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 |
|---|---|---|---|---|---|
Find the solution to the boundary value problem:
$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ in the region $0 < x < \pi$, $0 < y < \pi/4$,
subject to the boundary conditions:
$u(0,y) = u(\pi,y) = \sin(y)$,
$u(x,0) = 0$,
$u(x,\pi/4) = \sin(3x)$ | tiger_lab_math | [
-0.010558968409895897,
0.03424713388085365,
-0.04453207552433014,
-0.029888439923524857,
0.052983608096838,
0.01061920914798975,
-0.003305132733657956,
0.06413992494344711,
0.0668499618768692,
0.0012624072842299938,
0.01343163475394249,
0.0411829873919487,
-0.005262796767055988,
-0.0249137... | 0.825875 | 0.802484 | 0.023391 |
A mechanical engineer has to design a cantilever beam with a length of 30 cm and a height of 7 cm using a material with a Young's modulus of 2 x 10^5 MPa. The beam must withstand a uniformly distributed load of 1000 N at the free end.
The cross-section of the beam is a rectangle with a variable width (w) and a consta... | tiger_lab_math | [
0.012581391260027885,
0.028438083827495575,
-0.04592107981443405,
-0.05720711871981621,
0.04418232664465904,
-0.03337783366441727,
-0.04188981652259827,
0.08437298238277435,
0.028339484706521034,
-0.0029547226149588823,
0.02180730737745762,
0.03768733888864517,
-0.031005024909973145,
-0.00... | 0.816799 | 0.809957 | 0.006841 |
Find all the fixed points of the action of the group $\mathbb{Z}/3\mathbb{Z}$ on the projective plane $\mathbb{P}^2$ induced by the action given by $\zeta \cdot [x:y:z] = [\zeta x : \zeta^2 y : z]$ for $\zeta \in \mathbb{Z}/3\mathbb{Z}$. | tiger_lab_math | [
0.036154888570308685,
0.03863834589719772,
-0.042351655662059784,
-0.04630023613572121,
0.02749769203364849,
-0.021206289529800415,
0.009128126315772533,
0.03507247194647789,
0.04224829748272896,
-0.011787119321525097,
0.009252382442355156,
0.02693241275846958,
-0.029183536767959595,
-0.02... | 0.839268 | 0.817662 | 0.021606 |
Let $a, b, c$ be complex numbers, with $a$ real, such that \[a+b+c=ab+bc+ca=abc=3.\]Find $a$. | tiger_lab_math | [
0.0430360808968544,
0.03533778712153435,
-0.04192732647061348,
-0.04197528958320618,
0.0310788806527853,
-0.010916762985289097,
-0.00667934212833643,
0.05427214130759239,
0.031067976728081703,
-0.015533997677266598,
0.020630789920687675,
0.047723837196826935,
-0.04940740019083023,
-0.04692... | 0.849012 | 0.837555 | 0.011457 |
Let A = {1, 2, 3, 4} and B = {2, 3, 4, 5}. Find (A △ B) ∪ B, where △ represents the symmetric difference of two sets. | tiger_lab_math | [
0.03988746181130409,
0.011722471565008163,
-0.03544275462627411,
-0.02054566517472267,
0.03511078283190727,
-0.019607113674283028,
-0.022248798981308937,
0.0602952316403389,
0.06778208911418915,
-0.014585353434085846,
0.009114665910601616,
0.02805485390126705,
-0.02553899772465229,
-0.0050... | 0.833718 | 0.828293 | 0.005424 |
Here's a numerical problem for a math student to solve related to the Chinese Remainder Theorem:
Suppose you are given the system of congruences:
x ≡ 3 (mod 5)
x ≡ 4 (mod 7)
x ≡ 2 (mod 11)
Find the smallest positive integer solution for x using the Chinese Remainder Theorem. | tiger_lab_math | [
0.04030098021030426,
-0.006493410095572472,
-0.03921724110841751,
-0.044335316866636276,
0.04608947038650513,
-0.019395537674427032,
-0.019245987758040428,
0.051809411495923996,
0.043729573488235474,
-0.003569741966202855,
0.00977916270494461,
0.01643378660082817,
-0.03311329707503319,
-0.... | 0.833639 | 0.831702 | 0.001937 |
The length of the longer side of rectangle $R$ is $10$ percent more than the length of a side of square $S.$ The length of the shorter side of rectangle $R$ is $10$ percent less than the length of a side of square $S.$ What is the ratio of the area of rectangle $R$ to the area of square $S?$ Express your answer as a co... | tiger_lab_math | [
0.013441693969070911,
0.018856694921851158,
-0.05987972766160965,
-0.041954003274440765,
0.016056066378951073,
-0.030684826895594597,
-0.02900644950568676,
0.07354069501161575,
0.0463496595621109,
-0.000204465861315839,
0.02269255742430687,
0.030529391020536423,
-0.05489160493016243,
-0.00... | 0.819204 | 0.814341 | 0.004864 |
Suppose a farmer has 400 acres of land to plant two crops: wheat and corn. The profit per acre for growing wheat is $200 and for growing corn is $300. The farmer can spend at most $50,000 on farming operations. Each acre of wheat requires $150 of farming costs and each acre of corn requires $200 of farming costs. Assum... | tiger_lab_math | [
0.01739128679037094,
0.02355325222015381,
-0.03607936576008797,
-0.021936874836683273,
0.004688873887062073,
-0.02541307732462883,
-0.027701035141944885,
0.06220573931932449,
0.02726076729595661,
-0.007285831030458212,
-0.012249589897692204,
0.03553755581378937,
-0.03215960040688515,
-0.02... | 0.792364 | 0.789608 | 0.002756 |
A math student is asked to determine the symmetries of a 4-dimensional manifold with a metric tensor given by:
g = dx1⊗dx1 + dx2⊗dx2 + dx3⊗dx3 + dx4⊗dx4
Find the Lie algebra of the group of isometries of this manifold. | tiger_lab_math | [
0.03651687875390053,
0.018007639795541763,
-0.04093579947948456,
-0.0676165521144867,
0.042624332010746,
-0.029949387535452843,
-0.03271099179983139,
0.054355867207050323,
0.04856047034263611,
0.009319768287241459,
0.0028183087706565857,
0.02968241274356842,
-0.03435555845499039,
-0.035877... | 0.823243 | 0.813679 | 0.009564 |
If u is the real part of a function, and v is the imaginary part, then the Cauchy-Riemann equations for u and v take the following form in polar coordinates: r\frac{\partial u}{\partial r} = \frac{\partial v}{\partial \theta} and r\frac{\partial v}{\partial r} = -\frac{\partial u}{\partial \theta}. Is this argument Tru... | tiger_lab_math | [
0.036475975066423416,
0.03955099359154701,
-0.03588329628109932,
-0.045038193464279175,
0.02398240938782692,
-0.02635766752064228,
0.00966828316450119,
0.07643583416938782,
0.03073250688612461,
-0.026498829945921898,
0.01100896019488573,
0.02916700951755047,
-0.02029183879494667,
-0.025945... | 0.814541 | 0.810273 | 0.004269 |
"statement 1": The product of all the possible units digits of Sophie Germain primes greater than 3 is, "statement 2" :The product of all the possible units digits of Sophie Germain primes greater than more than 1 is, "options: " Entailment or contradiction or neutral? Let's program in Python in the response. | tiger_lab_math | [
0.030004916712641716,
-0.005350039340555668,
-0.038059819489717484,
-0.04348096251487732,
0.04454844817519188,
-0.042320284992456436,
-0.016473786905407906,
0.07714806497097015,
0.03473247215151787,
-0.004955441225320101,
0.021982135251164436,
0.04959962144494057,
-0.04621854051947594,
-0.... | 0.815812 | 0.809288 | 0.006524 |
Suppose there are two players, A and B, who are trying to decide on a strategy for a one-time game. A can choose between two strategies, strategy X or strategy Y, while B can choose between two strategies, strategy Z or strategy W. The payoff matrix for the game is given below:
| | Z (B's strategy) | W (B's str... | tiger_lab_math | [
0.011967290192842484,
0.011340375989675522,
-0.03859049454331398,
-0.04185172915458679,
0.034188494086265564,
-0.02398674376308918,
-0.013589517213404179,
0.06817591935396194,
0.026790795847773552,
-0.044223178178071976,
0.02844053879380226,
0.03479842469096184,
-0.02889515459537506,
-0.01... | 0.809239 | 0.800444 | 0.008795 |
Consider two criminals, A and B, who are arrested and imprisoned for a theft. The police have no evidence to convict them, so they decide to interrogate them separately. If both A and B confess to the crime, then they both will receive a 5-year sentence. If neither confesses, then they’ll both receive a 1-year sentence... | tiger_lab_math | [
0.008541064336895943,
0.005174028687179089,
-0.04182390123605728,
-0.02094244211912155,
0.022858107462525368,
-0.010928323492407799,
-0.015558925457298756,
0.06303390860557556,
0.04850860312581062,
-0.014986565336585045,
0.011261437088251114,
0.02303735539317131,
-0.03648023679852486,
-0.0... | 0.808497 | 0.794316 | 0.014182 |
Find the greatest natural number $n$ such that $n\leq 2008$ and $(1^2+2^2+3^2+\cdots + n^2)\left[(n+1)^2+(n+2)^2+(n+3)^2+\cdots + (2n)^2\right]$ is a perfect square.
