instruction_seed stringlengths 56 2.52k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.77 0.86 | mean_negative_score float64 0.76 0.85 | difference_score float64 0 0.03 |
|---|---|---|---|---|---|
Here's a precise numerical problem for a math student related to the Maximum Modulus Principle and Minimum Modulus Principle:
Let f(z) = 2z + 3 be a complex function, and let C be the circle |z - 1| = 2 in the complex plane. Find the maximum and minimum values of |f(z)| on C, and the points at which they are attained. | tiger_lab_math | [
0.04170066490769386,
0.013723142445087433,
-0.05819302797317505,
-0.035917870700359344,
0.04952508583664894,
-0.013699648901820183,
-0.015282555483281612,
0.03081541135907173,
0.025154124945402145,
-0.0023158853873610497,
0.03007475472986698,
0.03159215301275253,
-0.022678762674331665,
-0.... | 0.831746 | 0.822103 | 0.009644 |
Suppose that p represents the statement "It is sunny today" and q represents the statement "I will go to the beach". Using propositional logic, combination of p and q, find the truth value of the following statement: "It is sunny today and I will go to the beach, or it is not sunny today." | tiger_lab_math | [
0.0058442107401788235,
0.009666938334703445,
-0.05320054292678833,
-0.035902876406908035,
0.024596955627202988,
-0.051683660596609116,
-0.010471563786268234,
0.07048574835062027,
0.043952763080596924,
-0.03317166864871979,
0.011493606492877007,
0.039717819541692734,
-0.030853936448693275,
... | 0.811305 | 0.805744 | 0.005561 |
Let $\mathbf{a}$ and $\mathbf{b}$ be nonzero vectors such that
\[\|\mathbf{a}\| = \|\mathbf{b}\| = \|\mathbf{a} + \mathbf{b}\|.\]Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees. | tiger_lab_math | [
0.029681988060474396,
0.011203312315046787,
-0.03474948927760124,
-0.04387756809592247,
0.013276331126689911,
0.02141716331243515,
-0.018961690366268158,
0.047023579478263855,
0.04145127534866333,
-0.014021522365510464,
0.020691141486167908,
0.013922407291829586,
-0.0446634367108345,
-0.01... | 0.826385 | 0.815882 | 0.010502 |
A particle moves along the space curve given by the equation $r(t) = (t^2, \sin t, \cos t)$, where $t$ is the time. Find the curvature and torsion of this curve at $t = \frac{\pi}{2}$. | tiger_lab_math | [
0.010401583276689053,
0.04197331517934799,
-0.02658594585955143,
-0.029042823240160942,
0.038234371691942215,
-0.015121638774871826,
-0.04595869034528732,
0.06435106694698334,
0.038148801773786545,
-0.012571158818900585,
0.026184089481830597,
0.01785186491906643,
-0.034521784633398056,
-0.... | 0.813322 | 0.804579 | 0.008743 |
Consider a regular surface defined by the equation $f(x, y, z) = x^2 + y^2 + z^2 - 1 = 0$, calculate the sectional curvature at the point $(1/\sqrt{2}, 1/\sqrt{2}, 0)$ along the plane spanned by the tangent vectors $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$. | tiger_lab_math | [
0.0035242207814007998,
0.04611898958683014,
-0.044989366084337234,
-0.055978741496801376,
0.043299656361341476,
0.004708444699645042,
-0.012044518254697323,
0.06441139429807663,
0.015539203770458698,
-0.006650183815509081,
0.01594645529985428,
0.03584563732147217,
-0.02369515411555767,
-0.... | 0.82379 | 0.801013 | 0.022777 |
A student is designing a schedule for a music festival. There are 10 different bands and 6 different time slots available. The student wants to create a schedule where each band plays exactly once and each time slot has exactly 2 bands performing. How many ways can the student schedule the festival? | tiger_lab_math | [
0.030647916719317436,
0.010841764509677887,
-0.03807022422552109,
-0.022443413734436035,
0.030756667256355286,
-0.044546060264110565,
-0.038422416895627975,
0.04550759121775627,
0.02734297513961792,
-0.008144896477460861,
0.019493738189339638,
0.03917774185538292,
-0.024356670677661896,
-0... | 0.812132 | 0.810881 | 0.001251 |
Let u(x, y) be the harmonic function defined on the unit disk D = {z = x + iy: |z| < 1} such that u(x, 0) = x^2 - 1 for all |x| < 1. Find u(0, 1). | tiger_lab_math | [
0.010712509974837303,
0.026972796767950058,
-0.04941580444574356,
-0.03513064235448837,
0.030557088553905487,
0.0013346124906092882,
-0.0016089410055428743,
0.061124809086322784,
0.044585373252630234,
-0.0003761774569284171,
0.03392818942666054,
0.034775327891111374,
-0.02456609159708023,
... | 0.826996 | 0.809593 | 0.017402 |
Consider the surface given by the equation:
z = x^(2) + y^(2)
Find the mean curvature of this surface at the point (1,1,2). | tiger_lab_math | [
0.007184311747550964,
0.03304937854409218,
-0.055111709982156754,
-0.05138503760099411,
0.03047369234263897,
-0.006394635420292616,
-0.021089617162942886,
0.05658235400915146,
0.017768660560250282,
-0.0265608262270689,
0.005901406519114971,
0.029710985720157623,
-0.026403509080410004,
-0.0... | 0.832979 | 0.828318 | 0.00466 |
Determine the radius of convergence and interval of convergence for the power series:
$$\sum_{n=0}^\infty \frac{(2x+1)^n}{2^n n^2}$$ | tiger_lab_math | [
0.018430793657898903,
0.03842537850141525,
-0.031057877466082573,
-0.03736904636025429,
0.028027832508087158,
-0.014612814411520958,
-0.02044091373682022,
0.05726880952715874,
0.04387635737657547,
-0.016543608158826828,
0.02017245627939701,
0.0022404775954782963,
-0.032792236655950546,
-0.... | 0.837676 | 0.826691 | 0.010985 |
Define the sequence $a_1, a_2, a_3, \ldots$ by $a_n = \sum\limits_{k=1}^n \sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$. | tiger_lab_math | [
0.008801326155662537,
0.028553202748298645,
-0.02044682204723358,
-0.026455119252204895,
0.02070862241089344,
-0.004652421455830336,
-0.02970094047486782,
0.06339135766029358,
0.03439171239733696,
-0.009872272610664368,
0.022422434762120247,
0.005479602143168449,
-0.0507531501352787,
0.000... | 0.831667 | 0.821337 | 0.01033 |
The cubic polynomial $p(x)$ satisfies $p(2) = 1,$ $p(7) = 19,$ $p(15) = 11,$ and $p(20) = 29.$ Find
\[p(1) + p(2) + p(3) + \dots + p(21).\] Let's write a Python program to solve it. | tiger_lab_math | [
0.022411145269870758,
0.026553751900792122,
-0.06434210389852524,
-0.03943159431219101,
0.0337555818259716,
-0.047417182475328445,
-0.014225221239030361,
0.026791255921125412,
0.04415198788046837,
0.0006793172215111554,
0.023147858679294586,
0.038835082203149796,
-0.03240096569061279,
-0.0... | 0.833777 | 0.829284 | 0.004493 |
Sure thing! Here's a precise numerical problem for a math student to solve:
