instruction_seed stringlengths 56 2.52k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.77 0.86 | mean_negative_score float64 0.76 0.85 | difference_score float64 0 0.03 |
|---|---|---|---|---|---|
A marketing research firm wants to categorize a set of customer data based on their purchasing habits. The firm has collected data on the amount of money each customer spent on three types of products: groceries, clothing, and electronics, over the past three months. Use cluster analysis to divide 20 customers into thr... | tiger_lab_math | [
0.014151688665151596,
0.007014176808297634,
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0.021249372512102127,
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-0.0322931669652462,
0.042615681886672974,
0.023908989503979683,
0.004276691470295191,
0.012116859667003155,
0.026545187458395958,
-0.047931019216775894,
-0... | 0.787971 | 0.781817 | 0.006154 |
Statement 1 | Let G be a group and let a and b be two elements in G. If the order of a is 2 and the order of b is 3, then the order of the product ab is 6. Statement 2 | Let H be a group and n be a positive integer. If H is a cyclic group of order n, then H has exactly φ(n) generators, where φ is the Euler's phi functi... | tiger_lab_math | [
0.03708389028906822,
0.028149040415883064,
-0.03594047203660011,
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0.04265713691711426,
-0.03686385601758957,
-0.010603309608995914,
0.0730360597372055,
0.030580993741750717,
-0.011823037639260292,
0.004155751783400774,
0.03273125737905502,
-0.028611598536372185,
-0.035... | 0.823678 | 0.811239 | 0.012439 |
Statement 1 | The order of an element in a group is always less than or equal to the order of the group. Statement 2 | Every subgroup of an abelian group is abelian. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.033722251653671265,
0.021106332540512085,
-0.020103488117456436,
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0.03684354946017265,
-0.04828581586480141,
0.0018242975929751992,
0.05614549666643143,
0.04030056297779083,
-0.0054170736111700535,
0.006890674587339163,
0.004571084398776293,
-0.028559545055031776,
-0... | 0.80574 | 0.801706 | 0.004034 |
Let $a$, $b$, and $c$ be positive real numbers. What is the smallest possible value of $(a+b+c)\left(\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{b+c}\right)$? | tiger_lab_math | [
0.015783125534653664,
0.036267612129449844,
-0.05122801288962364,
-0.033010829240083694,
0.03494013473391533,
-0.0028295808006078005,
-0.010049010626971722,
0.054114289581775665,
0.05012379214167595,
-0.02806250751018524,
0.0011669308878481388,
0.025013195350766182,
-0.027135057374835014,
... | 0.853173 | 0.84533 | 0.007843 |
Bill is stocking the kitchenware section of the Walmart. He needs to stack 60 pots. On each shelf, he can stack five pots vertically and three sets of vertically stacked pots side-by-side. How many shelves does he need to stock all the pots? Let's write a program. | tiger_lab_math | [
0.03779292106628418,
0.0028898054733872414,
-0.0458376519382,
-0.0200770553201437,
0.04871651530265808,
-0.035670194774866104,
-0.04079078510403633,
0.06153273954987526,
0.042099982500076294,
-0.023552143946290016,
0.020127974450588226,
0.04346385598182678,
-0.02764906920492649,
-0.0295414... | 0.811163 | 0.810002 | 0.001161 |
The problem is: Determine if the following two graphs are isomorphic or not, and provide a graph isomorphism between them if they are isomorphic.
Graph A:
```
1
/ \
2 3
/ \ \
4 - 5 - 6
```
Graph B:
```
a
/ \
b c
/ \ \
d - e - f
```
Are Graph A and Graph B isomorph... | tiger_lab_math | [
0.02937271073460579,
0.01854005455970764,
-0.03533918783068657,
-0.012217171490192413,
0.037119824439287186,
-0.03363712877035141,
0.0026104990392923355,
0.07976566255092621,
0.028621694073081017,
-0.012353361584246159,
0.018435759469866753,
0.0025803567841649055,
-0.013690787367522717,
-0... | 0.820687 | 0.81453 | 0.006156 |
Consider a Riemannian manifold whose metric tensor is given by:
$$
g = \begin{pmatrix}
1 & 0 & 0 \\
0 & 2 & -1 \\
0 & -1 & 3
\end{pmatrix}
$$
Determine the Ricci curvature of this manifold at the point $(0,1,1)$. | tiger_lab_math | [
0.0052666086703538895,
0.030040185898542404,
-0.023710930719971657,
-0.05667255073785782,
0.029923439025878906,
-0.012754174880683422,
-0.019528580829501152,
0.060070667415857315,
0.015568387694656849,
-0.005145680624991655,
0.002610880183055997,
0.031548626720905304,
-0.046940937638282776,
... | 0.821688 | 0.810249 | 0.011439 |
Consider the conformal map $w = \frac{1}{2}(z^2 +1)$ that maps the upper half plane onto a region $\Omega$ bounded by a finite number of straight line segments and circular arcs. Find the image of the real axis under this map. | tiger_lab_math | [
0.017104124650359154,
0.043539710342884064,
-0.03616068884730339,
-0.038856033235788345,
0.028626972809433937,
-0.022545812651515007,
-0.010324754752218723,
0.05032193660736084,
0.04453406110405922,
0.002225936856120825,
0.024018991738557816,
0.011531289666891098,
-0.00910304207354784,
-0.... | 0.833087 | 0.812685 | 0.020402 |
Consider the complex partial differential equation given by:
$$ \frac{\partial^2 f}{\partial z \partial \bar{z}} - 3 \frac{\partial f}{\partial \bar{z}} = 2 $$
where $f(z,\bar{z})$ is a complex-valued function of two variables $z$ and $\bar{z}$.
Find the general solution of this equation. | tiger_lab_math | [
0.023306595161557198,
0.030677111819386482,
-0.06858973205089569,
-0.040309421718120575,
0.03377525135874748,
-0.007407409138977528,
0.004437468480318785,
0.06152257323265076,
0.0317472368478775,
0.003921527415513992,
0.03353722020983696,
0.03180282935500145,
-0.04429048299789429,
0.000111... | 0.81846 | 0.805394 | 0.013066 |
A function $f$ is defined for all real numbers and satisfies $f(2+x)=f(2-x)$ and $f(7+x)=f(7-x)$ for all $x.$ If $f(0) = 0,$ what is the least number of roots $f(x)=0$ must have in the interval $-1000\leq x \leq 1000$? | tiger_lab_math | [
0.03809673711657524,
0.020922347903251648,
-0.061422232538461685,
-0.037735648453235626,
0.019540470093488693,
-0.03229044750332832,
-0.012338034808635712,
0.051676053553819656,
0.022611333057284355,
-0.014210723340511322,
0.010351249016821384,
0.02287699468433857,
-0.028107620775699615,
-... | 0.84755 | 0.843085 | 0.004465 |
A bike messenger wants to find the fastest route to deliver an item to a destination 12 km away, with the speed limit being 30 km/hr. However, during the journey, the messenger has to climb a hill that is 3 km long and has a maximum slope of 10 degrees. What is the maximum time the messenger can take to reach the desti... | tiger_lab_math | [
0.0006178758922033012,
0.01310698315501213,
-0.035668760538101196,
-0.01753690466284752,
0.02417871542274952,
-0.014773424714803696,
-0.038441773504018784,
0.04559087008237839,
0.03878256306052208,
-0.019568223506212234,
0.021169982850551605,
0.041542693972587585,
-0.007957542315125465,
-0... | 0.810726 | 0.797954 | 0.012772 |
Suppose a sender wants to send a secret message (a plaintext) to a receiver using the ElGamal encryption scheme. The public parameters are (p=23, g=5, y=8) where p is a large prime number, g is a primitive root modulo p, and y=g^x mod p is the public key of the receiver where x is the secret private key. If the plainte... | tiger_lab_math | [
0.015095945447683334,
0.05546307563781738,
-0.02467428706586361,
-0.04609062150120735,
0.03389175608754158,
-0.03670225292444229,
-0.014533491805195808,
0.03844888508319855,
0.02003898285329342,
-0.040416765958070755,
0.018531586974859238,
0.026608938351273537,
-0.053828272968530655,
0.002... | 0.822142 | 0.816708 | 0.005434 |
Find the conformal map which transforms the unit disk $\{z: |z|<1\}$ onto the region bounded by the axis $x=0$ and the lines $x=1$, $y=1$, and $y=x$. Evaluate the derivative of this map at $z=0$. | tiger_lab_math | [