Let's write a Python program. | tiger_lab_math | [
0.01822070963680744,
0.04208562895655632,
-0.042350973933935165,
-0.04822663590312004,
0.0335225984454155,
-0.03480027988553047,
-0.031560882925987244,
0.04423833265900612,
0.029169848188757896,
-0.016181860119104385,
0.024488532915711403,
0.0282053891569376,
-0.03609635680913925,
-0.02551... | 0.84267 | 0.826184 | 0.016486 |
Find the value of $a$ so that the lines described by
\[\begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix} + t \begin{pmatrix} a \\ -2 \\ 1 \end{pmatrix}\]and
\[\begin{pmatrix} 1 \\ -3/2 \\ -5 \end{pmatrix} + u \begin{pmatrix} 1 \\ 3/2 \\ 2 \end{pmatrix}\]are perpendicular. | tiger_lab_math | [
0.020787924528121948,
0.02912973240017891,
-0.048504747450351715,
-0.033168308436870575,
0.05666384473443031,
-0.011193439364433289,
-0.020272500813007355,
0.0915389284491539,
0.03140094503760338,
-0.0016448041424155235,
0.009589992463588715,
0.04661779850721359,
-0.02202312834560871,
-0.0... | 0.834716 | 0.827822 | 0.006894 |
Let $f(z) = \frac{z^2-2z+3}{z^3-4}$ be a meromorphic function on the complex plane. Determine the location and order of any poles and zeroes of $f(z)$. | tiger_lab_math | [
0.02305401861667633,
0.03761657327413559,
-0.05012909695506096,
-0.04870324581861496,
0.015590474009513855,
0.006455724593251944,
-0.00006845292955404148,
0.03601972013711929,
0.03290392830967903,
-0.0018812303896993399,
0.023069757968187332,
0.029731865972280502,
-0.02440338209271431,
-0.... | 0.827731 | 0.819197 | 0.008534 |
Find all $a,$ $0^\circ < a < 360^\circ,$ such that $\cos a,$ $\cos 2a,$ and $\cos 3a$ form an arithmetic sequence, in that order. Enter the solutions, separated by commas, in degrees. | tiger_lab_math | [
0.029701121151447296,
0.029111038893461227,
-0.014438352547585964,
-0.03432951867580414,
0.019560258835554123,
-0.01952512189745903,
-0.020924031734466553,
0.06574055552482605,
0.03060278110206127,
-0.01415302138775587,
0.019187120720744133,
0.03147280961275101,
-0.032451193779706955,
-0.0... | 0.840506 | 0.827583 | 0.012923 |
Consider a graph with 6 vertices and the degree of each vertex is given by 2, 2, 3, 3, 4, and 4. Determine whether or not this graph can be constructed. | tiger_lab_math | [
0.025985250249505043,
0.025678414851427078,
-0.018636086955666542,
-0.02317654713988304,
0.029501020908355713,
-0.038681238889694214,
-0.01963881403207779,
0.054455142468214035,
0.02535391040146351,
-0.031468525528907776,
0.027018258348107338,
0.024735739454627037,
-0.012477654032409191,
-... | 0.828102 | 0.820211 | 0.007891 |
When we say that Ray is climbing up the stairs $m$ at a time, we mean that he starts on the floor (step $0$) then jumps to step $m$ and then to $2m$ and so on until the number of steps to the top is less than $m$. Ray climbs up a flight of stairs of $n$ steps in two ways. When he does it $4$ steps at a time, there are ... | tiger_lab_math | [
0.027847180142998695,
0.04635198786854744,
-0.030777592211961746,
-0.03183085098862648,
0.04369618371129036,
-0.008662018924951553,
-0.009804122149944305,
0.06565256416797638,
0.024930570274591446,
-0.012983156368136406,
0.02202664501965046,
0.04139406606554985,
-0.06189710646867752,
-0.01... | 0.838497 | 0.832371 | 0.006126 |
A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$.
| tiger_lab_math | [
0.011225638911128044,
0.01015026681125164,
-0.030580826103687286,
-0.023457206785678864,
0.04174729064106941,
-0.018435096368193626,
-0.03605623170733452,
0.07475020736455917,
0.030559590086340904,
-0.011800582520663738,
0.0041411868296563625,
0.019971253350377083,
-0.03887654095888138,
-0... | 0.84016 | 0.827105 | 0.013055 |
Sure! Here's a numerical problem for a math student to solve:
"A new disease has been spreading in a city. The number of people infected is growing at a rate proportional to the number of infected people and the number of people who are susceptible to the disease. The total population of the city is 10,000, and it is ... | tiger_lab_math | [
0.02973739430308342,
0.01524387113749981,
-0.041426435112953186,
-0.03577466309070587,
0.039700280874967575,
-0.03962545096874237,
-0.025343257933855057,
0.05139295384287834,
0.03275680914521217,
-0.02031451091170311,
0.011548579670488834,
0.03214826062321663,
-0.03009478747844696,
-0.0342... | 0.822199 | 0.820784 | 0.001414 |
Here's a precise numerical problem in Temporal Logic for a math student to solve:
Given the following statements:
- P: It is sunny today
- Q: It rained yesterday
Use the temporal operator "Next" to represent the following:
- It will be sunny tomorrow and it rained today
Write the compound proposition using P, Q,... | tiger_lab_math | [
0.029179809615015984,
0.029430795460939407,
-0.04061126336455345,
-0.050715286284685135,
0.038087520748376846,
-0.024493778124451637,
-0.02649674564599991,
0.06362535804510117,
0.04167743772268295,
-0.016999566927552223,
0.01802784390747547,
0.02384382300078869,
-0.013324580155313015,
-0.0... | 0.826112 | 0.818955 | 0.007157 |
Find the general solution to the complex partial differential equation:
$$\frac{\partial^2 u}{\partial z\partial \bar{z}} + 4\frac{\partial u}{\partial z} = 0$$
given that $u|_{z=0} = e^{i\theta}$ and $\frac{\partial u}{\partial z}|_{z=0}=0$, where $\theta$ is a constant. | tiger_lab_math | [
0.016699157655239105,
0.031031029298901558,
-0.037511326372623444,
-0.05086895823478699,
0.04408182576298714,
-0.016440030187368393,
0.005501332692801952,
0.05964731425046921,
0.056829988956451416,
-0.004041626583784819,
0.026651713997125626,
0.011159232817590237,
-0.023157214745879173,
-0... | 0.816124 | 0.804166 | 0.011958 |
Consider the boundary value problem given by:
u''(x) + 4u(x) = 0, for 0 < x < pi,
u(0) = 0, u(pi) = 2.
