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 father is building a playset for his son and needs to purchase lumber, nails, and fabric. When he started planning the project, the necessary lumber cost $450, the nails cost $30, and the fabric cost $80. However, recent economic inflation has caused the price of lumber to increase by 20%, the price of nails to inc... | tiger_lab_math | [
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Let M be a surface equipped with the metric ds^2 = du^2 + 4sinh^2(u/2)dv^2. Find all the isometries of M. | tiger_lab_math | [
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Consider a mechanism design problem where an auction is being held to sell a painting. There are 3 bidders, each with a private value for the painting. The private values are $100, $150, and $200. The auctioneer can choose between two auction formats: a sealed bid first-price auction or a second-price auction. In the s... | tiger_lab_math | [
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Suppose two companies, A and B, are the only producers in a market for a certain product. Both companies can produce the product at a constant marginal cost of $25 per unit. The market demand for the product is given by the inverse demand function p = 200 - Q, where p is the price (in dollars) and Q is the total output... | tiger_lab_math | [
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A furniture manufacturer wants to make a dining table with a maximum of 10 feet long and 6 feet wide. There are two models available, Model A and Model B, each made of oak or maple wood. Model A requires 5 hours of labor to produce and Model B requires 7 hours of labor. The manufacturer only has 30 labor hours availabl... | tiger_lab_math | [
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Let S be the set of natural numbers less than or equal to 20, and let P be the set of prime numbers less than or equal to 20. If we define a relation R on S x P as follows: (a, p) R (b, q) if and only if a + q = b + p, then how many pairs of elements are in the equivalence class [(1,2)]? | tiger_lab_math | [
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Consider a two-layer fully-connected neural network in which the hidden-unit nonlinear activation functions are given by logistic sigmoid functions. Does there exist an equivalent network in which the hidden unit nonlinear activation functions are given by hyperbolic tangent functions? Let's write a Python program. | tiger_lab_math | [
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Let S be the surface defined by the equation z = x^2 + y^2 between z = 0 and z = 4. Find the total Gaussian curvature of this surface. | tiger_lab_math | [
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Let S be a surface with constant Gaussian curvature K = 9 and total geodesic curvature of boundary curve C equal to 2π. Find the Euler characteristic χ(S) of S using Gauss-Bonnet theorem. | tiger_lab_math | [
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A steel company wants to produce two types of steel bars - Type A and Type B. The cost per ton of producing Type A steel is $1500 and for Type B steel is $2500. There are three raw materials required for the production - Raw Material 1, Raw Material 2, and Raw Material 3. The cost of raw material 1 per ton is $800, for... | tiger_lab_math | [
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Here is a precise numerical problem for a math student to solve in the area of Cheap Talk in Game Theory:
Two players are negotiating the price of a used car. Player 1 knows the true value of the car is $8000, and player 2 believes it could be worth either $5000 or $9000 with equal probability. Prior to negotiating, p... | tiger_lab_math | [
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Suppose a company wants to minimize the cost and maximize the revenue of a product. The cost function is given by C(x) = 0.01x^2 + 2x + 500, where x is the quantity produced, and the revenue function is given by R(x) = 10x - 0.1x^2. What is the optimal quantity that the company should produce to maximize revenue while ... | tiger_lab_math | [
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a telephone company needs to create a set of 3 - digit area codes . the company is entitled to use only digits 2 , 4 and 3 , which can be repeated . if the product of the digits in the area code must be even , how many different codes can be created ? Let's write a Python program to solve it. | tiger_lab_math | [
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Sure, here is a numerical optimization problem for the student to solve using simulated annealing method:
Consider the function f(x) = x^3 - 9x^2 + 24x on the interval [0, 4]. Use the simulated annealing method to find the global minimum of the function with an accuracy of 0.001. | tiger_lab_math | [
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In Set T, the average (arithmetic mean) equals the median W. Which of the following must be true?
I. Set T consists of evenly spaced numbers.
II. Set T consists of an odd number of terms.
III. Set T has no mode.
IV. None of the above.
Answer Choices: (A) I only (B) I and II (C) II and III (D) I, II, and III (E) IV only | tiger_lab_math | [
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Here's a precise numerical problem related to Abelian varieties:
Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with Cremona label 11a1, given by the Weierstrass equation $y^2=x^3+x$. Let $P=(-1,0)$ and $Q=(2,-2\sqrt{2})$ be two points on $E$. Find the coordinates of $R=P+Q$ on $E$. | tiger_lab_math | [
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The function $f(x)$ satisfies
\[f(xy) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(0) \neq 0.$ Find $f(10).$ Let's write a Python program to solve it. | tiger_lab_math | [
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Let's consider the modal logic formula "◊p → □p" where p stands for a mathematical proposition. If we interpret this formula as "It is possible that p implies it is necessarily true that p", Which of the following interpretations of p makes the formula true?
a) p is "2+2=4"
b) p is "1+1=3"
c) p is "x+2=5, where x is a... | tiger_lab_math | [
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A possible question for a math student to solve related to Elliptic Curves and their properties is:
"Let the elliptic curve $E$ be defined by the equation $y^2 = x^3 - 4x + 4$ over the field $\mathbb{F}_5$. Calculate the order $|E(\mathbb{F}_5)|$ of the group of points of $E$ over $\mathbb{F}_5$."