0.009665424935519695,
0.03858830779790878,
-0.031517013907432556,
-0.04592396691441536,
0.035920776426792145,
-0.00002329862218175549,
-0.007290848530828953,
0.038636937737464905,
0.032990824431180954,
0.0000892685420694761,
0.015538359992206097,
0.015406440943479538,
-0.02909071370959282,
... | 0.818891 | 0.807032 | 0.011858 |
Let M be a 2-dimensional manifold with a Riemannian metric g given in polar coordinates (r, θ) by
g = dr^2 + γ^2(θ)dθ^2
where γ(θ) is a smooth positive function. Find the Cartan connection 1-forms ω^i_j with respect to the orthonormal frame v_1=∂r and v_2 = 1/γ(θ) ∂θ on M. | tiger_lab_math | [
0.015262714587152004,
0.027225742116570473,
-0.03877240791916847,
-0.04734349250793457,
0.029905233532190323,
-0.0027648466639220715,
-0.023355677723884583,
0.054225772619247437,
0.04098290577530861,
0.00362965976819396,
0.0026195920072495937,
0.020657412707805634,
-0.031173592433333397,
-... | 0.816933 | 0.795761 | 0.021173 |
Suppose that f(z) = z^2 + 2iz is an analytic function defined on the complex plane. Find the value of f(i), where i is the complex number defined as i^2 = -1. | tiger_lab_math | [
0.03135792538523674,
0.03678583726286888,
-0.0503389798104763,
-0.04613777622580528,
0.047084540128707886,
-0.010894586332142353,
0.00599872088059783,
0.043017011135816574,
0.036715928465127945,
0.0006336123333312571,
0.025921355932950974,
0.02173551172018051,
-0.04228707775473595,
-0.0032... | 0.823147 | 0.818619 | 0.004528 |
A manufacturing company produces two types of products, A and B. The total available resources are 200 units of labor and 150 units of raw material. Each unit of product A requires 2 units of labor and 3 units of raw material to manufacture, while each unit of product B requires 4 units of labor and 2 units of raw mate... | tiger_lab_math | [
0.02925371378660202,
0.018662065267562866,
-0.06587905436754227,
-0.00002116948235197924,
0.018829580396413803,
-0.009888642467558384,
-0.05422664061188698,
0.09207582473754883,
0.028470944613218307,
-0.007936548441648483,
-0.0024132276885211468,
0.0359044075012207,
-0.007177402731031179,
... | 0.810326 | 0.794997 | 0.015329 |
Here's a precise numerical problem related to non-monotonic logic:
Suppose that John believes that if it's raining, then he will stay inside. However, if it's not raining, there's a 30% chance that he will stay inside anyway. If John is currently inside his house, what can we conclude about the weather outside?
Hint:... | tiger_lab_math | [
0.030456094071269035,
0.0031287935562431812,
-0.04720644652843475,
-0.013331705704331398,
0.03875057026743889,
-0.04488087445497513,
-0.023801522329449654,
0.058118198066949844,
0.030261917039752007,
-0.015170788392424583,
0.016296854242682457,
0.029018493369221687,
-0.0363687165081501,
-0... | 0.821156 | 0.809892 | 0.011263 |
If we let $f(n)$ denote the sum of all the positive divisors of the integer $n$, how many integers $i$ exist such that $1 \le i \le 2010$ and $f(i) = 1 + \sqrt{i} + i$? | tiger_lab_math | [
0.010747771710157394,
0.029841389507055283,
-0.04752162843942642,
-0.03181661292910576,
0.027122296392917633,
-0.01814662106335163,
-0.032989710569381714,
0.04229310527443886,
0.022919336333870888,
-0.001820717821829021,
0.017739109694957733,
0.032219938933849335,
-0.047705817967653275,
-0... | 0.850298 | 0.840282 | 0.010016 |
Let A and B be two sets with cardinalities 3 and 4 respectively. Find the number of second-order definable subsets of the Cartesian product A × B. | tiger_lab_math | [
0.03842654451727867,
0.019625674933195114,
-0.038338202983140945,
-0.02063687890768051,
0.033579226583242416,
-0.0043847220949828625,
-0.03033437207341194,
0.05197674036026001,
0.0561000294983387,
-0.008950281888246536,
0.021164679899811745,
0.022368276491761208,
-0.027926800772547722,
-0.... | 0.844216 | 0.831096 | 0.013119 |
Consider the metric given as $ds^2 = a^2\cos^2(x) \ dx^2 + b^2\sin^2(x) \ d\theta^2$, where $a, b$ are constants. Find the volume form of the given metric over the region $0 \leq x \leq \frac{\pi}{2}, \ 0 \leq \theta \leq 2\pi$. | tiger_lab_math | [
0.012994686141610146,
0.026848774403333664,
-0.05005254969000816,
-0.040354885160923004,
0.018344765529036522,
-0.022849004715681076,
-0.021071814000606537,
0.06864210963249207,
0.05409574881196022,
0.010057330131530762,
0.018553100526332855,
0.032110199332237244,
-0.00905921682715416,
-0.... | 0.828813 | 0.807106 | 0.021707 |
Here's a numerical problem for the student to solve:
Let X be an algebraic variety defined over the field of complex numbers with cohomology ring
$H^{*}(X; \mathbb{Z}) \cong \mathbb{Z}[x]/(x^{3}+3x^{2}+3x+1)$.
Find the cohomology groups $H^{*}(X;\mathbb{Z}/2\mathbb{Z})$ and $H^{*}(X;\mathbb{Z}/3\mathbb{Z})$. | tiger_lab_math | [
0.030674107372760773,
0.019836192950606346,
-0.04714116454124451,
-0.045261483639478683,
0.034205276519060135,
-0.020223084837198257,
-0.007103360258042812,
0.05521771311759949,
0.047327522188425064,
-0.007289977744221687,
0.016496915370225906,
0.026083214208483696,
-0.024161141365766525,
... | 0.840314 | 0.82344 | 0.016874 |
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no ... | tiger_lab_math | [
0.04424550011754036,
0.016910936683416367,
-0.05725117027759552,
-0.03845760598778725,
0.03144881874322891,
-0.02955315075814724,
-0.018532654270529747,
0.031037921085953712,
0.015684183686971664,
-0.013001867569983006,
-0.00033059457200579345,
0.032184451818466187,
-0.0442391000688076,
-0... | 0.826155 | 0.82082 | 0.005336 |
Clyde's four children wanted to go see their favorite band perform. The tickets were expensive so Clyde made a deal with his kids. If the average of their scores on their next math test was higher than 89, then he would buy them tickets to the concert. June earned a 97 on her math test, Patty earned an 85, Josh ear... | tiger_lab_math | [
0.015761839225888252,
0.012939486652612686,
-0.05698039382696152,
-0.0428108312189579,
0.042240459471940994,
-0.04081270098686218,
-0.023079579696059227,
0.01855165883898735,
0.04436546936631203,
-0.012103421613574028,
0.01172976940870285,
0.015361805446445942,
-0.021206315606832504,
-0.00... | 0.820409 | 0.817914 | 0.002495 |
Susan is playing a board game with 48 spaces from the starting space to the winning end space of the game. On the first turn, she moves forward eight spaces. On the second turn, she moves two spaces, but lands on a space that sends her back five spaces. On the third turn, she moves forward six more spaces. How many spa... | tiger_lab_math | [
0.011047212406992912,
0.021961120888590813,
-0.040645401924848557,
-0.040701910853385925,
0.02427002228796482,
-0.0120858708396554,
-0.038191694766283035,
0.044290438294410706,
0.04220644757151604,
-0.015501989983022213,
0.022469189018011093,
0.03554238751530647,
-0.03279280662536621,
-0.0... | 0.813005 | 0.809478 | 0.003526 |
A math student wants to calculate the probability of getting a heads in two flips of a fair coin. What is the probability of getting exactly one head in those two flips? | tiger_lab_math | [
0.019499896094202995,
0.01403855625540018,
-0.03652113676071167,
-0.03428085893392563,
0.021187961101531982,
-0.007190627045929432,
-0.016112947836518288,
0.045044369995594025,
0.047262322157621384,
-0.03123343549668789,
0.0021280476357787848,
0.012708048336207867,
-0.0016813765978440642,
... | 0.829315 | 0.825881 | 0.003434 |
Find all solutions in the interval [0, 2 π) to the equation `2sin(2x) = sqrt(2)`. | tiger_lab_math | [
0.006768370978534222,
0.03585708513855934,
-0.024308286607265472,
-0.04353224113583565,
0.028996845707297325,
-0.014957694336771965,
-0.020505167543888092,
0.056455712765455246,
0.0545344352722168,
-0.009483997710049152,
0.046962447464466095,
0.023061059415340424,
-0.014391114003956318,
-0... | 0.826451 | 0.821775 | 0.004676 |