A certain solid has a base given by the region bounded by the curves y = 2x and y = x^2. The solid intersects each plane perpendicular to the x-axis in a square. Using integration to find the volume of this solid. | tiger_lab_math | [
0.05745953321456909,
0.02771149016916752,
-0.041287098079919815,
-0.04109128937125206,
0.03452511131763458,
-0.019992060959339142,
-0.029986077919602394,
0.03801565617322922,
0.049343936145305634,
-0.01735314168035984,
0.032354194670915604,
0.03984428942203522,
-0.03783636540174484,
-0.031... | 0.831538 | 0.820519 | 0.011019 |
Let f(x) = x^3 - 6x^2 + 9x + 2. Find the critical points of the function f(x) and determine whether each point corresponds to a local maximum, local minimum, or neither. | tiger_lab_math | [
0.016941402107477188,
0.014069188386201859,
-0.05119374021887779,
-0.03189899027347565,
0.02621627226471901,
-0.03333652392029762,
-0.01762501336634159,
0.06142327934503555,
0.0213154349476099,
0.0007050115382298827,
0.024303626269102097,
0.036759357899427414,
-0.04562020301818848,
-0.0255... | 0.830993 | 0.826886 | 0.004107 |
Let M be a 2-sphere of radius 1 centered at the origin of 3-space. Using the standard Cartesian coordinates (x,y,z) of Euclidean three-space R³, determine the Hopf map of this manifold. | tiger_lab_math | [
0.023769263178110123,
0.022113803774118423,
-0.02072233334183693,
-0.05341322347521782,
0.03208448737859726,
-0.004987441003322601,
-0.012157302349805832,
0.04952492192387581,
0.025313226506114006,
-0.013205192051827908,
0.011849232017993927,
0.0308693815022707,
-0.04625061899423599,
-0.03... | 0.810544 | 0.797104 | 0.01344 |
Let $f(x) = \frac{x^3}{1 - x^2}$ be a real-valued function defined on the set $(-1, 1)$. Determine whether $f$ is a smooth function (i.e., infinitely differentiable) on $(-1, 1)$. | tiger_lab_math | [
0.013114629313349724,
0.03545057028532028,
-0.04763095825910568,
-0.04013259336352348,
0.046370189636945724,
-0.012412459589540958,
-0.012914115563035011,
0.07786449790000916,
0.0274721160531044,
-0.018923675641417503,
0.021495874971151352,
0.022726260125637054,
-0.03272756561636925,
-0.01... | 0.821224 | 0.810764 | 0.010461 |
Consider the following polynomial equation:
x^4 + y^4 + 4z^4 - 4x^2y^2 - 4x^2z^2 - 4y^2z^2 = 0
Find all solutions to this equation in which x, y, and z are non-negative integers. | tiger_lab_math | [
0.02748311683535576,
0.01983592100441456,
-0.035970427095890045,
-0.056902673095464706,
0.03066306933760643,
-0.018008887767791748,
0.005144248716533184,
0.0444033220410347,
0.041205815970897675,
-0.018057914450764656,
0.017420105636119843,
0.020371096208691597,
-0.01287330687046051,
-0.04... | 0.841114 | 0.831226 | 0.009888 |
Suppose we have the complex function f(z) = (z^2 - 3)/(z - 2). Find the analytic continuation of f(z) along the path from z = 2 to z = 5 + i, i.e., find a new function g(z) that agrees with f(z) on the segment [2, 5 + i] and is analytic on a domain containing the path. | tiger_lab_math | [
0.03720777854323387,
0.02473733201622963,
-0.04932115226984024,
-0.03957417979836464,
0.03883473947644234,
-0.010070623829960823,
-0.004375141113996506,
0.048315420746803284,
0.022036075592041016,
0.0012295183259993792,
0.014675981365144253,
0.00818021409213543,
-0.023777078837156296,
-0.0... | 0.824308 | 0.816841 | 0.007467 |
Here is a precise numerical problem related to using Fourier series to solve partial differential equations:
Determine the Fourier series representation for the periodic function
f(x) = x,
defined on the interval
0 ≤ x ≤ 2π.
Then, use this representation to solve the partial differential equation
u_t - u_xx ... | tiger_lab_math | [
0.009489375166594982,
0.024436889216303825,
-0.05141327902674675,
-0.029063796624541283,
0.028674257919192314,
-0.02175430953502655,
-0.004700176417827606,
0.06794407218694687,
0.05441728234291077,
0.01633877493441105,
0.025594525039196014,
0.04097473621368408,
-0.024275753647089005,
-0.02... | 0.821621 | 0.80883 | 0.012791 |
Consider a vector $\vec{v} = 3\hat{i} - 4\hat{j} + 12\hat{k}$ at point $(1,2,-3)$ along a curve $C$ given by $\vec{r}(t) = \cos(t)\hat{i} + \sin(t)\hat{j} + t\hat{k}$ for $0 \leq t \leq \pi/2$. Find the parallel transport of the vector $\vec{v}$ along the curve $C$ from $(1,2,-3)$ to $(0,1,-1)$. | tiger_lab_math | [
0.02820044942200184,
0.047269534319639206,
-0.039300985634326935,
-0.018279865384101868,
0.03425581008195877,
0.0008882802212610841,
-0.028120918199419975,
0.07611790299415588,
0.028121398761868477,
-0.0007185764843598008,
0.009772182442247868,
0.04818670079112053,
-0.02305680699646473,
0.... | 0.832731 | 0.81994 | 0.012791 |
Suppose X(t) is a stochastic process satisfying the following stochastic differential equation:
dX(t) = 2X(t)dt + 3dW(t)
where W(t) is a Wiener process. If X(0) = 1, find the expected value of X(1) and the variance of X(1). | tiger_lab_math | [
-0.009790129028260708,
-0.001053469255566597,
-0.0598134882748127,
-0.02670435979962349,
0.05570145696401596,
0.001345900702290237,
-0.035476263612508774,
0.08982840925455093,
0.0378279872238636,
-0.004839574918150902,
0.038634318858385086,
0.034515466541051865,
-0.0377902053296566,
-0.021... | 0.805145 | 0.802976 | 0.002169 |
Let $a,$ $b,$ $c$ be distinct, nonzero real numbers such that
\[a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}.\]Find $|abc|.$
Note: Intermediate Algebra writing problem, week 12. | tiger_lab_math | [
0.04336005449295044,
0.046013567596673965,
-0.03951416164636612,
-0.03368217125535011,
0.036212258040905,
-0.0104861781001091,
-0.0033805526327341795,
0.05818424001336098,
0.03820502385497093,
-0.024546582251787186,
0.00812805537134409,
0.009336796589195728,
-0.020402992144227028,
-0.04362... | 0.855097 | 0.830018 | 0.025079 |
Segment $AB$ has midpoint $C$, and segment $BC$ has midpoint $D$. Semi-circles are constructed with diameters $\overline{AB}$ and $\overline{BC}$ to form the entire region shown. Segment $CP$ splits the region into two sections of equal area. What is the degree measure of angle $ACP$? Express your answer as a decimal t... | tiger_lab_math | [
0.011501898989081383,
-0.0030748406425118446,
-0.039641235023736954,
-0.028632530942559242,
0.017565259709954262,
-0.028150424361228943,
-0.03761579468846321,
0.06154219061136246,
0.03508061543107033,
0.0007636483642272651,
0.024234671145677567,
0.03337966278195381,
-0.02000431716442108,
0... | 0.825932 | 0.813565 | 0.012367 |
Series D(n) is such that i*D(i) = j*D(j) for any pair of positive integers (i, j). If D(1) is a positive integer, which of the following is possible?
