height of $h$, then the length of its diagonal is equal to $\sqrt{l^2 + w^2 + h^2}$. Suppose $l = 3$ and $h = 12$; if the length of the diagonal is $13$, what is the width? Let's write a Python program to solve it. | tiger_lab_math | [
0.003007204970344901,
0.02782413735985756,
-0.04131317138671875,
-0.058108601719141006,
0.03423139825463295,
-0.020062847062945366,
-0.03390044718980789,
0.05589642748236656,
0.029849888756871223,
-0.005450312979519367,
0.02682582288980484,
0.03537330776453018,
-0.02831757813692093,
-0.010... | 0.834938 | 0.826998 | 0.00794 |
Suppose that there are two players, Player A and Player B, in a Signaling Game where Player A has two messages to send to Player B, "Yes" or "No". If Player A chooses to send the message "Yes", then Player B's payoff will be $5 if a certain event occurs, and $0 otherwise. If Player A chooses to send the message "No", t... | tiger_lab_math | [
-0.0038817767053842545,
0.018439291045069695,
-0.056112490594387054,
-0.018951542675495148,
0.03544387221336365,
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-0.011410163715481758,
0.048009008169174194,
0.028413275256752968,
-0.03639513999223709,
0.03188054636120796,
0.03660670667886734,
-0.01928628236055374,
-0... | 0.800302 | 0.793954 | 0.006348 |
Okay, here's an example of a problem:
The height of a right-angled triangle is 10 cm, and its base is 8 cm. Find its area using trigonometry. Round your answer to one decimal place. | tiger_lab_math | [
0.041105739772319794,
0.030307624489068985,
-0.02639840357005596,
-0.047180067747831345,
0.025933748111128807,
-0.024195954203605652,
-0.010005529969930649,
0.041747938841581345,
0.03177235648036003,
-0.007698115427047014,
0.015258544124662876,
0.03327954560518265,
-0.04162998124957085,
-0... | 0.812691 | 0.811813 | 0.000878 |
Sure, here's a math problem for an Optimal portfolio selection:
Suppose you have a portfolio of $100,000 that is invested in three different stocks: IBM, Apple, and Microsoft. You want to optimize your portfolio by finding the most profitable distribution of your investment across these three stocks. Here are some key... | tiger_lab_math | [
0.018720248714089394,
0.00011328493565088138,
-0.05345937982201576,
-0.029987305402755737,
0.02206195890903473,
-0.03644078969955444,
-0.03199850767850876,
0.03093578852713108,
0.036800410598516464,
-0.02027500979602337,
0.01693008653819561,
0.02610301412642002,
-0.03890452906489372,
-0.02... | 0.814968 | 0.80711 | 0.007859 |
Let A={1, 2, 3, 4}, B={2, 4, 6, 8}, and C={3, 5, 7, 9} be three sets, find the symmetric difference of A and (BΔC) using the associative property. | tiger_lab_math | [
0.04533771798014641,
0.0055064368061721325,
-0.04147418215870857,
-0.017788197845220566,
0.031925782561302185,
-0.03554491326212883,
-0.02152968756854534,
0.055851444602012634,
0.06752447038888931,
-0.015367027372121811,
0.001981687732040882,
0.015626301988959312,
-0.04712020233273506,
-0.... | 0.835648 | 0.826427 | 0.009221 |
Let $M$ be a manifold equipped with a metric $g$. Consider the following tensor field:
$$T = x^2 \sin(y) \partial_x \otimes \partial_y - \cos(y) \partial_y \otimes \partial_y + (2x+y) \partial_x \otimes \partial_x.$$
Find the covariant derivative of $T$ with respect to the Levi-Civita connection $\nabla$ of $g$ on $M$... | tiger_lab_math | [
0.009941346943378448,
0.03673604875802994,
-0.03935182839632034,
-0.03560059145092964,
0.031179487705230713,
0.0028961505740880966,
-0.017447970807552338,
0.0810651108622551,
0.022721311077475548,
0.01198516320437193,
0.004005111753940582,
0.0308231208473444,
-0.038444261997938156,
-0.0071... | 0.814642 | 0.798338 | 0.016304 |
A 10-by-9 inch piece of paper is used to form the lateral surface of a cylinder. If the entire piece of paper is used to make the lateral surface, which of the following must be true of the two possible cylinders that can be formed?
Answer Choices: (A) The volume of the cylinder with height 10 is 90/pi cubic inches gre... | tiger_lab_math | [
0.0061304401606321335,
0.029385318979620934,
-0.024142734706401825,
-0.010598980821669102,
0.03454078733921051,
-0.030227918177843094,
-0.040812134742736816,
0.09506746381521225,
0.006135860458016396,
-0.0035298443399369717,
0.02078479714691639,
0.04434359818696976,
-0.02785630151629448,
-... | 0.808302 | 0.800169 | 0.008133 |
Suppose a portfolio manager has invested in three different stocks: Stock A, Stock B, and Stock C. The manager can allocate a maximum of $100,000 among the three stocks. The expected returns and standard deviations of each stock, as well as the correlation between each pair of stocks, are provided below:
Stock A: Expe... | tiger_lab_math | [
0.016124654561281204,
0.0033252749126404524,
-0.06989576667547226,
-0.02755497582256794,
0.030580680817365646,
-0.0030193403363227844,
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0.042691830545663834,
0.03169930726289749,
-0.0019644838757812977,
0.017901841551065445,
0.04520415887236595,
-0.03363620489835739,
... | 0.796397 | 0.781323 | 0.015074 |
Two graphs G and H have the following adjacency matrices:
G = \begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0 \\
\end{bmatrix}
H = \begin{bmatrix}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0 \\
\end{bmatrix}
Are G and H isomorphic? If they are, provide an isomorphism between the two graphs. If they are not, exp... | tiger_lab_math | [
0.019114261493086815,
0.01923980936408043,
-0.02292890101671219,
-0.030092863366007805,
0.04510367289185524,
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0.07931740581989288,
0.021305887028574944,
-0.026751255616545677,
0.023297298699617386,
-0.005745160859078169,
-0.020415667444467545,
-0... | 0.805336 | 0.79284 | 0.012496 |
Solve the following boundary value problem:
$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+u = 0$,
subject to the boundary conditions
$u(0,y) = 1$ for $0\leq y \leq 1,$
$u(1,y) = 2$ for $0\leq y \leq 1,$
$u(x,0) = u(x,1) = 0$ for $0\leq x \leq 1.$
Find the solution $u(x,y)$ using the meth... | tiger_lab_math | [
-0.0021050518844276667,
0.03912363573908806,
-0.05251040682196617,
-0.0319044254720211,
0.04635836184024811,
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0.0015354988863691688,
0.07245010137557983,
0.061799872666597366,
-0.007455144543200731,
0.0028558040503412485,
0.03665084019303322,
-0.012579157948493958,
-0... | 0.819782 | 0.796243 | 0.023538 |
Karlanna places 600 marbles into $m$ total boxes such that each box contains an equal number of marbles. There is more than one box, and each box contains more than one marble. For how many values of $m$ can this be done? | tiger_lab_math | [
0.008671097457408905,
0.02731919102370739,
-0.036661017686128616,
-0.025098543614149094,
0.047544896602630615,