The lines $y=2$, $y=5$, $x=1$, and $x=a$ make a square. Find the product of the possible values for $a$. Let's write a Python program. | tiger_lab_math | [
0.020521165803074837,
0.04454278200864792,
-0.046137601137161255,
-0.04336438328027725,
0.04744679480791092,
-0.015787094831466675,
-0.030060602352023125,
0.07737651467323303,
0.05011686310172081,
-0.019394639879465103,
0.010171172209084034,
0.05082859843969345,
-0.029329359531402588,
-0.0... | 0.823814 | 0.822382 | 0.001432 |
jill ' s favorite number is even and has some repeating prime factors . john is guessing jills favorite number , and the only hint she gives is that 7 is a prime factor . what is johns best guess ? Let's write a program. | tiger_lab_math | [
0.038257449865341187,
0.013814345002174377,
-0.03362928330898285,
-0.028136305510997772,
0.04421280324459076,
-0.042676039040088654,
-0.017186855897307396,
0.06072580814361572,
0.032216526567935944,
-0.010208855383098125,
0.023573871701955795,
0.040149521082639694,
-0.03745942935347557,
-0... | 0.817357 | 0.815521 | 0.001837 |
Suppose a company hires an agent to perform a project that will generate $100,000 in revenue. The agent can either exert high effort, which will result in a successful project with probability 0.8, or low effort, which will result in a successful project with probability 0.3. If the project is successful, the agent rec... | tiger_lab_math | [
-0.003297604387626052,
0.0055991350673139095,
-0.03646615892648697,
-0.02337835170328617,
0.022198574617505074,
-0.006560411769896746,
-0.02428792044520378,
0.05576135218143463,
0.04013223946094513,
0.002730639185756445,
0.025156568735837936,
0.027737347409129143,
-0.03065570630133152,
-0.... | 0.788666 | 0.783322 | 0.005343 |
Let G be a group of order 40. Find the number of distinct conjugacy classes in G and use this information to determine whether G is isomorphic to C40, D20, or one of the eight non-abelian groups of order 40. | tiger_lab_math | [
0.00028647956787608564,
0.012694594450294971,
-0.02273939736187458,
-0.027707962319254875,
0.015551473014056683,
-0.05039459466934204,
-0.015599691309034824,
0.051953498274087906,
0.040614526718854904,
-0.011853610165417194,
0.004794558044523001,
0.006782574579119682,
-0.034925319254398346,
... | 0.816683 | 0.814016 | 0.002667 |
Let $X \sim N(0,1)$ and let the distortion measure be squared error. Here we do not allow block descriptions. Compute the minimum expected distortion for one bit quantization of $X$ using a squared error distortion measure. | tiger_lab_math | [
-0.011253556236624718,
0.013238301500678062,
-0.045816563069820404,
-0.030883049592375755,
0.037548717111349106,
-0.01604074239730835,
-0.033797215670347214,
0.05594129115343094,
0.023993704468011856,
-0.013529178686439991,
0.03242383897304535,
0.007486756891012192,
-0.0345383919775486,
-0... | 0.811013 | 0.802785 | 0.008228 |
A manufacturing company produces two types of products, A and B. Product A requires 3 hours of labor and 2 hours of materials, while product B requires 2 hours of labor and 4 hours of materials. The company has 100 labor hours and 120 materials hours available per day. If the profit for each unit of A is $5 and for eac... | tiger_lab_math | [
0.02906068228185177,
-0.000798453576862812,
-0.0568443238735199,
-0.00971249770373106,
0.024275248870253563,
-0.012533482164144516,
-0.051268789917230606,
0.08412627875804901,
0.030143052339553833,
-0.005527658853679895,
0.002542027272284031,
0.04159869998693466,
-0.002253410406410694,
-0.... | 0.815471 | 0.810306 | 0.005165 |
A manufacturing company wants to produce a cylindrical can with a volume of 500cm³. The can needs to have a minimum surface area to minimize the cost of materials. Using Newton's method, determine the radius and height of the can that minimizes the surface area while maintaining the required volume. | tiger_lab_math | [
0.011468340642750263,
0.03424522280693054,
-0.04671691358089447,
-0.025222940370440483,
0.04060902073979378,
-0.03565360605716705,
-0.041431672871112823,
0.07973325252532959,
0.019746625795960426,
0.00391948688775301,
0.016855698078870773,
0.035110559314489365,
-0.02446696162223816,
-0.031... | 0.803625 | 0.790911 | 0.012714 |
Consider the noncommutative polynomial ring $A=k\langle x,y\rangle/(xy-yx-1)$ over a field $k$.
Find the dimension of the vector space $V=\operatorname{Hom}_k(A,k)$ over $k$. | tiger_lab_math | [
0.03319704160094261,
0.024164259433746338,
-0.0248541422188282,
-0.04513981193304062,
0.027645906433463097,
-0.02814601920545101,
-0.004961022175848484,
0.052939459681510925,
0.03844984248280525,
-0.03690666705369949,
0.019668199121952057,
0.03357073292136192,
-0.021912241354584694,
-0.007... | 0.830027 | 0.817781 | 0.012245 |
a box contains either blue or red flags . the total number of flags in the box is an even number . a group of children are asked to pick up two flags each . if all the flags are used up in the process such that 60 % of the children have blue flags , and 45 % have red flags , what percentage of children have flags of bo... | tiger_lab_math | [
0.020769743248820305,
0.016050683334469795,
-0.05318693071603775,
-0.03160663694143295,
0.013677848502993584,
-0.04781633988022804,
-0.003918459638953209,
0.06526482850313187,
0.020612655207514763,
-0.03219910338521004,
0.019759658724069595,
0.03781447559595108,
-0.014438851736485958,
-0.0... | 0.826498 | 0.823042 | 0.003456 |
Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$
| tiger_lab_math | [
0.03141336515545845,
0.04152727499604225,
-0.05200584977865219,
-0.023106032982468605,
0.026464886963367462,
-0.02300635725259781,
-0.03752346336841583,
0.05818614363670349,
0.03714650496840477,
-0.00943522248417139,
0.01547712367027998,
0.026414504274725914,
-0.026441799476742744,
0.00911... | 0.843775 | 0.823355 | 0.020421 |
Suppose you are the manager of a grocery store and you need to decide how many units of a certain product to order from your supplier. You can order this product only in multiples of 20, and the cost per unit decreases as the order quantity increases (see table below). However, you also have limited shelf space to stor... | tiger_lab_math | [
0.030258655548095703,
0.0018960288725793362,
-0.046966295689344406,
-0.021937955170869827,
0.03567446395754814,
-0.039015475660562515,
-0.057723551988601685,
0.049647096544504166,
0.04555729404091835,
-0.018709350377321243,
-0.001063974224962294,
0.025047291070222855,
-0.04301287978887558,
... | 0.810937 | 0.804981 | 0.005956 |
Let X be a compact topological space and let H^(n)(X; Z) be the n-th cohomology group of X with coefficients in the integers. Suppose that H^(n)(X; Z) is non-zero for all n = 1, 2, 3. Find the smallest possible value of n such that H^(n)(X; Z) is infinite. | tiger_lab_math | [
0.02544078230857849,
0.02605215087532997,
-0.040508564561605453,
-0.03402644395828247,
0.030502136796712875,
0.0003433262463659048,
-0.013056410476565361,
0.05329341068863869,
0.0363258458673954,
-0.02203398197889328,
0.017689188942313194,
0.02663281187415123,
-0.03235451504588127,
-0.0198... | 0.841424 | 0.817961 | 0.023462 |
Let P[0,1] denotes all the polynomials on the interval [0,1]. Define the distance \rho(p, q)=\int_0^1|p(x)-q(x)| dx. Is (P[0,1],\rho) a complete space? Return 1 for yes and 0 for no. Please respond by writing a program in Python. | tiger_lab_math | [
0.026013722643256187,
0.03325176611542702,
-0.04874354973435402,
-0.029032552614808083,
0.026266200467944145,
-0.03224489465355873,
-0.007823423482477665,
0.037548430263996124,
0.043312881141901016,
-0.016186093911528587,
0.01262932363897562,
0.03256123512983322,
-0.009857493452727795,
-0.... | 0.832304 | 0.819224 | 0.01308 |
Consider the following undirected graph with vertices labeled from A to G and edge weights given in the parentheses. Determine whether or not this graph contains an Eulerian path and a Hamiltonian path, and if so, provide the respective path(s).
