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 list of five positive integers has all of the following properties:
$\bullet$ The only integer in the list that occurs more than once is $8,$
$\bullet$ its median is $9,$ and
$\bullet$ its average (mean) is $10.$
What is the largest possible integer that could appear in the list? | tiger_lab_math | [
0.04011258855462074,
0.015967341139912605,
-0.034188855439424515,
-0.04842662066221237,
0.04176412150263786,
-0.02761450596153736,
-0.04379060119390488,
0.051965560764074326,
0.0550929419696331,
-0.027181806042790413,
0.02683120407164097,
0.020012125372886658,
-0.03489559143781662,
-0.0321... | 0.836452 | 0.82438 | 0.012072 |
compute the line integral of \int_K xy dx, \int_L xy dx, where K is a straight line from (0,0) to (1,1) and L is the Parabola y=x^2 from (0,0) to (1,1). return the answer as a list | tiger_lab_math | [
0.01679699867963791,
0.03294927626848221,
-0.030772127211093903,
-0.05503000691533089,
0.04585582762956619,
-0.005862495396286249,
-0.028612889349460602,
0.05379915609955788,
0.046707820147275925,
-0.01954510435461998,
0.032867997884750366,
0.03928102180361748,
-0.018433082848787308,
-0.01... | 0.825546 | 0.823315 | 0.002231 |
A factory produces two types of goods, X and Y. The demand for X is normally distributed with a mean of 500 and a standard deviation of 50, while the demand for Y is normally distributed with a mean of 700 and a standard deviation of 70. The production cost of X is $20 per unit and the production cost of Y is $30 per u... | tiger_lab_math | [
0.00956452265381813,
0.010502398014068604,
-0.08409816771745682,
-0.008791537024080753,
0.015935955569148064,
-0.025220457464456558,
-0.05122687667608261,
0.08017255365848541,
0.04654495790600777,
-0.020845526829361916,
-0.0011857427889481187,
0.025389302521944046,
-0.02828403189778328,
-0... | 0.815559 | 0.797245 | 0.018314 |
Consider the forced van der Pol oscillator represented by the following differential equation:
$y'' -\mu(1-y^2)y' +y =Acos(\omega t)$
where $\mu$, $A$ and $\omega$ are constants.
Suppose that $\mu=0.7$, $A=1.5$, $\omega=2$ and initial conditions $y(0) = 0.2$ and $y'(0) = 0.3$. Determine the amplitude of the response,... | tiger_lab_math | [
0.054052580147981644,
0.021257536485791206,
-0.030625909566879272,
-0.03963603079319,
0.01626720279455185,
-0.024270592257380486,
-0.028718169778585434,
0.100132517516613,
0.03020540438592434,
0.001690084463916719,
0.019090717658400536,
0.02714882418513298,
-0.04277610778808594,
-0.0147301... | 0.816387 | 0.806472 | 0.009915 |
Consider the algebraic curve defined by the equation $y^2=x^3+7x+1$. Find the Jacobian of this curve at the point $(1, \sqrt{9})$. | tiger_lab_math | [
0.02252560667693615,
0.0140932472422719,
-0.06812290847301483,
-0.05230662226676941,
0.010300020687282085,
-0.04595896601676941,
-0.001995666418224573,
0.07374981790781021,
0.04188590124249458,
-0.01836271397769451,
0.003432515077292919,
0.024527303874492645,
-0.038230571895837784,
0.00122... | 0.838379 | 0.824333 | 0.014046 |
A factory produces two products A and B. Each unit of product A requires 3 hours of processing and generates a profit of $10 while each unit of product B requires 5 hours of processing and generates a profit of $15. The factory has a maximum of 80 hours of processing time available per day. How many units of each produ... | tiger_lab_math | [
0.0044443560764193535,
0.004717422649264336,
-0.05449274182319641,
0.011325889267027378,
0.0012532516848295927,
-0.021382715553045273,
-0.05531918257474899,
0.07706943154335022,
0.03910446912050247,
-0.012229083105921745,
-0.006838931702077389,
0.033713627606630325,
-0.007195689715445042,
... | 0.809845 | 0.803746 | 0.006099 |
Dean is playing a game with calculators. The 42 participants (including Dean) sit in a circle, and Dean holds 3 calculators. One calculator reads 1, another 0, and the last one -1. Dean starts by pressing the cube button on the calculator that shows 1, pressing the square button on the one that shows 0, and on the calc... | tiger_lab_math | [
0.029400357976555824,
0.04014216363430023,
-0.04572923481464386,
-0.041246768087148666,
0.0329347662627697,
-0.01872287690639496,
-0.017111465334892273,
0.04666617140173912,
0.026032019406557083,
-0.010395865887403488,
0.018108317628502846,
0.03971455991268158,
-0.030117198824882507,
-0.03... | 0.81514 | 0.810216 | 0.004924 |
Sets $A$ and $B$, shown in the Venn diagram, are such that the total number of elements in set $A$ is twice the total number of elements in set $B$. Altogether, there are 3011 elements in the union of $A$ and $B$, and their intersection has 1000 elements. What is the total number of elements in set $A$?
[asy]
label("$... | tiger_lab_math | [
0.03489568829536438,
0.009803840890526772,
-0.0407833494246006,
-0.02313534915447235,
0.01921483688056469,
-0.02004208043217659,
-0.04423943907022476,
0.04203937202692032,
0.03108859993517399,
-0.011200765147805214,
0.01654851995408535,
0.025553228333592415,
-0.0035985326394438744,
-0.0088... | 0.834848 | 0.824628 | 0.01022 |
In a Bertrand competition, two firms sell an identical product at a price that depends on the competitors' price. The cost of production for each firm is $10. Suppose the demand function is given by Q = 100 - P, where Q is the quantity demanded and P is the price charged. Calculate the Nash equilibrium price and profit... | tiger_lab_math | [
0.019372019916772842,
0.011733523570001125,
-0.045609183609485626,
-0.016621243208646774,
0.010848022997379303,
-0.015316860750317574,
-0.030442869290709496,
0.06368593871593475,
0.03630973398685455,
-0.02833852730691433,
0.007736975327134132,
0.03578401356935501,
-0.0417645201086998,
-0.0... | 0.806251 | 0.800771 | 0.00548 |
Assume that there are four volunteers who have agreed to install a new security system in their neighborhood. Each volunteer is considering whether they should install the security system themselves, or make a $200 contribution to a pool fund, which is currently worth $600, to hire a professional installer. If at least... | tiger_lab_math | [
0.015122607350349426,
0.01592426933348179,
-0.04159657284617424,
-0.021301912143826485,
0.041315916925668716,
-0.028453918173909187,
-0.025173526257276535,
0.030842147767543793,
0.06150420010089874,
-0.005123619921505451,
0.022511910647153854,
0.018628841266036034,
-0.04443821310997009,
-0... | 0.794821 | 0.787862 | 0.006959 |
The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$
| tiger_lab_math | [
0.016208119690418243,
0.039667997509241104,
-0.04639741778373718,
-0.02434663288295269,
0.014248637482523918,
0.0013050433481112123,
-0.04896083474159241,
0.0626937672495842,
0.03166916221380234,
-0.02112189307808876,
0.009601311758160591,
0.030953381210565567,
-0.03516804799437523,
-0.016... | 0.848637 | 0.834232 | 0.014406 |
A manufacturing company has three machines: A, B, and C.