-0.014766690321266651,
-0.024194439873099327,
0.06474864482879639,
0.024468662217259407,
-0.03024310991168022,
0.020728105679154396,
0.0441225990653038,
-0.023254791274666786,
-0.... | 0.831916 | 0.821902 | 0.010014 |
The lengths of two sides of a triangle are 33 units and 42 units. The third side also has an integral length. What is the least possible number of units in the perimeter of the triangle? | tiger_lab_math | [
0.018484096974134445,
0.0184834785759449,
-0.04137951508164406,
-0.0419577956199646,
0.03135819733142853,
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-0.05319617688655853,
0.03273218497633934,
0.03835992142558098,
-0.011975189670920372,
0.03785361722111702,
0.029907267540693283,
-0.02009628713130951,
-0.028459... | 0.829157 | 0.826514 | 0.002643 |
The equation of a digital filter is given by $y(n)=1 / 3(x(n)+x(n-1)+x(n-2))$, where $y(n)$ and $x(n)$ are, respectively, the nth samples of the output and input signals. Determine the pole(s) of the filter. | tiger_lab_math | [
0.021900376304984093,
0.005244950298219919,
-0.07076355814933777,
-0.03813554346561432,
0.03305858373641968,
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-0.029666535556316376,
0.057363368570804596,
0.03124467097222805,
-0.006638322491198778,
0.030110400170087814,
0.04523905739188194,
-0.0547252856194973,
-0.027... | 0.824341 | 0.82204 | 0.002301 |
I randomly pick an integer $p$ between $1$ and $10$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 4p - 2q = 2$? Express your answer as a common fraction. | tiger_lab_math | [
0.024406803771853447,
0.018853366374969482,
-0.04382820054888725,
-0.03227284550666809,
0.027453424409031868,
-0.034272920340299606,
-0.015814917162060738,
0.03452558070421219,
0.023880822584033012,
-0.013368644751608372,
0.016416458413004875,
0.029353221878409386,
-0.02242419682443142,
-0... | 0.847934 | 0.844305 | 0.00363 |
Suppose a company wants to maximize their profit while minimizing their production cost. They can produce two types of products that require different amounts of resources. Product A requires 2 units of labor and 3 units of material to be produced, while product B requires 3 units of labor and 2 units of material to be... | tiger_lab_math | [
0.02207818813621998,
0.009933351539075375,
-0.06640449166297913,
-0.006941006053239107,
0.014329271391034126,
-0.007198872044682503,
-0.052915725857019424,
0.08068262040615082,
0.021187040954828262,
-0.01584785059094429,
0.0046541946940124035,
0.0337466336786747,
-0.015330741181969643,
-0.... | 0.80766 | 0.791739 | 0.015921 |
set s contains exactly 10 numbers and has an average ( arithmetic mean ) of 6.2 . if one of the numbers in set s is increased by 3 , while all other numbers remain the same , what is the new average of set s ? Let's write a program. | tiger_lab_math | [
0.03946182131767273,
0.015242915600538254,
-0.04926489293575287,
-0.0170429777354002,
0.06431632488965988,
-0.04852309077978134,
-0.031189575791358948,
0.052158161997795105,
0.026307109743356705,
-0.02480691857635975,
0.02149847522377968,
0.009980421513319016,
-0.06141125038266182,
-0.0162... | 0.821893 | 0.81934 | 0.002554 |
Let set A = {1, 2, 3, 4, 5} and set B = {3, 4, 5, 6, 7}. Find the intersection of set A and set B. | tiger_lab_math | [
0.07198968529701233,
0.009934522211551666,
-0.03670637682080269,
-0.026635225862264633,
0.03588496893644333,
-0.026146408170461655,
-0.02847871743142605,
0.038144800812006,
0.05969837307929993,
-0.025572126731276512,
0.02480917237699032,
0.010473798960447311,
-0.0176575668156147,
-0.020577... | 0.828928 | 0.826915 | 0.002013 |
A manufacturing company produces cardboard boxes with a volume of 1000 cubic inches. The company wants to minimize the amount of cardboard used to make each box. Assuming that the top of the box is a square and that the length is twice the width, what are the dimensions of the box that will use the minimum amount of ca... | tiger_lab_math | [
0.011233995668590069,
0.03179129585623741,
-0.06537161767482758,
-0.0065065850503742695,
0.04140832647681236,
-0.027059238404035568,
-0.04518171399831772,
0.05361377075314522,
0.023099103942513466,
-0.0037480622995644808,
0.013173719868063927,
0.02016187645494938,
-0.022914094850420952,
-0... | 0.806535 | 0.799348 | 0.007187 |
Consider the proposition "it is necessary that A is true," where A is a mathematical statement. Let's say the mathematical statement A represents "The sum of two odd numbers is even."
What is the truth value for the proposition "it is necessary that A is true"? | tiger_lab_math | [
0.04006679356098175,
-0.0013606846332550049,
-0.04888293519616127,
-0.04397844523191452,
0.021342216059565544,
-0.03490936756134033,
-0.01528813224285841,
0.08031421899795532,
0.03150012344121933,
-0.023785073310136795,
-0.005460033658891916,
0.03271450847387314,
-0.024687286466360092,
-0.... | 0.822507 | 0.81397 | 0.008537 |
The sum of the first $20$ positive even integers is also the sum of four consecutive even integers. What is the largest of these four integers? Let's write a program. | tiger_lab_math | [
0.01282710675150156,
0.018557168543338776,
-0.03898940607905388,
-0.041714537888765335,
0.04581667110323906,
-0.011208121664822102,
-0.03613681718707085,
0.044620122760534286,
0.05575868859887123,
-0.03261087089776993,
0.01950635202229023,
0.046333104372024536,
-0.0566287524998188,
-0.0189... | 0.841234 | 0.83726 | 0.003973 |
Charlotte, Lisa, and Patricia have a joint baseball collection. One of their friends is selling a rare baseball card for $100 on his website. Not wanting to miss such a rare opportunity, they have pooled all their money to buy the card. Patricia has $6. Lisa has five times Patricia’s money, but double Charlotte’s. How ... | tiger_lab_math | [
0.009610974229872227,
0.015124645084142685,
-0.05677168816328049,
-0.014641872607171535,
0.0379067063331604,
-0.008867875672876835,
-0.024429911747574806,
0.041725993156433105,
0.046231355518102646,
-0.023582572117447853,
0.010139201767742634,
0.03109150007367134,
-0.04178252071142197,
-0.... | 0.827766 | 0.818249 | 0.009517 |
Consider the 2-dimensional manifold given by the surface of a sphere with radius 5 centered at the origin of the 3-dimensional Euclidean space. Determine the Laplacian of the function f(x,y,z) = x^2 + y^2 + z^2 on this manifold. | tiger_lab_math | [
-0.004515757318586111,
0.042866695672273636,
-0.043873682618141174,
-0.05235457047820091,
0.03683716058731079,
-0.0008571844664402306,
-0.022737940773367882,
0.047309376299381256,
0.01745012402534485,
-0.004725554957985878,
0.019810883328318596,
0.039209652692079544,
-0.027921875938773155,
... | 0.823875 | 0.811315 | 0.01256 |
The mean of set P does not exceed mean of any subset of set P. Which of the following must be true about set P?