Find the Fourier sine series representation of the solution. | tiger_lab_math | [
-0.014949913136661053,
0.03172377124428749,
-0.03878134861588478,
-0.03305533528327942,
0.03801743686199188,
-0.014769378118216991,
-0.011275864206254482,
0.06943459063768387,
0.07514244318008423,
0.004373478703200817,
0.0354451984167099,
0.03700585663318634,
-0.015615724958479404,
-0.0284... | 0.828386 | 0.825607 | 0.002779 |
Find all solutions to
\[\sqrt{x + 3 - 4 \sqrt{x - 1}} + \sqrt{x + 8 - 6 \sqrt{x - 1}} = 1.\] | tiger_lab_math | [
0.029342269524931908,
0.042944878339767456,
-0.016766000539064407,
-0.05554376170039177,
0.03635047376155853,
-0.021469520404934883,
-0.013857239857316017,
0.05944802984595299,
0.05330978333950043,
-0.019018923863768578,
0.015787072479724884,
0.009179639630019665,
-0.025930728763341904,
-0... | 0.840275 | 0.828504 | 0.011771 |
In a certain mathematical activity, we have five cards with five different prime numbers on them. We will distribute these five cards among three envelope: all could go in any envelope, or they could be broken up in any way among the envelopes. Then in each envelop, we find the product of all the cards in that envelope... | tiger_lab_math | [
0.033294957131147385,
0.0031678236555308104,
-0.01679910346865654,
-0.027298148721456528,
0.045827750116586685,
-0.035901911556720734,
-0.02043897844851017,
0.06239963695406914,
0.039179183542728424,
-0.0073252031579613686,
0.013123374432325363,
0.036202870309352875,
-0.037481263279914856,
... | 0.830379 | 0.829084 | 0.001295 |
Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2010$, and $a^2 - b^2 + c^2 - d^2 = 2010$. Find the number of possible values of $a.$ | tiger_lab_math | [
0.02958732470870018,
0.02096524089574814,
-0.044498831033706665,
-0.020073913037776947,
0.03988543525338173,
-0.024765871465206146,
-0.01912783272564411,
0.052552662789821625,
0.05642681196331978,
-0.019436264410614967,
0.012063409201800823,
0.03650341182947159,
-0.06063981354236603,
-0.05... | 0.843662 | 0.833355 | 0.010307 |
Here's a numerical problem that a math student can solve:
Let X be a topological space which satisfies T1 axiom, and A and B be disjoint closed sets in X. If U is an open set containing A, find an open set V containing B such that A and V are disjoint.
If X = {1, 2, 3, 4, 5}, and the topology on X is generated by the... | tiger_lab_math | [
0.040176600217819214,
0.009447176940739155,
-0.03915863856673241,
-0.03195429965853691,
0.050372540950775146,
0.00009471904195379466,
-0.035983867943286896,
0.05238842964172363,
0.05379967391490936,
-0.0019300157437101007,
0.02168326824903488,
0.01914716511964798,
-0.02865871973335743,
-0.... | 0.847163 | 0.818485 | 0.028678 |
There is only one value of $k$ for which the line $x=k$ intersects the graphs of $y=x^2+6x+5$ and $y=mx+b$ at two points which are exactly $5$ units apart. If the line $y=mx+b$ passes through the point $(1,6)$, and $b\neq 0$, find the equation of the line. Enter your answer in the form "$y = mx + b$". | tiger_lab_math | [
0.032965149730443954,
0.02472415380179882,
-0.05324279144406319,
-0.048630062490701675,
0.05683211609721184,
-0.03375925123691559,
-0.005566722713410854,
0.06951572000980377,
0.04888616502285004,
-0.02514192834496498,
0.03233418986201286,
0.03402482718229294,
0.0010732519440352917,
-0.0169... | 0.843668 | 0.82998 | 0.013688 |
Consider the Chirikov-Taylor map given by the equation:
x_(n+1) = x_n + y_n + (K/(2π))sin(2πx_n)
y_(n+1) = y_n + (K/(2π))sin(2πx_(n+1))
where 0 ≤ x_n,y_n ≤ 1 and K is the strength parameter.
If we start with initial conditions x_0 = 0.2 and y_0 = 0.6, and choose K = 0.8, what is the value of x_50, where x_50 is th... | tiger_lab_math | [
0.020566672086715698,
0.026995420455932617,
-0.0380755215883255,
-0.046021733433008194,
0.038000673055648804,
-0.01775873824954033,
-0.013499264605343342,
0.09836780279874802,
0.02489953488111496,
-0.018100375309586525,
-0.0018545978236943483,
0.01702704094350338,
-0.04801354557275772,
-0.... | 0.826446 | 0.818467 | 0.007979 |
What is the image of the upper half-plane under the transformation $f(z) = \frac{1+z}{1-z}$ using the Schwarz reflection principle? Specifically, find the image of the upper half of the unit circle $|z|=1$, and the lower half-plane below $y=0$. | tiger_lab_math | [
0.0014652960235252976,
0.03503173589706421,
-0.03751363977789879,
-0.037889134138822556,
0.041116151958703995,
-0.007493962999433279,
0.0038949446752667427,
0.06223474070429802,
0.02794136293232441,
0.0016192799666896462,
0.03182035684585571,
-0.0037122750654816628,
-0.018350841477513313,
... | 0.805306 | 0.7974 | 0.007906 |
In a moral hazard scenario, an automobile insurance policyholder has a 40% chance of getting in an accident that results in a $10,000 claim and a 60% chance of no accident. The policyholder must choose between adopting either a safe driving policy with a $500 premium or a risky driving policy with a $100 premium. The s... | tiger_lab_math | [
-0.009260392747819424,
-0.02451152913272381,
-0.013256936334073544,
-0.020868510007858276,
0.016427621245384216,
-0.00572426151484251,
-0.03145228698849678,
0.07731811702251434,
0.03862903267145157,
-0.01641751639544964,
0.007087821140885353,
0.0276009663939476,
-0.017770254984498024,
-0.0... | 0.781305 | 0.774485 | 0.00682 |
Consider a two-dimensional manifold given by the parametrization:
$x(u, v) = u\cos(v)$
$y(u, v) = u\sin(v)$
$z(u, v) = u^2$
Find the Riemann curvature tensor at the point $(u, v) = (1, 0)$. | tiger_lab_math | [
0.014155118726193905,
0.03735894337296486,
-0.029595287516713142,
-0.04275604337453842,
0.037811972200870514,
0.0020961437840014696,
-0.0003049058432225138,
0.05795373395085335,
0.023642774671316147,
-0.012620204128324986,
0.0056577883660793304,
0.034604914486408234,
-0.03332943841814995,
... | 0.825997 | 0.806398 | 0.019599 |
Consider a group of order 96. Find out whether the given group is isomorphic to the cyclic group C96, the alternating group A4 × C4 × C2 or one of the 196 non-abelian groups of order 96. If the group is isomorphic to one of them, determine which one it is. | tiger_lab_math | [
0.008070925250649452,
-0.0019499021582305431,
-0.03554556518793106,
-0.020120233297348022,
0.024169404059648514,
-0.05656237527728081,
0.0008255865541286767,
0.04871506616473198,
0.04125555232167244,
-0.0023995451629161835,
0.029614096507430077,
-0.000015465024262084626,
-0.04370910674333572... | 0.815388 | 0.811023 | 0.004365 |
Here's a precise numerical problem for a math student to solve:
A real estate agent wants to predict the selling price of a house based on its square footage. They've gathered the following data from recent sales:
Square footage (x): 1800, 2300, 1500, 2100, 2500
Selling price (y): $250,000, $320,000, $200,000, $280,0... | tiger_lab_math | [
0.004340124782174826,
0.03177795186638832,
-0.07589296996593475,
-0.05793602764606476,
0.04495614022016525,
0.005291799549013376,
-0.026756593957543373,
0.04467574134469032,
0.05439159274101257,
-0.012884420342743397,
0.010599825531244278,
0.0388106070458889,
-0.02636248990893364,
-0.01790... | 0.825394 | 0.821753 | 0.003641 |
Consider a graph G with 7 vertices and the following degree sequence: {3, 3, 3, 2, 2, 1, 1}. Is this degree sequence valid? If so, draw a possible graph G that corresponds to this degree sequence. If not, explain why not. | tiger_lab_math | [
0.022372981533408165,
0.02202346920967102,
-0.010418894700706005,
-0.04247894510626793,
0.03027597814798355,
-0.0335426889359951,
-0.021937817335128784,
0.06486977636814117,
0.015766989439725876,
-0.032769180834293365,
0.024050315842032433,
0.027548735961318016,
-0.02557695284485817,
-0.01... | 0.824435 | 0.815475 | 0.00896 |
Let G be a graph with four vertices a, b, c, and d and five edges {a, b}, {b, c}, {c, d}, {d, a}, and {a, c}.