Machine A can produce 150 units in a day, while machine B can produce 200 units in a day. Machine C can produce 300 units in a day. The company has a demand for 1500 units in a week.
What is the optimal schedule for using these three machines to meet the demand ... | tiger_lab_math | [
0.03163091093301773,
0.010732091031968594,
-0.055636562407016754,
-0.01518760621547699,
0.027906952425837517,
-0.014914081431925297,
-0.04643826186656952,
0.08066702634096146,
0.02528582513332367,
-0.00602406682446599,
0.010746914893388748,
0.01675829477608204,
-0.017750022932887077,
-0.02... | 0.806901 | 0.800071 | 0.00683 |
Consider the following graph:
```
A--B
| |\
| | \
C--D--E
```
Can you find a Hamiltonian cycle in the above graph? If yes, write down the sequence of vertices visited in the cycle. If no, explain why. | tiger_lab_math | [
0.010362640023231506,
0.0004231552011333406,
-0.007622340228408575,
-0.0252007395029068,
0.005162208341062069,
-0.04494531452655792,
-0.025526633486151695,
0.07193905860185623,
0.03165716677904129,
-0.015987517312169075,
0.024426624178886414,
0.019578784704208374,
-0.027996942400932312,
-0... | 0.819952 | 0.810974 | 0.008978 |
For how many values of $a$ is it true that:
(1) $a$ is a positive integer such that $a \le 50$.
(2) the quadratic equation $x^2 + (2a+1)x + a^2 = 0$ has two integer solutions? | tiger_lab_math | [
0.027268026024103165,
0.013398834504187107,
-0.05317150056362152,
-0.029418786987662315,
0.02837417460978031,
-0.027527065947651863,
-0.008134559728205204,
0.07474897056818008,
0.04217104613780975,
-0.04601519554853439,
0.0198526531457901,
0.036526717245578766,
-0.04345446452498436,
-0.025... | 0.843812 | 0.836052 | 0.00776 |
There are two boxes, one containing 10 red balls and the other containing 10 green balls. You are allowed to move the balls between the boxes so that when you choose a box at random and a ball at random from the chosen box, the probability of getting a red ball is maximized. This maximum probability is
Answer Choices: ... | tiger_lab_math | [
0.010762520134449005,
0.014478662051260471,
-0.04955769702792168,
-0.02054971270263195,
0.028847549110651016,
-0.02519766241312027,
-0.028994176536798477,
0.06432057917118073,
0.03724319115281105,
-0.023502295836806297,
0.02083166129887104,
0.033634137362241745,
-0.017498169094324112,
-0.0... | 0.816245 | 0.813685 | 0.002561 |
In a communication network consisting of 5 nodes connected by directed links, find the maximum flow that can be sent from node 1 to node 5 if the capacity of each link is as follows:
- capacity of the link from node 1 to node 2 is 5
- capacity of the link from node 1 to node 3 is 4
- capacity of the link from node 2 ... | tiger_lab_math | [
0.016660358756780624,
0.027263442054390907,
-0.03683173283934593,
-0.029408151283860207,
0.025871681049466133,
-0.022950420156121254,
-0.042041949927806854,
0.06335369497537613,
0.03167600929737091,
-0.0021115534473210573,
0.0234067440032959,
0.033445701003074646,
-0.02388869784772396,
-0.... | 0.813371 | 0.805524 | 0.007847 |
Let $b_1$, $b_2$, $b_3$, $c_1$, $c_2$, and $c_3$ be real numbers such that for every real number $x$, we have
\[
x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3).
\]Compute $b_1 c_1 + b_2 c_2 + b_3 c_3$. | tiger_lab_math | [
0.03341393917798996,
0.04810994118452072,
-0.038534361869096756,
-0.03216269612312317,
0.04367867112159729,
-0.03156140074133873,
-0.014253041706979275,
0.05775676295161247,
0.04406500235199928,
-0.01722785085439682,
0.010073430836200714,
0.037687644362449646,
-0.05276313051581383,
-0.0313... | 0.856015 | 0.837791 | 0.018224 |
A two-player zero-sum game has the following payoff matrix:
| | Player 2- Heads | Player 2- Tails |
|---|---|---|
| **Player 1- Heads** | 3 | -2 |
| **Player 1- Tails** | -1 | 4 |
Find the optimal strategies for both players and the value of the game. | tiger_lab_math | [
0.013927587307989597,
0.0219016894698143,
-0.050185322761535645,
-0.0223102904856205,
0.02982005849480629,
-0.0016305649187415838,
-0.009564503096044064,
0.05199502035975456,
0.03206411749124527,
-0.040831319987773895,
0.027529755607247353,
0.026487145572900772,
-0.028490792959928513,
-0.0... | 0.824724 | 0.815445 | 0.009279 |
Consider a Morse-Smale complex with 10 critical points, consisting of 6 index-0 critical points and 4 index-1 critical points. Suppose that the number of index-2 critical points is equal to the number of all other critical points combined. What is the total number of critical points in this Morse-Smale complex? | tiger_lab_math | [
0.009354663081467152,
0.027454134076833725,
-0.030616777017712593,
-0.03288784995675087,
0.046030741184949875,
-0.02412799373269081,
-0.032046519219875336,
0.04613320156931877,
0.013567596673965454,
-0.010074825026094913,
0.0020085591822862625,
0.022927969694137573,
-0.05078756809234619,
-... | 0.824133 | 0.82229 | 0.001843 |
Andrew is having two of his friends over for a sleepover. For one friend, Brian, he asks his mother to buy 3 donuts. He asks for his other friend, Samuel, to get the same. A day before the sleepover Andrew ends up inviting two more friends and asks his mother to buy them the same amount of donuts. Just in case, Andrew’... | tiger_lab_math | [
0.035044990479946136,
0.009928920306265354,
-0.035599421709775925,
-0.03490987792611122,
0.02169547602534294,
-0.009297442622482777,
-0.025073453783988953,
0.05543755367398262,
0.029286900535225868,
-0.024181239306926727,
0.005376609042286873,
0.03599104657769203,
-0.04524651914834976,
-0.... | 0.846845 | 0.834201 | 0.012644 |
Triangle $ABC$ is isosceles with angle $A$ congruent to angle $B$. The measure of angle $C$ is 30 degrees more than the measure of angle $A$. What is the number of degrees in the measure of angle $C$? | tiger_lab_math | [
0.018097691237926483,
0.0028379971627146006,
-0.03878583386540413,
-0.026091329753398895,
0.026639800518751144,
-0.01493395771831274,
-0.022072505205869675,
0.03407130762934685,
0.03394831344485283,
-0.01939593255519867,
0.02295941300690174,
0.031323451548814774,
-0.04358796030282974,
-0.0... | 0.823661 | 0.817939 | 0.005723 |
Let $f(x)=\frac{3x^2+5x+8}{x^2-x+4}$ and $g(x)=x-1$. Find $f(g(x))+g(f(x))$, evaluated when $x=1$. Please write a program to solve it | tiger_lab_math | [
0.019452976062893867,
0.0410439670085907,
-0.05008931830525398,
-0.053553055971860886,
0.04220813512802124,
-0.027495088055729866,
-0.018967539072036743,
0.07948385179042816,
0.028629282489418983,