I. Set P contains only one element
II. All elements in set P are equal
III. The median of set P equals the mean of set P
Answer Choices: (A) of the three qualities is necessary (B) II only (C) III only (D) II... | tiger_lab_math | [
0.031903743743896484,
0.006659322418272495,
-0.044532787054777145,
-0.03698877617716789,
0.03717190399765968,
-0.032087069004774094,
-0.0058794282376766205,
0.06326581537723541,
0.021337592974305153,
-0.03262284770607948,
0.021035069599747658,
0.013657580129802227,
-0.02658029831945896,
-0... | 0.813589 | 0.798559 | 0.015029 |
Here's a specific numerical problem for the math student to solve:
Consider the system of linear equations
$$
\begin{cases}
x + 2y + 3z = 6\\
4x + 5y + 6z = 15\\
7x + 8y + 9z = 24
\end{cases}
$$
Using sparse matrices, find the solution of the system of equations using Gauss-Seidel iteration method. Use an initial gu... | tiger_lab_math | [
0.022722601890563965,
0.03202136605978012,
-0.03790353238582611,
-0.04067827761173248,
0.059511907398700714,
-0.006033117417246103,
-0.015757469460368156,
0.04828216880559921,
0.034966375678777695,
-0.01656189374625683,
0.03211211785674095,
0.025294244289398193,
-0.026038087904453278,
-0.0... | 0.836915 | 0.834588 | 0.002327 |
A population of 1000 individuals are initially healthy. An infectious disease starts spreading in the population, with a daily infection rate of 0.03%. The infected individuals recover with a daily recovery rate of 0.1%. Assuming no immunity and no births or deaths during the period, use differential equations to model... | tiger_lab_math | [
0.026334170252084732,
0.020215414464473724,
-0.05578849837183952,
-0.03153333067893982,
0.031627535820007324,
-0.049808911979198456,
-0.0285496823489666,
0.07445710897445679,
0.02853703312575817,
-0.015184277668595314,
0.017263412475585938,
0.03196559473872185,
-0.02840345725417137,
-0.011... | 0.814225 | 0.807763 | 0.006462 |
Let $P$ be the plane passing through the origin with normal vector $\begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}.$ Find the matrix $\mathbf{P}$ such that for any vector $\mathbf{v},$ $\mathbf{P} \mathbf{v}$ is the projection of $\mathbf{v}$ onto plane $P.$ | tiger_lab_math | [
0.03266039863228798,
0.035915061831474304,
-0.041358571499586105,
-0.044261619448661804,
0.04756234958767891,
-0.017383795231580734,
-0.007172808516770601,
0.05626758560538292,
0.028115922585129738,
-0.02026626095175743,
0.014646321535110474,
0.022486943751573563,
-0.028292126953601837,
-0... | 0.833419 | 0.811572 | 0.021847 |
Consider the logistic map equation $x_{n+1} = r x_n (1-x_n)$, where $r=3.5$, and let $x_0=0.25$. Find the Feigenbaum constant $\delta$ and the associated value of $r_\infty$ for this system. | tiger_lab_math | [
0.00805725995451212,
0.004664397798478603,
-0.03507949784398079,
-0.04199722409248352,
0.03268558531999588,
-0.005886555649340153,
-0.03049554117023945,
0.07818415015935898,
0.019339077174663544,
-0.020754991099238396,
0.009634505026042461,
0.024578258395195007,
-0.03691175580024719,
-0.01... | 0.830629 | 0.82274 | 0.007889 |
A question for a math student to solve from the subtopic Conformal mapping and its applications in Complex analysis is:
Find the conformal map which maps the upper half plane $\{z\in\mathbb{C}:\text{Im}(z)>0\}$ to the unit disk $\{z\in\mathbb{C}:\|z\|<1\}$, such that the map sends $i$ to the origin and $-i$ to the poi... | tiger_lab_math | [
0.026386577636003494,
0.03393940627574921,
-0.03754124417901039,
-0.040563542395830154,
0.034803543239831924,
0.00853087566792965,
0.008901580236852169,
0.03671654313802719,
0.037546269595623016,
0.00787923764437437,
0.019019223749637604,
0.015360020101070404,
-0.019565315917134285,
-0.004... | 0.820619 | 0.79806 | 0.022559 |
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 write a Python program to solve it. | tiger_lab_math | [
0.0017861020751297474,
0.015409119427204132,
-0.05165448784828186,
-0.022123681381344795,
0.03438212722539902,
-0.01995609514415264,
-0.026589583605527878,
0.032215677201747894,
0.05142836645245552,
-0.0014765729429200292,
0.010599439963698387,
0.052559588104486465,
-0.017315486446022987,
... | 0.842926 | 0.829236 | 0.01369 |
A function $f$ is defined by $f(z) = i\overline{z}$, where $i^2 = -1$ and $\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$? | tiger_lab_math | [
0.01135994028300047,
0.025066271424293518,
-0.06754650175571442,
-0.0351155661046505,
0.041066888719797134,
-0.015169228427112103,
-0.0037502110935747623,
0.03564540296792984,
0.01915411278605461,
0.0011873828480020165,
0.01584557257592678,
0.02028900943696499,
-0.028371956199407578,
-0.01... | 0.836777 | 0.83404 | 0.002737 |
The expression
\[a(b - c)^3 + b(c - a)^3 + c(a - b)^3\]can be factored into the form $(a - b)(b - c)(c - a) p(a,b,c),$ for some polynomial $p(a,b,c).$ Find $p(a,b,c).$ | tiger_lab_math | [
0.03380822390317917,
0.03545618802309036,
-0.054213136434555054,
-0.0230520311743021,
0.03359745070338249,
-0.025195660069584846,
-0.021072885021567345,
0.06853996962308884,
0.04819281026721001,
-0.0058385166339576244,
0.0015516687417402864,
0.04848453402519226,
-0.05203602463006973,
-0.03... | 0.830859 | 0.825154 | 0.005705 |
Let X be a compact orientable surface of genus 2 with two boundary components. Find the intersection number of the two boundary components of X using Poincaré duality. | tiger_lab_math | [
-0.006068064831197262,
0.014140132814645767,
-0.03486061841249466,
-0.036860305815935135,
0.009851278737187386,
0.004651676397770643,
-0.020545322448015213,
0.03800717741250992,
0.046713605523109436,
-0.0012922434834763408,
0.024123547598719597,
0.026538060978055,
-0.019950084388256073,
-0... | 0.822727 | 0.809654 | 0.013073 |
Let A = {1,2} and B = {2,3}. Find the following sets:
1) (A x B) ∩ B
2) A x (B ∩ A)
3) (B ∩ A) x A
Verify whether the Cartesian product is associative with respect to intersection for these sets or not. | tiger_lab_math | [
0.07362273335456848,
0.021138975396752357,
-0.03116718865931034,
-0.0071497890166938305,
0.02137639746069908,
-0.019637655466794968,
0.00010000017209677026,
0.049457766115665436,
0.05057380720973015,
-0.004098409321159124,
0.021338796243071556,
0.01489394623786211,
-0.01857493631541729,
-0... | 0.832388 | 0.814838 | 0.01755 |
Let
\[f(a,b) = \left\{
\renewcommand{\arraystretch}{3}
\begin{array}{cl}
\dfrac{ab - a + 2}{2a} & \text{if $a + b \le 3$}, \\
\dfrac{ab - b - 2}{-2b} & \text{if $a + b > 3$}.
\end{array}
\renewcommand{\arraystretch}{1}
\right.\]Find $f(2,1) + f(2,4).$ Please write a program to solve it | tiger_lab_math | [
0.03385968878865242,
0.039258237928152084,
-0.04000897333025932,
-0.0501781664788723,
0.04147017374634743,
-0.0016370350494980812,
-0.02461569383740425,
0.0747726634144783,
0.019428158178925514,
0.002097640186548233,
0.02407727763056755,
0.046013325452804565,
-0.016977913677692413,
-0.0221... | 0.830929 | 0.821861 | 0.009068 |
Consider the elliptic curve defined by the equation $y^2 = x^3 + 2x + 1$ over the field $\mathbb{F}_{7}$.
Find the number of points on this elliptic curve over $\mathbb{F}_{7}$, i.e., the order of the group $\text{E}(\mathbb{F}_{7})$. Then use the Riemann-Roch theorem to determine the dimension of the space of regula... | tiger_lab_math | [
0.005089841317385435,
0.03482850268483162,
-0.03602663427591324,
-0.06259134411811829,
0.01543413009494543,
-0.023164570331573486,
-0.022170254960656166,
0.035178959369659424,
0.033202800899744034,
-0.029315195977687836,
-0.01430354081094265,
0.04895240068435669,
-0.048662543296813965,
-0.... | 0.835067 | 0.809498 | 0.025569 |
A closed surface S has a total Gaussian curvature of 3π. Furthermore, the principal curvatures at each point on this surface are given by k1 = 2 and k2 = 1. What is the Euler characteristic of the surface S? | tiger_lab_math | [
-0.007335663307458162,
0.0286470465362072,
-0.0444134846329689,
-0.06267102062702179,
0.04225419834256172,
-0.0019955646712332964,
-0.013076395727694035,
0.059721216559410095,
0.01653546839952469,
0.0008240244933404028,
0.02328745648264885,
0.0494673065841198,
-0.02210666611790657,
-0.0017... | 0.815886 | 0.810928 | 0.004958 |
You are interviewing two investment managers. Mr. Wong shows that the average return on his portfolio for the past 10 years has been 14%, with a standard deviation of 8% and a beta of 1.2. Ms. Petrov shows that the average return on her portfolio for the past 10 years has been 16%, with a standard deviation of 10% and ... | tiger_lab_math | [
-0.0010347168426960707,
0.028061222285032272,
-0.05909493565559387,
-0.037339821457862854,
0.03406520560383797,
-0.029395921155810356,
-0.0073159364983439445,
0.06407236307859421,
0.03259529918432236,
-0.01811659149825573,
0.010044384747743607,
0.022250426933169365,
-0.02636028453707695,
-... | 0.811767 | 0.80504 | 0.006727 |
A predator-prey model can be described by the differential equations:
dx/dt = 2x - 0.1xy
dy/dt = -0.3y + 0.002xy
where x represents the population of prey and y represents the population of predators.