I. 2*D(100) = D(99) + D(98)
II. D(1) is the only integer in the series
III. The series does not contain negative numbers
Answer Choices: (A) I only (B) II only (C) IIII o... | tiger_lab_math | [
0.022087514400482178,
0.007883830927312374,
-0.030803848057985306,
-0.04694082587957382,
0.01922730542719364,
-0.023914573714137077,
0.0035898739006370306,
0.0754881277680397,
0.04542084410786629,
-0.015626071020960808,
0.01993808150291443,
0.029736904427409172,
-0.038914673030376434,
-0.0... | 0.831501 | 0.828746 | 0.002755 |
A company produces three different types of chocolates: milk, dark, and white. The weight of each chocolate is recorded for a sample of 20 chocolates of each type. Conduct a one-way ANOVA with a significance level of 0.05 to determine if there is a significant difference in the mean weight of the three types of chocola... | tiger_lab_math | [
0.008561968803405762,
0.020133694633841515,
-0.04842670261859894,
-0.03565657511353493,
0.026420388370752335,
-0.009315749630331993,
-0.011015866883099079,
0.09168975055217743,
0.02061430737376213,
-0.014869199134409428,
0.0180902611464262,
0.04484843835234642,
-0.04309498891234398,
-0.022... | 0.798718 | 0.79715 | 0.001569 |
Consider the moduli space of smooth plane curves of degree 3, denoted by $\mathcal{M}_3$. Let $g_i$ be the genus of the $i^{th}$ curve in $\mathcal{M}_3$, and let $d_i$ be the number of lines that intersect the $i^{th}$ curve in $\mathcal{M}_3$. If $i$ ranges over all possible curves in $\mathcal{M}_3$, what is the sum... | tiger_lab_math | [
0.01642422378063202,
0.016960954293608665,
-0.04186546429991722,
-0.04224579408764839,
0.034559499472379684,
-0.015901638194918633,
-0.03970761224627495,
0.049909479916095734,
0.02901109866797924,
-0.027590708807110786,
0.009798816405236721,
0.04152724891901016,
-0.04335964843630791,
-0.02... | 0.840955 | 0.813609 | 0.027346 |
Suppose there are 20 students participating in a science project competition. The competition consists of 4 different activities: experiment, presentation, report writing, and quiz. Each student is required to participate in exactly 3 activities. What is the minimum number of judges needed to evaluate the students in s... | tiger_lab_math | [
-0.0037411455996334553,
-0.004204092547297478,
-0.050970081239938736,
-0.04739225283265114,
0.03758891671895981,
-0.025768468156456947,
-0.05441520735621452,
0.03753931447863579,
0.022716345265507698,
-0.010072282515466213,
-0.00263672205619514,
0.031595081090927124,
-0.02250806987285614,
... | 0.811788 | 0.810596 | 0.001192 |
A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(3,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $C$? | tiger_lab_math | [
-0.00003921819006791338,
0.03552280738949776,
-0.041196394711732864,
-0.026186030358076096,
0.030509930104017258,
-0.019511839374899864,
-0.02928875759243965,
0.01994202844798565,
0.03695657104253769,
-0.014126570895314217,
0.012978728860616684,
0.035922396928071976,
-0.05166604742407799,
... | 0.830384 | 0.820375 | 0.010009 |
What is the area of the main cardioid in the Mandelbrot set, which is formed by points that don't escape after a certain number of iterations? (Hint: the equation for the cardioid is given by $z = 1/4 (2 + 2\cos(\theta) + i\sin(\theta))$, where $z$ and $\theta$ are complex numbers.) | tiger_lab_math | [
0.027940422296524048,
0.03742513805627823,
-0.04811834543943405,
-0.048805829137563705,
0.01724383234977722,
-0.01940590888261795,
-0.019025547429919243,
0.044486261904239655,
0.03142128884792328,
-0.01693131774663925,
0.02439780719578266,
0.010536091402173042,
-0.03514522686600685,
-0.018... | 0.832741 | 0.828175 | 0.004566 |
Consider the following graph G:
```
0 --- 1 --- 2
| | |
3 --- 4 --- 5
```
Does G contain an Eulerian path? If yes, write down the sequence of vertices visited. If not, explain why not. | tiger_lab_math | [
-0.004778811242431402,
0.025478029623627663,
-0.002046172507107258,
-0.03382068872451782,
0.023469137027859688,
-0.028959931805729866,
-0.027562962844967842,
0.06191016733646393,
0.013859324157238007,
-0.0033437141682952642,
0.02418462745845318,
0.02866630256175995,
-0.00696105882525444,
-... | 0.817551 | 0.816524 | 0.001028 |
If I have four boxes arranged in a $2$ x $2$ grid, in how many distinct ways can I place the digits $1$, $2$, and $3$ in the boxes such that each box contains at most one digit? (I only have one of each digit, so one box will remain blank.) | tiger_lab_math | [
0.0027318408247083426,
0.022667476907372475,
-0.04243883118033409,
-0.030736910179257393,
0.032883420586586,
-0.025005586445331573,
-0.025718320161104202,
0.03492080047726631,
0.038740769028663635,
-0.013441386632621288,
0.00594350416213274,
0.031095294281840324,
-0.0373934730887413,
-0.01... | 0.841675 | 0.839575 | 0.0021 |
In a volunteer organization, there are 4 members who need to decide whether or not to help with a charity event. If none of them volunteer, the event won't happen and they will each receive 0 points. If one person volunteers, they will receive 50 points and the others will receive 0 points. If two people volunteer, the... | tiger_lab_math | [
-0.00045315086026676,
-0.01811159774661064,
-0.031176656484603882,
-0.029242485761642456,
0.021035201847553253,
-0.032302867621183395,
-0.014843763783574104,
0.03054327704012394,
0.04567379131913185,
-0.01100610289722681,
0.0076554911211133,
0.02070479840040207,
-0.030325734987854958,
-0.0... | 0.811308 | 0.798171 | 0.013137 |
In triangle $ABC$, $AB = AC = 100$, and $BC = 56$. Circle $P$ has radius $16$ and is tangent to $\overline{AC}$ and $\overline{BC}$. Circle $Q$ is externally tangent to $P$ and is tangent to $\overline{AB}$ and $\overline{BC}$. No point of circle $Q$ lies outside of $\triangle ABC$. The radius of circle $Q$ can be expr... | tiger_lab_math | [
0.020867029204964638,
0.03657497093081474,
-0.022641627117991447,
-0.036777839064598083,
0.03494374826550484,
-0.011883746832609177,
-0.014326045289635658,
0.042324189096689224,
0.04340535029768944,
-0.0016666995361447334,
0.03099249117076397,
0.03674333915114403,
-0.02075532078742981,
-0.... | 0.841814 | 0.810628 | 0.031186 |
Consider a simple graph with 6 vertices and the following adjacency matrix:
\begin{bmatrix}0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0\\ 1 & 1 & 0 & 1 & 1 & 1\\ 1 & 0 & 1 & 0 & 0 & 1\\ 0 & 1 & 1 & 0 & 0 & 1\\ 0 & 0 & 1 & 1 & 1 & 0\end{bmatrix}
Find whether the graph contains a Hamiltonian cycle or not. | tiger_lab_math | [
0.01861831732094288,
0.010021825321018696,
-0.02717985212802887,
-0.02893078327178955,
0.03472622111439705,
-0.03389274328947067,
-0.018922241404652596,
0.08546634018421173,
0.020192613825201988,
-0.024546533823013306,
0.011719288304448128,
0.0028036448638886213,
-0.013152118772268295,
-0.... | 0.822682 | 0.808889 | 0.013794 |
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 | [
0.029321476817131042,
0.03607024624943733,
-0.029933607205748558,
-0.0228402279317379,
0.0349179208278656,
-0.011175245977938175,
-0.025937598198652267,
0.05539274960756302,
0.04646914824843407,
0.001650360063649714,
0.018024517223238945,
0.04278898611664772,
0.0006762854172848165,
-0.0240... | 0.827406 | 0.819245 | 0.00816 |
Here's a question a math student can solve regarding Cryptanalysis of Playfair Cipher:
Suppose you intercepted an encrypted Playfair Cipher message with the letters "TUISTFAEZQ" and your goal is to determine the original message. You suspected that the letters "TH" were used consecutively in the original message. Usin... | tiger_lab_math | [
0.029865218326449394,
0.029186302796006203,
-0.035187069326639175,
-0.06975829601287842,
0.04336513578891754,
-0.01695813611149788,
-0.001219404162839055,
0.05419372767210007,
0.039305929094552994,
-0.022542236372828484,
0.005895825102925301,
0.007617447059601545,
-0.038219746202230453,
-0... | 0.820014 | 0.816889 | 0.003125 |
Let X be a projective variety of dimension two, with a morphism f: X → Y to a projective variety Y. Suppose that for every irreducible curve C contained in X, there exists an irreducible curve D contained in Y such that f(C) = D.
Suppose further that there exists an ample divisor H on Y such that f^*H is an ample div... | tiger_lab_math | [
0.020009709522128105,
0.036819156259298325,
-0.026040710508823395,
-0.03490206599235535,
0.02761787548661232,
-0.0200123880058527,
-0.0035569823812693357,
0.05248364433646202,
0.03962482139468193,
-0.012423500418663025,
0.0064994958229362965,
0.04287705942988396,
-0.04308624938130379,
-0.0... | 0.84613 | 0.818752 | 0.027379 |
Let A = {1, 2, 3} and B = {2, 3, 4}. Find the value of (A △ B) × A and (B △ A) × A, and determine whether the symmetric difference of two sets is associative with respect to Cartesian product. | tiger_lab_math | [
0.05723077431321144,
0.023412736132740974,
-0.04057949781417847,
-0.018498459830880165,
0.0332925021648407,
-0.017401671037077904,
-0.010461444966495037,
0.06570504605770111,
0.06419385224580765,
-0.007866842672228813,
0.006331180687993765,
0.02729298546910286,
-0.03445404767990112,
-0.015... | 0.836352 | 0.825775 | 0.010578 |
Consider the following system of linear equations:
\begin{align*}
3x_1 + 2x_2 + x_3 &= 9 \\
2x_1 + 4x_2 + 3x_3 &= 20 \\
x_1 + \frac{1}{2} x_2 + x_3 &= 5
\end{align*}
Use LU decomposition to solve for the value of $x_1$, $x_2$, and $x_3$. | tiger_lab_math | [
-0.001408549607731402,
0.0311573576182127,
-0.06439889222383499,
-0.05600154399871826,
0.05372795835137367,
-0.02591533213853836,
-0.0161174014210701,
0.06585239619016647,
0.0268161091953516,
-0.0038021549116820097,
0.01900271698832512,
0.030633393675088882,
-0.04632499814033508,
-0.038271... | 0.820867 | 0.818536 | 0.002331 |
A manufacturing company is producing widgets at a total cost of $400,000. The cost includes labor, materials, and overhead. The company wants to minimize the cost of producing widgets by adjusting the production levels of two factories, with factory one capable of producing 50 widgets per day and factory two capable of... | tiger_lab_math | [
-0.007510155439376831,
0.0198697317391634,
-0.05567526817321777,
-0.017805442214012146,
0.034135010093450546,
-0.027055930346250534,
-0.053562190383672714,
0.07342052459716797,
0.04112432152032852,
-0.0055359117686748505,
0.012017282657325268,
0.02257831022143364,
-0.008176064118742943,
-0... | 0.802599 | 0.791516 | 0.011083 |
In the diagram, $\angle PQR = 90^\circ$. What is the value of $x$?