Find the Tutte polynomial of G. | tiger_lab_math | [
0.027242733165621758,
0.02959979884326458,
-0.039131369441747665,
-0.030847257003188133,
0.037631556391716,
-0.04171896353363991,
-0.026018118485808372,
0.0342332199215889,
0.03179816156625748,
-0.009768268093466759,
0.015919024124741554,
0.030255382880568504,
-0.030850278213620186,
-0.025... | 0.827331 | 0.821099 | 0.006232 |
How many points are there in the Cantor set after the first n iterations of the construction process, where each iteration involves removing the middle 1/3 portion of each line segment? For example, after the first iteration, there are two line segments left, each 1/3 the length of the original, so there are two points... | tiger_lab_math | [
0.009796286933124065,
0.02490866556763649,
-0.037219543009996414,
-0.028920285403728485,
0.03033594973385334,
-0.006743174511939287,
-0.03748574107885361,
0.063236303627491,
0.03367532789707184,
0.0026462324894964695,
0.007222488522529602,
0.02049078978598118,
-0.04295498505234718,
-0.0206... | 0.83208 | 0.82664 | 0.00544 |
Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of maximum length? Please respond by writing a ... | tiger_lab_math | [
0.021773003041744232,
0.0026757975574582815,
-0.0376177579164505,
-0.027662033215165138,
0.017786722630262375,
-0.034491926431655884,
-0.03454413264989853,
0.05447140708565712,
0.03889306262135506,
-0.02423117496073246,
0.025555914267897606,
0.023976774886250496,
-0.04474674537777901,
-0.0... | 0.825906 | 0.823922 | 0.001985 |
According to Goldbach's Conjecture, every even integer greater than 2 can be expressed as the sum of two prime numbers. Can you find the prime numbers that add up to 126? | tiger_lab_math | [
0.021128660067915916,
0.0015705146361142397,
-0.0038381381891667843,
-0.04221787303686142,
0.02078634686768055,
-0.011648452840745449,
-0.003660472808405757,
0.04431173577904701,
0.022362368181347847,
-0.012623647227883339,
0.022189907729625702,
0.039624880999326706,
-0.033420663326978683,
... | 0.835208 | 0.826162 | 0.009047 |
Given the metric tensor in polar coordinates: $ds^2 = dr^2 + r^2d\theta^2$, calculate the all non-zero Christoffel symbols for this metric. | tiger_lab_math | [
0.009842129424214363,
0.01768960803747177,
-0.047268740832805634,
-0.05685107037425041,
0.02756681479513645,
-0.02905157394707203,
-0.02270791493356228,
0.04260949045419693,
0.05005419999361038,
-0.026272065937519073,
0.011855101212859154,
0.016659937798976898,
-0.02294071577489376,
-0.022... | 0.831155 | 0.823257 | 0.007898 |
Let X be a compact, connected, orientable, topological manifold of dimension 3. Suppose that the first and second homology groups of X are both isomorphic to Z.
Using Poincaré duality, determine the value of the intersection number of two closed, connected, orientable, non-intersecting, embedded 2-dimensional manifol... | tiger_lab_math | [
0.03195742890238762,
0.010349772870540619,
-0.03309784457087517,
-0.04178230091929436,
0.016326289623975754,
0.010536360554397106,
-0.011988797225058079,
0.04405181482434273,
0.04238737374544144,
-0.002259537810459733,
0.022391512989997864,
0.027442574501037598,
-0.02816944383084774,
-0.01... | 0.821422 | 0.796572 | 0.024849 |
A manufacturing company produces two types of products, Product A and Product B. The production process requires three resources: labor hours, machine hours, and raw materials. The company has a maximum of 900 labor hours, 600 machine hours, and 225 raw materials available per day. Each unit of Product A requires 3 lab... | tiger_lab_math | [
0.03077203594148159,
0.008071152493357658,
-0.06491926312446594,
0.004315591882914305,
0.013079462572932243,
-0.013916952535510063,
-0.060219656676054,
0.08826230466365814,
0.021939903497695923,
-0.00221449532546103,
0.002991076558828354,
0.03148985281586647,
-0.002621702617034316,
-0.0345... | 0.793727 | 0.776134 | 0.017594 |
Consider the smooth function $f(x,y) = x^2y$. Determine the partial derivatives $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ at point $(2,-3)$. | tiger_lab_math | [
0.02365664578974247,
0.04484650492668152,
-0.0655398964881897,
-0.036297332495450974,
0.04067089781165123,
-0.010928752832114697,
-0.021938802674412727,
0.06516171246767044,
0.04089004546403885,
-0.016824616119265556,
0.02096112072467804,
0.031885869801044464,
-0.04911961033940315,
-0.0081... | 0.823239 | 0.812048 | 0.011191 |
A grocery store sells 50 different types of fruits. The following list shows the quantity of each fruit sold during a week:
- 25 kg of apples
- 20 kg of bananas
- 15 kg of oranges
- 10 kg of grapes
Create a pie chart to represent the sales data using data mining techniques, and calculate the percentage of each fruit ... | tiger_lab_math | [
-0.005470480769872665,
0.010348614305257797,
-0.04806898906826973,
-0.03688070923089981,
0.027892284095287323,
-0.0279247984290123,
-0.035027045756578445,
0.054029062390327454,
0.03836126625537872,
0.0009846542961895466,
0.018333205953240395,
0.02299298532307148,
-0.03356894478201866,
0.01... | 0.787633 | 0.783043 | 0.004589 |
Let $x = (2 + \sqrt{3})^{1000},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$ Find
\[x(1 - f).\] | tiger_lab_math | [
0.02824065461754799,
0.033177025616168976,
-0.05167192220687866,
-0.021544402465224266,
0.028650658205151558,
-0.025149712339043617,
-0.026070309802889824,
0.06208028271794319,
0.0374009869992733,
0.002414699411019683,
0.03325123339891434,
0.03620191290974617,
-0.05449630692601204,
-0.0286... | 0.844268 | 0.839496 | 0.004772 |
The two squares shown share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
[asy] //code taken from thread for problem real alpha = 25; pair W=dir(225), X=dir(315),... | tiger_lab_math | [
0.018390875309705734,
0.017887506633996964,
-0.0593317411839962,
-0.059101007878780365,
0.029255686327815056,
-0.014281054958701134,
-0.017170075327157974,
0.06138350069522858,
0.05198810249567032,
-0.012496404349803925,
0.03143421560525894,
0.022359054535627365,
-0.028281325474381447,
-0.... | 0.843913 | 0.823715 | 0.020198 |
Let $M$ be a smooth manifold, and let $X$ be a smooth vector field on $M$ with components $X^1=2x$ and $X^2=x^2y$, where $(x,y)\in\mathbb{R}^2$ represents the coordinates of $M$. Given a smooth real-valued function $f$ on $M$ with $f(x,y)=x^3y+2x$, determine the Lie derivative $\mathcal{L}_Xf(p)$ of $f$ with respect to... | tiger_lab_math | [
0.01395297423005104,
0.040586359798908234,
-0.052962642163038254,
-0.05088340491056442,
0.02900529094040394,
0.00234900857321918,
-0.02198137529194355,
0.07288692891597748,
0.03178415447473526,
-0.005180456675589085,
0.011702395975589752,
0.0335063673555851,
-0.043776609003543854,
-0.01262... | 0.828011 | 0.808395 | 0.019616 |
Solve for x:
$\frac{sin^2(x) - cos^2(x)}{sin(x)cos(x)} = \frac{1}{cos(x)}$ | tiger_lab_math | [
0.005453773774206638,
0.04181357100605965,
-0.03457774221897125,
-0.035559628158807755,
0.03695232421159744,
-0.029841434210538864,
-0.0000933631818043068,
0.06487015634775162,
0.0569278821349144,
-0.014742754399776459,
0.023705990985035896,
0.009464402683079243,
-0.016136543825268745,
-0.... | 0.816977 | 0.814431 | 0.002546 |
Consider an RLC circuit with resistance R = 50 ohms, capacitance C = 0.1 microfarad and inductance L = 0.5 henry. The circuit has an alternating current source of frequency f = 50Hz and amplitude V = 110 volts.
(a) Write the second-order differential equation that describes the behavior of the circuit.