(4) A ----- B (2)
| / | \
| / | \
| ... | tiger_lab_math | [
0.01063279714435339,
0.0123550770804286,
-0.021611783653497696,
-0.02737981081008911,
0.025828281417489052,
-0.03422170504927635,
-0.01855601742863655,
0.07325907051563263,
0.03244965150952339,
-0.0010630183387547731,
0.03570021316409111,
0.03444467857480049,
0.002376085612922907,
-0.03238... | 0.811629 | 0.80785 | 0.003778 |
Let $R = (8,6)$. The lines whose equations are $8y = 15x$ and $10y = 3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. The length of $PQ$ equals $\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
| tiger_lab_math | [
0.03295418620109558,
0.012753957882523537,
-0.050153207033872604,
-0.05594811961054802,
0.03781214356422424,
-0.009020011872053146,
-0.013178560882806778,
0.07297131419181824,
0.03422018885612488,
-0.02281469665467739,
0.009434773586690426,
0.04065186157822609,
-0.030390692874789238,
-0.03... | 0.849825 | 0.828728 | 0.021097 |
The parabola $y^2 = 8x$ and the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at two points $A$ and $B.$ Find the distance $AB.$ | tiger_lab_math | [
0.029758909717202187,
0.010989013127982616,
-0.027069630101323128,
-0.04597371071577072,
0.03963834047317505,
-0.019367771223187447,
-0.03185737878084183,
0.06306808441877365,
0.057112209498882294,
-0.014508922584354877,
0.011377091519534588,
0.029921408742666245,
-0.011250683106482029,
-0... | 0.827499 | 0.824672 | 0.002827 |
In triangle $ABC$, we have $\angle A = 90^\circ$ and $\sin B = \frac{4}{7}$. Find $\cos C$. | tiger_lab_math | [
0.032748837023973465,
0.02821931429207325,
-0.03167881816625595,
-0.03709102421998978,
0.024129051715135574,
-0.017730796709656715,
-0.012187021784484386,
0.03926711902022362,
0.04928608238697052,
-0.022232316434383392,
0.026410715654492378,
0.028224138543009758,
-0.019413014873862267,
-0.... | 0.835984 | 0.828096 | 0.007887 |
In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. So the middle number in Row 2 is $2$ because $1+1=2.$ What is the sum of the numbers in Row 8 of Pascal's Triangle?
\begin{tabular}{rccccccccccc}
Row 0:& & & & & & 1\\\noalign{\smalls... | tiger_lab_math | [
0.05268791317939758,
0.01727067306637764,
-0.03785833343863487,
-0.048072464764118195,
0.03238198906183243,
-0.017210057005286217,
-0.010647563263773918,
0.05184300243854523,
0.022360187023878098,
-0.0053994557820260525,
0.013391716405749321,
0.0337042398750782,
-0.04324374347925186,
-0.03... | 0.830233 | 0.815399 | 0.014834 |
Our school's girls volleyball team has 14 players, including a set of 3 triplets: Alicia, Amanda, and Anna. In how many ways can we choose 6 starters if exactly one of the triplets is in the starting lineup? Let's write a Python program to solve it. | tiger_lab_math | [
0.014146959409117699,
0.014356271363794804,
-0.04804910346865654,
-0.030133722350001335,
0.04496489837765694,
-0.0367414765059948,
-0.0273376926779747,
0.03812691569328308,
0.030789591372013092,
-0.029856152832508087,
0.02940702624619007,
0.04092138260602951,
-0.012853436172008514,
-0.0348... | 0.821077 | 0.819026 | 0.00205 |
A manufacturing company needs to produce 500 units of a product using two machines. Machine 1 can produce 2 units in 1 hour and Machine 2 can produce 5 units in 1 hour. Each unit of the product requires 1 hour of processing time on Machine 1 and 0.5 hours on Machine 2. The company has only 700 hours of available proces... | tiger_lab_math | [
0.02278667874634266,
0.010623510926961899,
-0.06081719323992729,
-0.005705022718757391,
0.014879344031214714,
-0.005743990186601877,
-0.05166110768914223,
0.09053953737020493,
0.028994806110858917,
-0.0016994805773720145,
-0.0007874292205087841,
0.018404144793748856,
-0.0036071869544684887,
... | 0.811974 | 0.807221 | 0.004754 |
Consider the following two statements in the context of group theory: Statement 1: The intersection of two subgroups H and K of a group G is always a subgroup of G. Statement 2: Every subgroup of a cyclic group is cyclic. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.04423219710588455,
0.026448717340826988,
-0.026474522426724434,
-0.040017832070589066,
0.036764927208423615,
-0.041432879865169525,
0.0051557812839746475,
0.05937567725777626,
0.04478857293725014,
-0.030561326071619987,
0.01832723245024681,
0.00971224159002304,
-0.009780698455870152,
-0.... | 0.806549 | 0.799537 | 0.007013 |
Let X = {a, b, c, d} be a set equipped with the topology T = {Ø, {a}, {a, b, c, d}, {b, d}}. Determine whether this topological space is a Hausdorff space or not. | tiger_lab_math | [
0.017784753814339638,
0.020784741267561913,
-0.06748919188976288,
-0.0266084223985672,
0.041245315223932266,
-0.009930571541190147,
0.004936680197715759,
0.07328073680400848,
0.0408305786550045,
-0.014014729298651218,
0.003953816369175911,
0.010070249438285828,
-0.029386788606643677,
-0.03... | 0.825053 | 0.807036 | 0.018018 |
Mark wants to build a pyramid of soda cases that's four levels tall. Each level of the pyramid has a square base where each side is one case longer than the level above it. The top level is just one case. How many cases of soda does Mark need? Let's write a Python program to solve it. | tiger_lab_math | [
0.025685738772153854,
0.020312324166297913,
-0.037230826914310455,
-0.03378652408719063,
0.04276106506586075,
-0.022872500121593475,
-0.021789366379380226,
0.04612152278423309,
0.03142717853188515,
-0.014866978861391544,
0.021683119237422943,
0.036356937140226364,
-0.02932777628302574,
-0.... | 0.817455 | 0.812194 | 0.005261 |
Let $F_1 = (0,1)$ and $F_ 2= (4,1).$ Then the set of points $P$ such that
\[PF_1 + PF_2 = 6\]form an ellipse. The equation of this ellipse can be written as
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.\]Find $h + k + a + b.$ | tiger_lab_math | [
0.012208239175379276,
0.052448567003011703,
-0.04005024582147598,
-0.04184064641594887,
0.06366151571273804,
-0.020051322877407074,
-0.015575217083096504,
0.05118706449866295,
0.0450134314596653,
-0.008669803850352764,
0.0185763631016016,
0.05235259607434273,
-0.009350818581879139,
-0.0302... | 0.84104 | 0.817904 | 0.023135 |
Andrew is having two of his friends over for a sleepover. For one friend, Brian, he asks his mother to buy 3 donuts. He asks for his other friend, Samuel, to get the same. A day before the sleepover Andrew ends up inviting two more friends and asks his mother to buy them the same amount of donuts. Just in case, Andrew’... | tiger_lab_math | [
0.035044990479946136,
0.009928920306265354,
-0.035599421709775925,
-0.03490987792611122,
0.02169547602534294,
-0.009297442622482777,
-0.025073453783988953,
0.05543755367398262,
0.029286900535225868,
-0.024181239306926727,
0.005376609042286873,
0.03599104657769203,
-0.04524651914834976,
-0.... | 0.846845 | 0.834201 | 0.012644 |
A trapezium has two parallel sides of length 10 cm and 12 cm, and two non-parallel sides of length 5 cm each. Find the perimeter of the trapezium. | tiger_lab_math | [
0.023371540009975433,
0.009796353988349438,
-0.045542120933532715,
-0.041655078530311584,
0.027377458289265633,
-0.0196671299636364,
-0.03995215892791748,
0.05739520862698555,
0.022036563605070114,
-0.013993864879012108,
0.02762121707201004,
0.05061843618750572,
-0.024714041501283646,
