instruction_seed stringlengths 56 2.52k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.77 0.86 | mean_negative_score float64 0.76 0.85 | difference_score float64 0 0.03 |
|---|---|---|---|---|---|
A space curve is given by the equation x=t, y=t^2, and z=t^3 for t between 0 and 1. Find the curvature and torsion of the curve at t=1/2. | tiger_lab_math | [
0.013603964820504189,
0.039945926517248154,
-0.03657764196395874,
-0.04386705160140991,
0.03712305799126625,
-0.008763750083744526,
-0.022347144782543182,
0.05852420628070831,
0.0178313460201025,
-0.029035285115242004,
0.020176438614726067,
0.012477788142859936,
-0.041759245097637177,
-0.0... | 0.821741 | 0.818511 | 0.00323 |
Consider a group G of order 20. Determine whether G is isomorphic to the cyclic group C20, the dihedral group D10, or the non-abelian group of order 20, and provide a detailed explanation for your conclusion using the properties of these groups. | tiger_lab_math | [
0.00038960203528404236,
0.01845489628612995,
-0.03709648177027702,
-0.03347045183181763,
0.018062395974993706,
-0.06749118119478226,
-0.014041409827768803,
0.05489092692732811,
0.03276628628373146,
-0.008172470144927502,
0.018825171515345573,
0.017616601660847664,
-0.053231656551361084,
0.... | 0.809951 | 0.802765 | 0.007186 |
What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect cube? | tiger_lab_math | [
0.015135631896555424,
0.036793164908885956,
-0.053955450654029846,
-0.04398483783006668,
0.03825593367218971,
-0.008107360452413559,
-0.02243921533226967,
0.04193824157118797,
0.03711964190006256,
-0.02601662650704384,
0.030726298689842224,
0.026603221893310547,
-0.032411959022283554,
-0.0... | 0.847267 | 0.84592 | 0.001347 |
A company produces two types of products X and Y using machines A and B, respectively. The production time for product X on machine A is 3 minutes and on machine B is 5 minutes. The production time for product Y on machine A is 2 minutes and on machine B is 4 minutes. The company has 240 minutes of machine A time and 4... | tiger_lab_math | [
0.03148835897445679,
0.008858219720423222,
-0.05686536431312561,
-0.001957985805347562,
0.011926566250622272,
-0.007706973701715469,
-0.04833775386214256,
0.09330641478300095,
0.03530774265527725,
-0.008941729553043842,
0.008921516127884388,
0.034782614558935165,
-0.009124426171183586,
-0.... | 0.815903 | 0.803406 | 0.012496 |
To pass the time while she is waiting somewhere Carla likes to count things around her. While she is waiting for school to start on Monday she counts the tiles on the ceiling--38. While she is waiting for everyone to finish their tests after she has handed in hers, she counts the books in the room--75. On Tuesday Carla... | tiger_lab_math | [
0.022493740543723106,
0.014183693565428257,
-0.05149085074663162,
-0.04332590848207474,
0.04027549549937248,
-0.016707362607121468,
-0.039901264011859894,
0.06675634533166885,
0.01411000732332468,
-0.006747794337570667,
0.01141350343823433,
0.045280732214450836,
-0.0670386478304863,
-0.026... | 0.817187 | 0.814275 | 0.002913 |
Line $m$ is parallel to line $n$ and the measure of $\angle 1$ is $\frac 18$ the measure of $\angle 2$. What is the degree measure of $\angle 5$? [asy] size(100); defaultpen(linewidth(0.7)+fontsize(9));
path m = (-1.35,0.72)--(0.45,0.72), n = (-1,0)--(1,0), k = (-0.67,1.09)--(0.27,-0.48);
pair A = intersectionpoints(m... | tiger_lab_math | [
0.017646508291363716,
0.01725897751748562,
-0.04631759971380234,
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0.03952441364526749,
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0.044881727546453476,
0.030974190682172775,
0.005197480320930481,
0.012415427714586258,
0.018948551267385483,
-0.012120971456170082,
-0... | 0.826341 | 0.817417 | 0.008924 |
Consider the following quadratic programming problem:
\begin{align*}
\text{Minimize } &f(x_1, x_2) = 4x_1^2 + 5x_2^2 -2x_1x_2 -5x_1 -7x_2\\
\text{subject to } &x_1 + 2x_2 \geq 1\\
&x_1, x_2 \geq 0
\end{align*}
Use the KKT conditions to find the optimal values of $x_1$ and $x_2$ that minimize the function $f(x_1, x_2)... | tiger_lab_math | [
0.008896741084754467,
0.039217207580804825,
-0.06173459812998772,
-0.04568558558821678,
0.04759322106838226,
-0.013821682892739773,
-0.008539946749806404,
0.0717674121260643,
0.03223598003387451,
-0.006402279250323772,
0.0076477122493088245,
0.011527180671691895,
-0.02423759736120701,
0.02... | 0.825118 | 0.809594 | 0.015524 |
The points $(1, 7), (13, 16)$ and $(5, k)$, where $k$ is an integer, are vertices of a triangle. What is the sum of the values of $k$ for which the area of the triangle is a minimum? | tiger_lab_math | [
0.023768046870827675,
0.033919807523489,
-0.04948128014802933,
-0.03471064195036888,
0.05068105086684227,
-0.02218572050333023,
-0.007338294293731451,
0.03966573625802994,
0.051554225385189056,
-0.011044543236494064,
0.025360841304063797,
0.03926884010434151,
-0.014428640715777874,
-0.0167... | 0.847704 | 0.840488 | 0.007216 |
Find the Fourier transform of the function f(x) = x, defined in the range -π ≤ x ≤ π and use it to solve the boundary value problem:
y'' + y = x, -π ≤ x ≤ π,
y(-π) = y(π),
y'(-π) = y'(π). | tiger_lab_math | [
0.010611708275973797,
0.011849276721477509,
-0.043122682720422745,
-0.026234641671180725,
0.03183192014694214,
-0.02081466093659401,
-0.013742239214479923,
0.05147060751914978,
0.05792241171002388,
0.01130494475364685,
0.012463703751564026,
0.03939010947942734,
-0.0525965616106987,
-0.0167... | 0.820527 | 0.818881 | 0.001646 |
Let $f(z)=z^3 - 2z + 1$ be a complex-valued function in the closed unit disc $\{z: |z|\leq 1\}$. Find the maximum and minimum values of $|f(z)|$ for $z\in \{z: |z|= 1\}$. | tiger_lab_math | [
0.029565762728452682,
0.024085573852062225,
-0.04275790601968765,
-0.03685806319117546,
0.048278454691171646,
0.0018769917078316212,
-0.011158362030982971,
0.04857112467288971,
0.03445172309875488,
0.010411362163722515,
0.03419101610779762,
0.02996172569692135,
-0.03146151453256607,
-0.014... | 0.824456 | 0.807946 | 0.01651 |
Find constants $A,$ $B,$ and $C$ so that
\[\frac{4x}{(x - 5)(x - 3)^2} = \frac{A}{x - 5} + \frac{B}{x - 3} + \frac{C}{(x - 3)^2}.\]Enter the ordered triple $(A,B,C).$ | tiger_lab_math | [
0.04494282975792885,
0.029617024585604668,
-0.04730623960494995,
-0.028072578832507133,
0.041009604930877686,
-0.026664791628718376,
-0.013221317902207375,
0.0775034949183464,
0.02479017898440361,
-0.005780022591352463,
0.0008075578953139484,
0.030624693259596825,
-0.02222755178809166,
-0.... | 0.830856 | 0.823293 | 0.007562 |
Sure, here's a question for a math student to solve related to the Beauty Contest Game in Game Theory:
In a Beauty Contest Game where 100 players are asked to pick a number between 0 and 100, if the winning number is defined as two-thirds of the average of all chosen numbers, what would be the winning number in this g... | tiger_lab_math | [
0.024046193808317184,
0.004101315047591925,
-0.0538712739944458,
-0.028597593307495117,
0.024784957990050316,
-0.028949888423085213,
-0.0497114323079586,
0.020706163719296455,
0.01754327118396759,
-0.031844813376665115,
0.0078304847702384,
0.022529570385813713,
-0.0418781116604805,
-0.0324... | 0.82946 | 0.824084 | 0.005376 |
Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\},$ the second from $\{A,I,O\},$ and the third from $\{D,M,N,T\}.$