[asy]
size(100);
draw((0,1)--(0,0)--(1,0));
draw((0,0)--(.9,.47));
draw((0,.1)--(.1,.1)--(.1,0));
label("$P$",(0,1),N); label("$Q$",(0,0),SW); label("$R$",(1,0),E); label("$S$",(.9,.47),NE);
label("$2x^\circ$",(.15,.2)); label("$x^\circ$",(.32,-.02),... | tiger_lab_math | [
0.002482486655935645,
0.011504953727126122,
-0.04221266880631447,
-0.04664408043026924,
0.00968987587839365,
-0.028677400201559067,
-0.017222760245203972,
0.0578155592083931,
0.04430677369236946,
-0.00703767454251647,
0.023841217160224915,
0.02569013647735119,
-0.0278298482298851,
-0.02217... | 0.82752 | 0.82328 | 0.00424 |
Let f(x) = x^2 + 2x + 1 and g(x) = x + 1.
Find (f o g)(x), where o represents the function composition operator. | tiger_lab_math | [
0.029319239780306816,
0.01850440539419651,
-0.04567833989858627,
-0.04813320189714432,
0.04804384708404541,
-0.03947143256664276,
-0.0190614964812994,
0.08414949476718903,
0.043391115963459015,
-0.007415079511702061,
0.019631799310445786,
0.023337284103035927,
-0.03629975765943527,
-0.0358... | 0.827803 | 0.826301 | 0.001502 |
Let's say we have two algebraic equations:
x^3 + y^3 - 3xy = 0
and
2x^3 - 2y^3 - 5x + 5y = 0
Using Bezout's theorem, find the number of intersection points between these two curves in the plane. | tiger_lab_math | [
0.040655240416526794,
0.00410138675943017,
-0.05207003280520439,
-0.0480678416788578,
0.03055150806903839,
-0.006011486519128084,
-0.02193521521985531,
0.03155095502734184,
0.05756708234548569,
-0.025982528924942017,
-0.0002792687446344644,
0.024439608678221703,
-0.026657244190573692,
-0.0... | 0.84475 | 0.830599 | 0.01415 |
Suppose a signal is defined in the time domain by the function $f(t)$, where $t\in[0,2\pi]$ and
$$f(x)=\begin{cases}2\sin(x) & 0\leq x \leq \pi \\ 0 & \pi < x \leq 2\pi\end{cases}$$
Use the Fourier transform to determine the frequency domain representation of the signal. Specifically, find the complex Fourier series ... | tiger_lab_math | [
0.004508631303906441,
0.04963056743144989,
-0.05628577992320061,
-0.042220160365104675,
0.0475776307284832,
-0.013911159709095955,
-0.015506426803767681,
0.05291878804564476,
0.03746184706687927,
0.02041724883019924,
0.021256089210510254,
0.03356701880693436,
-0.03664829954504967,
-0.00914... | 0.804902 | 0.79712 | 0.007782 |
Consider an algebraic variety defined by the polynomial equation $f(x,y,z) = x^3+y^3+z^3-6xyz=0$. Determine the cohomology groups $H^i(X,\mathbb{Q})$ of $X$, where $X$ is the algebraic variety defined by this polynomial equation. | tiger_lab_math | [
0.028537508100271225,
0.025913570076227188,
-0.03749055787920952,
-0.039137158542871475,
0.020925680175423622,
-0.03736979886889458,
-0.006349919363856316,
0.056993916630744934,
0.04766175150871277,
-0.01682308316230774,
0.015030290931463242,
0.024883465841412544,
-0.020377954468131065,
-0... | 0.82892 | 0.806602 | 0.022318 |
A network flow problem:
Suppose you manage a shipping company that transports goods between cities. You have identified the following five cities that your company services, along with the maximum capacity of goods that can be transported between each pair of cities:
- City A can transport up to 50 units to city B, ... | tiger_lab_math | [
0.02808172069489956,
0.005093615036457777,
-0.027747655287384987,
-0.01992907002568245,
0.024456411600112915,
-0.012620124965906143,
-0.04391022399067879,
0.05826045200228691,
0.014828696846961975,
-0.010892036370933056,
0.03180055320262909,
0.02888353541493416,
-0.039159536361694336,
-0.0... | 0.816336 | 0.803692 | 0.012644 |
A sequence of numbers is defined by $a_1=1$ and $a_{n+1}=2a_n+1$ for $n\geq 1$. Find an explicit formula for $a_n$. | tiger_lab_math | [
0.04132703319191933,
0.03484507277607918,
-0.038190294057130814,
-0.034144069999456406,
0.03619835898280144,
-0.029502982273697853,
-0.026374682784080505,
0.07290395349264145,
0.03547180816531181,
-0.036496859043836594,
0.027716398239135742,
0.016040144488215446,
-0.03865763172507286,
-0.0... | 0.828132 | 0.8168 | 0.011332 |
Marcelle and Jaclyn each think of a polynomial. Each of their polynomials is monic, has degree 4, and has the same positive constant term and the same coefficient of $z$. The product of their polynomials is \[z^8 +3z^7 +z^6 +3z^5 +4z^4 +6z^3 +2z^2 +4.\]What is the constant term of Jaclyn's polynomial? | tiger_lab_math | [
0.019339533522725105,
0.020789379253983498,
-0.03665418177843094,
-0.04478280246257782,
0.04708389192819595,
-0.0176277793943882,
-0.012875535525381565,
0.037144504487514496,
0.05782494693994522,
-0.0013022103812545538,
-0.00236739544197917,
0.01451716385781765,
-0.011684788390994072,
-0.0... | 0.831186 | 0.820281 | 0.010904 |
Sure, here's an example question for a math student to solve related to meromorphic functions:
Let f(z) = (6z - 3) / (z - 2) be a meromorphic function defined on ℂ. Find the poles and zeros of f(z), and state whether f(z) is holomorphic at each of the poles and zeros. | tiger_lab_math | [
0.03548150509595871,
0.0178393442183733,
-0.04771509766578674,
-0.03676963225007057,
0.03503280505537987,
-0.020584383979439735,
-0.005606704857200384,
0.044707976281642914,
0.035193342715501785,
-0.0063124122098088264,
0.019146878272294998,
0.023691752925515175,
-0.024172192439436913,
-0.... | 0.821376 | 0.815978 | 0.005398 |
A circle centered at $O$ has radius 1 and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\angle
AOB=\theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, then express $OC$ in terms of $s$ and $c,$ where $s = \sin \theta$ and $c = \cos \theta.$
[asy]
pair A,B... | tiger_lab_math | [
0.010702489875257015,
0.026371067389845848,
-0.048068348318338394,
-0.03861169144511223,
0.02978551760315895,
-0.033003825694322586,
-0.03023320622742176,
0.042948272079229355,
0.03108689747750759,
0.0035474197939038277,
0.02879752404987812,
0.01594354398548603,
-0.010806450620293617,
0.00... | 0.832711 | 0.807076 | 0.025635 |
A manufacturing company produces two types of product A and B. The profit per unit for A is $8 and for B is $10. The company has 300 hours of production time available per week. Product A requires 2 hours of production time per unit, while product B requires 3 hours of production time per unit. Using ant colony optimiz... | tiger_lab_math | [
0.017551137134432793,
-0.001009316765703261,
-0.03939737007021904,
0.002619869541376829,
0.005936254281550646,
-0.030072063207626343,
-0.05830475687980652,
0.08630293607711792,
0.03056805580854416,
-0.008490356616675854,
0.003591216867789626,
0.034135427325963974,
-0.016672972589731216,
-0... | 0.800525 | 0.792629 | 0.007896 |
Thomas wants to throw a party for his best friend Casey. He needs to order enough chairs for all the party guests to sit in, one for each guest. First he orders 3 dozen chairs for 3 dozen guests he invites. Then he finds out that 1/3 of the guests want to bring a guest of their own, so Thomas needs to order more chairs... | tiger_lab_math | [