-0.0... | 0.812091 | 0.810535 | 0.001556 |
Here is a precise numerical problem for a math student to solve:
Using the Fourier transform, analyze the two-dimensional image below and determine the frequency of the lowest and highest intensity values:
```
[ 0, 0, 100, 255, 255]
[ 0, 100, 255, 0, 255]
[ 100, 255, 0, 255, 0]
[ 255, 0, 255, 100, 0]
... | tiger_lab_math | [
0.01335898693650961,
0.022223033010959625,
-0.04174027219414711,
-0.02511805109679699,
0.027298901230096817,
-0.006379228550940752,
-0.01131338719278574,
0.06970027089118958,
0.04714268073439598,
0.0063930172473192215,
0.025135522708296776,
0.013355457223951817,
-0.01920153759419918,
-0.01... | 0.823258 | 0.818107 | 0.005151 |
An elementary school teacher is making Halloween goodie bags for her class. She wants the bags to be personalized, so she surveys her students asking whether they'd like a vampire-themed bag or a pumpkin-themed bag. Of her 25 students, 11 indicate they want the vampire-themed bag and 14 indicate they want the pumpkin... | tiger_lab_math | [
0.021317750215530396,
0.00746035436168313,
-0.042522795498371124,
-0.04222678020596504,
0.028814900666475296,
-0.026730110868811607,
-0.03916968032717705,
0.028636503964662552,
0.02586028166115284,
-0.01820342056453228,
0.033546339720487595,
0.028347022831439972,
-0.02013411559164524,
-0.0... | 0.826552 | 0.822324 | 0.004227 |
Consider two companies producing identical products and competing in a market. Each company sets its price independently of the other. The demand function for the product is Q = 100 - P, where Q is the quantity demanded at price P. The companies have the same constant marginal cost of production of $20.
What price sho... | tiger_lab_math | [
0.017520900815725327,
0.018128303810954094,
-0.06441781669855118,
-0.02730012685060501,
0.03149167448282242,
-0.01012879516929388,
-0.01765328459441662,
0.06273262947797775,
0.02876366674900055,
-0.018851274624466896,
0.007396444212645292,
0.019503137096762657,
-0.03886427730321884,
-0.021... | 0.819259 | 0.803679 | 0.01558 |
Suppose a couple has to decide between two movie options: "The Notebook" and "John Wick". The husband prefers "John Wick" and the wife prefers "The Notebook". If they go to the movie that one of them prefers, they each receive a payoff of 3. If they go to the movie that both prefer equally, they each receive a payoff o... | tiger_lab_math | [
0.004454023204743862,
0.0028319931589066982,
-0.06488261371850967,
-0.02082662843167782,
0.02743547596037388,
-0.014654630795121193,
-0.0020605979952961206,
0.06565114110708237,
0.04487457871437073,
-0.03500896692276001,
0.01931704208254814,
0.028586776927113533,
-0.051428742706775665,
-0.... | 0.807394 | 0.802517 | 0.004877 |
Let $\overline{MN}$ be a diameter of a circle with diameter 1. Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\frac{3}5$. Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoin... | tiger_lab_math | [
0.010181219317018986,
0.03561612591147423,
-0.04470326006412506,
-0.03922668471932411,
0.029980821534991264,
-0.006054434459656477,
-0.03284689784049988,
0.05464473366737366,
0.049430832266807556,
-0.0018378812819719315,
0.024934692308306694,
0.03058636747300625,
-0.022519255056977272,
-0.... | 0.856708 | 0.822808 | 0.0339 |
Consider a projective algebraic curve $C$ defined by the equation $y^2z = x(x-z)(x-\lambda z)$, with $\lambda \neq 0,1$. Let $\mathcal{O}_C(1)$ denote the line bundle on $C$ associated to the divisor class of a hyperplane section. Compute the dimension $H^1(C,\mathcal{O}_C(1))$. | tiger_lab_math | [
0.0029598460532724857,
0.040047235786914825,
-0.03520580381155014,
-0.05388803407549858,
0.04343578219413757,
-0.014524632133543491,
-0.014950737357139587,
0.060298945754766464,
0.043065380305051804,
-0.0178554467856884,
-0.0007882979116402566,
0.03157902881503105,
-0.029109245166182518,
-... | 0.824211 | 0.797555 | 0.026656 |
Find the integer $n,$ $0 \le n \le 180,$ such that $\cos n^\circ = \cos 758^\circ.$ Let's write a Python program to solve it. | tiger_lab_math | [
-0.0012017268454656005,
0.035872090607881546,
-0.023343248292803764,
-0.0402548611164093,
0.023287272080779076,
-0.027095913887023926,
-0.02715630829334259,
0.036966484040021896,
0.03470888361334801,
0.00025751697830855846,
0.017688995227217674,
0.02402038313448429,
-0.03718527406454086,
-... | 0.833644 | 0.831778 | 0.001866 |
Let $E$ be a vector bundle over a compact, connected manifold $M$. Suppose the rank of the vector bundle $E$ is $r$ and that the total space of $E$ is homeomorphic to the $r$-dimensional Euclidean space $\mathbb{R}^r$. If $\chi(M)$ denotes the Euler characteristic of $M$, then what is the value of $c_r(E)$, the $r$th C... | tiger_lab_math | [
0.002276536077260971,
0.029301408678293228,
-0.013397884555161,
-0.05050799623131752,
0.033317502588033676,
-0.0029588204342871904,
-0.019459981471300125,
0.060898009687662125,
0.0320800356566906,
-0.006614443846046925,
-0.0051030609756708145,
0.04022561386227608,
-0.04699630290269852,
-0.... | 0.819192 | 0.807207 | 0.011985 |
Here's an epistemic logic problem for a math student:
Alice and Bob are playing a game. Alice is thinking of a number between 1 and 10, and Bob has to guess what it is. However, Bob doesn't know the number, and Alice knows that Bob doesn't know. Then, Alice says "I will give you a hint: the number is either 4 or 6." A... | tiger_lab_math | [
0.02546633593738079,
0.005415671039372683,
-0.03417205065488815,
-0.029309991747140884,
0.0321400985121727,
-0.03528348729014397,
-0.009901246055960655,
0.049432482570409775,
0.03604083135724068,
-0.03466669097542763,
0.02082550711929798,
0.038281526416540146,
-0.031401898711919785,
-0.026... | 0.828146 | 0.817495 | 0.010651 |
Determine all real numbers $ a$ such that the inequality $ |x^2 + 2ax + 3a|\le2$ has exactly one solution in $ x$. | tiger_lab_math | [
0.012599905952811241,
0.024151010438799858,
-0.03599052131175995,
-0.049860309809446335,
0.03726482018828392,
-0.016265537589788437,
0.0006700429366901517,
0.0832294374704361,
0.04791678115725517,
-0.028675377368927002,
0.01541341096162796,
0.027835121378302574,
-0.02741534635424614,
-0.02... | 0.857044 | 0.83259 | 0.024454 |
In a certain hyperbola, the center is at $(-2,0),$ one focus is at $(-2 + \sqrt{34},0),$ and one vertex is at $(-5,0).$ The equation of this hyperbola can be written as
\[\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1.\]Find $h + k + a + b.$ Let's write a program. | tiger_lab_math | [
0.021175945177674294,
0.03932936117053032,
-0.05826224014163017,
-0.034836020320653915,
0.04842362180352211,
-0.015850583091378212,
-0.020469995215535164,
0.059916310012340546,
0.05740373581647873,
-0.021172745153307915,
0.01835726760327816,
0.03449149429798126,
-0.012237396091222763,
-0.0... | 0.838654 | 0.829307 | 0.009347 |
A rectangular sheet of cardboard has dimensions of 30 cm by 40 cm. You want to cut squares of equal size from each corner of the cardboard to create an open box. If the volume of the box should be 3000 cm³, what is the maximum length of the square that can be cut out of each corner? | tiger_lab_math | [
0.028303135186433792,
0.028124123811721802,
-0.05171942338347435,
-0.03405103087425232,
0.032786909490823746,