When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added... | tiger_lab_math | [
0.02593197487294674,
0.004596425220370293,
-0.021872522309422493,
-0.02096281386911869,
0.03784632310271263,
-0.019983477890491486,
-0.020538408309221268,
0.058178842067718506,
0.05058293417096138,
-0.025898903608322144,
-0.006999708246439695,
0.03324073925614357,
-0.044664494693279266,
-0... | 0.830102 | 0.817419 | 0.012683 |
the visitors of a modern art museum who watched a certain picasso painting were asked to fill in a short questionnaire indicating whether they had enjoyed looking at the picture and whether they felt they had understood it . according to the results of the survey , all 100 visitors who did not enjoy the painting also d... | tiger_lab_math | [
-0.0017053320771083236,
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-0.0246295016258955,
0.058008864521980286,
-0.03857739269733429,
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0.018345355987548828,
0.04497186094522476,
-0.0032629920169711113,
0.019659923389554024,
0.021877026185393333,
-0.014920026063919067,
... | 0.814198 | 0.810609 | 0.00359 |
Given that the Fourier transform of a function $f(x)$ is defined by $\widehat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} dx$, solve the following:
Find the Fourier transform of $f(x) = e^{-ax^2}$ for some positive constant $a$. | tiger_lab_math | [
0.02073981985449791,
0.0341341532766819,
-0.026967007666826248,
-0.03452771529555321,
0.016629237681627274,
-0.016428954899311066,
-0.00144954991992563,
0.057105693966150284,
0.04069574922323227,
0.015666013583540916,
0.031537432223558426,
0.019099557772278786,
-0.018550099804997444,
-0.02... | 0.811179 | 0.807222 | 0.003957 |
Let $O$ be the origin. A variable plane has a distance of 1 from the origin, and intersects the $x$-axis, $y$-axis, and $z$-axis at $A,$ $B,$ and $C,$ respectively, all distinct from $O.$ Let $(p,q,r)$ be the centroid of triangle $ABC.$ Find
\[\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}.\] | tiger_lab_math | [
0.033014845103025436,
0.041976507753133774,
-0.04809868335723877,
-0.03835849463939667,
0.031731221824884415,
-0.005953311454504728,
-0.020505065098404884,
0.040454573929309845,
0.05139075964689255,
-0.012115659192204475,
0.029294339939951897,
0.027116073295474052,
-0.02835480496287346,
-0... | 0.848579 | 0.820759 | 0.02782 |
Let $\triangle XOY$ be a right-angled triangle with $m\angle XOY = 90^{\circ}$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Given that $XN=19$ and $YM=22$, find $XY$. | tiger_lab_math | [
0.018065573647618294,
0.009810706600546837,
-0.07473628222942352,
-0.05394555255770683,
0.01654825732111931,
-0.03071116842329502,
-0.017796190455555916,
0.042487721890211105,
0.046724118292331696,
-0.009096234105527401,
0.036210332065820694,
0.03862859308719635,
-0.04650026187300682,
-0.0... | 0.833732 | 0.821537 | 0.012195 |
the parameter of a square is equal to the perimeter of a rectangle of length 4 cm and breadth 2 cm . find the circumference of a semicircle whose diameter is equal to the side of the square . ( round off your answer to two decimal places ) ? Let's write a Python program. | tiger_lab_math | [
0.022725405171513557,
0.03538339212536812,
-0.03568527474999428,
-0.03149328753352165,
0.03544072434306145,
-0.021220259368419647,
-0.0307491235435009,
0.04178125038743019,
0.037898700684309006,
-0.012320333160459995,
0.019122330471873283,
0.05004921928048134,
-0.024605561047792435,
-0.013... | 0.816629 | 0.814706 | 0.001923 |
What is the Euler characteristic of the tangent bundle on the projective space $\mathbb{RP}^n$?
Hint: Recall that the Euler characteristic of a vector bundle $E$ over a space $X$ is defined as $\chi(X,E) = \sum_{i=0}^n (-1)^i \operatorname{rank}(H_i(X,E))$, where $n$ is the dimension of $X$. | tiger_lab_math | [
-0.0013557331403717399,
0.02075180597603321,
-0.007670976221561432,
-0.04219168424606323,
0.03548051044344902,
-0.009806574322283268,
-0.00905678328126669,
0.06206454709172249,
0.03587736934423447,
0.005137819331139326,
0.0006329260068014264,
0.04196065664291382,
-0.03484676033258438,
-0.0... | 0.807167 | 0.799554 | 0.007613 |
Find the least positive integer $n$ such that $$\frac 1{\sin 45^\circ\sin 46^\circ}+\frac 1{\sin 47^\circ\sin 48^\circ}+\cdots+\frac 1{\sin 133^\circ\sin 134^\circ}=\frac 1{\sin n^\circ}.$$ | tiger_lab_math | [
-0.006027217488735914,
0.04698965698480606,
-0.020302537828683853,
-0.04960649833083153,
0.029276492074131966,
-0.003866095095872879,
-0.018306078389286995,
0.050727054476737976,
0.03199775144457817,
0.0004998814547434449,
0.029360834509134293,
0.009385185316205025,
-0.0357750840485096,
-0... | 0.854675 | 0.845562 | 0.009113 |
Given quadrilateral $ABCD,$ side $\overline{AB}$ is extended past $B$ to $A'$ so that $A'B = AB.$ Points $B',$ $C',$ and $D'$ are similarly constructed.
[asy]
unitsize(1 cm);
pair[] A, B, C, D;
A[0] = (0,0);
B[0] = (2,0);
C[0] = (1.5,2);
D[0] = (0.2,1.5);
A[1] = 2*B[0] - A[0];
B[1] = 2*C[0] - B[0];
C[1] = 2*D[0] - ... | tiger_lab_math | [
0.031004885211586952,
0.021619625389575958,
-0.02589288167655468,
-0.03611452504992485,
0.02318406105041504,
-0.020695721730589867,
-0.0010663303546607494,
0.06489961594343185,
0.04389948397874832,
-0.008041824214160442,
0.01464426051825285,
0.029332110658288002,
-0.022022245451807976,
-0.... | 0.839274 | 0.815945 | 0.023329 |
Suppose a principal hires an agent to perform a task, and there are three possible outcomes: a successful outcome with a reward of $3000, an unsuccessful outcome with a penalty of $1000, and a mediocre outcome with no reward or penalty. The agent can either exert high effort, with a cost of $1500, or low effort, with a... | tiger_lab_math | [
0.0025302511639893055,
0.00212863995693624,
-0.052143119275569916,
-0.027956508100032806,
0.03131626173853874,
0.005149508826434612,
-0.02944728545844555,
0.05065442994236946,
0.025707919150590897,
-0.005483952350914478,
0.02137453481554985,
0.023945389315485954,
-0.029934024438261986,
-0.... | 0.801426 | 0.791897 | 0.00953 |
The function $y=\frac{x^3+8x^2+21x+18}{x+2}$ can be simplified into the function $y=Ax^2+Bx+C$, defined everywhere except at $x=D$. What is the sum of the values of $A$, $B$, $C$, and $D$? Please respond by writing a program in Python. | tiger_lab_math | [
0.011854822747409344,
0.03546817600727081,
-0.06367350369691849,
-0.0418761745095253,
0.03850751370191574,
-0.02956421673297882,
-0.007580908015370369,
0.06995119899511337,
0.04150785505771637,
-0.016502171754837036,
0.0017407400300726295,
0.0168304406106472,
-0.03089589625597,
-0.01874563... | 0.837606 | 0.835216 | 0.00239 |
Here is a numerical problem for a math student to solve:
A company has been recording its monthly sales data for the past year. The following table shows the company's monthly sales data for the year:
| Month | Sales (in thousands of dollars) |
|-------|---------------------------------|
| 1 | 100 ... | tiger_lab_math | [
0.02166680060327053,
0.009813761338591576,
-0.07069782167673111,
-0.06336026638746262,
0.049026429653167725,
-0.0027045973110944033,
-0.04679068922996521,
0.05936120077967644,
0.04586225375533104,
-0.0016217330703511834,
0.009161767549812794,
0.02752215415239334,
-0.0448717400431633,
-0.02... | 0.825304 | 0.82039 | 0.004914 |
Is triangle ABC obtuse angled?
1.a2 + b2 > c2.
2.The center of the circle circumscribing the triangle does not lie inside the triangle.