To solve this probl... | tiger_lab_math | [
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Let's say we have a 3-dimensional manifold described by the following parametric equation:
x(u,v) = ucos(v)
y(u,v) = usin(v)
z(u,v) = ln(u+2)
Find the scalar curvature of this manifold at the point where u = 1 and v = π/4. | tiger_lab_math | [
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A group of 6 students organized a fundraiser to go to Nicaragua for the summer. For them to go on the trip, each of them needs at least $450. On top of that, they need $3000 for their miscellaneous collective expenses. On the first day of their fundraiser, they receive $600. On the second day, they receive $900, and on... | tiger_lab_math | [
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A company wants to manufacture cylindrical cans of radius 5 cm and height 20 cm. The company wants to minimize the cost of material used to create these cans, which is proportional to the surface area of the can. The can must hold at least 400 mL of liquid. What is the minimum cost of material and how many cans can be ... | tiger_lab_math | [
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The Rotary Club is holding its annual fundraising Omelet Breakfast, with tickets sold in advance. The tickets come in different price levels, for young children, older children, adults, and seniors. This year they sold 53 small children tickets, 35 older children tickets, 75 adult tickets, and 37 senior tickets. To fig... | tiger_lab_math | [
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A factory produces two types of products: Product A and Product B. The profit for each unit of Product A is $5 and for each unit of Product B is $7. The factory has a production capacity of 70 units per day. In addition, it requires a total of 110 hours of labor to produce one unit of Product A and 145 hours for each u... | tiger_lab_math | [
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Consider the integral equation:
$$\phi(x) = \frac{1}{2} + \frac{1}{\pi} \int_{0}^{\pi} \cos(xt)\phi(t) \, dt$$
Find the solution of the above integral equation using complex analysis techniques. | tiger_lab_math | [
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Find the unit vector $\mathbf{v},$ lying in the $xz$-plane, which makes an angle of $45^\circ$ with $\begin{pmatrix} 2 \\ 2 \\ -1 \end{pmatrix},$ and an angle of $60^\circ$ with $\begin{pmatrix} 0 \\ 1 \\ - 1 \end{pmatrix}.$ | tiger_lab_math | [
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Let A, B, and C be matrices of order 2x2 and the following matrix equation holds: 2A - B = C. If A = [[1,2],[3,4]], B = [[0,1],[1,2]], and C = [[3,3],[5,6]], which of the following is true? Answer Choices: (A) The given equation holds. (B) The given equation does not hold. (C) A cannot be determined. (D) B cannot be de... | tiger_lab_math | [
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A company produces two products, A and B, using two resources, X and Y. Each unit of product A requires 5 units of resource X and 10 units of resource Y, while each unit of product B requires 2 units of resource X and 4 units of resource Y. The company has 120 units of resource X and 200 units of resource Y available f... | tiger_lab_math | [
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Let $g_{ij}$ be the metric given by $g_{11}=1, g_{22}=r^2, g_{33}=r^2 \sin^2 \phi$ where $r$ and $\phi$ are polar coordinates in 3D space. Find the Levi-Civita connection of this metric at a specific point, say $(r_0, \phi_0, z_0)$ for $r_0=2.5$, $\phi_0=\frac{\pi}{4}$ and $z_0=0.5$. | tiger_lab_math | [
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The quadratic equation $ax^2+20x+c=0$ has exactly one solution. If $a+c=29$, and $a<c$ find the ordered pair $(a,c)$. | tiger_lab_math | [
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-0.048842839896678925,
0.03812761232256889,
-0.025314467027783394,
0.0067245434038341045,
0.0714707002043724,
0.046222347766160965,
-0.032500747591257095,
0.019971400499343872,
0.02439895085990429,
-0.0313703790307045,
-0.02... | 0.844216 | 0.82865 | 0.015566 |
Consider the surface defined by the equation x^2 + 2y^2 + 3z^2 = 6 in Cartesian coordinates. Find the mean curvature of this surface at the point (1, 1, 1). | tiger_lab_math | [
0.006253522355109453,
0.04061760753393173,
-0.050530530512332916,
-0.05859265848994255,
0.04212280362844467,
-0.0057356152683496475,
-0.025317804887890816,
0.0509987436234951,
0.009254871867597103,
-0.028882602229714394,
0.02091268263757229,
0.027723055332899094,
-0.025348057970404625,
-0.... | 0.831651 | 0.82154 | 0.010111 |
Let S be the set of all even natural numbers. Determine whether the statement "Every natural number in S is greater than 5" is true or false in second-order logic. | tiger_lab_math | [
0.022693131119012833,
-0.004948827438056469,
-0.037854745984077454,
-0.04618944972753525,
0.0160114336758852,
-0.04205314815044403,
-0.010761604644358158,
0.05714692547917366,
0.038457322865724564,
-0.021913032978773117,
0.020871201530098915,
0.02984498254954815,
-0.028631772845983505,
-0.... | 0.826684 | 0.812016 | 0.014668 |
The figure shows three squares with non-overlapping interiors. The area of the shaded square is 1 square inch. What is the area of rectangle $ABCD$, in square inches?
[asy]size(100);
pair A = (0,0), D = (3,0),C = (3,2),B = (0,2);
draw(A--B--C--D--cycle);
draw(A--(1,0)--(1,1)--(0,1)--cycle);
filldraw(B--(1,2)--(1,1)--(... | tiger_lab_math | [
0.017963379621505737,
0.025279663503170013,
-0.053515661507844925,
-0.052866607904434204,
0.03386787697672844,
-0.013943971134722233,
-0.0007494493620470166,
0.05014737695455551,
0.03242310509085655,
-0.009764697402715683,
0.029491934925317764,
0.031202491372823715,
-0.03054405376315117,
-... | 0.829561 | 0.82128 | 0.008281 |
A cube has side length $6$. Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.)
[asy]
import three;
real t=-0.05;
triple A,B,C,D,EE,F,G,H;
A = (0,0,0);
B = (c... | tiger_lab_math | [
0.017413638532161713,
0.03873341903090477,
-0.024280335754156113,
-0.02791612036526203,
0.03471849858760834,
-0.024295371025800705,
-0.0376172810792923,
0.052109356969594955,
0.0302972923964262,
0.010755344294011593,
0.023151108995079994,
0.030627883970737457,
-0.00350210745818913,
-0.0070... | 0.830691 | 0.820615 | 0.010076 |
Consider the surface given by the equation $z = xy^2$. Determine the principal curvatures at the point $(1,2,4)$. | tiger_lab_math | [
0.0013971003936603665,
0.04029442369937897,
-0.037072986364364624,
-0.0470527783036232,
0.04087772220373154,
-0.009010037407279015,
-0.0005044901627115905,
0.0505925789475441,
0.022781720384955406,
-0.007795064244419336,
0.022798188030719757,
0.020663810893893242,
-0.026105815544724464,
-0... | 0.82954 | 0.817267 | 0.012273 |
The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\sqrt{2}$, what is the volume of the solid?