-0.019499916583299637,
0.019550716504454613,
0.02469097636640072,
-0.021773675456643105,
-0.0... | 0.842518 | 0.837368 | 0.00515 |
Consider the function f(x) = 3x^2 - 6x + 4. Find the minimum value of this function using the differential evolution optimization algorithm, with the initial search space ranging from x = -10 to x = 10 and a population size of 50. Use a mutation factor of 0.5 and a recombination constant of 0.7, and terminate the optim... | tiger_lab_math | [
0.019051266834139824,
0.014500653371214867,
-0.04242498800158501,
-0.03209086135029793,
0.046311475336551666,
-0.041962672024965286,
-0.03841787949204445,
0.0751776471734047,
0.04342983290553093,
-0.006955186370760202,
0.017589731141924858,
0.015057914890348911,
-0.02392570860683918,
-0.01... | 0.815863 | 0.814469 | 0.001394 |
A manufacturer produces two types of products, A and B. The products require two types of machines, X and Y, to be manufactured. Each unit of product A requires 2 hours of machine X time and 1 hour of machine Y time, while each unit of product B needs 1 hour of machine X time and 3 hours of machine Y time. The total we... | tiger_lab_math | [
0.018767712637782097,
0.006289191078394651,
-0.04391752555966377,
0.003081643721088767,
0.0036792962346225977,
-0.019094888120889664,
-0.05383621156215668,
0.08785183727741241,
0.02782391384243965,
-0.009364202618598938,
0.0025088274851441383,
0.03417013958096504,
-0.02598424442112446,
-0.... | 0.806423 | 0.788265 | 0.018158 |
Suppose there is a 4x4 Latin square filled with numbers 1, 2, 3, and 4 such that each row and each column has all the four numbers exactly once. Find the total number of distinct Latin squares possible for this 4x4 grid. | tiger_lab_math | [
0.0073232161812484264,
0.032613541930913925,
-0.028348365798592567,
-0.034796323627233505,
0.043570466339588165,
-0.03793707862496376,
-0.02645356021821499,
0.045929692685604095,
0.038707006722688675,
-0.0031022357288748026,
0.02240866981446743,
0.03213915601372719,
-0.042632170021533966,
... | 0.831471 | 0.825359 | 0.006112 |
Emily wants to find the zeros and poles of the complex function $f(z) = \frac{z^2+1}{(z-2)^2}$.
(a) Find the zeros of $f(z)$.
(b) Find the poles of $f(z)$.
(c) Determine the order of each pole.
(d) Sketch the poles and zeros of $f(z)$ on an Argand diagram. | tiger_lab_math | [
0.04715399444103241,
0.04051622375845909,
-0.033463627099990845,
-0.044890668243169785,
0.031829584389925,
-0.03070700354874134,
0.0015401527052745223,
0.03588531166315079,
0.019677115604281425,
0.01048707403242588,
0.02251727506518364,
0.010482840240001678,
-0.025752270594239235,
-0.02858... | 0.821301 | 0.810435 | 0.010866 |
Barney's grocery store sold out all of its items at the beginning of the pandemic, so they ordered extra items to restock the shelves. However, they ended up ordering far too much and have to keep the leftover items in the storeroom. If they ordered 4458 items, sold another 1561 items that day, and have 575 items in th... | tiger_lab_math | [
0.03733035549521446,
0.006968878209590912,
-0.03569938614964485,
-0.046255264431238174,
0.028922786936163902,
-0.02720819041132927,
-0.05405155569314957,
0.05075531080365181,
0.02440696209669113,
-0.02573571540415287,
0.02026679739356041,
0.03734521195292473,
-0.05400637537240982,
0.003407... | 0.823364 | 0.822459 | 0.000904 |
A water tank has a height of 3 meters and a diameter of 2 meters. Water is being pumped at a rate of 5 liters per second into the tank. Assuming the tank is initially empty and the water flows out of a hole of 5 cm diameter at the bottom of the tank, use the Navier-Stokes equations to determine the time it will take fo... | tiger_lab_math | [
0.03732401505112648,
0.03920173645019531,
-0.035283125936985016,
-0.0276618842035532,
0.04060354828834534,
-0.03840116411447525,
-0.047245725989341736,
0.07443243265151978,
0.00435809837654233,
0.007525333669036627,
0.010479363612830639,
0.02171815186738968,
-0.0435684509575367,
-0.0222070... | 0.789384 | 0.783377 | 0.006007 |
We have a triangle $\triangle ABC$ and a point $K$ on $BC$ such that $AK$ is an altitude of $\triangle ABC$. If $AC = 10,$ $BK = 7$, and $BC = 13,$ then what is the area of $\triangle ABC$? | tiger_lab_math | [
0.03226048871874809,
0.022296760231256485,
-0.05146711692214012,
-0.04592157527804375,
0.034555383026599884,
-0.02484637312591076,
-0.0028768274933099747,
0.04063010215759277,
0.05291302502155304,
-0.007195367943495512,
0.016274970024824142,
0.02811911143362522,
-0.04008699208498001,
-0.01... | 0.828203 | 0.821746 | 0.006457 |
Determine which group of order 8 is isomorphic to the following presentation:
G = {e, a, b, c, d, f, g, h | a^2 = b^2 = c^2 = d^2 = e, af = bf = cg = dg = e, ag = bh, ah = bg, fg = gf} | tiger_lab_math | [
0.02309536561369896,
0.02272649295628071,
-0.02022063545882702,
-0.05836256965994835,
0.029143204912543297,
-0.06430749595165253,
-0.005533193238079548,
0.06887350231409073,
0.03898153454065323,
-0.012166609056293964,
0.012437744997441769,
0.008981331251561642,
-0.03246741369366646,
-0.031... | 0.823404 | 0.821316 | 0.002088 |
Four people sit around a circular table, and each person will roll a standard six-sided die. What is the probability that no two people sitting next to each other will roll the same number after they each roll the die once? Express your answer as a common fraction. | tiger_lab_math | [
0.018888486549258232,
0.027210738509893417,
-0.03692583367228508,
-0.03244076296687126,
0.03280512988567352,
-0.02122114598751068,
0.001730831223540008,
0.040355440229177475,
0.03433681279420853,
-0.006529762409627438,
0.03467928245663643,
0.013531232252717018,
-0.0037488704547286034,
-0.0... | 0.826913 | 0.823751 | 0.003162 |
Let's say we have a composite function f(x) = (3x^2 + 5x - 2)^3, where g(x) = 3x^2 + 5x - 2 and h(x) = x^3. Use the chain rule to find f'(x) at x = 2. | tiger_lab_math | [
0.026156047359108925,
0.020393015816807747,
-0.02698957733809948,
-0.040965624153614044,
0.036290910094976425,
-0.029113298282027245,
-0.020748060196638107,
0.07245659828186035,
0.008492955006659031,
-0.0008484271820634604,
0.020844493061304092,
0.04365590959787369,
-0.029681246727705002,
... | 0.822932 | 0.817586 | 0.005346 |
A population of foxes and rabbits in a forest can be modeled using the Lotka-Volterra equations:
dR/dt = aR - bRF
dF/dt = -cF + dbRF
where R is the number of rabbits, F is the number of foxes, and a, b, c, and d are constants.