Answer Choices: (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. (B) Statement (2) ALONE is sufficient, but statemen... | tiger_lab_math | [
0.0273288581520319,
0.014503493905067444,
-0.012767578475177288,
-0.04262172058224678,
0.03156671300530434,
-0.031056763604283333,
-0.006911405362188816,
0.05760475993156433,
0.02817070670425892,
0.002335610333830118,
0.016367126256227493,
0.02511490136384964,
-0.03600016608834267,
-0.0086... | 0.815633 | 0.807442 | 0.008191 |
The students at Sherry Curtis School were able to vote for one or more events for their field day. Below are the top three choices. List the names of the choices from most popular to least popular, separated by commas.
\begin{tabular}{|c|c|c|c|} \hline
Event & Kickball & Picnic & Softball \\ \hline
& & &\\[-1.5ex]
Fr... | tiger_lab_math | [
0.013905412517488003,
-0.006887319963425398,
-0.048125606030225754,
-0.04927109554409981,
0.026243751868605614,
-0.044108204543590546,
-0.033834006637334824,
0.06303653120994568,
0.0642046108841896,
-0.005087350029498339,
0.04621382802724838,
0.0338129997253418,
-0.011062578298151493,
-0.0... | 0.787146 | 0.784316 | 0.002829 |
Consider the manifold given by the equation x^2 + y^2 + z^2 = 1. Determine the symmetries of this manifold and the corresponding Lie algebra. | tiger_lab_math | [
0.02080066129565239,
0.03638341277837753,
-0.03201284259557724,
-0.07171626389026642,
0.03959200903773308,
-0.014786957763135433,
-0.003860375378280878,
0.05849060043692589,
0.030614221468567848,
-0.01657462678849697,
0.0010517141781747341,
0.016337072476744652,
-0.045713603496551514,
-0.0... | 0.812583 | 0.797531 | 0.015053 |
Given a point $P$ on a triangular piece of paper $ABC,\,$ consider the creases that are formed in the paper when $A, B,\,$ and $C\,$ are folded onto $P.\,$ Let us call $P$ a fold point of $\triangle ABC\,$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36, AC=... | tiger_lab_math | [
0.03266096115112305,
0.042980145663022995,
-0.04279596358537674,
-0.03959012031555176,
0.03229153901338577,
-0.018175601959228516,
-0.009399114176630974,
0.04408030956983566,
0.038809966295957565,
-0.0034297029487788677,
0.02823767438530922,
0.0444701723754406,
-0.01928004063665867,
-0.036... | 0.842476 | 0.819136 | 0.02334 |
Here is a numerical problem for a math student to solve related to Semi-definite programming:
Find the optimal value for the following Semi-definite programming problem:
Maximize f(x) = -x1 - 2x2
Subject to
(x1)*[0 1; 1 2]*(x1;x2) <= 1
x1, x2 >= 0.
where [0 1; 1 2] represents a symmetric matrix. | tiger_lab_math | [
0.012842114083468914,
0.010898075997829437,
-0.06537819653749466,
-0.03164437785744667,
0.030340099707245827,
-0.000012231703294673935,
-0.02049882896244526,
0.06881195306777954,
0.052044253796339035,
-0.011935330927371979,
0.007705130148679018,
0.017472855746746063,
-0.013058079406619072,
... | 0.826468 | 0.816621 | 0.009846 |
Find all possible isomorphisms between the group G of order 20 and each of the following groups:
a) the cyclic group C20
b) the dihedral group D10
c) the non-abelian group of order 20. | tiger_lab_math | [
0.009766490198671818,
0.02055620402097702,
-0.03410736098885536,
-0.02708195336163044,
0.026344290003180504,
-0.05946002155542374,
-0.012145082466304302,
0.05026114359498024,
0.03443201258778572,
-0.030421007424592972,
0.016743989661335945,
0.013572286814451218,
-0.046176422387361526,
-0.0... | 0.807529 | 0.8026 | 0.004929 |
The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the $1$ at each end, row $4$ consists of only even numbers, as does row $2.$ How many of the first $20$ rows have this property? (Don't include row $0$ or row $1$). \begin{tabular}{ccccccccccc}
&&&&&1&&&&&\\
&&&&1&&1&&&&\\
&&&1... | tiger_lab_math | [
0.029487311840057373,
0.02750062569975853,
-0.03260311111807823,
-0.03611578047275543,
0.033726878464221954,
-0.027384059503674507,
-0.0061029791831970215,
0.061867132782936096,
0.0361604318022728,
0.0011435686610639095,
0.019505200907588005,
0.01923161931335926,
-0.02223794534802437,
-0.0... | 0.841297 | 0.823231 | 0.018066 |
Consider the open unit disk $D=\{z \in \mathbb{C}: |z|<1\}$ and a function $f(z) = \frac{1}{2}(z+1)$ defined on $D$. Apply the Riemann mapping theorem to find a biholomorphic map $F:D \to U$, where $U$ is the open unit disk centered at $3+i$ with radius 2, such that $F(0)=3+i$. What is the expression of $F(z)$? | tiger_lab_math | [
0.022548895329236984,
0.013819553889334202,
-0.05431319400668144,
-0.053774092346429825,
0.029669886454939842,
0.014444257132709026,
-0.012137024663388729,
0.06359642744064331,
0.042390450835227966,
0.013093843124806881,
0.019518639892339706,
0.02068314701318741,
-0.03187170997262001,
-0.0... | 0.831785 | 0.81002 | 0.021765 |
Sure, here's an example numerical problem related to the Principal-Agent Problem in Game Theory:
An agent is tasked with choosing one of two projects (project A or project B) to work on. Project A has a 40% chance of success, and if successful, will generate a profit of $120,000. Project B has a 70% chance of success,... | tiger_lab_math | [
0.011881832964718342,
0.0014821833465248346,
-0.03550231456756592,
-0.028259843587875366,
0.02064516954123974,
-0.01815476082265377,
-0.02840556390583515,
0.03357122093439102,
0.019795948639512062,
-0.012369506061077118,
0.02997463382780552,
0.024330327287316322,
-0.042194053530693054,
-0.... | 0.821746 | 0.815173 | 0.006574 |
Find $PQ$ in the triangle below.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (sqrt(3),0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$P$",P,S);
label("$Q$",Q,S);
label("$R$",R,N);
label("$9\sqrt{3}$",R/2,W);
label("$30^\circ$",(1.25,0),N);
[/asy] | tiger_lab_math | [
0.024678820744156837,
0.038708481937646866,
-0.04623539000749588,
-0.05715181678533554,
0.040764980018138885,
-0.02741340920329094,
-0.023375315591692924,
0.04531317949295044,
0.037152063101530075,
-0.004294456914067268,
0.03670724108815193,
0.045669425278902054,
-0.0020163445733487606,
-0... | 0.845514 | 0.835361 | 0.010153 |
A triangle has three sides of the following side lengths: $7$, $10$, and $x^2$. What are all of the positive integer values of $x$ such that the triangle exists? Separate your answers using commas and express them in increasing order. Let's write a Python program. | tiger_lab_math | [
0.02106636017560959,
0.013643908314406872,
-0.043296992778778076,
-0.035267483443021774,
0.0319240465760231,
-0.032386813312768936,
-0.01379076112061739,
0.05288384109735489,
0.041220005601644516,
-0.03267282992601395,
0.0328022837638855,
0.055571574717760086,
-0.04580288007855415,
-0.0339... | 0.839261 | 0.827695 | 0.011565 |
Consider the matrix A = [[3, -2], [1, 4]].
Find a diagonal matrix D and an invertible matrix P such that A = PDP^-1. | tiger_lab_math | [
0.032050129026174545,
0.0177298616617918,
-0.03223583102226257,
-0.03837208077311516,
0.03054538369178772,
-0.018262309953570366,
0.0026381639763712883,
0.06428546458482742,
0.04114338383078575,
-0.01916627772152424,
0.013780081644654274,
0.02566039003431797,
-0.02402498945593834,
-0.02313... | 0.839104 | 0.828234 | 0.01087 |
Consider the elliptic curve given by the equation $y^2 = x^3 + 5x + 7$.