[asy] size(180); import three; pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(... | tiger_lab_math | [
0.029638711363077164,
0.03190077468752861,
-0.05172889679670334,
-0.04389696195721626,
0.0461539626121521,
-0.03175900876522064,
-0.032498884946107864,
0.05336043983697891,
0.02786771021783352,
0.009586276486515999,
0.03222202509641647,
0.03420885279774666,
-0.02468864992260933,
-0.0418991... | 0.831256 | 0.824481 | 0.006775 |
A manufacturing company produces two products A and B. The profit from product A is $5 per unit, and the profit from product B is $7 per unit. The company has constraints on production capacity and requires to produce at least 100 units of the two products combined.
- The production of product A requires 2 hours of lab... | tiger_lab_math | [
0.025307035073637962,
0.01685764081776142,
-0.05156086012721062,
-0.009390613064169884,
0.013354940339922905,
-0.006110448390245438,
-0.044645968824625015,
0.07548423111438751,
0.02622048556804657,
-0.006180224474519491,
0.005666465498507023,
0.027322659268975258,
-0.015769584104418755,
-0... | 0.823735 | 0.807732 | 0.016002 |
A math student needs to find the slope of the tangent line to the curve of the function f(x) = x^2 at the point where x = 3. | tiger_lab_math | [
0.003479322651401162,
0.019129619002342224,
-0.058378901332616806,
-0.05581602454185486,
0.04110976308584213,
-0.016385171562433243,
-0.03394662216305733,
0.05907543748617172,
0.04196649417281151,
-0.01133788377046585,
0.015428892336785793,
0.03340880572795868,
-0.03678654506802559,
-0.016... | 0.83052 | 0.827173 | 0.003347 |
Suppose there are 10 farmers using a common piece of land for grazing their goats. The land can only support a maximum of 20 goats. Each farmer decides how many goats to graze on the land, with the goal of maximizing their own profit. The cost of 1 goat is $50, and the selling price for 1 goat is $100. If a farmer earn... | tiger_lab_math | [
0.0264375451952219,
0.018189581111073494,
-0.048525385558605194,
-0.02104354277253151,
0.005758072715252638,
-0.029212823137640953,
-0.02324897237122059,
0.060118868947029114,
0.01667189598083496,
-0.0218419898301363,
-0.006981750950217247,
0.032669149339199066,
-0.04694410413503647,
-0.01... | 0.83203 | 0.818497 | 0.013533 |
Consider the following optimization problem:
minimize f(x) = x^4 - 2x^2 + x subject to -10 <= x <= 10
Use the cuckoo search algorithm to find the minimum value of f(x) within the given domain. Use an initial population size of 20 and run the algorithm for 100 iterations. Report the solution you found and the corres... | tiger_lab_math | [
-0.0025208843871951103,
0.017286598682403564,
-0.03993697836995125,
-0.020815812051296234,
0.04032920300960541,
-0.03504057973623276,
-0.0207276102155447,
0.05399945378303528,
0.06662678718566895,
-0.012658682651817799,
-0.00393644766882062,
0.020841078832745552,
-0.017293570563197136,
-0.... | 0.820009 | 0.817817 | 0.002192 |
The Royal Hawaiian Hotel decorates its Rainbow Christmas Tree with non-flashing white lights and a series of colored flashing lights - red, blue, green, orange, and yellow. The red lights turn red every 20 seconds, the blue lights turn blue every 30 seconds, the green lights turn green every 45 seconds, the orange ligh... | tiger_lab_math | [
0.017331548035144806,
0.03298541530966759,
-0.03709792718291283,
-0.055295925587415695,
0.00805739127099514,
-0.03997541591525078,
-0.02559734508395195,
0.08220577985048294,
0.02227347157895565,
-0.013117562979459763,
0.02796277590095997,
0.0197844747453928,
-0.0239015631377697,
-0.0296410... | 0.811458 | 0.809945 | 0.001513 |
Compute
\[\sum_{n = 1}^\infty \frac{2n - 1}{n(n + 1)(n + 2)}.\] | tiger_lab_math | [
0.033911027014255524,
0.0523296482861042,
-0.012883429415524006,
-0.04695780575275421,
0.022845100611448288,
-0.01529715396463871,
-0.02995878830552101,
0.057866308838129044,
0.028170118108391762,
-0.015955869108438492,
0.015820955857634544,
0.024158600717782974,
-0.027460116893053055,
-0.... | 0.837926 | 0.836897 | 0.001028 |
Sure, here's a precise numerical problem for a math student to solve related to the Matching Pennies game in Game Theory:
Player A has two pennies; one with heads on both sides and the other with tails on both sides. Player B also has two pennies; one with heads and tails on opposite sides and the other with tails on ... | tiger_lab_math | [
0.02246834710240364,
0.0036375385243445635,
-0.04638248309493065,
-0.02240523137152195,
0.02477114275097847,
-0.013602305203676224,
-0.024037813767790794,
0.019624367356300354,
0.03338762745261192,
-0.02738858386874199,
0.022391606122255325,
0.02348232828080654,
-0.0172410998493433,
-0.031... | 0.836736 | 0.824774 | 0.011962 |
Consider the rational curve given by the parametrization $C: (t^2-t, t^3-t^2)$ in $\mathbb{R}^2$.