0.011237711645662785,
0.03404524177312851,
-0.0383637510240078,
-0.026380274444818497,
0.026913750916719437,
-0.03850848600268364,
-0.016733907163143158,
0.03206814453005791,
0.026647739112377167,
-0.01586216874420643,
0.033430859446525574,
0.03968510404229164,
-0.031962648034095764,
0.005... | 0.811232 | 0.805496 | 0.005736 |
"statement 1": Triangle STV has sides ST = TV = 26 , and SV = 20, "statement 2" :Triangle STV has sides ST = TV = less than 26 , and SV = 20, "options: " Entailment or contradiction or neutral? Let's write a Python program to solve it. | tiger_lab_math | [
0.02296667918562889,
-0.003436613827943802,
-0.04912747070193291,
-0.03400928154587746,
0.0668506994843483,
-0.029587281867861748,
-0.017406880855560303,
0.06775206327438354,
0.02357419766485691,
-0.005307129584252834,
0.03527320921421051,
0.04444711655378342,
-0.030053697526454926,
-0.010... | 0.816091 | 0.8144 | 0.001692 |
Here is a precise numerical problem for the math student to solve:
In a certain population, 30% of individuals are smokers and 20% of smokers have lung cancer. If 10% of non-smokers also have lung cancer, what is the probability that an individual selected at random from this population has lung cancer? | tiger_lab_math | [
0.014889206737279892,
0.010027860291302204,
-0.058343254029750824,
-0.045486122369766235,
0.045880578458309174,
-0.01438206061720848,
-0.024378366768360138,
0.04476676881313324,
0.03875869885087013,
-0.012027950026094913,
0.01894141547381878,
0.029404446482658386,
-0.021896323189139366,
-0... | 0.827445 | 0.821775 | 0.00567 |
Given a matrix A = [4 2; 1 3], diagonalize the matrix if possible. Find the diagonal matrix D and invertible matrix P such that A = PDP^-1. | tiger_lab_math | [
0.038989290595054626,
0.020981913432478905,
-0.02436807006597519,
-0.03814195469021797,
0.0374741367995739,
-0.019283656030893326,
0.005266712512820959,
0.07108967006206512,
0.031242508441209793,
-0.004860613960772753,
0.01434365101158619,
0.01950639672577381,
-0.02344907633960247,
-0.0289... | 0.820548 | 0.816966 | 0.003583 |
Consider the following sets:
A = {2, 4, 6}
B = {1, 3, 5}
Find:
a) A ∪ B
b) A ∩ B
c) A - B
d) B - A | tiger_lab_math | [
0.05676339566707611,
0.033505819737911224,
-0.03632006049156189,
-0.008042982779443264,
0.035410888493061066,
-0.024887461215257645,
-0.021095698699355125,
0.04170854017138481,
0.04635254293680191,
-0.03582281246781349,
0.017057200893759727,
0.026690050959587097,
-0.034076567739248276,
-0.... | 0.843404 | 0.838919 | 0.004484 |
Consider the sets A = {1, 2, 3} and B = {2, 3, 4}.
Find the symmetric difference of the sets (A - B) Δ (B ∩ A) and determine whether it is equal to the set (A Δ B) ∩ (A ∪ B). | tiger_lab_math | [
0.045167554169893265,
0.005126756615936756,
-0.03365454822778702,
-0.024870526045560837,
0.03317251428961754,
-0.021749941632151604,
-0.009062361903488636,
0.0630808174610138,
0.06057772785425186,
-0.015450847335159779,
0.011951747350394726,
0.023844322189688683,
-0.04692688584327698,
-0.0... | 0.834367 | 0.817733 | 0.016634 |
Find a conformal map that maps the outside of the unit circle $|z|<1$ onto the region outside the circle $|w-2i|=3$. | tiger_lab_math | [
0.014647792093455791,
0.034942805767059326,
-0.03126528486609459,
-0.051379550248384476,
0.0390779934823513,
-0.01283518597483635,
-0.016931939870119095,
0.03254668042063713,
0.022630518302321434,
-0.0028795890975743532,
0.025182029232382774,
0.02448735199868679,
-0.01720028556883335,
-0.0... | 0.828738 | 0.822602 | 0.006136 |
New York recorded 5000 new coronavirus cases on a particular week. In the second week, half as many new coronaviruses cases as the first week was recorded by the state. In the third week, 2000 more cases were recorded in the state. What is the total number of recorded new coronaviruses in the state after the three week... | tiger_lab_math | [
0.020167658105492592,
0.018945015966892242,
-0.06398465484380722,
-0.021805185824632645,
0.0397750586271286,
-0.014978399500250816,
-0.03933938592672348,
0.07446714490652084,
0.04708953946828842,
-0.010641002096235752,
0.01972629502415657,
0.04248298332095146,
-0.01851603016257286,
-0.0028... | 0.808969 | 0.807825 | 0.001143 |
If $x$ is a real number, find $(x+1)^2+2(x+1)(3-x)+(3-x)^2$. Let's write a program. | tiger_lab_math | [
0.031160185113549232,
0.02270880714058876,
-0.038763873279094696,
-0.03653724119067192,
0.04219929501414299,
-0.03519389033317566,
-0.017044829204678535,
0.06895122677087784,
0.052489519119262695,
-0.03278641402721405,
0.025730520486831665,
0.04090556129813194,
-0.04241834208369255,
-0.036... | 0.850142 | 0.845588 | 0.004554 |
Consider the periodic function $f(x)=\frac{x}{\pi}$ defined on $[-\pi,\pi]$. Find the Fourier series of $f(x)$ and determine whether it converges pointwise to $f(x)$ for all $x$ in $[-\pi,\pi]$. | tiger_lab_math | [
0.0015182270435616374,
0.03689328208565712,
-0.036969125270843506,
-0.029152866452932358,
0.025487391278147697,
-0.035509586334228516,
0.010793395340442657,
0.04773378372192383,
0.03706424683332443,
0.020637135952711105,
0.023323288187384605,
0.025039678439497948,
-0.024221165105700493,
-0... | 0.833153 | 0.820406 | 0.012747 |
A soft drink company claims that the average volume of its cans is 355 milliliters. To test this claim, a sample of 25 cans is taken and the volume of each can is recorded. The data shows a mean volume of 351 milliliters with a standard deviation of 4 milliliters. Does this data provide sufficient evidence to contradic... | tiger_lab_math | [
0.009443651884794235,
0.015144063159823418,
-0.052452124655246735,
-0.06331032514572144,
0.03116762265563011,
-0.046326182782649994,
-0.022126344963908195,
0.06883688271045685,
0.033156998455524445,
-0.008015101775527,
0.016624674201011658,
0.023408053442835808,
-0.029247388243675232,
-0.0... | 0.794669 | 0.792373 | 0.002296 |
Here is a numerical problem for a math student to solve in Decision theory:
A student has a certain amount of time to study for two exams: Math and English. If they allocate 4 hours to Math and 6 hours to English, they score a total of 80 points. If they allocate 5 hours to Math and 5 hours to English, they score a to... | tiger_lab_math | [
0.020043695345520973,
-0.02442973479628563,
-0.05612623691558838,
-0.03350994363427162,
0.026598770171403885,
-0.005664552561938763,
-0.038940683007240295,
0.03150857239961624,
0.04095875099301338,
-0.009606930427253246,
0.0047815232537686825,
0.010470717214047909,
-0.029020939022302628,
-... | 0.82527 | 0.8152 | 0.01007 |
Compute all values of $b$ for which the following system has a solution $(x,y)$ in real numbers:
\begin{align*}
\sqrt{xy} &= b^b, \\
\log_b (x^{\log_b y}) + \log_b (y^{\log_b x}) &= 4b^4.