1. Find the singular points of the curve.
2. Determine the genus of the curve.
3. Find the Weierstrass form of the curve.
4. Compute the period lattice of the curve.
5. Find the order of the Jacobian of the curve, i.e., the number of points on th... | tiger_lab_math | [
0.00452691363170743,
0.03765944764018059,
-0.0527416430413723,
-0.05260747671127319,
0.014731341041624546,
-0.03332885727286339,
-0.007022381294518709,
0.06374043971300125,
0.0298165176063776,
-0.010654726065695286,
-0.011347055435180664,
0.030806105583906174,
-0.04188317805528641,
-0.0131... | 0.825397 | 0.808036 | 0.017361 |
Consider a topological quantum field theory called the Chern-Simons theory defined on a closed 3-dimensional manifold M. Let K be a knot in M that can be represented as a closed loop. Calculate the link invariant of K associated with a quantum group G, where G is the simple Lie algebra of type A, of rank 2, and level k... | tiger_lab_math | [
0.043450165539979935,
0.024802271276712418,
-0.04113559424877167,
-0.04803202301263809,
0.04414265975356102,
-0.015895653516054153,
-0.022575296461582184,
0.0546349473297596,
0.018436700105667114,
0.012051264755427837,
0.019195156171917915,
0.039629001170396805,
-0.024152658879756927,
-0.0... | 0.795822 | 0.786193 | 0.00963 |
Consider the function $f(x,y) = x^3 - 3x^2y + y^3$. Determine the critical points of $f$ and classify them as local maxima, local minima, or saddle points. | tiger_lab_math | [
0.028461290523409843,
0.0186147540807724,
-0.05130995437502861,
-0.032456204295158386,
0.04176298528909683,
-0.021180177107453346,
-0.029624873772263527,
0.06448502838611603,
0.026418259367346764,
-0.008989027701318264,
0.016659554094076157,
0.039333514869213104,
-0.055333033204078674,
-0.... | 0.8274 | 0.820484 | 0.006916 |
Piravena must make a trip from $A$ to $B,$ then from $B$ to $C,$ then from $C$ to $A.$ Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of $3000\text{ km}$ from $A$ and with $B$ a distance of $3250\text{ km}$ fro... | tiger_lab_math | [
0.030201131477952003,
0.0358278714120388,
-0.03221065551042557,
-0.023722786456346512,
0.036140281707048416,
-0.0088130421936512,
-0.024805119261145592,
0.05790627747774124,
0.04648090898990631,
0.004547514952719212,
0.01860232464969158,
0.0413253977894783,
0.0014113843208178878,
-0.025404... | 0.827622 | 0.81733 | 0.010292 |
In a 4 person race, medals are awarded to the fastest 3 runners. The first-place runner receives a gold medal, the second-place runner receives a silver medal, and the third-place runner receives a bronze medal. In the event of a tie, the tied runners receive the same color medal. (For example, if there is a two-way ti... | tiger_lab_math | [
0.006267444230616093,
0.028024405241012573,
-0.03562448173761368,
-0.013642492704093456,
0.035963933914899826,
-0.03376169130206108,
-0.05981969088315964,
0.060563839972019196,
0.025184107944369316,
-0.005602965597063303,
0.013959824107587337,
0.027504999190568924,
-0.015015359967947006,
-... | 0.824752 | 0.820189 | 0.004563 |
Let $x$ and $y$ be real numbers such that $2(x^2 + y^2) = x + y.$ Find the maximum value of $x - y.$ Please write a program to solve it | tiger_lab_math | [
0.03738328441977501,
0.03549604117870331,
-0.04806750267744064,
-0.044319506734609604,
0.029913954436779022,
-0.023737559095025063,
-0.018700236454606056,
0.05948318541049957,
0.049936678260564804,
-0.051990997046232224,
0.027395134791731834,
0.05456390976905823,
-0.03489864990115166,
-0.0... | 0.84422 | 0.837112 | 0.007108 |
A road has four intersections. At the first intersection, there are 40 road signs showing different types of information mounted on the street poles. At the second intersection, 1/4 times more road signs than the first intersection are mounted on the street poles. The number of road signs at the third intersection is t... | tiger_lab_math | [
0.027951639145612717,
0.01580137014389038,
-0.03852298483252525,
-0.016796806827187538,
0.0341762974858284,
-0.02104993537068367,
-0.04777425155043602,
0.055905621498823166,
0.04310714453458786,
-0.0002626333443913609,
-0.0010707055917009711,
0.020946284756064415,
-0.03072105161845684,
-0.... | 0.821112 | 0.816908 | 0.004204 |
Here is a precise numerical problem for a math student to solve in quadratic programming:
A company manufactures two types of products, Product A and Product B. The profit per unit for Product A is $10 and for Product B is $12. The total production capacity of the company is 500 units. The company has to fulfill the f... | tiger_lab_math | [
0.028082937002182007,
0.013918157666921616,
-0.053898200392723083,
-0.025023439899086952,
0.03172504901885986,
-0.0048728385008871555,
-0.02774944342672825,
0.06391908973455429,
0.03796900063753128,
-0.014937999658286572,
-0.0006568020326085389,
0.03176277130842209,
-0.016016149893403053,
... | 0.831361 | 0.808191 | 0.02317 |
A factory produces two types of products: A and B. Each unit of product A requires 2 hours of processing time and 1 hour of assembly time, while each unit of product B requires 1 hour of processing time and 1.5 hours of assembly time. The factory has 90 hours of processing time and 75 hours of assembly time available p... | tiger_lab_math | [
0.021487273275852203,
0.0014545157318934798,
-0.04420190677046776,
0.007972009479999542,
0.000052350733312778175,
-0.025207461789250374,
-0.05623456463217735,
0.0813809484243393,
0.02974928915500641,
-0.009273757226765156,
0.008969845250248909,
0.029675550758838654,
-0.01654100976884365,
-... | 0.801705 | 0.785702 | 0.016003 |
Let $C$ be a smooth, projective, algebraic curve of genus $g$ over the field of complex numbers, and let $D$ be an effective divisor on $C$ with degree $d$. Then, the Riemann-Roch theorem states that the dimension of the space of global sections of $O_C(D)$ is given by:
$$
\mathrm{dim}(H^0(C,O_C(D))) = d-g+1+\mathrm{d... | tiger_lab_math | [
0.01094671431928873,
0.04621206596493721,
-0.041936978697776794,
-0.06086460128426552,
0.027228103950619698,
-0.01841684989631176,
-0.021343696862459183,
0.045726459473371506,
0.030976518988609314,
-0.018513765186071396,
0.0009390015038661659,
0.03429265692830086,
-0.03725331649184227,
-0.... | 0.830491 | 0.803901 | 0.02659 |
A study was conducted to compare the final exam scores of students in three different courses - Math, Science, and English. The mean scores and standard deviations for each course are obtained as follows:
Math: Mean = 73, SD = 8
Science: Mean = 81, SD = 12
English: Mean = 77, SD = 10
A new student has scored 75 on th... | tiger_lab_math | [
0.015559916384518147,
-0.0019152866443619132,
-0.052284836769104004,
-0.07103238999843597,
0.026904616504907608,
-0.033795975148677826,
-0.055096372961997986,
0.050995927304029465,
0.03229350224137306,
-0.014975144527852535,
0.009964101947844028,
0.010528916493058205,
-0.01999286748468876,
... | 0.796138 | 0.794072 | 0.002066 |
If the quadratic $x^2+4mx+m$ has exactly one real root, find the positive value of $m$. Let's write a program. | tiger_lab_math | [
0.01587321236729622,
0.027335114777088165,
-0.03892122581601143,
-0.04193194955587387,
0.0408034548163414,
-0.03635973483324051,
-0.009757847525179386,
0.0675206258893013,
0.05089784041047096,
-0.034106332808732986,
0.01420559361577034,
0.016779258847236633,
-0.02565113641321659,
-0.033521... | 0.834951 | 0.82938 | 0.00557 |
A manufacturing company produces two types of products, A and B. It has 8 hours of machine time daily and can produce 15 units of product A in 1 hour of machine time, and 10 units of product B in 1 hour of machine time. The company needs to produce at least 60 units of product A and 80 units of product B daily. If the ... | tiger_lab_math | [
0.029094386845827103,
0.0031334715895354748,
-0.052085958421230316,
0.00356440176256001,
0.016832267865538597,
-0.018057361245155334,
-0.052461888641119,
0.08719076961278915,
0.020424308255314827,
-0.010157415643334389,
0.0028932616114616394,
0.036000993102788925,
-0.006161294411867857,
-0... | 0.80922 | 0.803511 | 0.005709 |
Given a plaintext message "HELLO" converted into numerical format as [7 4 11 11 14], use ElGamal encryption and decryption to send it securely from sender Alice to receiver Bob. Assume that Alice's private key is p=23, g=5, and x=6, and that Bob's public key is (g^y) mod p=4. Find the ciphertext and verify that the dec... | tiger_lab_math | [
0.02364657260477543,
0.049909934401512146,
0.0014463606057688594,
-0.0616515651345253,
0.03223157301545143,