The initial population of rabbits is 100 and the initial population of foxes is 50. The va... | tiger_lab_math | [
0.009945672936737537,
0.022589145228266716,
-0.06943338364362717,
-0.020014608278870583,
0.02404281310737133,
-0.019119171425700188,
-0.028972862288355827,
0.06950006633996964,
0.03521471098065376,
-0.016274284571409225,
0.013123524375259876,
0.04318255931138992,
-0.05282163247466087,
-0.0... | 0.802773 | 0.799638 | 0.003135 |
Consider the reaction-diffusion equation for the concentration profile of a chemical species in one dimension given by:
∂C/∂t = D ∂²C/∂x² + kC - λC²,
where C(x, t) is the concentration of the chemical species at position x and time t, D is the diffusion coefficient, k is the reaction rate constant, and λ is the nonli... | tiger_lab_math | [
0.008856282569468021,
0.018875349313020706,
-0.05741206556558609,
-0.030296994373202324,
0.039873890578746796,
-0.03449259698390961,
-0.0376940593123436,
0.09380453079938889,
0.03664732724428177,
-0.0017969364998862147,
0.010013674385845661,
0.011437831446528435,
-0.019085906445980072,
-0.... | 0.780683 | 0.772029 | 0.008654 |
Every bag of Dummies (a chocolate candy) contains the same number of pieces.
The Dummies in one bag can't be divided equally among $9$ kids, because after each kid gets the same (whole) number of pieces, $7$ pieces are left over.
If the Dummies in three bags are divided equally among $9$ kids, what is the smallest nu... | tiger_lab_math | [
0.014368386939167976,
0.024666648358106613,
-0.02703518234193325,
-0.03966300189495087,
0.02735220640897751,
-0.02146567776799202,
-0.012302182614803314,
0.060868408530950546,
0.02714984118938446,
-0.030260305851697922,
0.03412648290395737,
0.027509085834026337,
-0.03579622507095337,
-0.02... | 0.856659 | 0.842114 | 0.014545 |
A manufacturing company produces three different types of products A, B, and C. The profit margins for each type of product are as follows:
- Product A has a profit margin of $5 per unit produced
- Product B has a profit margin of $7 per unit produced
- Product C has a profit margin of $9 per unit produced
The comp... | tiger_lab_math | [
0.028127441182732582,
0.012136219069361687,
-0.06024821475148201,
-0.010440298356115818,
0.023472266271710396,
-0.016126951202750206,
-0.038191504776477814,
0.08288128674030304,
0.016120795160531998,
-0.005907035898417234,
0.001110557815991342,
0.038787841796875,
-0.020046843215823174,
-0.... | 0.819626 | 0.810164 | 0.009462 |
A factory produces two products, X and Y. The total production time is limited to 16 hours. Product X requires 2 hours of production time and can be sold for $50 per unit, while product Y requires 3 hours of production time and can be sold for $60 per unit. The factory wants to maximize its profit. Using the Firefly Al... | tiger_lab_math | [
0.004109459463506937,
0.01251051016151905,
-0.045831382274627686,
-0.014542569406330585,
0.025351690128445625,
-0.01907961070537567,
-0.040493711829185486,
0.08865774422883987,
0.03602840006351471,
-0.02017403021454811,
0.008041520603001118,
0.020177694037556648,
-0.026310402899980545,
-0.... | 0.812416 | 0.805423 | 0.006992 |
Consider the function $f(z) = z^2-2z+2$. Determine if $f(z)$ is analytic in the region enclosed by the circle $|z-1| = 1.$ | tiger_lab_math | [
0.020211195573210716,
0.0395696684718132,
-0.03524443134665489,
-0.03704424202442169,
0.0361003503203392,
-0.0026981758419424295,
0.00628926046192646,
0.059409335255622864,
0.017354145646095276,
-0.006335626821964979,
0.024709220975637436,
0.009156924672424793,
-0.01458689197897911,
-0.005... | 0.834084 | 0.818203 | 0.015881 |
Let G be a group and a,b be elements of G. Which of the following statements is always true? I. If a*b = b*a for all a and b in G, then G is an abelian group. II. If a*b = b*a for some a and b in G, then G is an abelian group. III. If a*b = a for some a and b in G, then b is the identity element of G. Answer Choices: ... | tiger_lab_math | [
0.05536571517586708,
0.03949425369501114,
-0.024265632033348083,
-0.038530878722667694,
0.043669961392879486,
-0.07268276065587997,
0.01497136801481247,
0.07231757789850235,
0.02918994054198265,
-0.01000309269875288,
0.015991171821951866,
0.00866001844406128,
-0.03763271123170853,
-0.02568... | 0.810519 | 0.809673 | 0.000847 |
Let X be a compact manifold and let $H^{k}(X,\mathbb{Z})$ be its cohomology group with integer coefficients. Suppose there are 10 linearly independent elements in $H^{k}(X,\mathbb{Z})$. What is the minimum value of k? | tiger_lab_math | [
0.018342681229114532,
0.026456458494067192,
-0.041128721088171005,
-0.03344394266605377,
0.04446189105510712,
-0.021628549322485924,
-0.0026919811498373747,
0.05053522437810898,
0.041462138295173645,
-0.027354298159480095,
0.005082734860479832,
0.023727571591734886,
-0.03666060045361519,
-... | 0.832739 | 0.824794 | 0.007944 |
Consider a manifold with a metric tensor given by:
g = dx^2 + dy^2 + dz^2
Find the Riemann curvature tensor of the manifold at point (1, 2, -3). | tiger_lab_math | [
0.009705048985779285,
0.02832031436264515,
-0.031284403055906296,
-0.06216193363070488,
0.029388554394245148,
-0.005605733022093773,
-0.010524190030992031,
0.050118762999773026,
0.019610531628131866,
-0.02381875552237034,
0.00912381149828434,
0.03102225996553898,
-0.04674190282821655,
-0.0... | 0.830277 | 0.825569 | 0.004708 |
Consider the function $f(z) = \frac{4z-15}{(z-3)(z-5)}$. Find the Laurent series expansion of $f(z)$ centered at $z = 3$ for $2 < |z-3| < 4$. Determine the convergence of the series. | tiger_lab_math | [
0.017146585509181023,
0.04222364351153374,
-0.04029686003923416,
-0.03941969946026802,
0.040794454514980316,
0.018057765439152718,
0.0025485940277576447,
0.05787447839975357,
0.01555023342370987,
0.006915816105902195,
0.03122272714972496,
0.008168964646756649,
-0.035100918263196945,
-0.009... | 0.820318 | 0.818571 | 0.001747 |
Let $M$ be a closed, orientable two-dimensional surface. Calculate the first de Rham cohomology group $H^1_{dR}(M)$ of $M$. | tiger_lab_math | [
0.025673143565654755,
0.03407364711165428,
-0.02806459739804268,
-0.045303888618946075,
0.035120803862810135,
-0.026741979643702507,
-0.01896035671234131,
0.05639856308698654,