\end{align*} | tiger_lab_math | [
0.02889406681060791,
0.04770561680197716,
-0.03618663176894188,
-0.03526712954044342,
0.0495556965470314,
-0.029460815712809563,
-0.0010243133874610066,
0.057890553027391434,
0.07093919813632965,
-0.01912253350019455,
0.014036712236702442,
0.04198572784662247,
-0.037103764712810516,
-0.038... | 0.835295 | 0.823211 | 0.012085 |
In triangle $XYZ$, we have $\angle Z = 90^\circ$, $XY = 10$, and $YZ = \sqrt{51}$. What is $\tan X$? Please write a program to solve it | tiger_lab_math | [
0.028639044612646103,
0.03570534288883209,
-0.0372331328690052,
-0.04675484076142311,
0.028474869206547737,
-0.0149821937084198,
-0.02561635710299015,
0.037157170474529266,
0.03965683653950691,
-0.035287003964185715,
0.034125134348869324,
0.021110637113451958,
-0.04419465363025665,
-0.0042... | 0.849468 | 0.847534 | 0.001934 |
Consider a graph G with 6 vertices and the following edges: $(1,2)$, $(1,4)$, $(2,3)$, $(2,4)$, $(3,4)$, $(3,5)$, and $(4,6)$. Determine the chromatic polynomial of G. | tiger_lab_math | [
0.022469857707619667,
0.030126165598630905,
-0.03183935582637787,
-0.033506523817777634,
0.027262676507234573,
-0.055056121200323105,
-0.026169804856181145,
0.026422981172800064,
0.019870992749929428,
-0.015300421975553036,
0.031830839812755585,
0.03296753391623497,
-0.017528602853417397,
... | 0.83578 | 0.829682 | 0.006098 |
Let $a$ and $b$ be positive real numbers. Find the minimum value of
\[a^2 + b^2 + \frac{1}{(a + b)^2}.\] | tiger_lab_math | [
0.011760881170630455,
0.032610855996608734,
-0.04634447768330574,
-0.04156352952122688,
0.04671863839030266,
-0.023853827267885208,
-0.0031358071137219667,
0.04892613738775253,
0.059861354529857635,
-0.03445366024971008,
0.004220384638756514,
0.018849290907382965,
-0.016971109434962273,
-0... | 0.850086 | 0.842732 | 0.007354 |
Consider the function $f(x) = x^2$ defined on the interval $(-\pi,\pi]$, and let $S_n(x)$ denote the $n$th partial sum of its Fourier series. Determine the value of $n$ for which $S_n(x)$ converges pointwise to $f(x)$ for all $x\in (-\pi,\pi]$. | tiger_lab_math | [
0.004914253484457731,
0.04460739716887474,
-0.04589023441076279,
-0.025674674659967422,
0.027327315881848335,
-0.016209328547120094,
-0.007375308312475681,
0.05098410323262215,
0.041306592524051666,
0.014702013693749905,
0.030376458540558815,
0.027083417400717735,
-0.031159544363617897,
-0... | 0.83791 | 0.81221 | 0.0257 |
Find all possible isomorphisms between the group of symmetries of a square (D4) and the groups C8 and Q8. Specifically, determine if there are any isomorphisms between D4 and each of C8 and Q8, and if there are, describe all of these isomorphisms explicitly. | tiger_lab_math | [
0.02252710610628128,
0.02301892079412937,
-0.028270186856389046,
-0.04009387269616127,
0.029975123703479767,
-0.03160741180181503,
0.0026681458111852407,
0.06058782339096069,
0.05048619210720062,
-0.011120159178972244,
0.01849142275750637,
0.008064351044595242,
-0.04289359226822853,
-0.040... | 0.812404 | 0.798316 | 0.014088 |
Given the Hénon map formula:
x(n+1) = 1 - a * x(n)^2 + y(n)
y(n+1) = b * x(n)
where a = 1.4 and b = 0.3, and starting values x(0) = 0 and y(0) = 0. Calculate the first 10 iterations of the Hénon map and plot the results as (x(n), y(n)) points on a graph. | tiger_lab_math | [
0.02626856230199337,
0.03293360769748688,
-0.02028786391019821,
-0.052352942526340485,
0.041462406516075134,
-0.04853584244847298,
-0.02489723637700081,
0.0681292936205864,
0.03618646040558815,
-0.02414993941783905,
0.013203119859099388,
0.028592580929398537,
-0.03511961176991463,
-0.02875... | 0.825104 | 0.818515 | 0.006589 |
A wire carrying current of 2 A is bent into a rectangle as shown below. The sides of the rectangle measure 2 m and 4 m. Find the magnitude and direction of the magnetic field at the center of the rectangle if the wire is located in free space.
(Note: You may a... | tiger_lab_math | [
0.00989120826125145,
0.05285371094942093,
-0.04170863330364227,
-0.03967021405696869,
0.009418735280632973,
-0.029685936868190765,
-0.0382838174700737,
0.07840169966220856,
0.036123231053352356,
0.015926430001854897,
0.001480369595810771,
0.03406567499041557,
-0.00982219073921442,
-0.02458... | 0.793801 | 0.790512 | 0.003289 |
Two drivers, Tom and Jerry, are driving on a narrow road towards each other. If one of them swerves, he will avoid the crash, but the swerver will be called a "chicken." If neither swerves, they will both crash. Tom and Jerry know each other well and have played this game many times before. They both have the same inte... | tiger_lab_math | [
-0.0055276816710829735,
0.00905867014080286,
-0.027488302439451218,
-0.02694282867014408,
0.02372545190155506,
0.004227738827466965,
-0.016407206654548645,
0.07245729863643646,
0.048200253397226334,
-0.03656027466058731,
-0.006155718583613634,
0.024972762912511826,
-0.041764914989471436,
-... | 0.816366 | 0.808052 | 0.008314 |
Here's a question for a math student on Singularities of algebraic varieties:
Find the singular point(s) for the algebraic variety V defined by the equation f(x,y,z) = x^2 + y^3 + z^3 + xyz = 0. | tiger_lab_math | [
0.028832800686359406,
0.03048340231180191,
-0.05179065838456154,
-0.04848873242735863,
0.02817756123840809,
-0.016443634405732155,
-0.011554790660738945,
0.05027138441801071,
0.051029447466135025,
-0.012245056219398975,
0.003995169419795275,
0.02156880311667919,
-0.018396632745862007,
-0.0... | 0.832876 | 0.825943 | 0.006932 |
A city in the US has a basketball league with three basketball teams, the Aziecs, the Braves and the Celtics. A sports writer notices that the tallest player of the Aziecs is shorter than the shortest player of the Braves. The shortest of the Celtics is shorter than the shortest of the Aziecs, while the tallest of the ... | tiger_lab_math | [
0.028425663709640503,
0.021586617454886436,
-0.03700565919280052,
-0.010164947248995304,
0.051588594913482666,
-0.022153811529278755,
-0.008093340322375298,
0.05943628028035164,
0.03562883287668228,
-0.02649638056755066,
0.01659642718732357,
0.050349850207567215,
-0.028780126944184303,
-0.... | 0.823432 | 0.82246 | 0.000972 |
Suppose you have a set of 5 beads, each of a different color, and you want to make a necklace using all of them. In how many different ways can you create the necklace (assuming that two necklaces are equivalent if one can be rotated to look like the other)? Apply Polya's enumeration theorem to solve the problem. | tiger_lab_math | [
0.041306059807538986,
0.00497626094147563,
-0.0393967479467392,
-0.019880015403032303,
0.023376906290650368,
-0.02757059410214424,
-0.03373058885335922,
0.056082502007484436,
0.029364293441176414,
-0.017528094351291656,
0.0015698353527113795,
0.04346555471420288,
-0.023258473724126816,
-0.... | 0.830299 | 0.819875 | 0.010424 |
Consider the elliptic curve defined by the equation $y^2 = x^3 + 2x - 5$ over the field of integers modulo 11. Find all the points on this curve that have order 5. | tiger_lab_math | [
0.006684591993689537,
0.01404916774481535,
-0.0402650386095047,
-0.0603618361055851,
0.03241759166121483,
-0.04560314118862152,
-0.02553301677107811,
0.044832225888967514,
0.034325502812862396,
-0.01968638226389885,
-0.00008981677092378959,
0.04428248479962349,
-0.03571208566427231,
0.0069... | 0.847397 | 0.835575 | 0.011822 |
If $A$ is an angle such that $\tan A + \sec A = 2,$ enter all possible values of $\cos A,$ separated by commas. | tiger_lab_math | [
0.03443753346800804,
0.016892720013856888,
-0.026729049161076546,
-0.028559019789099693,
0.021638354286551476,
-0.026700183749198914,
-0.021099459379911423,
0.04469845071434975,
0.03976542130112648,
-0.009867263957858086,
0.0304753165692091,
0.03886675462126732,
-0.030799616128206253,
-0.0... | 0.838892 | 0.825462 | 0.013429 |
Circle $O$ is located on the coordinate plane with center at $(2,3)$. One endpoint of a diameter is at $(-1,-1)$. What are the coordinates of the other endpoint of this diameter? Express your answer as an ordered pair. Please write a program to solve it | tiger_lab_math | [
0.015446063131093979,
0.044534023851156235,
-0.049820903688669205,
-0.055248651653528214,
0.029280899092555046,
-0.007265339605510235,
-0.03921458497643471,
0.03689401224255562,
0.03301634266972542,
-0.029028410091996193,
0.030555360019207,
0.03998628631234169,
-0.025976315140724182,
-0.01... | 0.843656 | 0.839619 | 0.004037 |
In a game of Stag Hunt, two players must choose to either hunt a stag, which will yield a payoff of 5 points to each player, or a hare, which will yield a payoff of 2 points to each player. However, if one player chooses to hunt the stag and the other chooses to hunt the hare, the player who hunts the hare will receive... | tiger_lab_math | [
0.002087571192532778,
0.009662885218858719,
-0.04885216802358627,
-0.03284089267253876,
0.021998219192028046,
-0.02512502484023571,
-0.02240275777876377,
0.04176384210586548,
0.04357447102665901,
-0.028146114200353622,
0.010944169946014881,
0.031024198979139328,
-0.028543390333652496,
-0.0... | 0.788384 | 0.785541 | 0.002843 |
Milton spilled some ink on his homework paper. He can't read the coefficient of $x$, but he knows that the equation has two distinct negative, integer solutions. What is the sum of all of the distinct possible integers that could be under the ink stain?