-0.0371888242661953,
-0.002817717147991061,
0.03939306363463402,
0.024544568732380867,
-0.026875901967287064,
0.014017961919307709,
0.01809137687087059,
-0.03900976851582527,
-0.0101... | 0.806974 | 0.803432 | 0.003542 |
Consider the topological space X = {1,2,3} with the topology τ = {∅,X,{1},{2,3}}. Let f: X → X be defined by f(x) = x². Determine whether f is a continuous function or not using the concept of convergence. | tiger_lab_math | [
0.021803222596645355,
0.025283964350819588,
-0.059460338205099106,
-0.01864050328731537,
0.029148275032639503,
-0.017670048400759697,
-0.015805011615157127,
0.07344909012317657,
0.02513670176267624,
-0.0005073603824712336,
0.013432578183710575,
0.0020163189619779587,
-0.027762437239289284,
... | 0.818078 | 0.802393 | 0.015685 |
Consider a two-dimensional manifold with the following metric tensor:
g = [(2x-y)² 2]
[ 2 1]
Determine the Ricci curvature of this manifold at the point (1,2). | tiger_lab_math | [
0.014011458493769169,
0.03058769926428795,
-0.016865089535713196,
-0.05766576901078224,
0.01840686798095703,
-0.012610314413905144,
-0.017941344529390335,
0.060557395219802856,
0.027119817212224007,
-0.0063763814978301525,
0.004178948234766722,
0.03929450362920761,
-0.038504231721162796,
-... | 0.81999 | 0.812339 | 0.007651 |
Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a smooth function with only two critical points $(0,0)$ and $(2,2)$, and the Hessian matrix of $f$ has distinct eigenvalues at both critical points. Construct a Morse-Smale complex for $f$ and compute its homology groups. | tiger_lab_math | [
0.01960279978811741,
0.035395342856645584,
-0.041044704616069794,
-0.042626965790987015,
0.033267851918935776,
-0.005020917393267155,
-0.033915381878614426,
0.05561769753694534,
0.024706952273845673,
0.001990673365071416,
0.0015369593165814877,
0.020234184339642525,
-0.022993357852101326,
... | 0.822845 | 0.798801 | 0.024045 |
Calculate $\sqrt{10p} \cdot \sqrt{5p^2} \cdot \sqrt{6p^4}$ . Express your answer in simplest radical form in terms of $p$.
Note: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\sqrt{14}$ as "sqrt(14)" or "sqrt{14}". | tiger_lab_math | [
0.03698825463652611,
0.06029561907052994,
-0.023918217048048973,
-0.03824697434902191,
0.043945733457803726,
-0.0336851105093956,
-0.004714156500995159,
0.05532272905111313,
0.04739689081907272,
-0.008768754079937935,
0.014804210513830185,
0.018616056069731712,
-0.034559465944767,
-0.05699... | 0.824006 | 0.821994 | 0.002011 |
Here's a problem for the student to solve in Combinatorics - Network flow problems:
A factory produces three types of products - A, B, and C. It has a daily supply of 200 units of product A, 250 units of product B, and 150 units of product C. The factory wants to distribute these products to three warehouses - X, Y, ... | tiger_lab_math | [
0.038739848881959915,
0.012289401143789291,
-0.04285217449069023,
-0.00176427629776299,
0.023008283227682114,
-0.0006677242927253246,
-0.04394633695483208,
0.07185559719800949,
0.010905925184488297,
-0.010361049324274063,
0.011613557115197182,
0.035860657691955566,
-0.02144395187497139,
-0... | 0.820889 | 0.801931 | 0.018958 |
Suppose a 3x3 Hill cipher encryption key is given by the matrix K = [6 24 1; 13 16 10; 20 17 15]. Perform an encryption of the plaintext message "MATHEMATICS" using this key. | tiger_lab_math | [
0.04579746350646019,
0.0370476134121418,
-0.030121609568595886,
-0.04802795872092247,
0.053582288324832916,
-0.03966283053159714,
-0.022113872691988945,
0.05622338876128197,
0.01996081881225109,
-0.028392182663083076,
0.01611446402966976,
0.013770471327006817,
-0.03365669026970863,
-0.0084... | 0.823826 | 0.819644 | 0.004182 |
Consider the following metric for the two-dimensional plane:
$$ds^2 = 4(dx^2 + dy^2)$$
Calculate the Christoffel symbols $\Gamma^k_{ij}$ for this metric. | tiger_lab_math | [
0.016041364520788193,
0.03301020339131355,
-0.0401470847427845,
-0.06131908297538757,
0.04539661481976509,
-0.023366782814264297,
-0.0206823218613863,
0.052089713513851166,
0.05477656424045563,
-0.002061387524008751,
0.013651730492711067,
0.035122282803058624,
-0.02586841955780983,
-0.0157... | 0.831738 | 0.818061 | 0.013677 |
Find the largest value of $c$ such that $1$ is in the range of $f(x)=x^2-5x+c$. Let's write a Python program to solve it. | tiger_lab_math | [
0.013471066951751709,
0.03807873651385307,
-0.0548742413520813,
-0.03564492613077164,
0.036467473953962326,
-0.034005071967840195,
-0.011385763995349407,
0.06149277836084366,
0.029404232278466225,
-0.02793864533305168,
0.014437890611588955,
0.036037374287843704,
-0.01673860102891922,
-0.00... | 0.833643 | 0.826718 | 0.006925 |
In a Battle of the Sexes game, a husband and wife must decide where to go on their date. The husband prefers to go see an action movie, while the wife would like to go watch a romantic comedy. If they both go to the same movie, they will each get 4 units of pleasure. However, if they go to different movies, the husband... | tiger_lab_math | [
-0.0023764551151543856,
0.003841974539682269,
-0.06439577788114548,
-0.027437333017587662,
0.037462763488292694,
-0.004441177938133478,
-0.01935633085668087,
0.053950171917676926,
0.05594667047262192,
-0.03940020501613617,
0.0170778539031744,
0.04270587116479874,
-0.05422421544790268,
-0.0... | 0.820324 | 0.813549 | 0.006775 |
If $\|\mathbf{a}\| = 3$ and $\|\mathbf{b}\| = 6,$ then find $(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}).$ Please write a program to solve it | tiger_lab_math | [
0.0408024787902832,
0.04790740832686424,
-0.060169514268636703,
-0.04527406021952629,
0.04241502657532692,
-0.009490776807069778,
-0.03221375495195389,
0.06477943062782288,
0.03334249183535576,
-0.029050413519144058,
0.020818788558244705,
0.048157621175050735,
-0.03873590752482414,
-0.0411... | 0.844324 | 0.843435 | 0.00089 |
Let A = {1, 2} and B = {3, 4, 5}, and C = {6, 7}. Find the following products:
(A × B) × C and A × (B × C) and verify that they are equal. | tiger_lab_math | [
0.06090649589896202,
0.0349615179002285,
-0.035775281488895416,
-0.036810502409935,
0.05549857020378113,
-0.03236592933535576,
-0.007760566659271717,
0.06388772279024124,
0.03252628445625305,
-0.027835601940751076,
-0.0036476231180131435,
0.0225597582757473,
-0.03198380395770073,
-0.037039... | 0.831691 | 0.825285 | 0.006405 |
A company produces two types of products, A and B. The profit per unit of type A is $10 and for type B it is $15. Each unit of A takes 2 hours for processing while each unit of B takes 3 hours to process. The company has 300 hours of processing time available in a week. What is the optimal production plan that maximize... | tiger_lab_math | [
0.019534381106495857,
0.004244188778102398,
-0.06538945436477661,
-0.0023592920042574406,
0.006782211363315582,
-0.020700901746749878,
-0.04556220769882202,
0.09153667837381363,
0.03127136453986168,
-0.007543145678937435,
0.019422613084316254,
0.022880109027028084,
-0.004373189993202686,
-... | 0.80074 | 0.796809 | 0.003931 |
A population of 100,000 people has recently experienced an outbreak of a disease. The rate at which people are being infected is proportional to the number of infected individuals and the number of susceptible individuals. A mathematical model has been developed to describe the spread of disease in this population as:
... | tiger_lab_math | [
0.013158786110579967,
0.023811865597963333,
-0.04633871093392372,
-0.03149197995662689,
0.03707679733633995,
-0.039400119334459305,
-0.02441372349858284,
0.07216820865869522,
0.03566543012857437,
-0.02120635099709034,
0.015580895356833935,
0.03070911206305027,
-0.017612336203455925,
-0.001... | 0.826174 | 0.816265 | 0.009909 |
Find the perimeter of a triangle XYZ, where XY = 10 cm, angle YXZ = 60 degrees, and angle XYZ = 80 degrees. Use trigonometry to solve the problem. | tiger_lab_math | [
0.016279803588986397,
0.016346490010619164,
-0.014000930823385715,
-0.03925329074263573,
0.013826101087033749,
-0.014802987687289715,
-0.03095417283475399,
0.03523462265729904,
0.025923501700162888,
-0.030659103766083717,
0.01385465357452631,
0.033337775617837906,
-0.038322046399116516,
-0... | 0.803154 | 0.798225 | 0.004929 |
Here's a numerical problem on the subtopic "Moduli spaces of algebraic curves":
Problem:
Consider the moduli space $\mathcal{M}_2$ of algebraic curves of genus 2. Let $C$ be the algebraic curve defined by the equation $y^2 + y = x^5 - 1$. Determine whether $C$ belongs to $\mathcal{M}_2$.