0.04963650181889534,
-0.001745015848428011,
0.009094874374568462,
0.028852427378296852,
-0.02992234192788601,
-0.0... | 0.822455 | 0.80806 | 0.014395 |
A metal plate is 100 cm wide and 50 cm high. A circular hole with a diameter of 20 cm is to be cut out from the middle of the plate. Determine the maximum area of the remaining metal plate after the circular hole has been cut out by applying topological optimization. | tiger_lab_math | [
0.01766049675643444,
0.02200908772647381,
-0.05679957568645477,
-0.037688858807086945,
0.024564391002058983,
-0.02911774255335331,
-0.04800152778625488,
0.052012160420417786,
0.04519933834671974,
-0.00991416908800602,
-0.0019464829238131642,
0.036909062415361404,
-0.04702645167708397,
0.00... | 0.816471 | 0.810222 | 0.006249 |
Consider an auction where a seller wants to sell a painting to one of three bidders. The three bidders' valuations of the painting are $V_1 = 50$, $V_2 = 80$, and $V_3 = 100$. If the seller uses a second-price sealed-bid auction (also known as a Vickrey auction), how much should each bidder bid in order to maximize the... | tiger_lab_math | [
-0.01829877495765686,
0.005108888726681471,
-0.04232092946767807,
-0.01471329852938652,
0.03088114596903324,
-0.012451516464352608,
-0.02851063571870327,
0.042860839515924454,
0.04308977350592613,
-0.024741383269429207,
0.028781183063983917,
0.021395031362771988,
-0.03774008899927139,
-0.0... | 0.812514 | 0.802951 | 0.009563 |
Here's a possible numerical problem for a math student on the Feigenbaum constant and universality in chaotic systems:
Suppose we have a logistic map given by the function f(x) = rx(1-x), where x is the population density and r is a control parameter. Starting with an initial condition of x0 = 0.5, we iterate the map ... | tiger_lab_math | [
0.0067385099828243256,
0.013132991269230843,
-0.044994644820690155,
-0.03739519044756889,
0.03866257891058922,
-0.028179703280329704,
-0.03650016710162163,
0.05141233280301094,
0.022664884105324745,
-0.010891102254390717,
0.007767970208078623,
0.0381055511534214,
-0.028447993099689484,
-0.... | 0.828715 | 0.816029 | 0.012687 |
Consider a surface in 3-dimensional Euclidean space having parametric equations:
x(u,v) = u
y(u,v) = v
z(u,v) = u^2 - v^2
Find the sectional curvature K at the point (1, 1, 0) of the surface. | tiger_lab_math | [
0.008064521476626396,
0.03607388213276863,
-0.03937477618455887,
-0.05599498003721237,
0.045638736337423325,
0.0023059165105223656,
-0.001866688602603972,
0.06408954411745071,
0.012392327189445496,
-0.012866314500570297,
0.03271397203207016,
0.037760667502880096,
-0.022461775690317154,
0.0... | 0.832321 | 0.816674 | 0.015647 |
Consider the following undirected graph:
```
A -- B -- C
| | |
D -- E -- F
```
Does this graph have an Eulerian circuit? If so, what is the Eulerian circuit? Does it have a Hamiltonian circuit? If so, what is the Hamiltonian circuit? | tiger_lab_math | [
-0.004118484444916248,
0.011911294423043728,
-0.025909705087542534,
-0.022894131019711494,
0.025566337630152702,
-0.03946797177195549,
-0.02760358154773712,
0.0681992620229721,
0.032585158944129944,
-0.008608702570199966,
0.02374975197017193,
0.032386116683483124,
-0.009208595380187035,
-0... | 0.811658 | 0.806452 | 0.005206 |
Let K be a simplicial complex with a vertex set {v1, v2, v3, v4} and simplices {v1, v2, v3}, {v2, v3, v4}, and {v1, v3, v4}. Find the number of maximal simplices in K. | tiger_lab_math | [
0.029231790453195572,
0.024614017456769943,
-0.031852833926677704,
-0.04577375575900078,
0.036857958883047104,
-0.01399451307952404,
-0.011811564676463604,
0.038132671266794205,
0.041950639337301254,
-0.0020435303449630737,
0.029322998598217964,
0.013652080669999123,
-0.014482737518846989,
... | 0.83294 | 0.819927 | 0.013013 |
Let rectangle R = [1, 2.5] * [1, 2]. Calculate the Riemann Sum S_{3,2} for \int \int_{R} xy dA for the integral, using the lower-left vertex of rectangles as sample points. Please write a program to solve it | tiger_lab_math | [
0.014972148463129997,
0.04037443548440933,
-0.03923609480261803,
-0.04734085127711296,
0.01501664612442255,
-0.00004434120637597516,
-0.01533924974501133,
0.04882092401385307,
0.01999276503920555,
-0.002766910009086132,
0.014259668998420238,
0.044634707272052765,
-0.02783825807273388,
-0.0... | 0.832156 | 0.819472 | 0.012684 |
Consider a two-dimensional torus defined by the equation (cos(u)+3)² + (sin(u)-4)² + (cos(v)+1)² + (sin(v)-2)² = 4. Determine the group of isometries that preserve this torus. | tiger_lab_math | [
0.0387723408639431,
0.0360339879989624,
-0.042425405234098434,
-0.031138520687818527,
0.04774419218301773,
-0.02997317910194397,
-0.02465713582932949,
0.06475798785686493,
0.029919298365712166,
-0.0006230337312445045,
0.012491635978221893,
0.025647465139627457,
-0.02438194677233696,
-0.007... | 0.835031 | 0.822142 | 0.012889 |
What is the total number of cubes that make up an n-th iteration of the Menger sponge construction, where n is a positive integer?
For example, for the first iteration (n=1), the Menger sponge consists of 20 cubes. For the second iteration (n=2), it consists of 400 cubes. | tiger_lab_math | [
0.022547122091054916,
0.01869591511785984,
-0.03954664245247841,
-0.023338699713349342,
0.040201250463724136,
-0.033181264996528625,
-0.05968360975384712,
0.047549765557050705,
0.032608650624752045,
-0.0032794876024127007,
0.013264153152704239,
0.026452751830220222,
-0.03849102556705475,
-... | 0.800296 | 0.79807 | 0.002225 |
Let U be the universal set such that U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and let A be a subset of U such that A = {1, 3, 5, 7, 9}. Find the complement of set A, and then find the union of set A and its complement. Verify that the result is equal to the universal set U. | tiger_lab_math | [
0.027835482731461525,
0.02557917684316635,
-0.01831420324742794,
-0.03547630086541176,
0.029314430430531502,
-0.043083835393190384,
-0.006448233500123024,
0.04618865251541138,
0.032999392598867416,
-0.018050793558359146,
0.022678863257169724,
0.026312481611967087,
-0.03730767220258713,
-0.... | 0.830202 | 0.817737 | 0.012465 |
Suppose $f(x,y) = \sqrt{x^2 + y^2}$ is a function from $\mathbb{R}^2$ to $\mathbb{R}$. Determine if $f$ is a smooth map when $\mathbb{R}^2$ is given the standard differential structure. | tiger_lab_math | [