Find the rational function $f(x,y)$ such that $C$ is the zero set of $f$, i.e., $C = \{(x,y)\in\mathbb{R}^2:f(x,y)=0\}$. | tiger_lab_math | [
0.030289679765701294,
0.04132401943206787,
-0.04959661513566971,
-0.023152699694037437,
0.040100838989019394,
-0.014530349522829056,
-0.011871580965816975,
0.06921509653329849,
0.0409327857196331,
-0.03371862694621086,
0.007280647289007902,
0.01765281893312931,
-0.032990653067827225,
-0.00... | 0.837633 | 0.812113 | 0.02552 |
Consider a one-shot game between two players: player 1 and player 2. Each player has two choices: cooperate (C) or defect (D). The payoff matrix for the game is as follows: if both players cooperate, player 1 gets 3 and player 2 gets 3; if player 1 defects and player 2 cooperates, player 1 gets 4 and player 2 gets 1; i... | tiger_lab_math | [
0.03100167214870453,
-0.004511998500674963,
-0.05308803915977478,
-0.01971568912267685,
0.022383317351341248,
-0.0021603675559163094,
-0.019112324342131615,
0.07136015594005585,
0.02581349015235901,
-0.031372684985399246,
0.018931087106466293,
0.018793253228068352,
-0.047115400433540344,
-... | 0.815275 | 0.80194 | 0.013335 |
Here's a precise numerical problem for a math student to solve:
Suppose you are the owner of a small online store and are considering investing in a new delivery service to ship your products. However, you want to make sure that the packaging of the products is robust enough to withstand any possible damage during tra... | tiger_lab_math | [
0.025483114644885063,
0.02507943846285343,
-0.03743588179349899,
-0.04609516263008118,
0.04110139235854149,
-0.012017736211419106,
-0.04093221202492714,
0.041006214916706085,
0.031866755336523056,
-0.013673155568540096,
0.010740665718913078,
0.034892961382865906,
-0.04164271801710129,
-0.0... | 0.834313 | 0.815206 | 0.019107 |
In triangle $ABC,$ $AB = 9,$ $BC = 10,$ and $AC = 11.$ If $D$ and $E$ are chosen on $\overline{AB}$ and $\overline{AC}$ so that $AD = 4$ and $AE = 7,$ then find the area of triangle $ADE.$
[asy]
unitsize (1 cm);
pair A, B, C, D, E;
A = (2,3);
B = (0,0);
C = (6,0);
D = interp(A,B,0.4);
E = interp(A,C,3/5);
draw(A--... | tiger_lab_math | [
0.03825942054390907,
0.0054022204130887985,
-0.04378221184015274,
-0.05183111131191254,
0.01765221729874611,
-0.03702615946531296,
-0.011228603310883045,
0.048484593629837036,
0.03415308892726898,
0.009085826575756073,
0.03491225838661194,
0.030009740963578224,
-0.035936303436756134,
-0.00... | 0.834375 | 0.817419 | 0.016956 |
A math student needs to determine the equation of an elliptic curve defined by the point (0, 1) and the tangent line at that point passing through (2,3) in the field of real numbers. | tiger_lab_math | [
0.01022362895309925,
0.035331353545188904,
-0.046278875321149826,
-0.05764706805348396,
0.03971269726753235,
-0.02462678775191307,
-0.02187199890613556,
0.050558991730213165,
0.02145407535135746,
-0.025126773864030838,
0.0009148942772299051,
0.03102862648665905,
-0.030323131009936333,
-0.0... | 0.82973 | 0.813898 | 0.015832 |
Let $S$ be a region in the plane with area 10. When we apply the matrix
\[\begin{pmatrix} 2 & 1 \\ 7 & -3 \end{pmatrix}\]to $S,$ we obtain the region $S'.$ Find the area of $S'.$ Please write a program to solve it | tiger_lab_math | [
0.036493659019470215,
0.03892649710178375,
-0.06805247813463211,
-0.047193869948387146,
0.04043407365679741,
-0.013918949291110039,
-0.018021829426288605,
0.06147054582834244,
0.0227105263620615,
-0.016518348827958107,
0.016939552500844002,
0.045789990574121475,
-0.04610490798950195,
-0.01... | 0.824045 | 0.821252 | 0.002793 |
Consider a complex manifold $M$ with first Chern class $c_1(M) = -3a + 2b$, where $a$ and $b$ are classes in $H^2(M,\mathbb{Z})$. If $c_2(M) = 10a^2 + 20ab - 12b^2$, compute the third Chern class $c_3(M)$. | tiger_lab_math | [
0.025285029783844948,
0.029929328709840775,
-0.024127019569277763,
-0.047160811722278595,
0.04005486145615578,
-0.017088672146201134,
-0.011893297545611858,
0.057338546961545944,
0.026955008506774902,
-0.017327547073364258,
0.0102164251729846,
0.035627469420433044,
-0.050172630697488785,
-... | 0.828833 | 0.813139 | 0.015693 |
A company produces two types of products; Product A and Product B. Each unit of Product A generates a profit of $4 and requires 3kg of raw material. Each unit of Product B generates a profit of $6 but requires 4kg of raw material. The company has 300kg of raw material available for production. If the company wants to m... | tiger_lab_math | [
0.03831375762820244,
0.015978509560227394,
-0.06329157948493958,
-0.014210818335413933,
0.018118156120181084,
-0.01749766431748867,
-0.044972144067287445,
0.08095038682222366,
0.018215108662843704,
-0.0144783491268754,
0.007411785423755646,
0.046398721635341644,
-0.014708509668707848,
-0.0... | 0.809965 | 0.806355 | 0.00361 |
For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+... | tiger_lab_math | [
0.021303711459040642,
0.021762201562523842,
-0.033322907984256744,
-0.030764080584049225,
0.025187546387314796,
-0.015355790965259075,
-0.014296979643404484,
0.046892449259757996,
0.038597624748945236,
-0.011407527141273022,
0.017992518842220306,
0.0198843851685524,
-0.0643201693892479,
0.... | 0.826168 | 0.80788 | 0.018288 |
Consider the Riemann surface defined by the equation $y^2=x^3-3x$. Let $\Theta(z)$ denote the theta function associated to this Riemann surface. Find the value of $\Theta(0)$, i.e., evaluate the theta function at the origin. | tiger_lab_math | [