-0.0131661556661129,
-0.03256908804178238,
0.04300091415643692,
0.026074174791574478,
-0.026166947558522224,
0.030065326020121574,
0.04127738997340202,
-0.047785449773073196,
-0.00... | 0.807108 | 0.80269 | 0.004418 |
A $\frac 1p$ -array is a structured, infinite, collection of numbers. For example, a $\frac 13$ -array is constructed as follows:
\begin{align*} 1 \qquad \frac 13\,\ \qquad \frac 19\,\ \qquad \frac 1{27} \qquad &\cdots\\ \frac 16 \qquad \frac 1{18}\,\ \qquad \frac{1}{54} \qquad &\cdots\\ \frac 1{36} \qquad \frac 1{108}... | tiger_lab_math | [
0.01714269444346428,
0.05388858541846275,
-0.022930670529603958,
-0.0394662544131279,
0.04841781407594681,
-0.03618434816598892,
-0.010816005058586597,
0.06956694275140762,
0.03607818856835365,
0.00228783767670393,
0.011457491666078568,
0.020928358659148216,
-0.03206792101264,
-0.022425908... | 0.823632 | 0.81253 | 0.011102 |
A math student wants to study geometric invariant theory within algebraic geometry. Here's a numerical problem for the student to solve:
Let G be the group of all degree 2 linear transformations of the plane generated by the matrices
a = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} ,
b = \begin{pmatrix} -1 & 0 \\... | tiger_lab_math | [
0.03979727253317833,
0.031665120273828506,
-0.05276961624622345,
-0.051009126007556915,
0.032259754836559296,
-0.021345961838960648,
-0.01375013217329979,
0.05114806070923805,
0.043790385127067566,
-0.009318937547504902,
-0.01105028297752142,
0.0031642154790461063,
-0.01901121810078621,
-0... | 0.831729 | 0.803445 | 0.028284 |
Let S be a sphere with a radius of 3 units. Find the total curvature of S using the Gauss-Bonnet theorem for a surface. | tiger_lab_math | [
-0.005246995482593775,
0.018253019079566002,
-0.03447709605097771,
-0.04816337302327156,
0.04013243317604065,
-0.021584810689091682,
-0.025492701679468155,
0.045849986374378204,
0.016704875975847244,
-0.02374393492937088,
0.022697461768984795,
0.04923291876912117,
-0.03532019257545471,
-0.... | 0.819706 | 0.809175 | 0.010531 |
Let M be a smooth manifold with local coordinates (x,y) on an open set U in R^2. Suppose that the vector field X = (x+y, y) is defined on U.
Suppose also that f: M → R is a smooth function. Determine the Lie derivative of X along f:
L_X f = ? | tiger_lab_math | [
0.006938879378139973,
0.022138141095638275,
-0.049088191241025925,
-0.05657810717821121,
0.039555732160806656,
-0.005429750308394432,
-0.03301965817809105,
0.06830254197120667,
0.03788251057267189,
-0.012854151427745819,
0.007691612932831049,
0.03238046169281006,
-0.05254529044032097,
-0.0... | 0.811202 | 0.806336 | 0.004865 |
Alma is taking a walk and comes across four goats standing behind a fence. Alma has a bag of baby carrots she brought to have for a snack and decides to feed the goats with them. Alma wants to make sure all the goats get the same amount of carrots, so she starts counting them out. She has 47 baby carrots. If she wants ... | tiger_lab_math | [
0.034083183854818344,
0.026241851970553398,
-0.028381580486893654,
-0.027105480432510376,
0.011579379439353943,
-0.027566859498620033,
-0.018432892858982086,
0.07152119278907776,
0.04276258870959282,
-0.025625191628932953,
0.004266043659299612,
0.03822558745741844,
-0.04733748361468315,
-0... | 0.844831 | 0.836659 | 0.008172 |
Find the smallest positive integer $b$ for which $x^2 + bx + 2008$ factors into a product of two polynomials, each having integer coefficients. | tiger_lab_math | [
0.010729140602052212,
0.03942194953560829,
-0.04492131248116493,
-0.03362862020730972,
0.023449473083019257,
-0.013283824548125267,
-0.01473775040358305,
0.0665707141160965,
0.051011405885219574,
-0.008290432393550873,
0.006690256297588348,
0.03997354581952095,
-0.026926087215542793,
-0.04... | 0.850062 | 0.843541 | 0.006521 |
Sure! Here's an example of a problem for a math student to solve:
A square image has a width and height of 256 pixels. We want to apply a 2D Gaussian filter with a standard deviation of 2 pixels to the image using partial differential equations. Write the partial differential equation for the Gaussian filter and solve... | tiger_lab_math | [
0.006726530380547047,
0.021401654928922653,
-0.03702012822031975,
-0.04604463651776314,
0.037586063146591187,
-0.02725355140864849,
-0.02392176352441311,
0.04567242041230202,
0.03295906260609627,
-0.015164941549301147,
0.014800776727497578,
0.02273162454366684,
-0.027905061841011047,
-0.02... | 0.813812 | 0.805934 | 0.007878 |
Here's a precise numerical problem on finding the maximum flow in a network:
Consider the following flow network with capacities on each edge and a source node `S` and a sink node `T`:
```
S ---3---> A --4--> C
/ | | / ^ / |
5 2 1/ |2/ 4|
\ v v/ 6 \/ |
B ---2---> D --3--> T
`... | tiger_lab_math | [
0.0368361696600914,
0.020229347050189972,
-0.04008905962109566,
-0.006727331783622503,
0.03215155005455017,
-0.03616471588611603,
-0.0477452352643013,
0.05602583661675453,
0.02792828157544136,
-0.0012573592830449343,
0.04134361818432808,
0.03689468279480934,
-0.024832911789417267,
-0.02188... | 0.817492 | 0.813087 | 0.004405 |
A bowling ball is a solid ball with a spherical surface and diameter 30 cm. To custom fit a bowling ball for each bowler, three holes are drilled in the ball. Bowler Kris has holes drilled that are 8 cm deep and have diameters of 2 cm, 2 cm, and 3 cm. Assuming the three holes are right circular cylinders, find the numb... | tiger_lab_math | [
-0.003467304864898324,
0.04667403921484947,
-0.06015872582793236,
-0.033141735941171646,
0.04850385710597038,
-0.021766165271401405,
-0.04028534144163132,
0.020976686850190163,
0.03365510702133179,
-0.012326492927968502,
0.013897811062633991,
0.044066477566957474,
-0.021014142781496048,
-0... | 0.814037 | 0.811671 | 0.002366 |
A problem in hyperbolic geometry:
A hyperbolic triangle on the hyperbolic plane has angles of 60 degrees, 70 degrees, and 50 degrees. Find the area of the hyperbolic triangle. | tiger_lab_math | [
0.01873578503727913,
0.011119425296783447,
-0.026700958609580994,
-0.0363842211663723,
0.01091095246374607,
-0.010986237786710262,
-0.026165388524532318,
0.05328553169965744,
0.03939099237322807,
-0.03068646974861622,
0.01740334928035736,
0.027244074270129204,
-0.03284028172492981,
-0.0313... | 0.821623 | 0.814164 | 0.007459 |
Let $a_1,a_2,\ldots$ be a sequence determined by the rule $a_n= \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 2008$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$? | tiger_lab_math | [
-0.0016404546331614256,
0.023520687595009804,
-0.04549841210246086,
-0.02829904481768608,
0.02004869282245636,
-0.0005808832356706262,
-0.0365782268345356,
0.06499452143907547,
0.0364295057952404,
-0.025082865729928017,
-0.0038914743345230818,
0.023841826245188713,
-0.04530356451869011,
-0... | 0.850146 | 0.836898 | 0.013249 |
Suppose we have a toric variety defined as the closure of the orbit of the torus action on $\mathbb{A}^4$ with weights $(1,2,3,4)$ and $f(x,y,z,w) = x^3y^2z^2w$. Compute the Mori cone of this toric variety. | tiger_lab_math | [
0.04871952533721924,
0.04582840949296951,
-0.025777241215109825,
-0.01270272210240364,
0.018930958583950996,
-0.020341360941529274,