0.002343679778277874,
0.02767559513449669,
-0.0443047471344471,
-0.04964644834399223,
0.04617321491241455,
-0.009652037173509598,
-0.03051047772169113,
0.06790737062692642,
0.0307463426142931,
-0.02251916192471981,
0.017360111698508263,
0.025813955813646317,
-0.036518342792987823,
-0.01663... | 0.812896 | 0.795964 | 0.016932 |
Consider the function f(x) = x^2 + 2x + 5, where x ∈ [-5,5]. Use quasi-Newton methods to find the minimum value of f(x) and the value of x at which the minimum occurs within a tolerance of 0.01. | tiger_lab_math | [
0.009333314374089241,
0.008279534988105297,
-0.04040885344147682,
-0.0321192741394043,
0.04931265860795975,
-0.029345490038394928,
-0.02095705457031727,
0.06404218077659607,
0.02145383134484291,
0.0017412389861419797,
0.01020805723965168,
0.02143511176109314,
-0.01878037303686142,
-0.01528... | 0.822232 | 0.818009 | 0.004224 |
In triangle $ABC$, $AB$ is congruent to $AC$, the measure of angle $ABC$ is $72^{\circ}$ and segment $BD$ bisects angle $ABC$ with point $D$ on side $AC$. If point $E$ is on side $BC$ such that segment $DE$ is parallel to side $AB$, and point $F$ is on side $AC$ such that segment $EF$ is parallel to segment $BD$, how m... | tiger_lab_math | [
0.02637391909956932,
0.004581455606967211,
-0.03439675271511078,
-0.048878513276576996,
0.02811175025999546,
-0.022939283400774002,
-0.009673123247921467,
0.04921865090727806,
0.02815054915845394,
0.01597100868821144,
0.024454418569803238,
0.023537365719676018,
-0.021276229992508888,
-0.01... | 0.836909 | 0.825013 | 0.011896 |
Find the value of the integral $\int_C z^2 \cos(z)\ dz$, where $C$ is the circle $|z-1| = 2$ traversed counterclockwise. | tiger_lab_math | [
0.013391261920332909,
0.04005979374051094,
-0.03544243052601814,
-0.03052101843059063,
0.031514547765254974,
-0.00643080985173583,
-0.010919962078332901,
0.027564404532313347,
0.03363937884569168,
-0.010018338449299335,
0.036651741713285446,
0.004428823012858629,
-0.018747948110103607,
-0.... | 0.826187 | 0.822837 | 0.00335 |
At the 2007 Math Olympics, Team Canada won $17$ out of a possible $100$ medals. Which one of the following is closest to the fraction of medals that they won? $$
\frac{1}{4} \qquad \frac{1}{5} \qquad \frac{1}{6} \qquad \frac{1}{7} \qquad \frac{1}{8}
$$ Let's write a Python program. | tiger_lab_math | [
0.033761948347091675,
0.03832545131444931,
-0.050506968051195145,
-0.05044996365904808,
0.033779535442590714,
-0.027082527056336403,
-0.0269374568015337,
0.04871983081102371,
0.027890153229236603,
-0.018361635506153107,
0.00357438949868083,
0.02665671706199646,
-0.018597198650240898,
-0.00... | 0.842616 | 0.840212 | 0.002404 |
A metal sphere with a radius of 5 cm is placed in a region of space where the electrostatic potential satisfies Laplace's equation. The potential on the surface of the sphere is 25 V. Determine the potential at a distance of 10 cm from the center of the sphere. Assume that the potential elsewhere on the surface is zero... | tiger_lab_math | [
0.009311669506132603,
0.03713592141866684,
-0.03740581125020981,
-0.04845922440290451,
0.03949440270662308,
-0.047248724848032,
-0.015420623123645782,
0.05919082090258598,
0.04784121736884117,
-0.010683574713766575,
0.018957221880555153,
0.04755439609289169,
-0.03335154801607132,
-0.024130... | 0.795797 | 0.789509 | 0.006288 |
In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. So the middle number in Row 2 is $2$ because $1+1=2.$ What is the sum of the numbers in Row 8 of Pascal's Triangle?
\begin{tabular}{rccccccccccc}
Row 0:& & & & & & 1\\\noalign{\smalls... | tiger_lab_math | [
0.054220303893089294,
0.01793539710342884,
-0.03922721743583679,
-0.049786634743213654,
0.030001286417245865,
-0.01567809283733368,
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0.012310431338846684,
0.03012032061815262,
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-0.... | 0.831932 | 0.817109 | 0.014823 |
A furniture company wants to cluster their customers based on their spending habits. They recorded the amount spent by each customer in the past 6 months, in thousands of dollars:
Customer 1: 5, 3, 4, 6, 2, 5
Customer 2: 7, 5, 6, 8, 4, 5
Customer 3: 1, 2, 3, 4, 5, 6
Customer 4: 2, 4, 1, 3, 2, 5
Customer 5: 6, 8, 7, 9,... | tiger_lab_math | [
0.016117218881845474,
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0.000... | 0.804254 | 0.797374 | 0.006879 |
Find the maximum value of the expression $x^{2} - xy + y^{2}$ subject to the constraint $x + y \leq 5$ and $x \geq 0, y \geq 0$ using semi-definite programming. | tiger_lab_math | [
0.003440654603764415,
0.02547142282128334,
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0.04089701920747757,
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0.06120830774307251,
0.06913723796606064,
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0.020246092230081558,
0.0322941392660141,
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0.00... | 0.825801 | 0.817222 | 0.008579 |
A math student is given the following specific problem to solve based on the subtopic Theta functions and their applications in Algebraic geometry:
Suppose you have a Riemann surface of genus 3 with a basis of cycles A1, A2, A3 and B1, B2, B3. Compute the period lattice and the corresponding lattice in the complex pla... | tiger_lab_math | [
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0.0348043330013752,
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Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some citie... | tiger_lab_math | [
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... | 0.833062 | 0.820184 | 0.012878 |
$\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | tiger_lab_math | [
0.02707977406680584,
0.051999688148498535,
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0.038024891167879105,
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0.06987963616847992,
0.05203951895236969,
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0.015792254358530045,
0.032976310700178146,
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Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?