0.012264545075595379,
0.03642476350069046,
-0.059333886951208115,
-0.0407421812415123,
0.030442288145422935,
-0.009888160042464733,
-0.009916701354086399,
0.06316125392913818,
0.03349054977297783,
-0.01445457711815834,
0.017688829451799393,
0.025513941422104836,
-0.04536982998251915,
-0.02... | 0.828902 | 0.816797 | 0.012104 |
Let $m$ and $n$ be positive integers satisfying the conditions
$\quad\bullet\ \gcd(m+n,210)=1,$
$\quad\bullet\ m^m$ is a multiple of $n^n,$ and
$\quad\bullet\ m$ is not a multiple of $n.$
Find the least possible value of $m+n.$
| tiger_lab_math | [
0.0260126031935215,
0.017395051196217537,
-0.02691357582807541,
-0.058012232184410095,
0.04075106978416443,
-0.02636033482849598,
-0.017394570633769035,
0.06217656657099724,
0.03533731400966644,
-0.014237932860851288,
0.01565459370613098,
0.036136284470558167,
-0.0367245227098465,
-0.02075... | 0.85237 | 0.837538 | 0.014831 |
A rocket is being launched into space and its trajectory is being controlled through optimal control. The rocket has a maximum thrust of 500 N and a mass of 10,000 kg. The goal is to get the rocket to a velocity of 10,000 m/s while minimizing the fuel consumption. The initial velocity of the rocket is 5,000 m/s and it ... | tiger_lab_math | [
0.026419920846819878,
0.034078724682331085,
-0.047953445464372635,
-0.025901583954691887,
0.027462070807814598,
-0.02029424160718918,
-0.017880791798233986,
0.08503171801567078,
0.054512858390808105,
-0.013675293885171413,
0.008281528018414974,
0.014557727612555027,
-0.0046287281438708305,
... | 0.772028 | 0.763554 | 0.008474 |
A is a set containing 8 different numbers. B is a set containing 7 different numbers, all of which are members of A . Which of the following statements CANNOT be true?
Answer Choices: (A) The range of A is less than the range of B. (B) The mean of A is greater than the mean of B . (C) The range of A is equal to the ... | tiger_lab_math | [
0.04761934280395508,
0.015108607709407806,
-0.03416502848267555,
-0.03148594871163368,
0.04780108109116554,
-0.044394172728061676,
-0.025602202862501144,
0.06266490370035172,
0.05136523395776749,
-0.03306559845805168,
0.002782993484288454,
0.029612498357892036,
-0.03861335664987564,
-0.027... | 0.821103 | 0.817471 | 0.003632 |
A company wants to optimize its production using integer programming. They want to produce two products, A and B. The production of each product requires two machines, X and Y. Machine X is available for 45 hours and machine Y is available for 30 hours per week. Product A earns a profit of $10 per unit, while product B... | tiger_lab_math | [
0.022406049072742462,
0.009529031813144684,
-0.06238044425845146,
-0.0017420429503545165,
0.01567039079964161,
-0.0035577528178691864,
-0.042220067232847214,
0.08719027787446976,
0.030940579250454903,
-0.01709590293467045,
0.01210981234908104,
0.02702721394598484,
-0.01758796162903309,
-0.... | 0.810111 | 0.792911 | 0.017201 |
The deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, chicken, cheddar cheese, and white bread are each offered at the deli. If Al never orders a sandwich with a ham/cheddar cheese combination nor a sandwic... | tiger_lab_math | [
0.02110930345952511,
0.01754075288772583,
-0.01890137977898121,
-0.02313932776451111,
0.02147497609257698,
-0.011010068468749523,
-0.04191657155752182,
0.03613603860139847,
0.03481178358197212,
-0.03137761354446411,
0.025095298886299133,
0.02542228437960148,
-0.029685128480196,
-0.01400196... | 0.815722 | 0.814434 | 0.001288 |
Two cars are driving straight at each other on a road. Each driver must choose whether to swerve to the left or the right. If both drivers swerve to the same side, there will be a collision, but if one driver swerves left while the other swerves right, the one who swerved left will win and the one who swerved right wil... | tiger_lab_math | [
-0.00508488342165947,
0.01469885278493166,
-0.037135154008865356,
-0.02851845510303974,
0.023993415758013725,
0.008618483319878578,
-0.014656158164143562,
0.05460219830274582,
0.04237176850438118,
-0.016218626871705055,
0.0038402555510401726,
0.026709159836173058,
-0.01980660855770111,
-0.... | 0.820439 | 0.804394 | 0.016045 |
Square each integer $n$ in the range $1\le n\le 10$ and find the remainders when the squares are divided by $11$. Add up all the distinct results and call it $m$. What is the quotient when $m$ is divided by $11$? Let's write a Python program to solve it. | tiger_lab_math | [
0.03760227560997009,
0.010937493294477463,
-0.028734128922224045,
-0.047289371490478516,
0.04062790423631668,
-0.04032384976744652,
-0.014472558163106441,
0.05120336264371872,
0.0340854786336422,
-0.0040226830169558525,
0.022166578099131584,
0.04933828115463257,
-0.06414145231246948,
0.002... | 0.836421 | 0.829626 | 0.006794 |
Let M be the 3-dimensional unit sphere in R^4, i.e., M = {(x,y,z,w) ∈ R^4 : x^2 + y^2 + z^2 + w^2 = 1}. Determine the Hopf map of M. | tiger_lab_math | [
0.017653124406933784,
0.02740735374391079,
-0.02645018883049488,
-0.05970558151602745,
0.054372638463974,
-0.029414625838398933,
-0.022902943193912506,
0.044337306171655655,
0.03379925712943077,
-0.00962574128061533,
0.0124349445104599,
0.032302577048540115,
-0.022612612694501877,
-0.04508... | 0.824397 | 0.81457 | 0.009826 |
Sure! Here is a precise numerical problem for a math student to solve:
The equation x^2 + xy + y^2 = 5 defines a curve. Find the derivative of y with respect to x when x = 1.