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
fill((0,0)--(3,0)-... | tiger_lab_math | [
0.020274214446544647,
0.024302203208208084,
-0.028391459956765175,
-0.04194100946187973,
0.03803477808833122,
-0.02596965990960598,
-0.009509595111012459,
0.06829903274774551,
0.04208940640091896,
-0.01153283566236496,
0.030724285170435905,
0.04362772777676582,
-0.009092974476516247,
-0.03... | 0.855157 | 0.845194 | 0.009963 |
Let A = {1, 2} and B = {a, b}. Find (A x B) ∪ (A x {c}) and simplify the resulting Cartesian product. | tiger_lab_math | [
0.04397376999258995,
0.04160220921039581,
-0.03660949692130089,
-0.02758597955107689,
0.03399353101849556,
-0.02328253537416458,
-0.01656636781990528,
0.06426849216222763,
0.05168923735618591,
-0.028692195191979408,
0.0035436595790088177,
0.028139881789684296,
-0.014272485859692097,
-0.035... | 0.833432 | 0.827743 | 0.00569 |
A block of wood has the shape of a cube with sides of length 10 cm. A cylindrical hole with a radius of 3 cm is drilled straight through the center of the cube along one of its diagonals. What is the shortest distance between two opposite corners of the block after the hole is drilled? | tiger_lab_math | [
0.022472884505987167,
0.036149390041828156,
-0.05696018412709236,
-0.02680676244199276,
0.02632751688361168,
-0.023914022371172905,
-0.037922974675893784,
0.05012936890125275,
0.025136489421129227,
-0.012109729461371899,
0.026338575407862663,
0.027932899072766304,
-0.03347591310739517,
0.0... | 0.826039 | 0.820943 | 0.005096 |
A telecommunications company needs to lay out fiber optic cables to connect 12 cities. The cost of laying fiber optic between each city is different and the company wants to minimize the total cost of laying out the cables. Use ant colony optimization to solve and find the cheapest layout for the fiber optic cables. | tiger_lab_math | [
0.011323398910462856,
0.02830432541668415,
-0.037340790033340454,
-0.03138623386621475,
0.02459179423749447,
-0.01928301714360714,
-0.034365925937891006,
0.05391119047999382,
0.03864053264260292,
-0.014242221601307392,
0.02169528231024742,
0.016707755625247955,
-0.0206910353153944,
-0.0306... | 0.79681 | 0.790573 | 0.006238 |
Given a topological space X consisting of two circles intersecting at exactly two points, compute the total number of ways to assign colors (black or white) to each point of intersection such that no two adjacent points have the same color. | tiger_lab_math | [
0.0047583384439349174,
0.017041463404893875,
-0.032004743814468384,
-0.02099418453872204,
0.008358965627849102,
-0.0016937359469011426,
-0.038297295570373535,
0.046416595578193665,
0.04656220227479935,
-0.02160247601568699,
0.017418449744582176,
0.028142379596829414,
-0.007361544296145439,
... | 0.842945 | 0.824543 | 0.018402 |
Consider the 2D heat equation given by:
∂u/∂t = k(∂²u/∂x² + ∂²u/∂y²)
where u(x,y,t) is the temperature at point (x,y) and time t, and k is a constant.
Suppose a thin metal plate occupies the region -1 ≤ x ≤ 1 and -1 ≤ y ≤ 1, and is initially at a temperature of u(x,y,0) = sin(πx)cos(πy).
Find the temperature distri... | tiger_lab_math | [
0.003574177622795105,
0.017040515318512917,
-0.05194962024688721,
-0.03423937410116196,
0.03561898693442345,
-0.026068540289998055,
-0.034198176115751266,
0.07581466436386108,
0.038339823484420776,
-0.0000843808229546994,
0.01766115054488182,
0.03926385939121246,
-0.013004392385482788,
0.0... | 0.80725 | 0.800317 | 0.006933 |
Consider two travelers who accidentally break their identical antique vases while traveling. The airline suggests a compensation amount to each traveler, but the travelers have to independently decide on their claim amount. The compensation offered ranges between $1-100. However, if both travelers claim the same amount... | tiger_lab_math | [
0.015347148291766644,
0.005619168281555176,
-0.03714266046881676,
-0.022272363305091858,
0.04647276923060417,
-0.005295938812196255,
-0.022116733714938164,
0.05876188725233078,
0.03176099807024002,
-0.020410964265465736,
0.03537067025899887,
0.031212501227855682,
-0.02983167953789234,
-0.0... | 0.814551 | 0.795723 | 0.018828 |
Consider a graph with 8 vertices that has the following adjacency matrix:
\begin{matrix}
0 & 1 & 0 & 1 & 0 & 0 & 0 & 1\\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 1\\
0 & 1 & 0 & 1 & 1 & 0 & 0 & 0\\
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0\\
0 & 0 & 1 & 1 & 0 & 1 & 1 & 0\\
0 & 0 & 0 & 1 & 1 & 0 & 1 & 1\\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1\\
1 & 1 ... | tiger_lab_math | [
0.015938395634293556,
0.01773480512201786,
-0.014770802110433578,
-0.03294166177511215,
0.0196343082934618,
-0.022226853296160698,
-0.030287718400359154,
0.08559383451938629,
0.025572624057531357,
0.0014732015551999211,
0.0023032668977975845,
0.020915530622005463,
-0.013978070579469204,
-0... | 0.82263 | 0.802146 | 0.020484 |
In a trust game, player A receives $10 and can decide to pass any amount between $0 and $10 to player B. The amount given to player B is then tripled, and player B can choose to keep or return a portion of the tripled amount to player A. If player B decides to keep the tripled amount, what is the maximum amount of mone... | tiger_lab_math | [
0.024148685857653618,
0.006885678507387638,
-0.03545049577951431,
-0.010146965272724628,
0.0233035609126091,
-0.019639810547232628,
-0.008873381651937962,
0.046322233974933624,
0.027564827352762222,
-0.04013371840119362,
0.031853221356868744,
0.04222502186894417,
-0.05215635895729065,
-0.0... | 0.807231 | 0.802852 | 0.004379 |
Sure, here's a problem:
A toy manufacturer produces two types of cars: compact and luxury. Each compact car brings in a profit of $5,000 while each luxury car brings in a profit of $8,000. The manufacturer has the capacity to produce 200 cars per month, but can only produce a maximum of 100 luxury cars due to producti... | tiger_lab_math | [
0.022240689024329185,
0.014742235653102398,
-0.04353988543152809,
-0.011423608288168907,
0.008396023884415627,
-0.008132249116897583,
-0.03448052331805229,
0.062303327023983,
0.029289202764630318,
-0.013988585211336613,
0.008689329028129578,
0.038059547543525696,
-0.026737846434116364,
-0.... | 0.826823 | 0.816095 | 0.010728 |
Consider the following situation: A farmer has 101 chickens, 102 ducks, and 103 pigs. The farmer wants to sell them all in smaller groups of equal size, with no animal left out. What is the largest number of animals the farmer can sell in each group? Use the Chinese remainder theorem to solve the problem. | tiger_lab_math | [
0.028443992137908936,
0.003338536946102977,
-0.05476287752389908,
-0.031209789216518402,
0.02242286130785942,
-0.018284961581230164,
-0.02217561937868595,
0.05464663356542587,
0.0252203568816185,
-0.015654178336262703,
0.00224789185449481,
0.04040605574846268,
-0.04643835499882698,