(b) Find the g... | tiger_lab_math | [
-0.005555031821131706,
0.035085201263427734,
-0.04413701221346855,
-0.043562471866607666,
0.0241683442145586,
-0.029785772785544395,
-0.026952048763632774,
0.09307494759559631,
0.03694617748260498,
-0.011638771742582321,
0.01843402162194252,
0.012539486400783062,
-0.03811589628458023,
0.00... | 0.800131 | 0.792656 | 0.007475 |
There were students from three different schools taking part in a debate organized by the town's mayor on a stage. The first school had twice as many students as the second school, and the second school had 40 more students than the third school. If the third school had 200 students, and after the debate was over, ever... | tiger_lab_math | [
0.04847123473882675,
0.007793533615767956,
-0.05895170569419861,
-0.04722139611840248,
0.038274217396974564,
-0.036585185676813126,
-0.03608165681362152,
0.05033913254737854,
0.026095790788531303,
-0.021239718422293663,
0.015003674663603306,
0.051455359905958176,
-0.048356544226408005,
-0.... | 0.820436 | 0.819567 | 0.000869 |
a student committee on academic integrity has 42 ways to select a president and vice president from a group of candidates . the same person can not be both president and vice president . how many candidates are there ? Let's write a program. | tiger_lab_math | [
0.015270350500941277,
-0.0020440365187823772,
-0.010177575051784515,
-0.02285096049308777,
0.024571683257818222,
-0.04649060592055321,
-0.05446980148553848,
0.02844594046473503,
0.0161419864743948,
-0.02373236045241356,
0.03160972148180008,
0.03684478998184204,
-0.017167141661047935,
-0.02... | 0.782297 | 0.778552 | 0.003745 |
Here's a question for a math student to solve related to game theory:
"Alice and Bob are playing a game where they each choose a number between 1 and 10. If their numbers add up to an even number, Alice wins, and if they add up to an odd number, Bob wins. What is the optimal strategy for each player to maximize their ... | tiger_lab_math | [
0.019035417586565018,
0.004435931798070669,
-0.05693360045552254,
-0.035993847995996475,
0.020205672830343246,
-0.01896471157670021,
-0.023640092462301254,
0.035645049065351486,
0.04093356430530548,
-0.028579795733094215,
0.004221235401928425,
0.015011008828878403,
-0.043375708162784576,
-... | 0.830931 | 0.815102 | 0.015828 |
Consider the language L = {+, . , 0, 1} and let F be the field of complex numbers. Define a model A = (F, +, . , 0, 1) of L where the interpretations of '+' and '.' are the usual ones for complex numbers.
Now, write down a formula in L which is true in the model A if and only if -1 does not have a square root in F. | tiger_lab_math | [
0.02132425270974636,
0.01399462390691042,
-0.0363784059882164,
-0.05275784060359001,
0.04594675451517105,
-0.042569972574710846,
-0.01083739660680294,
0.05514639616012573,
0.024302585050463676,
-0.03750195354223251,
0.010486713610589504,
0.04866913706064224,
-0.053200267255306244,
-0.02689... | 0.831794 | 0.820399 | 0.011395 |
Suppose that $f(z)$ is a holomorphic function in the upper half-plane $\mathbb{H}=\{z\in\mathbb{C}: \text{Im}(z)>0\}$ such that $|f(z)|\leqslant 1$ for all $z\in\mathbb{H}$. Use the Schwarz reflection principle to find the maximum value of $|f(2+3i)|$. | tiger_lab_math | [
0.020495105534791946,
0.04011886566877365,
-0.0450504869222641,
-0.02838350273668766,
0.03206570819020271,
0.0048812152817845345,
-0.005109166726469994,
0.05754707381129265,
0.03671525791287422,
0.011624682694673538,
0.03337191417813301,
0.02893371693789959,
-0.019485894590616226,
-0.00303... | 0.822044 | 0.798943 | 0.023101 |
Suppose there are three closed boxes on a table, and each box contains either a red or a blue ball. An agent knows that at least one box contains a blue ball, but doesn't know which one. The agent makes the following statements:
- Box 1 does not contain a blue ball.
- If Box 2 contains a blue ball, then Box 3 contains... | tiger_lab_math | [
0.008819542825222015,
0.01306921523064375,
-0.03881348296999931,
-0.024266880005598068,
0.03814181312918663,
-0.005376755725592375,
0.002339606638997793,
0.045667994767427444,
0.044532276690006256,
0.0036743958480656147,
0.011904763989150524,
0.03603824973106384,
-0.0268716923892498,
-0.01... | 0.827363 | 0.818762 | 0.008601 |
Sure, here's a numerical problem related to the Principal-Agent Problem in game theory:
A CEO wants to incentivize his sales team to maximize profits, and decides to offer them a bonus based on the company's earnings. The sales team can choose between two strategies: a high-effort strategy that has a 60% chance of gen... | tiger_lab_math | [
0.00513166980817914,
-0.005118656903505325,
-0.03180057555437088,
-0.033212754875421524,
0.02040518820285797,
-0.014688495546579361,
-0.017618807032704353,
0.03521278500556946,
0.0168995950371027,
-0.01552724838256836,
0.013271961361169815,
0.023631250485777855,
-0.049366265535354614,
-0.0... | 0.825894 | 0.82174 | 0.004154 |
Find the Gauss-Bonnet theorem for a surface if the curvature of the surface is given by $K = \frac{1}{4}(2x^2 + 4y^2 + 9z^2)$ and the surface is bounded by a curve with length $L = 8\pi$. Find the total curvature of the surface. | tiger_lab_math | [
-0.011301805265247822,
0.034937534481287,
-0.03665876388549805,
-0.06508797407150269,
0.03972836956381798,
-0.018556734547019005,
-0.018753277137875557,
0.04631796106696129,
0.020005561411380768,
-0.005163549445569515,
0.024521050974726677,
0.04241214692592621,
-0.0283367857336998,
0.00003... | 0.822491 | 0.813771 | 0.00872 |
Using the finite element method, find the numerical solution of the following problem:
$$\frac{d}{dx}\left(k(x)\frac{du(x)}{dx}\right) + f(x) = 0,$$
with the following boundary conditions:
$$u(0) = 0, \quad u(1) = 1,$$
where $k(x) = 1+\sin(x^2)$ and $f(x) = 2\sin(\pi x)$. Use a uniform mesh of 10 elements to disc... | tiger_lab_math | [
0.008055346086621284,
0.03883129358291626,
-0.023489220067858696,
-0.053329434245824814,
0.041913218796253204,
-0.0036340539809316397,
-0.002844111295416951,
0.0787779912352562,
0.0360003300011158,
0.024347485974431038,
0.020595528185367584,
0.02629309892654419,
-0.014469129964709282,
-0.0... | 0.820247 | 0.807216 | 0.013031 |
A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and this point is colored purple. What is the probability that the number of the purple point is greater than the number of the green point, but les... | tiger_lab_math | [
-0.008159425109624863,
0.005835161544382572,
-0.0451524555683136,
-0.0463729165494442,
0.022166868671774864,
-0.024207893759012222,
-0.020452963188290596,
0.04843360185623169,
0.04273068904876709,
0.0037530940026044846,
0.002502798568457365,
0.03557306155562401,
-0.023721782490611076,
-0.0... | 0.830113 | 0.825345 | 0.004768 |
Consider the entire function $f(z)= z^3 - z^2 + 4z - 1$. Find the number of zeros of this function in the open unit disc $D(0,1)$. | tiger_lab_math | [
0.01612686552107334,
0.018738090991973877,
-0.052242208272218704,
-0.03288782387971878,
0.035351578146219254,
0.0019371503731235862,
-0.012404825538396835,
0.049439769238233566,
0.04005458578467369,
-0.007831305265426636,
0.02046523056924343,
0.019573774188756943,
-0.021915778517723083,
-0... | 0.843372 | 0.83482 | 0.008552 |
A math student is working on Mori Theory and needs to solve the following problem:
Find the number of rational curves passing through exactly 6 general points in the projective plane.