If initially there are 1000 prey and 100 predators:
a) Find the solution for x(t) and y(t) using numerical methods... | tiger_lab_math | [
0.005535493139177561,
0.007759239058941603,
-0.036059457808732986,
-0.03823436424136162,
0.03234502300620079,
-0.03275139257311821,
-0.04864359647035599,
0.08840987086296082,
0.040910154581069946,
-0.008340395987033844,
0.006697396747767925,
0.040011025965213776,
-0.04791850969195366,
-0.0... | 0.799296 | 0.797457 | 0.00184 |
A math student is planning to buy a new phone. The phone comes in 3 different colors - black, white, and silver. The student has a budget of $500 and wants to buy at least 2 phones, but not more than 4 phones. How many ways can the student buy the phones if each phone costs $150? Use generating functions to solve the p... | tiger_lab_math | [
0.015505671501159668,
0.006857700180262327,
-0.057086531072854996,
-0.04854612797498703,
0.03472070023417473,
-0.010605312883853912,
-0.03645487502217293,
0.06235736981034279,
0.040754321962594986,
-0.014359886758029461,
0.018225478008389473,
0.04456251114606857,
-0.023564541712403297,
-0.... | 0.833437 | 0.827649 | 0.005789 |
Given the algebraic curve defined by the equation $y^2=x^3-4x^2$, find the singular points of the curve and determine their types. | tiger_lab_math | [
0.01056209858506918,
0.03108399361371994,
-0.04527241736650467,
-0.047566089779138565,
0.03411274403333664,
-0.03818770870566368,
0.0025067334063351154,
0.06908605992794037,
0.05408239737153053,
-0.01888275519013405,
0.010449113324284554,
0.027941610664129257,
-0.018659180030226707,
-0.004... | 0.824874 | 0.812671 | 0.012202 |
Suppose that $f(z) = z^3 + 2z^2 + 3z + 4$ and $g(z) = 2z^2 + 5z + 1$ are complex-valued functions defined on the closed disc $D(0,4)$. Find the maximum and minimum values of $|f(z) / g(z)|$ on the boundary of $D(0,4)$. | tiger_lab_math | [
0.02249656617641449,
0.034033335745334625,
-0.05726160854101181,
-0.0360490120947361,
0.05343356728553772,
-0.0036580658052116632,
-0.0062631224282085896,
0.04949839413166046,
0.046371035277843475,
0.016587505117058754,
0.027038943022489548,
0.0228499174118042,
-0.02439090795814991,
-0.015... | 0.833855 | 0.818453 | 0.015403 |
If $\det \mathbf{A} = 5,$ then find $\det (\mathbf{A^3}).$ | tiger_lab_math | [
0.05487951636314392,
0.017890846356749535,
-0.04470827430486679,
-0.028239170089364052,
0.045534152537584305,
-0.02252206578850746,
-0.02062246762216091,
0.0647304505109787,
0.033864159137010574,
-0.008399732410907745,
0.017215358093380928,
0.01471225917339325,
-0.04882790893316269,
-0.007... | 0.819994 | 0.81791 | 0.002084 |
Sure, here is a numerical problem for a math student in Statistics, Monte Carlo simulation:
Suppose a company called XYZ wants to estimate the probability of having at least 5 customers in their store during a given hour. The average number of customers they usually receive during that hour is 4. To estimate this prob... | tiger_lab_math | [
0.02237052470445633,
0.001435269950889051,
-0.03330555185675621,
-0.040835365653038025,
0.032741766422986984,
-0.012542358599603176,
-0.04663711413741112,
0.03831380605697632,
0.03855867683887482,
-0.010650659911334515,
0.01620345376431942,
0.029556360095739365,
-0.02571459300816059,
-0.02... | 0.82763 | 0.819899 | 0.007731 |
Jessica was trying to win a gift card to her favorite store. To win, she had to guess the total number of red & white jelly beans in the bowl of mixed color jelly beans. She figured it would take three bags of jelly beans to fill up the fishbowl. She assumed that each bag of jellybeans had a similar distribution of c... | tiger_lab_math | [
-0.009030339308083057,
0.02425856702029705,
-0.05189165100455284,
-0.029465125873684883,
0.031770121306180954,
-0.030418217182159424,
-0.055753983557224274,
0.056931931525468826,
0.028132325038313866,
-0.01448861788958311,
0.01180860586464405,
0.02488371543586254,
-0.030695950612425804,
-0... | 0.806074 | 0.801831 | 0.004243 |
Consider the trefoil knot, which is a knot that looks like three intertwined loops as shown in this image: https://en.wikipedia.org/wiki/Trefoil_knot#/media/File:Trefoil_knot_construction.gif
Suppose you have a piece of rope that is 2 meters long and you want to tie it into a trefoil knot shape. What is the minimum di... | tiger_lab_math | [
0.012253888882696629,
0.03327099606394768,
-0.03151090815663338,
-0.05166267976164818,
0.020585108548402786,
-0.026681842282414436,
-0.027916189283132553,
0.042079925537109375,
-0.0009125145734287798,
-0.003924521151930094,
0.012692908756434917,
0.022502917796373367,
-0.025969071313738823,
... | 0.814133 | 0.811834 | 0.002299 |
the megatek corporation is displaying its distribution of employees by department in a circle graph . the size of each sector of the graph representing a department is proportional to the percentage of total employees in that department . if the section of the circle graph representing the manufacturing department take... | tiger_lab_math | [
-0.0024087969213724136,
0.01316246297210455,
-0.06024736538529396,
-0.039463989436626434,
0.03958887979388237,
-0.027235757559537888,
-0.037147071212530136,
0.0341908298432827,
0.03938305005431175,
0.000788428820669651,
0.01915910467505455,
0.02130253240466118,
-0.0040063196793198586,
-0.0... | 0.815645 | 0.814527 | 0.001118 |
Let X be a topological space defined by the closed interval [0, 1]. Suppose f is a continuous function from X to the real numbers R. If f(0) = 5 and f(1) = 10, find the value of c in the interval [0, 1] such that f(c) = 8. | tiger_lab_math | [
0.030428718775510788,
0.035728126764297485,
-0.04238160699605942,
-0.021694617345929146,
0.03659705072641373,
-0.008061171509325504,
-0.02547590620815754,
0.07496264576911926,
0.023009950295090675,
-0.0103178471326828,
0.003942738752812147,
0.03207423537969589,
-0.04933866113424301,
-0.001... | 0.831976 | 0.818552 | 0.013424 |
Consider the two-dimensional torus $T^2$, i.e., a doughnut-like object with a hole in the middle. Suppose we impose a Morse function $f:T^2\rightarrow\mathbb{R}$ that has exactly two critical points: one minimum and one maximum, with critical values $f(p_1)=0$ and $f(p_2)=1$, respectively.
Calculate the Morse homolog... | tiger_lab_math | [
0.03335755690932274,
0.03752671554684639,
-0.020668601617217064,
-0.036229103803634644,
0.020130008459091187,
-0.004254300147294998,
-0.03682074695825577,
0.055890511721372604,
0.004007362294942141,
0.005773537792265415,
0.002613070886582136,
0.036434102803468704,
-0.011998617090284824,
-0... | 0.831728 | 0.804301 | 0.027427 |
Consider the projective variety given by the equation $xy - z^2 = 0$ in the projective space $\mathbb{P}^2(\mathbb{C})$. What are the coordinates of the three intersection points of this variety with the line given by the equations $x = 0$ and $y = 0$? | tiger_lab_math | [
0.01409667544066906,
0.02647152915596962,
-0.04201836511492729,
-0.03923758864402771,
0.025797855108976364,
-0.012186969630420208,
0.0015351466136053205,
0.04329648241400719,
0.05275727063417435,
-0.02763616107404232,
0.007360971067100763,
0.03277665376663208,
-0.021816134452819824,
-0.036... | 0.836505 | 0.82091 | 0.015595 |
Two players play the game of matching pennies. Each player simultaneously flips a penny and hides the result. If the pennies match (i.e., both heads or both tails), player 1 gets $1 from player 2. If the pennies do not match, player 2 gets $1 from player 1.