Please respond by writing a program in Python. | tiger_lab_math | [
0.010606043972074986,
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0.007930692285299301,
0.0541851744055748,
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Alice and Bob, two volunteers at a hospital, have the option to work either the morning shift or the evening shift. If they both work the same shift, they each receive a volunteer certificate. However, if they work opposite shifts, they each receive a smaller certificate. The payoffs for each scenario are as follows:
... | tiger_lab_math | [
0.021462803706526756,
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0.000724084151443094,
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0.06890231370925903,
0.041240062564611435,
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0.019457412883639336,
0.0285316314548254,
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... | 0.817736 | 0.808747 | 0.008989 |
A circle centered at $A$ with a radius of $1$ and a circle centered at $B$ with a radius of $4$ are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle? [asy]
draw((-3,0)--(7.5,0));
draw(Circle((-1,1),1),linewidth(0... | tiger_lab_math | [
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0.017660751938819885,
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Let X be the algebraic variety defined by the equation x^3 + y^3 - 3xy = 0. Determine the singular point(s) of X and classify them as either ordinary or non-ordinary. | tiger_lab_math | [
0.003525734879076481,
0.012817319482564926,
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0.025918947532773018,
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0.009200412780046463,
0.025953255593776703,
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-0.0... | 0.832735 | 0.825379 | 0.007356 |
An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction. | tiger_lab_math | [
0.028358688578009605,
0.008295713923871517,
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0.02092650532722473,
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0.05218065530061722,
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0.029524609446525574,
0.03853660076856613,
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-0.02... | 0.816855 | 0.813882 | 0.002973 |
A graph contains 7 vertices. Each vertex is colored either blue or green. If each vertex is colored randomly with equal probability of being either blue or green, what is the probability that the graph contains at least one blue vertex and at least one green vertex? | tiger_lab_math | [
-0.007644997909665108,
0.011456887237727642,
-0.03514188528060913,
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0.007460413035005331,
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0.03671535849571228,
0.03444407135248184,
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0.01350067276507616,
0.018920689821243286,
0.003020102623850107,
-0.01... | 0.817209 | 0.814619 | 0.002591 |
Statement 1 | The set of all integers under addition is an abelian group. Statement 2 | If G is a group and H is a subgroup of G, then the order of H divides the order of G. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
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0.0015396513044834137,
0.011349222622811794,
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-0.0127... | 0.811824 | 0.807832 | 0.003993 |
Consider the circle defined as the set of all points in the plane at a distance of 1 unit from the origin. Let's define a smooth function $f:S^1\rightarrow\mathbb{R}$, which assigns to each point on the circle a real number. Determine the number of critical points of function $f$ on the circle. | tiger_lab_math | [
0.008286870084702969,
0.03848862648010254,
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0.04283081740140915,
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0.0417693629860878,
0.025367392227053642,
0.002663422841578722,
0.007590807043015957,
0.03866332769393921,
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-0.01685... | 0.836582 | 0.807052 | 0.02953 |
Let $0 \le a,$ $b,$ $c \le 1.$ Find the maximum value of
\[\sqrt{abc} + \sqrt{(1 - a)(1 - b)(1 - c)}.\] | tiger_lab_math | [
0.036812227219343185,
0.04517868161201477,
-0.04137337580323219,
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0.040754273533821106,
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0.05329224839806557,
0.02671855315566063,
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0.01640155538916588,
0.04234422743320465,
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-0.02... | 0.848641 | 0.833372 | 0.015268 |
Consider the intuitionistic logic proposition P: "There exists a natural number n such that n + 1 = 4." Is P provable in intuitionistic logic? Why or why not? | tiger_lab_math | [
0.019368121400475502,
0.01440507173538208,
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0.03422561287879944,
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0.027580270543694496,
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0.018139982596039772,
0.033500734716653824,
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-0... | 0.8073 | 0.804541 | 0.002758 |
Suppose there are two types of cars: high-quality and low-quality. The seller knows the type of each car, but the buyer cannot distinguish between them. The value of a high-quality car to the buyer is $10,000, and the value of a low-quality car is $6,000. The seller knows that half of the cars are high-quality and half... | tiger_lab_math | [
0.0014450079761445522,
0.006267800461500883,
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0.0057014236226677895,
0.0280904658138752,
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0.07139914482831955,
0.04636675491929054,
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0.023687096312642097,
0.04445027932524681,
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-0.... | 0.804174 | 0.792281 | 0.011892 |
A group has an order of 35 which is not cyclic. Determine the possible values of p and q, where p and q are distinct primes, such that this group is isomorphic to the direct product of two cyclic groups of orders p and q. | tiger_lab_math | [
0.027256837114691734,
0.01774817518889904,
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0.01104107778519392,
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0.03303654491901398,
0.016409214586019516,
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-0.0... | 0.826947 | 0.819705 | 0.007242 |
A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in s... | tiger_lab_math | [
0.014987212605774403,
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0.02998616173863411,
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0.04051467031240463,
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0.00007497942715417594,
0.019243422895669937,
0.01873021014034748,
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A math student is performing an experiment to measure the heights of 10 students in their class. However, they notice that one student has a height that is significantly different from the other measurements. Using robust statistics, they want to find the median height of the remaining 9 students.
Here are the 10 hei... | tiger_lab_math | [
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0.000525998417288065,
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0.03324307128787041,
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0.005233599338680506,
0.0193285271525383,
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-0.01... | 0.830892 | 0.821531 | 0.009361 |
Let A = {2, 3, 4, 5, 6} and B = {3, 4, 5, 6, 7}. Find the symmetric difference of A and B, and verify that it is commutative. | tiger_lab_math | [
0.05453244969248772,
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0.041492633521556854,
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0.06958069652318954,
0.057495418936014175,
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0.006109704729169607,
0.019889552146196365,
-0.0318962037563324,
-0.018... | 0.83524 | 0.83241 | 0.00283 |
A researcher wants to test the effect of fertilizer on the growth of a certain type of plant. The plants are randomly divided into two groups: one group will receive fertilizer and the other group will not. After 4 weeks, the heights of both groups of plants are measured. The heights (in cm) of the plants that received... | tiger_lab_math | [
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0.02614734321832657,
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0.01... | 0.801097 | 0.79724 | 0.003857 |
Consider the following optimization problem:
Minimize f(x,y) = 3x^2 + 5y^2 - 2x - 4y + 10
Subject to the following constraints:
x^2 + y^2 <= 25
3x + 4y <= 20
Solve the above problem using particle swarm optimization algorithm and find the minimum value of the objective function f(x,y). | tiger_lab_math | [
-0.0033397485967725515,
0.012780999764800072,
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0.04674351215362549,
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0.04198216646909714,
0.0019299782579764724,
0.006519254297018051,
0.025158626958727837,
-0.03122708760201931,
-0... | 0.817386 | 0.813162 | 0.004224 |
Suppose we have a set of 10 numbers, and we want to determine if there exist three distinct numbers in the set such that their sum is equal to zero. What is the minimum possible number of comparisons between elements in the set that are needed to determine if such a triple exists? | tiger_lab_math | [
0.02035314030945301,
0.005209813825786114,
-0.04039108008146286,
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0.046965450048446655,
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0.008975294418632984,
0.01437594834715128,
-0.04430583119392395,
-0.02... | 0.838882 | 0.837197 | 0.001685 |
Suppose two players, A and B, each have a deck of 10 cards numbered from 1 to 10. They each choose a card from their deck without revealing it to the other player. If player A's number is greater than player B's number, then player A wins. If player B's number is greater or equal to player A's number, then player B win... | tiger_lab_math | [
0.02130340412259102,
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0.018610257655382156,
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0.034169621765613556,
0.03297208994626999,
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0.01576702669262886,
0.03101402521133423,
-0.04366997256875038,
-0.00238... | 0.818503 | 0.811731 | 0.006772 |
A train is traveling with a speed of 25 m/s. Suddenly, a passenger realizes that he has dropped his phone at the back of the train, which is 50 meters from where he is sitting. He decides to run towards the back of the train to retrieve it. Assume that the passenger runs with a speed of 5 m/s relative to the train. Use... | tiger_lab_math | [
0.021116383373737335,
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0.007219261024147272,
0.011190183460712433,
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-0.018... | 0.818546 | 0.81613 | 0.002417 |
A candy company makes two types of candies: chocolate and fruit flavored. The company has 150 pounds of chocolate and 120 pounds of fruit-flavored candies. Making a pound of chocolate candies requires 4 units of sugar and 3 units of milk, while making a pound of fruit-flavored candies requires 2 units of sugar and 1 un... | tiger_lab_math | [
0.018487293273210526,
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0.03335219621658325,
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0.010420828126370907,
0.022804053500294685,
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-0.03... | 0.819841 | 0.809125 | 0.010715 |
Consider the message "SECRET" and the keyword "MATH". Encrypt the message using the Vigenere cipher and provide the resulting ciphertext. | tiger_lab_math | [
0.04745987430214882,
0.03632340952754021,
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0.042929407209157944,
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0.028329750522971153,
0.018644891679286957,
-0.03517106547951698,
-0.01... | 0.802003 | 0.800852 | 0.001151 |
Seven points are evenly spaced out on a circle and connected as shown below to form a 7-pointed star. What is the sum of the angle measurements of the seven tips of the star, in degrees? One such angle is marked as $\alpha$ below.