(Note: The answer is a positive integer.) | tiger_lab_math | [
0.031198056414723396,
0.023004556074738503,
-0.025823405012488365,
-0.049587082117795944,
0.035667773336172104,
-0.01175111997872591,
-0.030712097883224487,
0.023824919015169144,
0.036185186356306076,
-0.037049807608127594,
0.00398146640509367,
0.020405471324920654,
-0.01435732189565897,
-... | 0.834907 | 0.822456 | 0.012451 |
Let $\mathbf{a} = \begin{pmatrix} 3 \\ p \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 2 \\ 1 \\ q \end{pmatrix}$ be vectors that are equal in magnitude, and orthogonal. Enter the ordered pair $(p,q).$ | tiger_lab_math | [
0.048111312091350555,
0.030579010024666786,
-0.0523863323032856,
-0.03527257218956947,
0.04018725827336311,
0.004988350439816713,
0.003273946465924382,
0.05834123119711876,
0.03603838011622429,
-0.009506077505648136,
0.0019504798110574484,
0.025231996551156044,
-0.04295976459980011,
-0.026... | 0.828916 | 0.807395 | 0.021521 |
A salesman wants to visit 6 cities, starting and ending at his home city. The distances between each city are as follows:
- Home to City 1: 10 km
- Home to City 2: 15 km
- Home to City 3: 12 km
- Home to City 4: 18 km
- Home to City 5: 20 km
- City 1 to City 2: 5 km
- City 1 to City 3: 8 km
- City 1 to City 4: 15 km
-... | tiger_lab_math | [
0.01308588869869709,
0.02914384752511978,
-0.027190346270799637,
-0.03756219521164894,
0.05375136807560921,
-0.013298523612320423,
-0.024966979399323463,
0.029240284115076065,
0.01841730810701847,
-0.008617791347205639,
0.01659112609922886,
0.052083998918533325,
-0.01980164460837841,
-0.03... | 0.814094 | 0.811993 | 0.002101 |
a geometric shape is obtained by combining a square of side 8 cm and a semicircle of diameter 8 cm . what is the perimeter of the shape obtained ? Let's write a Python program to solve it. | tiger_lab_math | [
0.007952610962092876,
0.02966134063899517,
-0.027113843709230423,
-0.050412677228450775,
0.02671431191265583,
-0.00879910308867693,
-0.04282434284687042,
0.06371740251779556,
0.03120337799191475,
-0.024762487038969994,
0.02050735428929329,
0.0426325760781765,
-0.03487210348248482,
-0.02163... | 0.820485 | 0.818179 | 0.002306 |
Given a 2-dimensional manifold $M$ with a metric tensor $g_{ij}=\begin{pmatrix} 1 & 0 \\ 0 & 4 \end{pmatrix}$ and a tensor field $T_{ij}=\begin{pmatrix} xy & y^2 \\ x^2 & 2xy \end{pmatrix}$ defined on $M$, find the covariant derivative $\nabla_k T_{ij}$ for $k=1,2$. | tiger_lab_math | [
0.012963919900357723,
0.03045565076172352,
-0.04000047221779823,
-0.04232775419950485,
0.04528238996863365,
-0.01520391646772623,
-0.024371182546019554,
0.06759268790483475,
0.0272925216704607,
0.019387908279895782,
0.00013034856237936765,
0.021135719493031502,
-0.028152015060186386,
-0.00... | 0.808834 | 0.80013 | 0.008704 |
A student is going on a hike and has a knapsack that can hold a maximum weight of 4 kg. They have 5 items that they want to take with them, each with a weight and a value as given in the table below.
| Item | Weight (kg) | Value ($) |
|------|-------------|-----------|
| A | 2 | 12 |
| B | 1 ... | tiger_lab_math | [
0.037749987095594406,
0.02196250855922699,
-0.04990996792912483,
-0.03655832260847092,
0.03988698869943619,
-0.0504169799387455,
-0.011666909791529179,
0.06109858676791191,
0.03809022158384323,
-0.011237219907343388,
0.023968810215592384,
0.03967734053730965,
-0.04289446398615837,
-0.01259... | 0.82301 | 0.819592 | 0.003418 |
Let $\star (x)$ be the sum of the digits of a positive integer $x$. $\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\mathcal{S}$, we have that $\star (n)=12$ and $0\le n< 10^{7}$. If $m$ is the number of elements in $\mathcal{S}$, compute $\star(m)$.
| tiger_lab_math | [
0.037807758897542953,
0.01806650124490261,
-0.041229043155908585,
-0.02870016172528267,
0.05161333456635475,
-0.03855198994278908,
-0.028383852913975716,
0.056114718317985535,
0.03781024366617203,
-0.005047332029789686,
0.02128038927912712,
0.04658256471157074,
-0.053541164845228195,
-0.02... | 0.840286 | 0.823901 | 0.016385 |
Let the function Q(n) represent the product of the first n prime numbers, where n > 0. If x = Q(n) + 1, which of the following must be true?
(i) x is always odd
(ii) x is always prime
(iii) x is never the square of an integer
Answer Choices: (A) ii only (B) iii only (C) i and ii only (D) i and iii only (E) ii and iii o... | tiger_lab_math | [
0.024546140804886818,
0.026296978816390038,
-0.02678755484521389,
-0.02815089002251625,
0.03111334890127182,
-0.028733201324939728,
-0.0002492319035809487,
0.07353745400905609,
0.030179372057318687,
-0.016944732517004013,
0.025224236771464348,
0.013302245177328587,
-0.034762799739837646,
-... | 0.834837 | 0.828781 | 0.006056 |
Statement 1 | A group G is abelian if and only if its group operation is commutative. Statement 2 | The order of a group is always equal to the number of its elements. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.03345206752419472,
0.023002078756690025,
-0.0292192529886961,
-0.042146410793066025,
0.034523215144872665,
-0.05601508170366287,
-0.010858029127120972,
0.05899670720100403,
0.02429969608783722,
-0.019267279654741287,
0.0026957616209983826,
0.0018952543614432216,
-0.026727156713604927,
-0... | 0.805611 | 0.803675 | 0.001936 |
A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\frac{1}{2}$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consist... | tiger_lab_math | [
0.01615642011165619,
0.012847686186432838,
-0.057094257324934006,
-0.032016441226005554,
0.010579489171504974,
0.000027811212930828333,
-0.015589441172778606,
0.04747030884027481,
0.02620399184525013,
-0.022402843460440636,
0.011003542691469193,
0.03104346990585327,
-0.017821528017520905,
... | 0.838605 | 0.825348 | 0.013257 |
Let's assume that there are two mathematicians, Alice and Bob, and they both make a statement:
Alice says "I know that 2+2=4"
Bob says "I don't know that 2+2=4"
Assuming that both statements are true, what can you infer about their knowledge in epistemic logic? | tiger_lab_math | [
0.0197579525411129,
0.0009711850434541702,
-0.03211165592074394,
-0.03713603317737579,
0.03499307483434677,
-0.034789007157087326,
-0.010250328108668327,
0.06441731005907059,
0.03840230032801628,
-0.022517655044794083,
0.015216399915516376,
0.030634811148047447,
-0.018788274377584457,
-0.0... | 0.800294 | 0.794813 | 0.005481 |
In a certain country, there are 4 major political parties – Party A, Party B, Party C, and Party D. If the probability of Party A winning the election is 0.5, the probability of Party B winning the election is 0.3, the probability of Party C winning the election is 0.15, and the probability of Party D winning the elect... | tiger_lab_math | [
0.00867497269064188,
0.004808614030480385,
-0.03687935695052147,
-0.028182031586766243,
0.03377198800444603,
-0.02725951373577118,
-0.010040562599897385,
0.043052367866039276,
0.03650978207588196,
0.0013998993672430515,
0.014735428616404533,
0.02948790229856968,
-0.00010393228876637295,
-0... | 0.803298 | 0.800698 | 0.0026 |
Here is a precise numerical problem for a math student to solve:
A digital image of size 256x256 pixels is corrupted by salt-and-pepper noise. The noisy image can be modeled as $u(x,y) = f(x,y) + n(x,y)$, where $f(x,y)$ is the original image and $n(x,y)$ is the noise signal that takes the values 0 and 1 with equal pro... | tiger_lab_math | [
-0.014378441497683525,
0.04468206688761711,
-0.034690652042627335,
-0.027884218841791153,
0.04010603949427605,
0.00013057951582595706,