Hint: To differentiate an implicitly defined function involving both x and y, you will need to use the chain rule. | tiger_lab_math | [
0.010954177007079124,
0.01966109685599804,
-0.04445047676563263,
-0.03952905908226967,
0.04847394675016403,
-0.023497389629483223,
-0.01850436069071293,
0.04806612432003021,
0.03342096880078316,
-0.020939277485013008,
0.008776459842920303,
0.021359490230679512,
-0.028307825326919556,
-0.01... | 0.827372 | 0.819493 | 0.007878 |
A particle is moving along a curve defined by the equation $r(t) = (3\cos(t),2\sin(t),t)$. Suppose a vector $\vec{v}_0 = (1,0,0)$ is initially pointing in the direction of the positive $x$-axis at $t=0$. What is the parallel transport of $\vec{v}_0$ along the curve $r(t)$ when $t=\frac{\pi}{2}$? | tiger_lab_math | [
0.006718943826854229,
0.0443945974111557,
-0.0314718596637249,
-0.017432061955332756,
0.027981474995613098,
0.002821768634021282,
-0.04383948817849159,
0.07415040582418442,
0.03228673338890076,
-0.01179686188697815,
0.0013042642967775464,
0.03260742872953415,
-0.017870794981718063,
-0.0209... | 0.822724 | 0.815632 | 0.007091 |
Here is a problem for the student to solve:
Let S be the set of all natural numbers that can be expressed as a sum of cubes of other natural numbers, i.e., S = {x | x = a³ + b³ + c³, for some natural numbers a, b, and c}. Find the smallest number in S that is not the sum of cubes of three smaller natural numbers. | tiger_lab_math | [
0.01943194307386875,
0.013038615696132183,
-0.057041142135858536,
-0.037411291152238846,
0.04254394769668579,
-0.026541877537965775,
-0.02378554455935955,
0.050527799874544144,
0.045743346214294434,
-0.007555459160357714,
0.027308788150548935,
0.021080007776618004,
-0.0449429526925087,
-0.... | 0.836618 | 0.825607 | 0.011011 |
The vertices of a square are the centers of four circles as shown below. Given each side of the square is 6cm and the radius of each circle is $2\sqrt{3}$cm, find the area in square centimeters of the shaded region. [asy]
fill( (-1,-1)-- (1,-1) -- (1,1) -- (-1,1)--cycle, gray);
fill( Circle((1,1), 1.2), white);
fill( C... | tiger_lab_math | [
0.026601005345582962,
0.0490778423845768,
-0.037513602524995804,
-0.039200957864522934,
0.04070911929011345,
-0.029237238690257072,
-0.025925463065505028,
0.04480326920747757,
0.030527839437127113,
-0.0011914789211004972,
0.031189685687422752,
0.03358945995569229,
-0.02121083252131939,
-0.... | 0.818961 | 0.807699 | 0.011262 |
Here is a precise numerical optimization problem for a math student to solve using tabu search:
The management of a manufacturing company needs to determine the optimal production levels of two products, A and B, to maximize the total profit subject to the following constraints:
- Product A requires 3 units of labor ... | tiger_lab_math | [
0.03288360312581062,
0.01472077239304781,
-0.06210503727197647,
-0.015577946789562702,
0.026025664061307907,
-0.011970055289566517,
-0.035089924931526184,
0.06772185117006302,
0.036330562084913254,
-0.001458447310142219,
-0.0005343013326637447,
0.023065311834216118,
-0.020914433524012566,
... | 0.824007 | 0.808624 | 0.015384 |
Let A, B, and C be nonempty sets, and let h: A -> B and k: B -> C be functions such that the function k ∘ h : A -> C is onto (surjective). Which of the following must be true? Answer Choices: (A) h is one-to-one. (B) h is onto. (C) k is one-to-one. (D) k is onto., | tiger_lab_math | [
0.04405735433101654,
0.03424504026770592,
-0.028123419731855392,
-0.010811333544552326,
0.045786771923303604,
-0.01182534545660019,
-0.008544723503291607,
0.05772622302174568,
0.03364007920026779,
-0.03685741499066353,
0.013248446397483349,
0.020293034613132477,
-0.012046551331877708,
-0.0... | 0.82915 | 0.818613 | 0.010537 |
Consider a Cournot competition between two firms. Firm 1 has a cost function of C1 = 200 + 2Q1, where Q1 is the quantity produced by firm 1, and Firm 2 has a cost function of C2 = 150 + 3Q2, where Q2 is the quantity produced by firm 2. The market demand function is P = 100 - Q, where P is the product price and Q is the... | tiger_lab_math | [
0.021712884306907654,
0.01901870034635067,
-0.04922918975353241,
-0.04108612611889839,
0.0227429810911417,
-0.0046371761709451675,
-0.03380090743303299,
0.07101953029632568,
0.03015969879925251,
-0.02090148627758026,
0.011506710201501846,
0.03981737792491913,
-0.04422849044203758,
-0.01936... | 0.817037 | 0.802693 | 0.014344 |
S is a set containing 10 different positive odd primes. T is a set containing 8 different numbers, all of which are members of S. Which of the following statements CANNOT be true?