[asy]
dotfactor=4;
draw(Circle((0,0),1));
real x = 2*pi/7;
pair A,B,C,D,E,F,G;
A=(cos(... | tiger_lab_math | [
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0.036606088280677795,
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0.04395605996251106,
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... | 0.834818 | 0.82403 | 0.010787 |
A factory produces two types of products: product A and product B. Each unit of product A requires 1 hour of processing time and generates a profit of $150, while each unit of product B requires 2 hours of processing time and generates a profit of $300. The factory has a maximum of 60 hours available for processing. Ho... | tiger_lab_math | [
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0.01706661842763424,
0.03234998136758804,
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-0.... | 0.802256 | 0.798189 | 0.004067 |
A triangle has one angle that measures 30 degrees, one angle that measures 60 degrees, and a side opposite the 60 degree angle that measures 10 inches. Find the perimeter of the triangle using trigonometry. | tiger_lab_math | [
0.023444002494215965,
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0.03008793666958809,
0.04731522500514984,
-0.0418965108692646,
-0.032234... | 0.799821 | 0.794877 | 0.004944 |
Statement 1 | A group with a prime number of elements is cyclic. Statement 2 | In a field, the additive identity and the multiplicative identity are the same. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.036021918058395386,
0.02260681800544262,
-0.025344956666231155,
-0.045732706785202026,
0.047910500317811966,
-0.046413447707891464,
0.0006012310623191297,
0.06237768009305,
0.041686031967401505,
-0.033635206520557404,
0.010724937543272972,
0.006605041213333607,
-0.02745087817311287,
-0.0... | 0.813712 | 0.810659 | 0.003053 |
On the unit sphere $S^3$, determine the Hopf map $h:S^3\rightarrow S^2$ defined by $h(z,w)=(|z|^2-|w|^2, 2z\bar{w}, 2z\bar{w}i)$. Find the preimage of the point $(1,0,0)\in S^2$ under the Hopf map. | tiger_lab_math | [
0.015111755579710007,
0.016304336488246918,
-0.03967061638832092,
-0.047440361231565475,
0.035601187497377396,
-0.034185778349637985,
-0.007180893328040838,
0.06617994606494904,
0.03586844727396965,
-0.002653982723131776,
0.03296419605612755,
0.028171805664896965,
-0.025684695690870285,
-0... | 0.821904 | 0.805048 | 0.016856 |
in δ pqs above , if pq = 2 and ps = 3 , then ? Let's write a Python program. | tiger_lab_math | [
0.01778545416891575,
0.008366136811673641,
-0.038841791450977325,
-0.031037475913763046,
0.02459033392369747,
-0.009158439934253693,
-0.017384294420480728,
0.05076004937291145,
0.03483986482024193,
-0.008462456986308098,
0.01956944167613983,
0.043856289237737656,
-0.04469795152544975,
-0.0... | 0.831139 | 0.829916 | 0.001223 |
Let A = {1, 5, 7} , B = {1, 2, 3} and C = {2, 3, 7, 8}. Find (A △ B) △ C, where △ represents the symmetric difference of two sets. | tiger_lab_math | [
0.047784071415662766,
0.017448443919420242,
-0.04454668238759041,
-0.023209311068058014,
0.037737634032964706,
-0.016491372138261795,
-0.03111198917031288,
0.058408237993717194,
0.05496129393577576,
-0.032543063163757324,
0.0020168526098132133,
0.02589675597846508,
-0.03370152786374092,
-0... | 0.835244 | 0.830499 | 0.004745 |
Find the Laurent series expansion of the function f(z) = 1/z centered at z = 2 and determine the range of z-values for which the series converges. | tiger_lab_math | [
0.008411634713411331,
0.043154265731573105,
-0.05189131945371628,
-0.04229475557804108,
0.03714339807629585,
-0.0021565821953117847,
-0.005271300207823515,
0.033207524567842484,
0.022448405623435974,
-0.018182916566729546,
0.03092833235859871,
0.0227652657777071,
-0.03580582141876221,
-0.0... | 0.821657 | 0.818953 | 0.002704 |
Consider the following minimization problem for a 2-dimensional function:
Minimize: f(x,y) = (x-2)^2 + (y-3)^2
Subject to constraints:
- x + y >= 4
- x,y >= 0
Use a genetic algorithm to find the minimum value of the function f(x,y) and the values of x and y that achieve this minimum, subject to the given constraint... | tiger_lab_math | [
0.0020792519208043814,
0.012521136552095413,
-0.050949402153491974,
-0.03334445133805275,
0.051787491887807846,
-0.021672414615750313,
-0.057703759521245956,
0.06544049829244614,
0.037380896508693695,
-0.012089184485375881,
0.009931397624313831,
0.02970774658024311,
-0.01500745303928852,
0... | 0.81182 | 0.800487 | 0.011332 |
The points $A$, $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$. It is given that $AB=13$, $BC=14$, $CA=15$, and that the distance from $O$ to $\triangle ABC$ is $\frac{m\sqrt{n}}k$, where $m$, $n$, and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by t... | tiger_lab_math | [
0.020211486145853996,
0.04497760161757469,
-0.05246270075440407,
-0.04465248063206673,
0.038187175989151,
-0.021840857341885567,
-0.02187793329358101,
0.040858425199985504,
0.04710664972662926,
-0.0016720402054488659,
0.019175492227077484,
0.042571064084768295,
-0.028368918225169182,
-0.00... | 0.853564 | 0.82745 | 0.026113 |
A traveling salesman wants to visit 5 different cities. He knows the distance between each pair of cities given below. What is the shortest possible route that covers all cities and returns to the starting point?
City A to City B: 10 km
City A to City C: 15 km
City A to City D: 20 km
City A to City E: 25 km
City B to ... | tiger_lab_math | [
0.011560694314539433,
0.020954517647624016,
-0.031143542379140854,
-0.033367130905389786,
0.03774412348866463,
-0.018843669444322586,
-0.02721099928021431,
0.036280445754528046,
0.022505616769194603,
-0.015704594552516937,
0.019349733367562294,
0.04947243258357048,
-0.0183438491076231,
-0.... | 0.810438 | 0.806619 | 0.003819 |
Determine whether the formula "◊(p → q)" is true or false in a temporal model where p is true for all moments and q is true for some moments, but not all. | tiger_lab_math | [
0.007572777569293976,
0.016101937741041183,
-0.04284479841589928,
-0.03276311233639717,
0.027619432657957077,
-0.03253331780433655,
-0.015059143304824829,
0.08395761996507645,
0.026423001661896706,
-0.020711606368422508,
0.028477540239691734,
0.030605698004364967,
-0.017337288707494736,
-0... | 0.811412 | 0.810463 | 0.00095 |
A student wants to buy a new set of pens. He wants to buy at least 3 pens but no more than 7 pens. If he has $20 to spend and each pen costs either $2 or $3, how many different numbers of pens can he buy? Use recurrence relations to solve the problem. | tiger_lab_math | [
0.012465161271393299,
0.020362138748168945,
-0.05012372508645058,
-0.028934238478541374,
0.032556336373090744,
-0.019802141934633255,
-0.014600872993469238,
0.04259255900979042,
0.036691274493932724,
-0.0061767143197357655,
0.028474753722548485,
0.03763442113995552,
-0.02852100133895874,
-... | 0.834043 | 0.829693 | 0.00435 |
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