If player 1 decides to always flip a head, what is the best ... | tiger_lab_math | [
-0.004758227616548538,
0.01716991327702999,
-0.046235423535108566,
-0.019772207364439964,
0.005833081901073456,
-0.007240590173751116,
-0.0061703515239059925,
0.03890536352992058,
0.037206534296274185,
-0.04834771156311035,
0.02170705422759056,
0.02570871263742447,
-0.02611931785941124,
-0... | 0.811165 | 0.800761 | 0.010404 |
Suppose f(z) is analytic in the annulus $1<|z|<2$ with poles at z=2 and z=3. Also, assume that Res(f,2) = 3 and Res(f,3) = -4/3. Calculate the value of the following integral:
$\int_{|z|=5/2}\frac{z^5}{f(z)}dz$ | tiger_lab_math | [
0.025519875809550285,
0.04392224922776222,
-0.049407340586185455,
-0.03952082619071007,
0.03162522241473198,
0.012370498850941658,
-0.00981322955340147,
0.04912075027823448,
0.02446919120848179,
0.019652126356959343,
0.01622960716485977,
0.019381560385227203,
-0.02830745466053486,
-0.01141... | 0.825283 | 0.816824 | 0.008459 |
A fluid has a dynamic viscosity of 0.01 Pa.s and a density of 1000 kg/m³. The flow of the fluid is modeled by the Navier-Stokes equations. If the flow velocity at a point in the fluid is given by u = 0.1 sin(πy/L) m/s, where y is the vertical distance from the bottom of a channel and L is the height of the channel, fin... | tiger_lab_math | [
0.008340778760612011,
0.02747271955013275,
-0.03301122039556503,
-0.022738365456461906,
0.034928098320961,
-0.035328011959791183,
-0.03798960521817207,
0.07312119007110596,
0.009363151155412197,
0.0007387098739854991,
0.020452970638871193,
0.03511633351445198,
-0.02979114279150963,
-0.0275... | 0.795622 | 0.790963 | 0.00466 |
Suppose two individuals, A and B, are offered the same job that pays a salary of $100,000. Each individual has a choice between a high-effort or a low-effort level of work. If both individuals choose the high-effort level, the total output is $200,000. If both individuals choose the low-effort level, the total output i... | tiger_lab_math | [
-0.006505115889012814,
0.005074997898191214,
-0.05676429718732834,
-0.017164787277579308,
0.02324441261589527,
0.0056115989573299885,
-0.0156411100178957,
0.06904633343219757,
0.037748053669929504,
-0.02755879983305931,
0.02501511014997959,
0.012735930271446705,
-0.017002269625663757,
-0.0... | 0.807664 | 0.799497 | 0.008167 |
Let's say we have a differential equation given by:
𝑑𝑦/𝑑𝑥 = 𝑥/𝑦
with the initial condition 𝑦(1) = 2.
Determine the value of 𝑦 when 𝑥 = 3 using the optimal control approach. | tiger_lab_math | [
0.014514503069221973,
0.007611126638948917,
-0.05543294548988342,
-0.04460805654525757,
0.020177703350782394,
-0.021229272708296776,
-0.026195913553237915,
0.07503688335418701,
0.05926850810647011,
-0.015629196539521217,
0.008261973969638348,
0.023613637313246727,
-0.033584415912628174,
-0... | 0.820148 | 0.813951 | 0.006197 |
two sets of 7 consecutive positive integers have exactly one integer in common . the sum of the integers in the set with greater numbers is how much greater than the sum of the integers in the other set ? Please respond by writing a program in Python. | tiger_lab_math | [
0.026549868285655975,
0.020164132118225098,
-0.03890799358487129,
-0.042034223675727844,
0.039202980697155,
-0.019507743418216705,
-0.024059046059846878,
0.03532308340072632,
0.05056537687778473,
-0.04151042178273201,
0.017286060377955437,
0.022492028772830963,
-0.04865538701415062,
-0.011... | 0.829527 | 0.826983 | 0.002544 |
Consider the surface S given by the equation:
z = x^3 + y^3
Find the mean curvature of the surface S at the point (-1, 1, -2) using the formula K = (H1 + H2)/2, where H1 and H2 are the principal curvatures. | tiger_lab_math | [
0.003741743741557002,
0.026237722486257553,
-0.06314709037542343,
-0.06167052313685417,
0.04218835383653641,
-0.015285255387425423,
-0.012697008438408375,
0.06247580796480179,
0.012753032147884369,
-0.008718456141650677,
0.027777675539255142,
0.03218116611242294,
-0.03878702223300934,
-0.0... | 0.815662 | 0.812787 | 0.002875 |
A roll of quarters contains $40$ quarters and a roll of dimes contains $50$ dimes. James has a jar that contains $83$ quarters and $159$ dimes. Lindsay has a jar that contains $129$ quarters and $266$ dimes. James and Lindsay pool these quarters and dimes and make complete rolls with as many of the coins as possible. ... | tiger_lab_math | [
0.011334296315908432,
0.03588298708200455,
-0.03247187286615372,
-0.02474418841302395,
0.02004820667207241,
-0.024748625233769417,
-0.013136194087564945,
0.049077779054641724,
0.04248615354299545,
-0.021695218980312347,
0.021163135766983032,
0.026309354230761528,
-0.04399597644805908,
-0.0... | 0.83422 | 0.831164 | 0.003056 |
Let A = {2, 4, 6, 8} be a set. Determine whether the empty set (Ø) is a subset of A or not. | tiger_lab_math | [
0.02916182018816471,
0.019990835338830948,
-0.024566076695919037,
-0.028893960639834404,
0.048637572675943375,
-0.03258240595459938,
-0.0033103066962212324,
0.055631425231695175,
0.04109526053071022,
-0.01726577617228031,
0.012989013455808163,
0.0104052210226655,
-0.022619126364588737,
-0.... | 0.825188 | 0.824291 | 0.000897 |
Consider a sealed-bid auction where two bidders, A and B, are interested in buying a painting. Bidder A values the painting at $800 and bidder B values it at $500. The auction is a second-price sealed-bid auction, which means that the highest bidder wins the painting but only pays the second-highest bid. Assume that bo... | tiger_lab_math | [
-0.004144039936363697,
0.014649257063865662,
-0.03951649367809296,
-0.005201089195907116,
0.04177378863096237,
-0.007785972207784653,
-0.03354550525546074,
0.054951347410678864,
0.05400833860039711,
-0.010224252939224243,
0.03591043874621391,
0.028795000165700912,
-0.03265257179737091,
-0.... | 0.788118 | 0.783214 | 0.004904 |
Sure, here's a specific problem for a student in control theory and optimal control:
Suppose we have a simple linear system given by:
dx/dt = ax + bu
where a and b are constants. We want to design a feedback control law of the form u = -kx, where k is a constant, that minimizes the cost function:
J = ∫[0, T] (x² + ... | tiger_lab_math | [
0.003352249041199684,
0.02459542825818062,
-0.0583757609128952,
-0.044147636741399765,
0.03345635160803795,
-0.02634471282362938,
-0.030377009883522987,
0.05637305602431297,
0.03911714255809784,
-0.011624105274677277,
0.020054863765835762,
0.012876015156507492,
-0.031475577503442764,
-0.02... | 0.817777 | 0.813433 | 0.004344 |
There is a committee composed of eight women and two men. When they meet, they sit in a row---the women in indistinguishable rocking chairs and the men on indistinguishable stools. How many distinct ways are there for me to arrange the eight chairs and two stools for a meeting? Let's write a Python program to solve it... | tiger_lab_math | [
0.022187797352671623,
0.012417600490152836,
-0.050684988498687744,
-0.007810747716575861,
0.033012766391038895,
-0.025728870183229446,
-0.01963290013372898,
0.03545762598514557,
0.03987298533320427,
-0.020586183294653893,
0.014030691236257553,
0.021861324086785316,
-0.027728311717510223,
-... | 0.82319 | 0.81803 | 0.00516 |
A math student needs to determine if two players, Player A and Player B, know that each of them has an integer between 1 and 5 inclusive. Player A knows the sum of the two integers, and Player B knows the product of the two integers. They have the following conversation:
Player A: "I don't know what your number is."