Answer Choices: (A) The median of S is prime. (B) The median of T is prime (C) The median of S is equal to the median of T. (D) The sum ... | tiger_lab_math | [
0.04707144573330879,
0.011119885370135307,
-0.04023896902799606,
-0.030329786241054535,
0.04168584942817688,
-0.045163460075855255,
-0.014386082999408245,
0.07943513989448547,
0.03444356471300125,
-0.022321710363030434,
0.012732516042888165,
0.014699744060635567,
-0.044955775141716,
-0.033... | 0.829562 | 0.820822 | 0.00874 |
Consider a signaling game in which the sender has two possible types, A or B, each occurring with equal probability. The receiver can either choose Action 1 or Action 2. When the sender is of type A, the payoff matrix is:
| | Action 1 | Action 2 |
|-------|----------|----------|
| Signal A | (4, 0) | (0, 4) ... | tiger_lab_math | [
0.01766383647918701,
-0.0025273591745644808,
-0.050335220992565155,
-0.03957526013255119,
0.02349669300019741,
-0.03434973210096359,
-0.028073636814951897,
0.0834801122546196,
0.03662226349115372,
-0.04564833641052246,
0.020893791690468788,
0.051118459552526474,
-0.026602694764733315,
-0.0... | 0.816186 | 0.803975 | 0.012211 |
Statement 1 | The order of an element in a finite group is always less than or equal to the order of the group. Statement 2 | Every group of prime order is cyclic. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.0393381342291832,
0.022741928696632385,
-0.027377240359783173,
-0.05675793066620827,
0.03734677657485008,
-0.0498078390955925,
-0.004406113643199205,
0.05515305697917938,
0.03785359859466553,
-0.01995590701699257,
0.018722478300333023,
0.013213610276579857,
-0.04003215581178665,
-0.00339... | 0.806882 | 0.803735 | 0.003147 |
Consider a graph G with 7 vertices and the following edge set: {(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(5,6),(5,7)}. Determine the largest clique that exists in G. | tiger_lab_math | [
0.02951282635331154,
0.02813536487519741,
-0.018474025651812553,
-0.0455508828163147,
0.02505517192184925,
-0.037804845720529556,
-0.027300996705889702,
0.03400307148694992,
0.014364236034452915,
-0.026442335918545723,
0.030606821179389954,
0.02487804926931858,
-0.04502693563699722,
-0.001... | 0.828755 | 0.821659 | 0.007096 |
Consider a graph with 7 vertices given by {(1,2),(2,3),(2,4),(3,4),(4,5),(6,7)}. What is the maximum number of vertices that can be selected such that no two selected vertices share an edge? | tiger_lab_math | [
0.04322613403201103,
0.017315518110990524,
-0.04505322128534317,
-0.02562784217298031,
0.01967201940715313,
-0.03284936770796776,
-0.02537216618657112,
0.023850491270422935,
0.020142214372754097,
-0.04387141019105911,
0.023658933117985725,
0.021010519936680794,
-0.015313123352825642,
-0.00... | 0.825316 | 0.82195 | 0.003366 |
Suppose an educator wants to determine whether there is a significant difference in the mean scores of three groups of students who completed a mathematics quiz. The educator collected the following data:
- Group A: 85, 87, 88, 89, 90
- Group B: 84, 82, 86, 78, 92
- Group C: 79, 81, 83, 80, 78
Using a significance le... | tiger_lab_math | [
0.008834870532155037,
0.0023260079324245453,
-0.058029960840940475,
-0.06497882306575775,
0.0347035676240921,
-0.023381821811199188,
-0.018760038539767265,
0.06732586026191711,
0.032958485186100006,
-0.019610272720456123,
0.006615761201828718,
0.020892204716801643,
-0.0333445705473423,
-0.... | 0.806897 | 0.803987 | 0.00291 |
Let set A = {2, 3, 5, 7} and set B = {3, 5, 8, 9}. Find the symmetric difference of set A and set B and verify whether it is commutative or not. | tiger_lab_math | [
0.050683021545410156,
0.010398810729384422,
-0.04210349917411804,
-0.03300043195486069,
0.03424901142716408,
-0.040604911744594574,
-0.012525743804872036,
0.06994963437318802,
0.06189623475074768,
-0.033486224710941315,
0.01425265334546566,
0.016681719571352005,
-0.03781870752573013,
-0.01... | 0.825995 | 0.821019 | 0.004976 |
Consider a population of 1000 bacteria that doubles every hour. Write a differential equation representing this population growth and find the solution for the population after 5 hours. | tiger_lab_math | [
0.015644973143935204,
0.009590260684490204,
-0.03672904521226883,
-0.021857893094420433,
0.02108369953930378,
-0.039890605956315994,
-0.03981418162584305,
0.05067465826869011,
0.03694188594818115,
-0.011432125233113766,
0.01383067388087511,
0.028995582833886147,
-0.027146855369210243,
-0.0... | 0.82669 | 0.824385 | 0.002305 |
We have a triangle $\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$? Please write a program to solve it | tiger_lab_math | [
0.032634373754262924,
0.006603121757507324,
-0.04412000626325607,
-0.043102163821458817,
0.019952531903982162,
-0.02086365781724453,
-0.030501393601298332,
0.043707624077796936,
0.039147503674030304,
-0.03525449335575104,
0.021891014650464058,
0.033049967139959335,
-0.03482005372643471,
-0... | 0.84364 | 0.841897 | 0.001742 |
Find the Laurent series expansion of the function
f(z) = ln(z)
centered at
z0 = 1
that converges in the annulus
1/2 < |z-1| < 3/2. | tiger_lab_math | [
-0.003392595797777176,
0.045305170118808746,
-0.03787050023674965,
-0.04978817701339722,
0.03540533781051636,
0.01438002847135067,
-0.018155425786972046,
0.03978067636489868,
0.014676012098789215,
-0.008947479538619518,
0.034227076917886734,
0.01735159568488598,
-0.03342636302113533,
-0.00... | 0.825531 | 0.819555 | 0.005976 |
$ABCD$, a rectangle with $AB = 12$ and $BC = 16$, is the base of pyramid $P$, which has a height of $24$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ denote the center of the circumsphere of $F$, and let $T$ denote the apex of $P$. If the volume ... | tiger_lab_math | [