-0.017588045448064804,
0.0568014420568943,
0.03706113621592522,
0.00530200032517314,
0.017404790967702866,
0.029896298423409462,
-0.025263862684369087,
-0.022... | 0.830168 | 0.817776 | 0.012392 |
Statement 1 | The set of all 2x2 matrices with real entries forms a group under matrix addition. Statement 2 | In a group, the product of two elements is always unique. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.03962424024939537,
0.030471058562397957,
-0.029822034761309624,
-0.03802437335252762,
0.05487842112779617,
-0.04556695371866226,
-0.008557025343179703,
0.07961080968379974,
0.04762565344572067,
-0.015052043832838535,
-0.0015830873744562268,
0.020205175504088402,
-0.021145688369870186,
-0... | 0.811597 | 0.802958 | 0.008639 |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ -2 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \end{pmatrix}.\]A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -8 \\ 12 \end{pmatrix} + u \begin{pmatrix} 1 \\ 3 \end{pmatrix}.\]If $\theta$ is... | tiger_lab_math | [
0.034757133573293686,
0.02561851218342781,
-0.04558014124631882,
-0.04004059359431267,
0.04782688617706299,
-0.02330736070871353,
-0.015975579619407654,
0.07458238303661346,
0.045164600014686584,
0.0013513641897588968,
0.02414260432124138,
0.04618862271308899,
-0.016755597665905952,
-0.015... | 0.819413 | 0.808017 | 0.011396 |
Let triangle $ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C.$ If $\frac{DE}{BE} = \frac{8}{15},$ then find $\tan B.$ | tiger_lab_math | [
0.0426168367266655,
0.03469293564558029,
-0.03374802693724632,
-0.03816613927483559,
0.03057771548628807,
-0.012898333370685577,
-0.01002440880984068,
0.041991349309682846,
0.03256463631987572,
-0.0014110917691141367,
0.019495826214551926,
0.02152453176677227,
-0.029921401292085648,
-0.008... | 0.840943 | 0.81897 | 0.021974 |
Let X be an algebraic curve defined by the equation y^2 = x^3 - x^2 - 4x + 4 over the field of rational numbers Q. What is the dimension of the first cohomology group H^1(X,Q) of X with coefficients in Q? | tiger_lab_math | [
0.006703621242195368,
0.02441488951444626,
-0.040693432092666626,
-0.0573599711060524,
0.029570987448096275,
-0.025807131081819534,
-0.010268132202327251,
0.05718276649713516,
0.049721796065568924,
-0.027743982151150703,
-0.0018254818860441446,
0.025056691840291023,
-0.031519658863544464,
... | 0.822124 | 0.819629 | 0.002495 |
Consider the following bipartite graph. It represents seven students and six courses that they could take. Find a perfect matching in this graph if each student can only take one course.
```
+--------+
/ | | \
S1 S2 S3 S4
\ | | /
+--------+
/ \
C1 C2
| ... | tiger_lab_math | [
0.019387681037187576,
0.010549073107540607,
-0.05358845740556717,
-0.03109070099890232,
0.03350844234228134,
-0.03854609653353691,
-0.017662089318037033,
0.03676346316933632,
0.03640994057059288,
-0.034386370331048965,
0.024011006578803062,
0.013544945977628231,
0.00003982648559031077,
-0.... | 0.838811 | 0.823981 | 0.014829 |
Let G be a group of order 16. Determine whether G is isomorphic to the cyclic group C16, the dihedral group D8, or the non-abelian group of order 16. If it is isomorphic to one of these groups, provide an explicit isomorphism. If it is not isomorphic to any of these groups, give a brief explanation. | tiger_lab_math | [
0.005431349854916334,
0.011067471466958523,
-0.02081790380179882,
-0.035471782088279724,
0.0390462726354599,
-0.05291786417365074,
-0.0016339715803042054,
0.06152636930346489,
0.03132355958223343,
-0.0066914306953549385,
0.037609100341796875,
0.005946348421275616,
-0.03268952667713165,
-0.... | 0.808376 | 0.801822 | 0.006554 |
Let M be a two-dimensional manifold with coordinates (u,v) defined by the following equations:
x = (u-v)cos(u+v)
y = (u+v)sin(u-v)
Calculate the Cartan connection on M at the point (1,1). | tiger_lab_math | [
0.02398991957306862,
0.034662555903196335,
-0.02937834896147251,
-0.05178495869040489,
0.039335962384939194,
-0.010879462584853172,
-0.002258406952023506,
0.04970443993806839,
0.039678800851106644,
0.011771629564464092,
0.0041950964368879795,
0.04202846437692642,
-0.025811420753598213,
-0.... | 0.813197 | 0.803615 | 0.009582 |
in a certain parallelogram the degree measure of one angle exceeds that of the other by 70 what is the degree measure of the smaller angle ? Let's write a Python program to solve it. | tiger_lab_math | [
0.017375366762280464,
-0.00011060063116019592,
-0.03403567150235176,
-0.039860934019088745,
0.014618003740906715,
0.0007810188108123839,
-0.021319469437003136,
0.04572167992591858,
0.04587401822209358,
-0.029029106721282005,
0.015635114163160324,
0.03678524121642113,
-0.04209906607866287,
... | 0.829411 | 0.827213 | 0.002197 |
Consider the metric given by $ds^2 = 2(dx^2 + dy^2 + dz^2)$. Compute the Killing vector fields of this metric. | tiger_lab_math | [
0.008499852381646633,
0.04314099997282028,
-0.03293413668870926,
-0.05574880540370941,
0.03286248445510864,
-0.02040087804198265,
-0.033153291791677475,
0.04327845200896263,
0.04365897923707962,
-0.011130372062325478,
0.013293505646288395,
0.014357438310980797,
-0.025585543364286423,
-0.01... | 0.826071 | 0.821749 | 0.004322 |
Consider a game between two players, player A and player B. The payoff matrix for the game is given as follows:
```
Player B
L R
Player A U (2,2) (6,0)
D (0,6) (4,4)
```
The first number in each pair represents the payoff for player A, and the second number represents t... | tiger_lab_math | [
0.0533817782998085,
0.008480680175125599,
-0.046930793672800064,
-0.02289336360991001,
0.032911792397499084,
-0.01749618910253048,
-0.021132642403244972,
0.05997516214847565,
0.047370973974466324,
-0.046376973390579224,
0.028783362358808517,
0.04891449213027954,
-0.04110260680317879,
-0.03... | 0.815887 | 0.811295 | 0.004592 |
Sure, here's a precise numerical problem for a math student to solve related to solving differential equations in calculus:
Find the solution to the differential equation dy/dx = x + y, with initial condition y(0) = 1. | tiger_lab_math | [
0.024887721985578537,
0.010379590094089508,
-0.03765079379081726,
-0.04144589602947235,
0.038436103612184525,
-0.003578943433240056,
-0.015531591139733791,
0.053960565477609634,
0.0567469596862793,
-0.021682914346456528,
0.022135790437459946,
0.02177164889872074,
-0.02988998033106327,
-0.0... | 0.824269 | 0.818653 | 0.005616 |
Keisha's basketball team must decide on a new uniform. The seventh-graders will pick the color of the shorts (black or gold) and the eighth-graders will pick the color of the jersey (black, white, or gold), but the two groups of players will not confer together. If, for both garments, each possible color is equally l... | tiger_lab_math | [
0.006488000508397818,
0.017823124304413795,
-0.05556653067469597,
-0.03158797696232796,
0.018837854266166687,
-0.032046929001808167,
-0.02330688200891018,
0.053581010550260544,
0.029439449310302734,
-0.01916201040148735,
0.019833695143461227,
0.029347391799092293,
-0.013918094336986542,
-0... | 0.826613 | 0.820724 | 0.005889 |
Let X be a complex manifold given by the equation z^3 + w^3 - 3z^2w = 0 in C^2. Find the first Chern class of X. | tiger_lab_math | [
0.015013464726507664,