P... | tiger_lab_math | [
0.04144427552819252,
0.029762962833046913,
-0.06124008819460869,
-0.03345530107617378,
0.0348811149597168,
-0.008800942450761795,
0.002863470232114196,
0.04976082220673561,
0.04064349830150604,
-0.04153219610452652,
0.006592265330255032,
0.04505639895796776,
-0.01749752089381218,
-0.034030... | 0.837313 | 0.824868 | 0.012445 |
Two friends, Alex and Taylor, are trying to decide whether to watch a romantic movie or a comedy movie. Alex wants to watch the romantic movie, while Taylor wants to watch the comedy movie. They assign points to each movie as follows:
- If they both choose the romantic movie, Alex gets 4 points and Taylor gets 3 point... | tiger_lab_math | [
0.0280505008995533,
0.007403760217130184,
-0.048174817115068436,
-0.027658669278025627,
0.0341549851000309,
-0.024914491921663284,
-0.016156211495399475,
0.053184252232313156,
0.028436491265892982,
-0.01948014833033085,
0.008960185572504997,
0.009413053281605244,
-0.0358365960419178,
-0.00... | 0.820684 | 0.811579 | 0.009105 |
Consider the following undirected graph with 6 vertices and 7 edges:
```
1
/ \
/ \
2 3
/ \ / \
4 5 6 7
```
Find the automorphism group of this graph, i.e., the set of all permutations of the vertices that preserve the adjacency relation of the graph. | tiger_lab_math | [
0.03825581073760986,
0.026471156626939774,
-0.026480047032237053,
-0.022249331697821617,
0.03713969141244888,
-0.032694391906261444,
-0.00654068635776639,
0.05019170418381691,
0.035380780696868896,
-0.030858127400279045,
0.006305113434791565,
0.008624042384326458,
-0.018108535557985306,
-0... | 0.827388 | 0.810311 | 0.017077 |
A math student needs to count the number of ways 5 different books can be arranged on a shelf such that two specific books are never adjacent. How many unique arrangements are possible? | tiger_lab_math | [
0.048753295093774796,
0.0003637798654381186,
-0.048903290182352066,
-0.0383068323135376,
0.03446566313505173,
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-0.038518745452165604,
0.037235796451568604,
0.05491644889116287,
-0.026035238057374954,
-0.007728603668510914,
0.01559117529541254,
-0.02624323032796383,
-... | 0.830843 | 0.824876 | 0.005967 |
In Perfectville, the streets are all $20$ feet wide and the blocks they enclose are all squares of side length $400$ feet, as shown. Sarah runs around the block on the $400$-foot side of the street, while Sam runs on the opposite side of the street. How many more feet than Sarah does Sam run for every lap around the ... | tiger_lab_math | [
0.0189922247081995,
0.05033808574080467,
-0.04701249673962593,
-0.018470482900738716,
0.04779171198606491,
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0.052330683916807175,
0.03798716515302658,
-0.015537496656179428,
0.010610794648528099,
0.03244917094707489,
-0.028911899775266647,
-0.019... | 0.838907 | 0.829119 | 0.009788 |
A is a set containing 8 different positive odd numbers. B is a set containing 7 different numbers, all of which are members of A. Which of the following statements CANNOT be true?
Answer Choices: (A) The range of B is even. (B) The mean of A is even. (C) The mean of B is even. (D) The range of A is equal to the range o... | tiger_lab_math | [
0.05066624656319618,
0.010350236669182777,
-0.03795196861028671,
-0.03884555771946907,
0.042818114161491394,
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-0.026369882747530937,
0.06043346971273422,
0.05687115713953972,
-0.03657902032136917,
0.00006692232273053378,
0.02403596229851246,
-0.03172749653458595,
-0.037... | 0.826651 | 0.820571 | 0.00608 |
Suppose two players, Alice and Bob, are about to play a trust game. Alice starts with $10 and decides how much to send to Bob, who receives triple the amount Alice sends. Bob then decides how much of the received amount to return to Alice.
If Alice sends $4 to Bob, how much should Bob send back to maximize the total a... | tiger_lab_math | [
0.01858283393085003,
0.010194124653935432,
-0.044324662536382675,
-0.02088472619652748,
0.025645287707448006,
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-0.005020231939852238,
0.041134558618068695,
0.03833422809839249,
-0.031387753784656525,
0.03595656901597977,
0.03620737046003342,
-0.0482792891561985,
-0.02... | 0.814476 | 0.809354 | 0.005122 |
suppose you have three identical prisms with congruent equilateral triangles as the end - polygons . suppose you attach them by the rectangular faces so they are perfectly aligned . there will be some large faces created by two or more co - planar faces of the individual prisms : count each such large face as one . giv... | tiger_lab_math | [
0.03790806978940964,
0.03041231445968151,
-0.04417944699525833,
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0.022139018401503563,
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0.07549844682216644,
0.016270682215690613,
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0.013608311302959919,
0.05344674363732338,
-0.024183809757232666,
-0.0... | 0.837333 | 0.826164 | 0.011169 |
Let M be a 2-dimensional compact manifold with genus 2. Determine the Euler characteristic of M. | tiger_lab_math | [
0.0013513860758394003,
0.028135430067777634,
-0.012485221959650517,
-0.055288199335336685,
0.03607883304357529,
-0.013255968689918518,
-0.02504703961312771,
0.06324375420808792,
0.036919303238391876,
0.0004917753976769745,
-0.0002100974670611322,
0.04389166459441185,
-0.021212512627243996,
... | 0.81905 | 0.815885 | 0.003166 |
Christine and Janice are having a contest to see who can throw a ball the highest. They each get three throws. On her first throw, Christine throws it 20 feet high. Janice's throw is 4 feet lower than Christine's. On their second throw, Christine throws it ten feet higher than her first throw and Janice throws it twice... | tiger_lab_math | [
0.0005559829296544194,
0.025765391066670418,
-0.04810154065489769,
-0.039454344660043716,
0.039575863629579544,
-0.015131169930100441,
-0.02530018799006939,
0.07106807827949524,
0.04339296743273735,
-0.009463776834309101,
0.021939029917120934,
0.02900964766740799,
-0.04723469540476799,
-0.... | 0.816117 | 0.813783 | 0.002334 |
What is the maximum number of independent vertices that can be selected from the following graph?
```
A --- B
/|\ |
/ | \ |
C--D--E--F
|
|
G
```
(Note: "Independent" means that the selected vertices cannot be adjacent, i.e., they are not connected by an edge.) | tiger_lab_math | [
0.04491344094276428,
0.01484761480242014,
-0.033529192209243774,
-0.01843172125518322,
0.025815928354859352,
-0.04576397314667702,
-0.03463152423501015,
0.03431679680943489,
0.030892767012119293,
-0.013384515419602394,
0.025296218693256378,
0.026699677109718323,
-0.014773422852158546,
-0.0... | 0.817988 | 0.815853 | 0.002135 |
Regions I, II and III are bounded by squares. The perimeter of region I is 12 units and the perimeter of region II is 24 units. What is the ratio of the area of region I to the area of region III? Express your answer as a common fraction.
[asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0)--cycle,linewidth(2));
draw((9,0)--(... | tiger_lab_math | [
0.043318379670381546,
0.02430613525211811,
-0.05681910365819931,
-0.037093956023454666,
0.020519649609923363,
-0.024779461324214935,
-0.013249277137219906,
0.07365679740905762,
0.02459675259888172,
0.0017662537284195423,
0.0214698426425457,
0.031596168875694275,
-0.026705335825681686,
-0.0... | 0.829523 | 0.823858 | 0.005665 |
Consider the wave equation:
$$u_{tt} = c^2 u_{xx}$$
where $u(x,t)$ represents the displacement of a string at position $x$ and time $t$, and $c$ is the wave speed.