-0.03962210565805435,
0.08294626325368881,
0.022025009617209435,
0.00026834322488866746,
0.02180054411292076,
0.022962000221014023,
-0.02111695520579815,
-... | 0.822046 | 0.807388 | 0.014658 |
Sure, here is a precise numerical problem for a math student to solve:
A company produces two models of smartphones: Model X and Model Y. Each Model X earns a profit of $40 and each Model Y earns a profit of $50. The company can produce a maximum of 5000 smartphones per week due to limited resources. Model X requires ... | tiger_lab_math | [
0.009183273650705814,
0.014350151643157005,
-0.05119205638766289,
-0.02134271152317524,
0.023184554651379585,
-0.017497599124908447,
-0.047646667808294296,
0.05831054225564003,
0.04106539860367775,
-0.023709917441010475,
-0.00898168608546257,
0.04075971245765686,
-0.009447179734706879,
-0.... | 0.81884 | 0.804248 | 0.014592 |
A two-dimensional image f(x,y) is defined on the region [0,1] x [0,1]. Assuming that f(x,y) satisfies the following partial differential equation:
∂f/∂t = ∂²f/∂x² + ∂²f/∂y²
If the initial condition is given by f(x,y,0) = xy(1-x)(1-y), find the solution of f(x,y,0.5) at (x,y) = (0.5,0.5) up to 4 decimal places. | tiger_lab_math | [
0.0011025905841961503,
0.01571752317249775,
-0.05350049212574959,
-0.039271146059036255,
0.04846819490194321,
-0.009664097800850868,
-0.02683122269809246,
0.09113016724586487,
0.04024269059300423,
-0.006400223821401596,
0.02196732722222805,
0.015970146283507347,
-0.020706333220005035,
-0.0... | 0.819986 | 0.811136 | 0.00885 |
In a sealed bid first-price auction, two bidders are bidding for a painting. Bidder A values the painting at $5000 and Bidder B values it at $7000. What bid should each bidder submit, assuming they have complete information about each other's valuations, in order to maximize their chances of winning the auction? The hi... | tiger_lab_math | [
-0.003417432541027665,
0.029988406226038933,
-0.02789071388542652,
0.00015193727449513972,
0.041383564472198486,
-0.027022501453757286,
-0.02650957927107811,
0.03239214047789574,
0.0485963374376297,
-0.01562089566141367,
0.03629561513662338,
0.025413449853658676,
-0.03498115390539169,
-0.0... | 0.793814 | 0.786646 | 0.007168 |
Find the smallest positive integer $n$ such that
\[\begin{pmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & \frac{1}{2} \end{pmatrix}^n = \mathbf{I}.\] | tiger_lab_math | [
0.03215703368186951,
0.039805687963962555,
-0.042319830507040024,
-0.03840738534927368,
0.04763608053326607,
-0.013768975622951984,
-0.018396979197859764,
0.04935816302895546,
0.03967614844441414,
-0.010931890457868576,
0.014485918916761875,
0.01291381660848856,
-0.03162166103720665,
-0.03... | 0.855952 | 0.843552 | 0.0124 |
Consider the following system of linear equations:
$x_1 + 2x_2 + x_3 = 4$
$4x_1 - x_2 + 2x_3 = 2$
$3x_1 + x_2 + 4x_3 = 6$
Use the Jacobi iterative method to find the solution of the system of equations, with initial guess $x_0 = [0, 0, 0]$, and iterate until the relative error is less than $10^{-3}$. | tiger_lab_math | [
0.024825748056173325,
0.02185571938753128,
-0.04409528896212578,
-0.04001409560441971,
0.04220353439450264,
-0.03229152411222458,
-0.013073800131678581,
0.07146497815847397,
0.03421225771307945,
-0.010952371172606945,
0.012170393951237202,
0.020689303055405617,
-0.0390482135117054,
-0.0223... | 0.825464 | 0.817694 | 0.00777 |
Sure, here's a question on the double pendulum that a math student could solve:
Assuming the masses of the two pendulums are equal and each rod has a length of 1 meter, find the initial conditions required for the double pendulum system to enter into chaotic behavior. Provide numerical values for the angles and angula... | tiger_lab_math | [
0.027629991993308067,
-0.007837141864001751,
-0.040201157331466675,
-0.04748869687318802,
0.03245837241411209,
-0.03841080889105797,
-0.02475067973136902,
0.05155149847269058,
0.0147052351385355,
-0.018737804144620895,
-0.006157096941024065,
0.019107088446617126,
-0.027788087725639343,
-0.... | 0.807583 | 0.802128 | 0.005456 |
Consider a linear logic system consisting of three variables A, B, and C. If A implies B and B implies C, what is the maximum number of variables that can be true at any given time, assuming that only one variable can be true at a time? | tiger_lab_math | [
0.01813673786818981,
0.015042051672935486,
-0.06402545422315598,
-0.020471740514039993,
0.041762374341487885,
-0.023137066513299942,
-0.04884946718811989,
0.055240198969841,
0.026702264323830605,
-0.03733869642019272,
0.008672489784657955,
0.04173272103071213,
-0.016715355217456818,
-0.029... | 0.828734 | 0.821304 | 0.00743 |
Let $p$ and $q$ be the two distinct solutions to the equation $$(x-5)(2x+9) = x^2-13x+40.$$What is $(p + 3)(q + 3)$? Let's write a Python program to solve it. | tiger_lab_math | [
0.02199031226336956,
0.019564080983400345,
-0.05319112539291382,
-0.03274501860141754,
0.03366033360362053,
-0.032034989446401596,
0.0009595596347935498,
0.06848267465829849,
0.05043303593993187,
-0.015425451099872589,
0.02264130488038063,
0.027632957324385643,
-0.041320253163576126,
-0.01... | 0.833969 | 0.83274 | 0.001229 |
Find the Laurent series expansion of f(z) = sin(z)/(z^2 - 4) valid in the annulus 2 < |z| < ∞ and determine the values of z where the series converges absolutely, converges conditionally, and diverges. | tiger_lab_math | [
-0.00676459725946188,
0.04562155902385712,
-0.03409157320857048,
-0.041876696050167084,
0.038396093994379044,
-0.005848802160471678,
-0.009526824578642845,
0.046856362372636795,
0.029554417356848717,
0.0070922281593084335,
0.03397804871201515,
0.0052419425919651985,
-0.015771448612213135,
... | 0.82654 | 0.817957 | 0.008583 |
Consider a classroom with 30 students. Each student can pick 1, 2 or 3 languages from a set of 8 languages: English, Spanish, French, German, Chinese, Japanese, Hindi, and Arabic. Assuming that all students have made a choice, how many different language patterns are possible for the classroom? Use generating functions... | tiger_lab_math | [
0.017579108476638794,
0.004801977425813675,
-0.034333258867263794,
-0.03234771639108658,
0.037946801632642746,
-0.04173111915588379,
-0.03858819603919983,
0.05151894688606262,
0.031225046142935753,
-0.02588830515742302,
0.026472965255379677,
0.028193484991788864,
-0.03454282507300377,
-0.0... | 0.810577 | 0.809456 | 0.00112 |
Let A = {1, 3, 5, 7, 9} and B = {2, 3, 4, 6, 7}. Find (A △ B) ∩ {3, 4, 5, 6}. | tiger_lab_math | [
0.04574303328990936,
0.014173840172588825,
-0.03622167184948921,
-0.026110153645277023,
0.03173566982150078,
-0.02105044759809971,
0.0041624074801802635,
0.03581578657031059,
0.05213211849331856,
-0.030083557590842247,
0.013650194741785526,
0.022242628037929535,
-0.029870392754673958,
-0.0... | 0.834503 | 0.833154 | 0.001349 |
Consider a graph with 10 vertices, where the distance between each pair of vertices is known. Determine the diameter of the graph. | tiger_lab_math | [
0.019351128488779068,
0.0150475287809968,
-0.03380660340189934,
-0.02093818224966526,
0.02564391680061817,
-0.03329657390713692,
-0.03569676727056503,
0.0359627939760685,
0.02725762128829956,
-0.03764558583498001,
0.022296210750937462,
0.03381528705358505,
-0.031039047986268997,
-0.0031298... | 0.818151 | 0.812615 | 0.005537 |
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 sphere is cut into four congruent wedges. The circumference of the sphere is $12\pi$ inches. What is the number of cubic inches in the volume of one wedge? Express your answer in terms of $\pi$.