(Note: The genus of an algebr... | tiger_lab_math | [
0.024366626515984535,
0.0243269894272089,
-0.04026971012353897,
-0.05232509598135948,
0.040439341217279434,
-0.021230559796094894,
-0.009260531514883041,
0.0657634288072586,
0.0314362496137619,
-0.013987962156534195,
-0.011803939007222652,
0.01741960272192955,
-0.019692696630954742,
-0.015... | 0.836479 | 0.814985 | 0.021493 |
Consider the following two graphs:
Graph A: has 5 vertices and 7 edges
Graph B: has 5 vertices and 7 edges
Are Graph A and Graph B isomorphic? Explain your answer by identifying the corresponding vertices and edges between the two graphs. | tiger_lab_math | [
0.03571893647313118,
0.02468082308769226,
-0.023566437885165215,
-0.025557877495884895,
0.017775386571884155,
-0.03898336738348007,
-0.012742438353598118,
0.05365528538823128,
0.033329401165246964,
-0.02204754389822483,
0.020020166411995888,
0.007177386432886124,
-0.02195342443883419,
-0.0... | 0.810377 | 0.802445 | 0.007932 |
The graphs of $y=|x|$ and $y=-x^2-3x-2$ are drawn. For every $x$, a vertical segment connecting these two graphs can be drawn as well. Find the smallest possible length of one of these vertical segments. | tiger_lab_math | [
0.0030633551068603992,
0.025827327743172646,
-0.04816179722547531,
-0.03617643564939499,
0.027790233492851257,
-0.018614381551742554,
-0.01666978932917118,
0.06762399524450302,
0.044734567403793335,
-0.01641860045492649,
0.014469905756413937,
0.034250956028699875,
-0.0022557235788553953,
-... | 0.844175 | 0.832447 | 0.011728 |
Here's a question for a math student in Complexity theory:
Suppose you have a problem where the input size is n, and the running time of your algorithm is given by 5n^3 + 2^n. Determine the largest input size that can be solved within 10 seconds. | tiger_lab_math | [
0.02984611503779888,
0.021159017458558083,
-0.044077590107917786,
-0.029892312362790108,
0.024124907329678535,
-0.0034862004686146975,
-0.044514451175928116,
0.05123395100235939,
0.036973148584365845,
-0.027884962037205696,
0.018988588824868202,
0.014969231560826302,
-0.024849068373441696,
... | 0.833401 | 0.831161 | 0.00224 |
A sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ is defined recursively by $a_1 = 1,$ $a_2 = 1,$ and for $k \ge 3,$
\[a_k = \frac{1}{3} a_{k - 1} + \frac{1}{4} a_{k - 2}.\]Evaluate $a_1 + a_2 + a_3 + \dotsb.$ | tiger_lab_math | [
0.02834601141512394,
0.04811224341392517,
-0.04011659324169159,
-0.04138690605759621,
0.04270341992378235,
-0.013884901069104671,
-0.013261280953884125,
0.07146017998456955,
0.037674903869628906,
-0.018052594736218452,
0.015481575392186642,
0.012735567055642605,
-0.040095508098602295,
-0.0... | 0.837071 | 0.825151 | 0.01192 |
Find all the possible groups of order 27, where 27 is a prime power. Which of these groups are cyclic and which ones are isomorphic to the direct product of a cyclic group of order 3 and a group of order 9? | tiger_lab_math | [
0.027302222326397896,
0.015288795344531536,
-0.03877772018313408,
-0.05523179844021797,
0.012428171001374722,
-0.040331579744815826,
-0.008556156419217587,
0.027071431279182434,
0.02286108396947384,
-0.027317380532622337,
0.028728803619742393,
0.017922203987836838,
-0.04305582493543625,
-0... | 0.821017 | 0.817647 | 0.003369 |
A factory produces two types of products, A and B, which require raw materials of varying quantities. Each unit of product A requires 3 units of material X, 4 units of material Y, and 5 units of material Z. Each unit of product B requires 2 units of material X, 5 units of material Y, and 7 units of material Z. The maxi... | tiger_lab_math | [
0.03462965413928032,
0.012568814679980278,
-0.04739271104335785,
-0.0009156055166386068,
0.012122848071157932,
-0.026058468967676163,
-0.05218266323208809,
0.0778733640909195,
0.014778916724026203,
-0.012638700194656849,
-0.0033211142290383577,
0.023766836151480675,
-0.013014331459999084,
... | 0.800379 | 0.779495 | 0.020884 |
The function $f(x)$ satisfies
\[f(xy) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(0) \neq 0.$ Find $f(10).$ Please respond by writing a program in Python. | tiger_lab_math | [
0.031794752925634384,
0.04113059490919113,
-0.05245295539498329,
-0.031194133684039116,
0.03060079552233219,
-0.028118319809436798,
-0.008944511413574219,
0.05088319629430771,
0.03546501323580742,
-0.02844177559018135,
0.021166471764445305,
0.02254297025501728,
-0.02976486086845398,
-0.011... | 0.829934 | 0.821758 | 0.008176 |
Suppose you have access to a large vat of distilled water, several gallons large. You have two precise measuring pipettes, one to measure exactly 1/2 of an ounce and one to measure exactly 1/3 of an ounce. You can pour precisely measured amounts into a beaker, which initially is empty. You can use either pipette to rem... | tiger_lab_math | [
0.021412594243884087,
0.037246450781822205,
-0.027331458404660225,
-0.016859663650393486,
0.04116716608405113,
-0.0630345270037651,
-0.022519486024975777,
0.08397429436445236,
0.02659635804593563,
-0.0034243501722812653,
0.03817078843712807,
0.02981860563158989,
-0.030904831364750862,
-0.0... | 0.821123 | 0.819012 | 0.00211 |
Consider the following graph G:
Determine whether the graph has a perfect matching. If it does, write down the edges that are part of the perfect matching. If it does not, explain why. | tiger_lab_math | [
0.0028194915503263474,
0.011004854924976826,
-0.024488866329193115,
-0.03490809351205826,
0.03793608024716377,
-0.041857823729515076,
-0.014072592370212078,
0.03199581056833267,
0.030985167250037193,
-0.01836256869137287,
0.010845839977264404,
-0.007359666284173727,
-0.013187088072299957,
... | 0.818163 | 0.805195 | 0.012968 |
Let $\bold{v} = \begin{pmatrix} 5 \\ -3 \end{pmatrix}$ and $\bold{w} = \begin{pmatrix} 11 \\ -2 \end{pmatrix}$. Find the area of the parallelogram with vertices $\bold{0}$, $\bold{v}$, $\bold{w}$, and $\bold{v} + \bold{w}$. | tiger_lab_math | [
0.03606323152780533,
0.008968926966190338,
-0.055636581033468246,
-0.05437877029180527,
0.05378037318587303,
-0.009695107117295265,
-0.020670350641012192,
0.07418076694011688,
0.03214579448103905,
0.013945544138550758,
0.007401186041533947,
0.03580785542726517,
-0.034813813865184784,
-0.01... | 0.820909 | 0.817173 | 0.003736 |
Here's a precise numerical problem for a math student to solve related to The Lyapunov exponent and its role in characterizing chaos:
Consider the dynamical system described by the equation: x_(n+1) = 3.9x_n(1 - x_n), where x_0 = 0.6. Calculate the Lyapunov exponent for this system up to time n = 100, and use this exp... | tiger_lab_math | [
0.03084079921245575,
-0.006950392387807369,
-0.043873582035303116,
-0.04549575597047806,
0.0431443490087986,
-0.01931590959429741,
-0.030005700886249542,
0.06686476618051529,
0.01750633865594864,
-0.019720597192645073,
0.004578839056193829,
0.03169727325439453,
-0.046837218105793,
-0.02137... | 0.821741 | 0.816923 | 0.004818 |
Suppose there is a network of 6 cities and each city is connected to every other city by road. The distance between city 1 and city 6 is 8 units, while the distances between the other cities are as follows: city 1 to city 2 is 3 units, city 1 to city 3 is 5 units, city 2 to city 3 is 2 units, city 2 to city 4 is 2 unit... | tiger_lab_math | [
0.013138280250132084,
0.023243887349963188,
-0.029506023973226547,
-0.02566402032971382,
0.031112078577280045,
-0.017179537564516068,
-0.03761400282382965,
0.03446543216705322,
0.03429034724831581,
-0.030247341841459274,
0.014589040540158749,
0.036691103130578995,
-0.015930326655507088,
-0... | 0.817895 | 0.807497 | 0.010398 |
Suppose $f(z)=z^2+z+1$ is an entire function. Use the Maximum Modulus Principle to find the maximum value of $|f(z)|$ for $|z| \leqslant 1$. | tiger_lab_math | [
0.03424261510372162,
0.026847288012504578,
-0.03736484795808792,
-0.03239376097917557,
0.04597464203834534,
-0.028967682272195816,
0.0030954594258219004,
0.0401274673640728,
0.024179909378290176,
-0.011818044818937778,
0.019523456692695618,
0.024301309138536453,
-0.03443822264671326,
-0.01... | 0.826923 | 0.806814 | 0.02011 |
Suppose there are six volunteers in a community, and each one was willing to contribute $10 to a food drive project. However, the project needs a minimum of $40 to be successful. If the project is successful, the volunteers will receive a reward of $30 each, but if it fails, no one gets any reward. Assume that the deci... | tiger_lab_math | [
-0.00392192741855979,
0.008591331541538239,
-0.04192842170596123,
-0.014339646324515343,
0.03900331258773804,
-0.03024766780436039,
-0.010715974494814873,
0.037343256175518036,
0.025067565962672234,
-0.016665661707520485,
0.009292840957641602,
0.008538498543202877,
-0.030017584562301636,
-... | 0.823258 | 0.811986 | 0.011272 |
A rocket is launched from Earth and needs to reach a height of 5000 meters within 60 seconds. The rocket's acceleration is limited to 20 m/s^2 and its maximum velocity is 200 m/s. Determine the optimal control function for the rocket's acceleration that will minimize the time it takes to reach the desired height, while... | tiger_lab_math | [
0.02172713167965412,
0.03541918098926544,
-0.04477605223655701,
-0.03616466373205185,
0.03056339919567108,
-0.025792323052883148,
-0.036673709750175476,
0.09437451511621475,
0.04848632588982582,
-0.003328966675326228,
-0.0016500760102644563,
0.017333855852484703,
-0.01301254890859127,
-0.0... | 0.800726 | 0.793528 | 0.007198 |
Let $a_1,a_2,\ldots$ be a sequence determined by the rule $a_n= \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2008$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$? Let's write a Python program. | tiger_lab_math | [
-0.002687581814825535,
0.02675960771739483,
-0.04477537423372269,
-0.025685619562864304,
0.02298169955611229,
-0.0060693747363984585,
-0.03411855176091194,
0.05667515844106674,
0.02969190664589405,
-0.026308082044124603,
0.0017403049860149622,
0.029598306864500046,
-0.03928334265947342,
-0... | 0.848196 | 0.834206 | 0.013989 |
A manufacturing company produces two types of products, A and B. The profit per unit of product A is $8 and that of product B is $10. The production process requires a certain amount of two resources, resource X and resource Y. Company has a total of 1000 units of resource X and 2000 units of resource Y. To produce a u... | tiger_lab_math | [
0.02488093450665474,
0.005328788887709379,
-0.05951074883341789,
-0.002366309752687812,
0.011772691272199154,
-0.02410712093114853,
-0.05184522271156311,
0.08836124837398529,
0.02898462302982807,
0.00009222043445333838,
0.00552253145724535,
0.04028697311878204,
-0.020808150991797447,
-0.03... | 0.815621 | 0.799239 | 0.016382 |
A factory has to produce two types of products, A and B. The factory has 100 machines and a limited amount of raw materials. Each unit of product A requires 3 hours of machine time and 2 hours of raw materials, while each unit of product B requires 2 hours of machine time and 1 hour of raw materials. The profit for eac... | tiger_lab_math | [
0.03816920518875122,
0.00918567180633545,
-0.05201460421085358,
-0.00015076719864737242,
0.003971375524997711,
-0.007254285737872124,
-0.04691411182284355,
0.08376718312501907,
0.023693736642599106,
-0.009220905601978302,
0.001978498650714755,
0.037454333156347275,
-0.013951902277767658,
-... | 0.819777 | 0.805047 | 0.01473 |
Consider the system of two equations:
x^2 - y + e^x = 0
xsin(y) + y^2 - 1 = 0
Use Newton's method to approximate the solutions (x,y) to this system to the precision of 10^-4 with the initial guess (1,1). | tiger_lab_math | [
0.039093054831027985,
0.01948624663054943,
-0.02960851788520813,
-0.028001852333545685,
0.03833044320344925,
-0.0485730841755867,
-0.02078847400844097,
0.06241818144917488,
0.037663593888282776,
-0.01685980334877968,
0.028055481612682343,
0.03972622752189636,
-0.03722048178315163,
-0.03837... | 0.824963 | 0.823526 | 0.001438 |
A manufacturing company produces a particular item at a cost of $10 per unit in the absence of any quality issues. However, as the quality of the item decreases, the cost of production increases. Suppose that the cost of producing a unit of the item is given by the function:
C(x) = 10 + 100/x
where x represents the q... | tiger_lab_math | [
0.011355768889188766,
0.015633568167686462,
-0.059801824390888214,
-0.0023836048785597086,
0.04136669635772705,
-0.031598083674907684,
-0.04128279164433479,
0.09844899922609329,
0.03351965919137001,
-0.00026362089556641877,
0.01750860922038555,
0.022041110321879387,
-0.0394618958234787,
-0... | 0.816134 | 0.807489 | 0.008645 |
Consider a simple undirected graph with 7 vertices as shown below. Find the smallest possible size of a vertex cover for this graph.