Suppose the string is fixed at both ends ($u(0,t) = u(L,t) = 0$), and the initial displacement and velocity are given by:
$$u(x, 0) = \sin\left(\fra... | tiger_lab_math | [
0.004930615890771151,
0.046555306762456894,
-0.0451885387301445,
-0.025873759761452675,
0.0468931645154953,
-0.026559846475720406,
-0.029173079878091812,
0.08847561478614807,
0.033359624445438385,
0.00464964983984828,
0.03492879122495651,
0.008153428323566914,
-0.03078318201005459,
-0.0302... | 0.816191 | 0.798268 | 0.017923 |
In the diagram, $AB$ and $CD$ are straight lines. What is the value of $x?$ [asy]
draw((0,0)--(12,0));
draw((0,5)--(12,5));
draw((3,0)--(5,5)--(9,0));
label("$60^\circ$",(5,4.5),W);
label("$50^\circ$",(5.5,4.5),E);
label("$A$",(0,5),W);
label("$C$",(0,0),W);
label("$B$",(12,5),E);
label("$D$",(12,0),E);
label("$120^\ci... | tiger_lab_math | [
0.008648833259940147,
0.003795880824327469,
-0.05008332431316376,
-0.03931935131549835,
0.04196684807538986,
-0.05245911329984665,
-0.02262921631336212,
0.05874277278780937,
0.056931380182504654,
0.002299319952726364,
0.012997493147850037,
0.028809113427996635,
-0.006979854311794043,
-0.02... | 0.835289 | 0.833379 | 0.001911 |
In this diagram, both polygons are regular. What is the value, in degrees, of the sum of the measures of angles $ABC$ and $ABD$?
[asy]
draw(10dir(0)--10dir(60)--10dir(120)--10dir(180)--10dir(240)--10dir(300)--10dir(360)--cycle,linewidth(2));
draw(10dir(240)--10dir(300)--10dir(300)+(0,-10)--10dir(240)+(0,-10)--10dir(24... | tiger_lab_math | [
0.005488860420882702,
0.00562818581238389,
-0.03939685970544815,
-0.039599280804395676,
0.026049038395285606,
-0.03217624872922897,
-0.02985931560397148,
0.04721555486321449,
0.03452521562576294,
-0.005047639831900597,
0.001376593834720552,
0.033753227442502975,
-0.021624021232128143,
-0.0... | 0.832087 | 0.826288 | 0.005798 |
Consider a set S = {2, 4, 6, 8, x, y} with distinct elements. If x and y are both prime numbers and 0 < x < 40 and 0 < y < 40, which of the following MUST be true?
I. The maximum possible range of the set is greater than 33.
II. The median can never be an even number.
III. If y = 37, the average of the set will be grea... | tiger_lab_math | [
0.0410546138882637,
0.021994946524500847,
-0.0442301370203495,
-0.029969658702611923,
0.027067972347140312,
-0.03127104789018631,
-0.018930448219180107,
0.07175521552562714,
0.0354759581387043,
-0.03103429079055786,
0.02679373323917389,
0.02688741497695446,
-0.044485487043857574,
-0.026950... | 0.835505 | 0.824932 | 0.010573 |
Consider the following graph G:
Does the graph G have a Hamiltonian cycle? If yes, provide one cycle that visits all of the vertices of the graph exactly once. If not, provide a brief explanation of why. | tiger_lab_math | [
-0.006153108086436987,
0.015162307769060135,
-0.019088173285126686,
-0.03601657971739769,
0.02505199797451496,
-0.055577728897333145,
-0.02110321633517742,
0.05569402873516083,
0.015135527588427067,
-0.01645432412624359,
0.027704767882823944,
0.01161267701536417,
-0.023332826793193817,
-0.... | 0.811849 | 0.796938 | 0.014911 |
The LCM of three numbers is four times their GCF. Which of the following must be true of the numbers?
I. At least one of the numbers is odd.
II. Two of the three numbers must be same.
III. At least one number is the same as GCF.
Answer Choices: (A) only (B) option II only (C) and III only (D) and II only (E) I and III ... | tiger_lab_math | [
-0.00219343532808125,
0.01596451923251152,
-0.026740606874227524,
-0.049122605472803116,
0.05391518771648407,
-0.008753892034292221,
-0.020046014338731766,
0.056073855608701706,
-0.002942995633929968,
-0.01918879896402359,
0.007270330563187599,
0.01928872987627983,
-0.02155567705631256,
-0... | 0.823596 | 0.822306 | 0.00129 |
you have a 6 - sided cube and 6 cans of paint , each a different color . you may not mix colors of paint . how many distinct ways can you paint the cube using a different color for each side ? ( if you can reorient a cube to look like another cube , then the two cubes are not distinct . ) ? Let's program in Python in t... | tiger_lab_math | [
0.018338551744818687,
0.03526022285223007,
-0.020016362890601158,
0.0046903956681489944,
0.028119370341300964,
-0.028160858899354935,
-0.019853655248880386,
0.04106874018907547,
0.03781285509467125,
-0.02116788923740387,
0.034042712301015854,
0.041824180632829666,
-0.012112992815673351,
-0... | 0.819307 | 0.811428 | 0.007879 |
A factory needs to determine the best combination of production levels for three products in order to maximize profits. The production of each product requires different amounts of labor, resources, and time. Using Particle Swarm Optimization, find the production levels that will maximize the profits of the factory, gi... | tiger_lab_math | [
0.018417762592434883,
0.004052636679261923,
-0.051826395094394684,
-0.007906120270490646,
0.015079707838594913,
-0.020676376298069954,
-0.05199723690748215,
0.08123727142810822,
0.02271999605000019,
0.0011783967493101954,
0.00587227288633585,
0.01607590541243553,
-0.015069959685206413,
-0.... | 0.780362 | 0.771377 | 0.008985 |
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,
-0.05673585459589958,
-0.03302033618092537,
0.025152046233415604,
-0.03334195911884308,
-0.045062191784381866,
0.056995272636413574,
0.03207092359662056,
-0.008327421732246876,
0.010420872829854488,
0.01865435391664505,
-0.04989265277981758,
-0.03... | 0.813352 | 0.812161 | 0.001191 |
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| tiger_lab_math | [
0.026079300791025162,
0.02514449879527092,
-0.03334717079997063,
-0.04456004127860069,
0.05127829685807228,
-0.012738160789012909,
-0.023618068546056747,
0.05420904979109764,
0.0480462871491909,
-0.004088965710252523,
0.010756878182291985,
0.023465652018785477,
-0.014228911139070988,
-0.03... | 0.844068 | 0.828928 | 0.01514 |
Compute the double integrals over indicated rectangles $\iint\limits_{R}{{2x - 4{y^3}\,dA}}$, $R = [-5,4] \times [0, 3] | tiger_lab_math | [
0.016989054158329964,
0.03576434403657913,
-0.023872466757893562,
-0.030991273000836372,
0.02428952045738697,
-0.005662343464791775,
-0.02197558805346489,
0.06554841250181198,
0.055220793932676315,
0.014907111413776875,
0.02008220925927162,
0.046099819242954254,
-0.024469949305057526,
-0.0... | 0.821195 | 0.81746 | 0.003735 |
Find the positive integer $n$ such that
$$\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.$$ | tiger_lab_math | [
0.030453501269221306,
0.048394251614809036,
-0.04208299145102501,
-0.043785687536001205,
0.03914114832878113,
-0.014778366312384605,
-0.02214791812002659,
0.03914007544517517,
0.03906690329313278,
0.009723375551402569,
0.01002220343798399,
0.0117173558101058,
-0.04089631512761116,
-0.02073... | 0.851159 | 0.839311 | 0.011848 |
Find all possible isomorphisms between the group of order 8 that you are given and C8, Q8, and D4, and explain why other groups of order 8 cannot be isomorphic to any of these. | tiger_lab_math | [
0.019928913563489914,
0.022403571754693985,
-0.010361452586948872,
-0.0411074124276638,
0.015824077650904655,
-0.04167138785123825,
-0.004631739109754562,
0.04283294454216957,
0.046432387083768845,
-0.016787763684988022,
0.02330896258354187,
-0.004876977298408747,
-0.03748847916722298,
-0.... | 0.809359 | 0.803957 | 0.005401 |
Sure, here's a numerical problem for a mathematics student related to the topic of Heat transfer and diffusion equations:
A metal rod of length 1 meter and diameter 2 cm initially has a temperature of 200 degrees Celsius. The surface of the rod is exposed to air at a constant temperature of 25 degrees Celsius. The the... | tiger_lab_math | [
0.02193031646311283,
0.0010671090567484498,