0.020719589665532112,
-0.02103051170706749,
-0.045727651566267014,
0.02943705953657627,
-0.016113460063934326,
0.0006031992961652577,
0.05885912850499153,
0.031845107674598694,
-0.022495800629258156,
0.02444317936897278,
0.022303827106952667,
-0.03818544000387192,
-0.... | 0.827792 | 0.824705 | 0.003087 |
A manufacturing company produces two types of products, Product A and Product B. The profit earned per unit of Product A is $10 and per unit of Product B is $15. The total production capacity is 120 units. Moreover, each unit of Product A takes 3 hours of machine time while each unit of Product B takes 5 hours of machi... | tiger_lab_math | [
0.031761910766363144,
0.0062425872310996056,
-0.05069785937666893,
0.001697911648079753,
0.017468012869358063,
-0.02177773229777813,
-0.06286770850419998,
0.09787874668836594,
0.027226252481341362,
-0.005115648731589317,
0.0035178742837160826,
0.04217258468270302,
-0.009782702662050724,
-0... | 0.797307 | 0.788526 | 0.008781 |
Consider the complex differential equation $z''+2iz'-2z=0$ with initial conditions $z(0)=1$ and $z'(0)=2i$. Find the solution $z(t)$ of this equation for $t \in [0,\frac{\pi}{4}]$. | tiger_lab_math | [
0.02702878974378109,
0.035461027175188065,
-0.03715901076793671,
-0.02900436334311962,
0.03954688087105751,
-0.000842286623083055,
-0.008422830142080784,
0.05502566322684288,
0.038714248687028885,
-0.0039234706200659275,
0.030176395550370216,
0.0044518690556287766,
-0.02656998671591282,
-0... | 0.824213 | 0.815879 | 0.008334 |
A tank in the shape of an inverted cone has a height of 10 meters and a base radius of 5 meters. Find the volume of water required to fill the tank to a depth of 8 meters. (Hint: The area under the curve can be represented by the definite integral of the function of the tank's cross-sectional area with respect to the h... | tiger_lab_math | [
0.031561221927404404,
0.04628777503967285,
-0.04902489855885506,
-0.01742660626769066,
0.036062758415937424,
-0.03973452374339104,
-0.024200530722737312,
0.06596803665161133,
0.012792364694178104,
0.00302313594147563,
0.004731634631752968,
0.028570445254445076,
-0.0515788309276104,
-0.0079... | 0.815562 | 0.806196 | 0.009366 |
If x, y and z are three different positive integers whose sum is prime, which of the following statements could be true?
Answer Choices: (A) x + y = z (B) Each of x + 3, y + 3, and z + 3 is prime. (C) Each of x + y, x + z, and y + z is prime. (D) The average (arithmetic mean) of x, y, and z is prime. (E) Each of x, y... | tiger_lab_math | [
0.04414984583854675,
0.008235163055360317,
-0.02340044640004635,
-0.03454217687249184,
0.03838924691081047,
-0.030578961595892906,
-0.009820771403610706,
0.05477485433220863,
0.04030349850654602,
-0.013554323464632034,
0.024920523166656494,
0.014824493788182735,
-0.05273490771651268,
-0.03... | 0.824849 | 0.820969 | 0.00388 |
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 $a_n < 0$. Please respond by writing a program in Python. | tiger_lab_math | [
0.006753452122211456,
0.03521908074617386,
-0.021827368065714836,
-0.02890164591372013,
0.023356463760137558,
-0.005437744781374931,
-0.02775990217924118,
0.05372440442442894,
0.03139284625649452,
-0.011477906256914139,
0.02313840016722679,
0.0068694525398314,
-0.05199761316180229,
0.00121... | 0.824952 | 0.816865 | 0.008087 |
Consider the following boundary value problem:
$y''(x) + 2y'(x) + 2y(x) = e^{3x},\quad 0 < x < 1,$
$y(0) = 0,\quad y(1) = 2.$
Find the solution $y(x)$ to the boundary value problem. | tiger_lab_math | [
0.009596007876098156,
0.026898551732301712,
-0.041858673095703125,
-0.033160727471113205,
0.028065381571650505,
-0.021174637600779533,
-0.017512844875454903,
0.08730562031269073,
0.061481695622205734,
-0.008766764774918556,
0.00014239043230190873,
0.03296612575650215,
-0.04054037854075432,
... | 0.822433 | 0.814474 | 0.007958 |
A man is trying to maximize the amount of money he saves each month. In particular, he is trying to decide between two different apartments. The first apartment costs $800 per month in rent and will cost an additional $260 per month in utilities. The second apartment costs $900 per month and will cost an additional $20... | tiger_lab_math | [
0.024219002574682236,
0.028532294556498528,
-0.040139585733413696,
-0.021603403612971306,
0.029292987659573555,
-0.017296409234404564,
-0.023378552868962288,
0.06347241997718811,
0.05254211649298668,
-0.004694770090281963,
0.008380706422030926,
0.016792533919215202,
-0.05193979665637016,
-... | 0.808598 | 0.807093 | 0.001505 |
A metal factory produces rectangular metal sheets with a length of 20 cm and a width of 10 cm. The factory wants to reduce the length of the metal sheet by keeping the same area of the sheet. What should be the new length of the sheet that minimizes the cost of metal used per sheet? The cost of metal is proportional to... | tiger_lab_math | [
0.009740784764289856,
0.02565777115523815,
-0.06401361525058746,
-0.035016100853681564,
0.03275289386510849,
-0.03242933750152588,
-0.05064082518219948,
0.07014823704957962,
0.0488506443798542,
-0.003804584266617894,
0.0010756824631243944,
0.029731586575508118,
-0.04776185750961304,
0.0008... | 0.81131 | 0.806781 | 0.004529 |
Consider the reaction-diffusion equation:
$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2} + \alpha u - \beta u^2$
where $u(x,t)$ is the concentration of a chemical substance in a medium, $D$ is the diffusivity, $\alpha$ and $\beta$ are reaction rate constants.
Find the steady-state solution for $... | tiger_lab_math | [
0.007911475375294685,
0.025424644351005554,
-0.047947973012924194,
-0.02447025664150715,
0.04062170162796974,
-0.03542448952794075,
-0.02543402835726738,
0.10268330574035645,
0.028047194704413414,
-0.0158968698233366,
0.014807537198066711,
0.011994194239377975,
-0.019763486459851265,
-0.02... | 0.807954 | 0.800073 | 0.007881 |
On a 2-dimensional manifold M, let the tensor field T be defined by $T = x^2 y\, \partial/\partial x \otimes dx + xy^2 \, \partial/\partial y \otimes dy$. Find the covariant derivative of T with respect to the metric $g = dx^2 + 2dxdy + dy^2$ at the point $(1,1)$. | tiger_lab_math | [
0.0035839451011270285,
0.026952972635626793,
-0.04378774017095566,
-0.0557221919298172,
0.030452454462647438,
-0.012421974912285805,
-0.028211142867803574,
0.0742548406124115,
0.04144006967544556,
0.007379954215139151,
-0.006697362754493952,
0.030147092416882515,
-0.03514863923192024,
-0.0... | 0.817382 | 0.806137 | 0.011245 |
A manufacturing company produces two products, A and B. Each unit of product A requires 2 hours of labor and 3 hours of machine time, while each unit of product B requires 1 hour of labor and 2 hours of machine time. The company has 480 hours of labor available and 600 hours of machine time available. If the profit on ... | tiger_lab_math | [
0.02084510773420334,
0.0051381574012339115,
-0.0625375509262085,
-0.014834599569439888,
0.025784911587834358,
-0.003002412850037217,
-0.04333018511533737,
0.08239834010601044,
0.030520455911755562,
0.000868114409968257,
0.0036307736299932003,
0.029560038819909096,
-0.002211080165579915,
-0... | 0.82015 | 0.810672 | 0.009478 |
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