```
1 ----- 2
/|\ |\
/ | \ | \
3 | 4 | 5
\ | / | /
\| / |/
6 ----- 7
```
(Note: A vertex cover of a graph is a subset of vertices such that every edg... | tiger_lab_math | [
0.006808631122112274,
0.03188531845808029,
-0.024272138252854347,
-0.04047437757253647,
0.020718062296509743,
-0.02125464379787445,
-0.021175701171159744,
0.050665680319070816,
0.025029083713889122,
-0.022268932312726974,
0.036545414477586746,
0.015962326899170876,
0.0015153700951486826,
-... | 0.822075 | 0.814227 | 0.007848 |
A furniture company produces two types of tables: standard and deluxe. Each standard table requires 4 hours of carpentry work and 2 hours of painting, while each deluxe table requires 3 hours of carpentry work and 4 hours of painting. The company has 20 carpenters and 16 painters available for the production of tables.... | tiger_lab_math | [
0.01109516341239214,
0.015083187259733677,
-0.06779461354017258,
-0.02137887105345726,
0.040491700172424316,
-0.0050400374457240105,
-0.02234775386750698,
0.07464668154716492,
0.038535356521606445,
-0.011261125095188618,
0.02307523414492607,
0.03570934385061264,
-0.019179372116923332,
-0.0... | 0.8217 | 0.806782 | 0.014918 |
Let $\omega$ be a complex number such that $\omega^7 = 1$ and $\omega \ne 1.$ Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6.$ Then $\alpha$ and $\beta$ satisfy the quadratic
\[x^2 + ax + b = 0\]for some real numbers $a$ and $b.$ Enter the ordered pair $(a,b).$ Let's write a ... | tiger_lab_math | [
0.042891863733530045,
0.04504252225160599,
-0.05945059657096863,
-0.03793416917324066,
0.038418643176555634,
-0.017004020512104034,
-0.0012773839989677072,
0.05442001298069954,
0.040153659880161285,
-0.03802230954170227,
0.0182638019323349,
0.03563843294978142,
-0.04198867455124855,
-0.036... | 0.855338 | 0.83257 | 0.022768 |
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14,$ find the smallest possible value of $|w^3+z^3|.$ | tiger_lab_math | [
0.010308708995580673,
0.04672741889953613,
-0.03476130962371826,
-0.05654766410589218,
0.0648573562502861,
0.0013214125065132976,
-0.0006873121601529419,
0.04851216450333595,
0.04073614627122879,
0.0025245260912925005,
0.027212753891944885,
0.018737761303782463,
-0.027357308194041252,
-0.0... | 0.842437 | 0.836539 | 0.005898 |
Each of the first eight prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number? | tiger_lab_math | [
0.016048992052674294,
0.0106199961155653,
-0.0340728685259819,
-0.031161481514573097,
0.031421687453985214,
-0.01097620464861393,
-0.001081180525943637,
0.05374407395720482,
0.04465537890791893,
-0.025452006608247757,
-0.0045987325720489025,
0.020846592262387276,
-0.01705031841993332,
-0.0... | 0.835813 | 0.828772 | 0.007041 |
Statement 1 | Every finite integral domain is a field. Statement 2 | The intersection of two subgroups of a group is also a subgroup. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.02769792079925537,
0.020343756303191185,
-0.02322106808423996,
-0.0454869419336319,
0.04859578609466553,
-0.036868251860141754,
-0.005662564188241959,
0.03661932051181793,
0.05145934969186783,
-0.024533940479159355,
0.016479218378663063,
0.02731906808912754,
-0.015857376158237457,
-0.028... | 0.81272 | 0.80913 | 0.00359 |
The number of cans in the layers of a display in a supermarket form an arithmetic sequence. The bottom layer has 28 cans; the next layer has 25 cans and so on until there is one can at the top of the display. How many cans are in the entire display? Please write a program to solve it | tiger_lab_math | [
0.04103563353419304,
0.03390396758913994,
-0.034264713525772095,
-0.03801391273736954,
0.039296410977840424,
-0.013532621785998344,
-0.04365008324384689,
0.06757371127605438,
0.01119767501950264,
-0.0296152513474226,
0.02992045134305954,
0.0427878312766552,
-0.04319024085998535,
-0.0117288... | 0.829115 | 0.826758 | 0.002358 |
Let's say the domain of discourse is {1, 2, 3, 4} and the predicate P(x) is defined as "x is an even number". Write a higher-order logic formula for the statement "There exists a function f such that, for every even number x, P(f(x)) is true." | tiger_lab_math | [
0.021609418094158173,
0.02273968607187271,
-0.03620254248380661,
-0.03355250880122185,
0.032307785004377365,
-0.03242439031600952,
-0.030266163870692253,
0.06440819799900055,
0.03500021994113922,
-0.021102212369441986,
0.0021542285103350878,
0.034275371581315994,
-0.010253758169710636,
-0.... | 0.827641 | 0.80648 | 0.02116 |
If $a>0$ and $b>0,$ a new operation $\nabla$ is defined as follows: $$a \nabla b = \frac{a + b}{1 + ab}.$$For example, $$3 \nabla 6 = \frac{3 + 6}{1 + 3 \times 6} = \frac{9}{19}.$$Calculate $(1 \nabla 2) \nabla 3.$ | tiger_lab_math | [
0.036777425557374954,
0.03343713656067848,
-0.04033639281988144,
-0.0036741646472364664,
0.04990292340517044,
-0.0069303493946790695,
-0.02809206023812294,
0.08249591290950775,
0.025662928819656372,
-0.0000918586811167188,
0.007262436207383871,
0.013755416497588158,
-0.042035412043333054,
... | 0.821539 | 0.816912 | 0.004628 |
Consider the reaction-diffusion equation:
$$ u_t = D u_{xx} + u(1-u)(\alpha - u) $$
with $D=0.1$ and $\alpha=0.5$. Assume that $u(x,0) = \cos(\pi x)$ and $u(0,t) = u(1,t) = 0$. Solve the equation for $t\in [0,2]$ and $x\in[0,1]$ using a numerical method of your choice. Plot the initial condition and the solution at... | tiger_lab_math | [
-0.0036722151562571526,
0.02983371913433075,
-0.04661723971366882,
-0.013500817120075226,
0.03323740139603615,
-0.02051464095711708,
-0.031495388597249985,
0.08802615851163864,
0.035783182829618454,
0.005911066196858883,
0.028560791164636612,
0.01248040609061718,
-0.017417486757040024,
-0.... | 0.832362 | 0.818502 | 0.01386 |
Consider the boundary value problem given by:
$$\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 0$$
with boundary conditions
$$y(0) = 0, \quad y(1) + y'(1) = 1.$$
Find the general solution of the differential equation and use the boundary conditions to determine the particular solution. Then, find the value of $y(1/2)$. | tiger_lab_math | [
-0.0016537209739908576,
0.020070770755410194,
-0.05534416437149048,
-0.0374942384660244,
0.030743375420570374,
-0.017079131677746773,
-0.006971980910748243,
0.07411685585975647,
0.042608536779880524,
-0.009881509467959404,
0.0019510637503117323,
0.03407230228185654,
-0.027916494756937027,
... | 0.821427 | 0.806654 | 0.014773 |
A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The prob... | tiger_lab_math | [
0.014340897090733051,
0.03598721697926521,
-0.027240369468927383,
-0.025318913161754608,
0.0447281189262867,
-0.008039234206080437,
-0.04814622923731804,
0.0625738799571991,
0.03816773369908333,
0.004351567942649126,
0.023801811039447784,
0.022573523223400116,
-0.01615886762738228,
-0.0193... | 0.840916 | 0.821383 | 0.019532 |
A company produces two types of products, P1 and P2, and has limited resources. The profit per unit of P1 is $20 while the profit per unit of P2 is $30. The company has 800 units of labor, 700 units of material, and 120 units of machine time available. The production of P1 takes 4 units of labor, 3 units of material, a... | tiger_lab_math | [
0.021540090441703796,
0.02366497367620468,
-0.03156711906194687,
-0.008608960546553135,
0.012398608960211277,
-0.022151900455355644,
-0.05056043341755867,
0.086399145424366,
0.018070576712489128,
0.0021481066942214966,
0.0013340669684112072,
0.0313696563243866,
-0.00991873349994421,
-0.021... | 0.808341 | 0.792656 | 0.015685 |
Consider a graph G with 8 vertices and 12 edges. Determine if G has an Eulerian path and if it has a Hamiltonian path. | tiger_lab_math | [
0.005071201361715794,
0.002221684670075774,
-0.012473165057599545,
-0.037902362644672394,
0.022724293172359467,
-0.054584864526987076,
-0.030041376128792763,
0.06507594138383865,
0.021763073280453682,
-0.016430038958787918,
0.018341396003961563,
0.029097190126776695,
-0.012694583274424076,
... | 0.824209 | 0.818634 | 0.005575 |
Let $x,$ $y,$ and $z$ be positive real numbers such that $xyz = 32.$ Find the minimum value of
\[x^2 + 4xy + 4y^2 + 2z^2.\] | tiger_lab_math | [
0.015540357679128647,
0.032810334116220474,
-0.05092906951904297,
-0.04437708854675293,
0.0428706519305706,
-0.016422564163804054,
-0.01842753030359745,
0.028674721717834473,
0.03695663809776306,
-0.030100025236606598,
0.024931950494647026,
0.032317496836185455,
-0.02318562939763069,
-0.02... | 0.847445 | 0.843235 | 0.00421 |
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