Note: To measure the circumference, take the largest circle on the surface of the sphere. Please respond by writing a prog... | tiger_lab_math | [
0.013300153426826,
0.05058404803276062,
-0.03823692724108696,
-0.020946281030774117,
0.0428120419383049,
-0.02862999215722084,
-0.031582724303007126,
0.028748346492648125,
0.03193177282810211,
0.0007868708344176412,
0.01709846779704094,
0.018831497058272362,
-0.007464781869202852,
-0.01825... | 0.828188 | 0.822935 | 0.005253 |
Suppose you want to estimate the value of π using Monte Carlo simulation. Use a program to randomly generate 10,000 points within a unit square (square with sides of length 1) centered at the origin. Count how many of these points are also inside the unit circle centered at the origin. Use this information to estimate ... | tiger_lab_math | [
-0.0008965788874775171,
0.01718161813914776,
-0.039079487323760986,
-0.03111816942691803,
0.041296545416116714,
-0.033385392278432846,
-0.017976218834519386,
0.03646470606327057,
0.03353739529848099,
-0.01127245556563139,
0.014404341578483582,
0.02945862151682377,
-0.0380525104701519,
-0.0... | 0.811749 | 0.802657 | 0.009092 |
the dimensions of a rectangular solid are 6 inches , 5 inches , and 8 inches . if a cube , a side of which is equal to one of the dimensions of the rectangular solid , is placed entirely within thespherejust large enough to hold the cube , what the ratio of the volume of the cube to the volume within thespherethat is n... | tiger_lab_math | [
0.010364504531025887,
0.046242181211709976,
-0.052533358335494995,
-0.027480870485305786,
0.03216079995036125,
-0.02899697795510292,
-0.023421263322234154,
0.05051660165190697,
0.037445805966854095,
-0.015256724320352077,
0.015564258210361004,
0.047310084104537964,
-0.02459825575351715,
-0... | 0.826324 | 0.819835 | 0.00649 |
Consider the partial differential equation:
∂u/∂t + i∂u/∂x = 0
where u(x,t) is a complex-valued function of x and t.
If u(x,0) = sin(x) + 2i and u(0,t) = 1 for all t≥0, find the complex-valued function u(x,t) that satisfies the partial differential equation and the initial and boundary conditions. | tiger_lab_math | [
-0.0012692187447100878,
0.02562420815229416,
-0.05002701282501221,
-0.028228528797626495,
0.04038955271244049,
-0.006674899719655514,
-0.007564759347587824,
0.07638723403215408,
0.0450618676841259,
0.003948994912207127,
0.029483996331691742,
0.0308211799710989,
-0.03172138333320618,
-0.019... | 0.814256 | 0.795842 | 0.018414 |
Let $F_n$ be the Fibonacci sequence, that is, $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$. Compute
\[\sum_{n=0}^{\infty} \frac{F_n}{10^n}.\] | tiger_lab_math | [
0.015366229228675365,
0.034634239971637726,
-0.025060441344976425,
-0.028114549815654755,
0.04304489865899086,
-0.020951244980096817,
-0.03528571128845215,
0.07081268727779388,
0.02228245884180069,
-0.021014200523495674,
0.010025113821029663,
0.01412142999470234,
-0.05866524204611778,
-0.0... | 0.840249 | 0.828867 | 0.011382 |
Let X be a topological space with the homology groups H_1(X) and H_2(X) quivalent to Z and 0 respectively. Find the value of H_1(X x S^2), where S^2 is the 2-dimensional sphere. | tiger_lab_math | [
0.008948060683906078,
0.013882341794669628,
-0.04391028732061386,
-0.03932256996631622,
0.025693364441394806,
-0.024230366572737694,
-0.017712246626615524,
0.04524987190961838,
0.04909954592585564,
-0.018085800111293793,
0.015933359041810036,
0.036560796201229095,
-0.02911861054599285,
-0.... | 0.829568 | 0.819747 | 0.009821 |
set s contains exactly 10 numbers and has an average ( arithmetic mean ) of 6.2 . if one of the numbers in set s is increased by 4 , while all other numbers remain the same , what is the new average of set s ? Please write a program to solve it | tiger_lab_math | [
0.036619942635297775,
0.026816612109541893,
-0.042504020035266876,
-0.02277297154068947,
0.06264601647853851,
-0.05136493965983391,
-0.031226687133312225,
0.05457394942641258,
0.0336640439927578,
-0.025951894000172615,
0.027014316990971565,
0.004655527416616678,
-0.04829208925366402,
-0.01... | 0.826229 | 0.825151 | 0.001078 |
On a compact two-dimensional surface of genus 2, consider a topological quantum field theory (TQFT) with the following properties:
- Assigns a vector space of dimension 4 to any point on the surface.
- Assigns a linear map of determinant 1 to any one-dimensional oriented manifold embedded in the surface.
- Satisfies th... | tiger_lab_math | [
0.008720772340893745,
0.042176514863967896,
-0.038557346910238266,
-0.0632406622171402,
0.036657847464084625,
-0.011040037497878075,
-0.03009210340678692,
0.06119074672460556,
0.02533387579023838,
0.001853763940744102,
0.014405905269086361,
0.0348375178873539,
-0.029112255200743675,
0.0035... | 0.807218 | 0.789726 | 0.017492 |
a basket contains 9 apples , of which 1 is spoiled and the rest are good . if we select 2 apples from the basket simultaneously and at random , what is the probability that the 2 apples selected will include the spoiled apple ? Let's write a Python program to solve it. | tiger_lab_math | [
0.007307786028832197,
0.0066715385764837265,
-0.03474722057580948,
-0.013726687990128994,
0.03294685110449791,
-0.028428103774785995,
-0.01616576872766018,
0.03905365616083145,
0.016922758892178535,
0.0018552885157987475,
0.016196422278881073,
0.02978968806564808,
-0.026625679805874825,
0.... | 0.815935 | 0.812515 | 0.00342 |
The cost of building a certain house in an area is 100,000 more than the construction cost of each of the houses in the area. But it sells for 1.5 times as much as the other houses, which sell at $320,000 each. How much more profit is made by spending the extra money to build? | tiger_lab_math | [
0.02188848704099655,
0.015446599572896957,
-0.02973787672817707,
-0.012743505649268627,
0.028565090149641037,
-0.01599995791912079,
-0.03713013976812363,
0.05637179687619209,
0.03768998384475708,
-0.027367105707526207,
0.028353821486234665,
0.03912712261080742,
-0.048907071352005005,
-0.03... | 0.793606 | 0.791275 | 0.00233 |
Suppose you have a fuzzy set A defined on the universe of discourse X = {1, 2, 3, 4, 5} where the membership function of A is given by:
µA(1) = 0.4
µA(2) = 0.6
µA(3) = 0.8
µA(4) = 0.6
µA(5) = 0.4
Find the complement of fuzzy set A. | tiger_lab_math | [
0.018309231847524643,
0.020358731970191002,
-0.023929443210363388,
-0.0311077069491148,
0.04516247659921646,
-0.03933853283524513,
-0.022578414529561996,
0.02890896610915661,
0.04157121852040291,
-0.010047728195786476,
0.006162585224956274,
0.025164185091853142,
-0.02036423049867153,
-0.00... | 0.822573 | 0.821139 | 0.001433 |
Consider the algebraic curve given by the equation $y^2 + xy + x^3 = 0$ over the field $\mathbb{Q}$ of rational numbers.
Find the Hodge diamond for the cohomology of this curve. | tiger_lab_math | [
0.01220668200403452,
0.02181319333612919,
-0.039413996040821075,
-0.04444924741983414,
0.018003612756729126,
-0.027994489297270775,
-0.00850263237953186,
0.06734899431467056,
0.04699451103806496,
-0.03932245075702667,
0.008046465925872326,
0.019208824262022972,
-0.023681413382291794,
-0.01... | 0.83471 | 0.816068 | 0.018641 |
"statement 1": Susan , John , Peter , Jack , Daisy , Tim , Matt and Kim need to be seated in 8 identical chairs in straight line so that Susan is seated always left to Tim, "statement 2" :Susan , John , Peter , Jack , Daisy , Tim , Matt and Kim need to be seated in more than 4 identical chairs in straight line so that... | tiger_lab_math | [
0.026360249146819115,
0.011512215249240398,
-0.05522933602333069,
-0.040078725665807724,
0.03910459950566292,
-0.0321909636259079,
-0.015797730535268784,
0.05940546840429306,
0.021149400621652603,
-0.019754553213715553,
0.019065381959080696,
0.03445420041680336,
-0.010065155103802681,
-0.0... | 0.820288 | 0.812107 | 0.008182 |
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