-0.040335897356271744,
-0.03717601299285889,
0.024783585220575333,
-0.05237622931599617,
-0.03806304186582565,
0.05525319650769234,
0.03123142011463642,
-0.010290328413248062,
0.008062831126153469,
0.033858656883239746,
-0.021939674392342567,
-0.... | 0.808241 | 0.805808 | 0.002433 |
Calculate the contour integral ∮C (z^2 + 2z - 1) dz, where C is the circle of radius 3 centered at the origin, using Cauchy's theorem. | tiger_lab_math | [
0.03370308130979538,
0.028212938457727432,
-0.030446087941527367,
-0.02784961275756359,
0.03836493194103241,
-0.004229552112519741,
-0.01648414321243763,
0.04151890426874161,
0.029062360525131226,
-0.008772781118750572,
0.03345518186688423,
0.02629968151450157,
-0.03758340701460838,
-0.015... | 0.816787 | 0.809957 | 0.006831 |
Suppose you own a company that produces two products: product A and product B. The profit gained from each unit of product A is $4, and each unit of product B is $6. Every day, your company can produce up to 150 units of product A and 200 units of product B. However, the production of both products requires a certain a... | tiger_lab_math | [
0.02126215398311615,
0.012594485655426979,
-0.04390857741236687,
-0.015233354642987251,
0.01024932786822319,
-0.023340782150626183,
-0.04835851117968559,
0.074764184653759,
0.025744153186678886,
-0.017001694068312645,
-0.0010338262654840946,
0.03076964244246483,
-0.013579332269728184,
-0.0... | 0.818297 | 0.802912 | 0.015384 |
Consider the mirror symmetry between a quintic threefold and a Calabi-Yau hypersurface. Let X be the quintic threefold in projective space P⁵ with equation:
X: x₁⁵ + x₂⁵ + x₃⁵ + x₄⁵ + x₅⁵ + t(x₁x₂x₃x₄x₅) = 0,
where t is a complex parameter. Let Y be the Calabi-Yau hypersurface of degree 5 in P⁴ with equation:
Y: y... | tiger_lab_math | [
0.009513087570667267,
0.02430790290236473,
-0.036972399801015854,
-0.06327404081821442,
0.046473901718854904,
-0.009857729077339172,
-0.020327549427747726,
0.06435778737068176,
0.04154595732688904,
-0.009387890808284283,
0.004767754580825567,
0.03393739089369774,
-0.030637549236416817,
-0.... | 0.81941 | 0.801872 | 0.017538 |
Sure, here's a problem that involves Integer programming:
A cake shop has 100 pounds of sugar, 80 pounds of flour, and 40 pounds of butter. They make two types of cake - vanilla cake and chocolate cake. To make one pound of vanilla cake, they need 0.5 pounds of sugar, 0.25 pounds of flour, and 0.15 pounds of butter. T... | tiger_lab_math | [
0.02896868996322155,
0.022152485325932503,
-0.03582100570201874,
-0.022891830652952194,
0.032182082533836365,
-0.012838589958846569,
-0.029263628646731377,
0.04962341487407684,
0.029501143842935562,
-0.025964280590415,
0.021505769342184067,
0.027249764651060104,
-0.02648344822227955,
-0.02... | 0.831659 | 0.823125 | 0.008534 |
Let X be a square with vertices (0,0), (1,0), (1,1), and (0,1) and let f:X → R be a Morse function such that:
1. f has two critical points, both of index 1.
2. One critical point is located at (1/2,1/2) and has function value 0.
3. The other critical point is located on the boundary of X and has function value 1.
Com... | tiger_lab_math | [
0.03305989131331444,
0.02821529284119606,
-0.03625451773405075,
-0.025747250765562057,
0.020896006375551224,
-0.010359132662415504,
-0.03743722662329674,
0.059490084648132324,
0.019924215972423553,
0.010553604923188686,
0.00641501322388649,
0.03540368378162384,
-0.022532885894179344,
-0.02... | 0.816195 | 0.804928 | 0.011267 |
Let the manifold be a sphere with radius 2 and let the function be f(x,y,z) = xyz. Find the Laplacian of f on the sphere. | tiger_lab_math | [
-0.0008949214243330061,
0.04144968092441559,
-0.043258436024188995,
-0.04497243091464043,
0.028091955929994583,
-0.004352452699095011,
-0.019686328247189522,
0.035147782415151596,
0.02626316249370575,
-0.01010893564671278,
0.019599368795752525,
0.037389516830444336,
-0.0381709523499012,
-0... | 0.823182 | 0.817978 | 0.005204 |
The positive difference between the two roots of the quadratic equation $3x^2 - 7x - 8 = 0$ can be written as $\frac{\sqrt{m}}{n}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$. Please write a program to solve it | tiger_lab_math | [
0.04419396072626114,
0.016416078433394432,
-0.045325592160224915,
-0.054866891354322433,
0.02899656444787979,
-0.035317011177539825,
-0.013173408806324005,
0.06658129394054413,
0.05052966624498367,
-0.04367036744952202,
0.011866838671267033,
0.03869810700416565,
-0.03311607986688614,
-0.02... | 0.838877 | 0.831214 | 0.007663 |
A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$. What is the probability of the arrow stopping on region $C$? Express your answer as a common fraction. | tiger_lab_math | [
0.021265145391225815,
0.007966472767293453,
-0.05608218163251877,
-0.027022987604141235,
0.039980482310056686,
-0.01778930425643921,
-0.024995107203722,
0.05742701515555382,
0.041367992758750916,
-0.009999257512390614,
0.011167939752340317,
0.015198673121631145,
-0.018993979319930077,
-0.0... | 0.828632 | 0.819508 | 0.009124 |
Find the Hodge polynomial of the complex projective variety defined by the equation $x^4 + y^4 + z^4$ = 0. | tiger_lab_math | [
0.009808167815208435,
0.032385874539613724,
-0.02282077632844448,
-0.054553233087062836,
0.03581398352980614,
-0.028705982491374016,
-0.007546198554337025,
0.037287209182977676,
0.051678817719221115,
-0.025397492572665215,
0.026683086529374123,
0.02367931418120861,
-0.010339248925447464,
-... | 0.824553 | 0.821792 | 0.002761 |
Let X be the projective plane curve defined by the equation $x^2+y^2+z^2=0$, and let C be the rational curve on X defined by the parameterization:
$$C: t \rightarrow [t^2-1:t^2+1:-2t]$$
Find the degree of C and the genus of X. | tiger_lab_math | [
0.009961085394024849,
0.030887771397829056,
-0.0446968674659729,
-0.03721172735095024,
0.03852749243378639,
-0.029363473877310753,
-0.015258059836924076,
0.0466344989836216,
0.03003351390361786,
-0.0430203452706337,
0.015533911064267159,
0.03722789138555527,
-0.03796614333987236,
-0.000245... | 0.836777 | 0.819208 | 0.017569 |
Consider a team of 5 students who are going to participate in a Math Olympiad competition. If they have to select 4 questions out of 10 different questions, how many different ways can they choose the questions to attempt? Use combinatorial optimization to find the optimal solution. | tiger_lab_math | [
0.024582523852586746,
-0.0048363348469138145,
-0.042495012283325195,
-0.027404606342315674,
0.041800130158662796,
-0.03174728527665138,
-0.03661014512181282,
0.031062310561537743,
0.03978589549660683,
-0.014471864327788353,
-0.013261216692626476,
0.013808004558086395,
-0.0314515084028244,
... | 0.823851 | 0.814696 | 0.009155 |
Let $n$ be the smallest positive integer that is a multiple of $75$ and has exactly $75$ positive integral divisors, including $1$ and itself. Find $\frac{n}{75}$.
| tiger_lab_math | [
0.030476994812488556,
0.03376638516783714,
-0.01943312957882881,
-0.04399413242936134,
0.020193513482809067,
-0.015206594951450825,
-0.009409039281308651,
0.03647325187921524,
0.028989510610699654,
-0.02253500558435917,
0.00855183694511652,
0.02093491144478321,
-0.040749046951532364,
-0.01... | 0.849679 | 0.839491 | 0.010188 |
If $x^2 + 2x + 5$ is a factor of $x^4 + Px^2 + Q,$ find $P + Q.$ | tiger_lab_math | [
0.015048365108668804,
0.031764935702085495,
-0.04517854377627373,
-0.035230834037065506,
0.03877502679824829,
-0.029344070702791214,
-0.0055687609128654,
0.06931297481060028,
0.045463453978300095,
-0.014636090025305748,
0.010366666130721569,
0.03190174326300621,
-0.043778374791145325,
-0.0... | 0.836456 | 0.835515 | 0.000941 |
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