-0.0020... | 0.826862 | 0.822642 | 0.00422 |
A conducting sphere of radius 10 cm has a charge of 5 microcoulombs uniformly distributed over its surface. Find the potential at a distance of 20 cm from the center of the sphere, assuming the region outside the sphere is at zero potential. Use Laplace's equation for electrostatics. | tiger_lab_math | [
-0.009213857352733612,
0.0410754531621933,
-0.034728001803159714,
-0.04077716916799545,
0.03616178780794144,
-0.0439985990524292,
-0.02887582778930664,
0.04888519272208214,
0.043417789041996,
-0.021641254425048828,
0.021259622648358345,
0.03941509500145912,
-0.02934185415506363,
-0.0212204... | 0.797468 | 0.795647 | 0.001822 |
Suppose there are two players, Player 1 and Player 2, playing a game of Cheap Talk. Player 1 can communicate a message to Player 2 before they make their simultaneous moves. Suppose the payoff matrix for the game is as follows:
```
| L | R |
-----------------------------------
U | 2, 2 | 0,... | tiger_lab_math | [
0.0011224595364183187,
0.027757365256547928,
-0.03909837082028389,
-0.0377696231007576,
0.0329616405069828,
-0.021213272586464882,
-0.01927172765135765,
0.05597993731498718,
0.02988719753921032,
-0.05299454927444458,
0.030013641342520714,
0.03835219889879227,
-0.028311774134635925,
-0.0193... | 0.82244 | 0.818801 | 0.003639 |
Consider an industry in which two firms, A and B, compete for market share by deciding how many units to produce. Assume the market demand function is given by P = 50 - Q, where P is the price of the product and Q is the total quantity produced by both firms. The cost functions for firms A and B are given by C₁(Q₁) = 1... | tiger_lab_math | [
0.004738091491162777,
0.030523590743541718,
-0.061754994094371796,
-0.029756858944892883,
0.023484276607632637,
0.001505766180343926,
-0.03369379788637161,
0.07420533150434494,
0.02698611468076706,
-0.0149855837225914,
0.005100097507238388,
0.030259888619184494,
-0.041042231023311615,
-0.0... | 0.807195 | 0.785427 | 0.021769 |
Here is a question for a math student to solve related to Meromorphic functions and their properties:
Find all meromorphic functions $f(z)$ defined on the complex plane such that $f(z+1)+f(z-1)=\frac{1}{z}$ for all $z \in \mathbb{C}\setminus\left\{0,\pm 1, \pm 2, \ldots\right\}$. | tiger_lab_math | [
0.015008621849119663,
0.042960453778505325,
-0.05188322067260742,
-0.06583372503519058,
0.04295729100704193,
0.01150418445467949,
-0.008544913493096828,
0.05292442440986633,
0.04826821759343147,
-0.009525207802653313,
0.015236256644129753,
0.010070772841572762,
-0.030541813001036644,
-0.01... | 0.841135 | 0.818682 | 0.022453 |
there are 8 players in a chess group , and each player plays each of the others once . given that each game is played by two players , how many total games will be played ? Let's write a Python program to solve it. | tiger_lab_math | [
0.014215415343642235,
0.008941218256950378,
-0.059439290314912796,
-0.0350271500647068,
0.02872512862086296,
-0.03033251315355301,
-0.018981533125042915,
0.029876219108700752,
0.03196112439036369,
-0.02871193177998066,
0.014907094649970531,
0.02142169699072838,
-0.03786623105406761,
-0.014... | 0.807998 | 0.806705 | 0.001293 |
Let G be a group and a, b be elements of G. Statement 1 | If a and b commute (i.e., ab = ba), then the order of ab is equal to the order of a times the order of b. Statement 2 | If G is a cyclic group generated by an element g, then every element of G can be expressed as a power of g. Answer Choices: (A) True, True (B... | tiger_lab_math | [
0.049931641668081284,
0.028660619631409645,
-0.02864212729036808,
-0.0518638975918293,
0.032275937497615814,
-0.053212277591228485,
-0.0035501825623214245,
0.06625548005104065,
0.02421039342880249,
-0.025605371221899986,
0.015072664245963097,
0.025787321850657463,
-0.04026319086551666,
-0.... | 0.80711 | 0.800368 | 0.006742 |
A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign painter paints house n... | tiger_lab_math | [
0.009476253762841225,
0.03300001472234726,
-0.042493052780628204,
-0.02054067887365818,
0.03928361088037491,
-0.017735622823238373,
-0.03182346001267433,
0.04285724088549614,
0.03978925943374634,
-0.024996111169457436,
0.017034098505973816,
0.043492235243320465,
-0.03595921769738197,
-0.01... | 0.857347 | 0.845517 | 0.01183 |
The graph of the polynomial
$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$
has five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?
$\textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$ | tiger_lab_math | [
0.024510294198989868,
0.03600631281733513,
-0.02276894450187683,
-0.04016033932566643,
0.02646815776824951,
-0.031219327822327614,
-0.008988359943032265,
0.0657227486371994,
0.04135146737098694,
-0.017751459032297134,
0.011726247146725655,
0.010158239863812923,
-0.009481503628194332,
-0.02... | 0.843765 | 0.829172 | 0.014594 |
The graph of the parabola defined by the equation $y=(x-2)^2+3$ is rotated 180 degrees about its vertex, then shifted 3 units to the left, then shifted 2 units down. The resulting parabola has zeros at $x=a$ and $x=b$. What is $a+b$? Please write a program to solve it | tiger_lab_math | [
0.020447948947548866,
0.03279905021190643,
-0.04326851665973663,
-0.02803918346762657,
0.021188287064433098,
-0.009192591533064842,
-0.023551184684038162,
0.07571671158075333,
0.0343545563519001,
-0.037035513669252396,
0.00595810217782855,
0.03529928997159004,
-0.03149662911891937,
-0.0284... | 0.846311 | 0.841474 | 0.004837 |
Suppose Alice wants to send a secret message to Bob using ElGamal encryption. She chooses a prime number p=17 and a primitive root α=3. She also chooses her secret key as a=4. Bob's public key is β=13. If Alice's plaintext message is m=7, what is the encrypted message (c1, c2) that she sends to Bob? | tiger_lab_math | [
0.02419792115688324,
0.045973844826221466,
-0.03122825175523758,
-0.059018272906541824,
0.031993597745895386,
-0.04371647536754608,
-0.008378819562494755,
0.03621978312730789,
0.025862006470561028,
-0.025360209867358208,
0.02521943487226963,
0.01897970214486122,
-0.04682202637195587,
-0.02... | 0.824517 | 0.823006 | 0.001511 |
In the diagram below, lines $k$ and $\ell$ are parallel. Find the measure of angle $x$ in degrees.
[asy]
size(200);
import markers;
pair A = dir(-22)*(0,0);
pair B = dir(-22)*(4,0);
pair C = dir(-22)*(4,2);
pair D = dir(-22)*(0,2);
pair F = dir(-22)*(0,1.3);
pair G = dir(-22)*(4,1.3);
pair H = dir(-22)*(2,1);
//mark... | tiger_lab_math | [
0.013249434530735016,
0.010120399296283722,
-0.02668796479701996,
-0.04538032039999962,
0.02852095663547516,
-0.004427350591868162,
-0.02670351043343544,
0.04304992035031319,
0.044374655932188034,
0.001039962749928236,
0.024257628247141838,
0.0390782430768013,
-0.01606966368854046,
0.00353... | 0.828753 | 0.815808 | 0.012945 |
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