0.027845120057463646,
0.032560985535383224,
-0.03377602994441986,
-0.03847990930080414,
0.021411269903182983,
-0.031602513045072556,
-0.009562395513057709,
0.050553224980831146,
0.04202685505151749,
0.0033369422890245914,
0.016529420390725136,
0.0476248525083065,
-0.02692405879497528,
-0.0... | 0.842403 | 0.814019 | 0.028384 |
Consider the following non-homogeneous system of linear equations:
\begin{align*}
2x + 3y - z &= 5\\
-x + 2y + z &= 3\\
3x + 2y + 2z &= 1
\end{align*}
Solve this system of linear equations. | tiger_lab_math | [
0.03098275698721409,
0.03472977876663208,
-0.0461033396422863,
-0.061549603939056396,
0.05746721103787422,
-0.017541037872433662,
-0.013946016319096088,
0.06360433995723724,
0.021982461214065552,
-0.010382276028394699,
0.032519299536943436,
0.031651973724365234,
-0.04129454120993614,
-0.04... | 0.819138 | 0.818067 | 0.001071 |
In the Rail fence cipher, a plaintext message "MATHEMATICS" is encrypted with a rail fence of depth 3. If we read the cipher text row by row from top to bottom, we get "MTAIHSEMTC". Find the original plaintext message. | tiger_lab_math | [
0.05278150364756584,
0.022419827058911324,
-0.022602161392569542,
-0.04503251612186432,
0.04617020860314369,
-0.04729868844151497,
-0.04111224040389061,
0.05581308528780937,
0.030667340382933617,
-0.0246447641402483,
0.016242429614067078,
-0.0008968317415565252,
-0.02649545669555664,
-0.01... | 0.815346 | 0.813457 | 0.00189 |
For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+... | tiger_lab_math | [
0.031234094873070717,
0.018747147172689438,
-0.03058166615664959,
-0.0332048125565052,
0.024029936641454697,
-0.01387706771492958,
-0.012089605443179607,
0.04360909387469292,
0.04014233872294426,
-0.00294951768592,
0.016462836414575577,
0.016960877925157547,
-0.06603582203388214,
0.0033771... | 0.83336 | 0.808929 | 0.024431 |
A math student is given the following problem on graphing trigonometric functions:
"Consider the function f(x) = 2sin(3x), where x is in radians. Graph this function, including the x and y-intercepts, maximum and minimum points, and at least one full period. What is the period of this function? What are the amplitude ... | tiger_lab_math | [
0.028348075225949287,
0.03833291679620743,
-0.0589541420340538,
-0.040831007063388824,
0.028081292286515236,
-0.01770252361893654,
-0.014705337584018707,
0.05153968185186386,
0.045951876789331436,
0.018602736294269562,
0.02790578454732895,
0.023501267656683922,
-0.02042408101260662,
-0.008... | 0.80235 | 0.784193 | 0.018157 |
Consider the following undirected graph with 7 vertices and 10 edges:
```
1---2---3
|\ | / |
| \ | / |
| \|/ |
4---5----6
| /
| /
| /
| /
7
```
Determine whether this graph has an Eulerian path, a Eulerian circuit, a Hamiltonian path, or a Hamiltonian cycle. If so, provide the path or ... | tiger_lab_math | [
0.006849908735603094,
0.016325658187270164,
-0.02269946038722992,
-0.033425189554691315,
0.014800205826759338,
-0.0356217622756958,
-0.03074285387992859,
0.07183048874139786,
0.021526474505662918,
-0.011572309769690037,
0.02377418987452984,
0.019434364512562752,
-0.011137651279568672,
-0.0... | 0.820239 | 0.814353 | 0.005886 |
Suppose a manufacturing company produces light bulbs that have a lifespan of 1000 hours. The company wants to test the bulbs before they are released to customers. A sample of 50 bulbs is taken from the production line and tested. The average lifespan of these bulbs is found to be 980 hours with a standard deviation of... | tiger_lab_math | [
-0.008341667242348194,
0.0019136543851345778,
-0.04302377626299858,
-0.03350337594747543,
0.035951174795627594,
-0.04439176246523857,
-0.04998132213950157,
0.082188181579113,
0.02296762354671955,
-0.0065964218229055405,
0.007649997249245644,
0.0289597287774086,
-0.017748897895216942,
-0.00... | 0.796653 | 0.790213 | 0.00644 |
An abundant number is a positive integer such that the sum of its proper divisors is greater than the number itself. The number 12 is an abundant number since $1 + 2 + 3 + 4 + 6 > 12$. What is the smallest abundant number that is not a multiple of 6? Let's write a program. | tiger_lab_math | [
0.03530479222536087,
0.023694731295108795,
-0.018775630742311478,
-0.021098610013723373,
0.05044611915946007,
-0.047609858214855194,
-0.025026556104421616,
0.04482625797390938,
0.02947082556784153,
-0.022898659110069275,
0.015945574268698692,
0.04329567402601242,
-0.040495455265045166,
-0.... | 0.809332 | 0.806522 | 0.00281 |
Alright, here is a precise numerical problem for you:
"Suppose you have a sequence of six numbers: 1, 2, 3, 4, 5, and 6. In how many ways can you arrange these numbers such that no three consecutive numbers are increasing?"
Hint: The answer involves using the Catalan number C3, which is equal to 5. | tiger_lab_math | [
0.0293594803661108,
0.019894788041710854,
-0.03074766881763935,
-0.01417053584009409,
0.04406927153468132,
-0.01652524806559086,
-0.020820895209908485,
0.042726825922727585,
0.030223943293094635,
-0.0233394056558609,
0.011721809394657612,
0.033189620822668076,
-0.0482327863574028,
-0.01532... | 0.840919 | 0.836345 | 0.004574 |
Here's a possible question for a math student to solve:
Consider the function $f(z) = \frac{e^{3z}}{(z+1)^3}$ on the complex plane. What is the residue of $f$ at the pole $z=-1$?
Hint: To compute the residue of a function $f(z)$ at a pole $z=z_0$, we can expand $f(z)$ as a Laurent series around $z_0$ and look for the... | tiger_lab_math | [
0.03315072879195213,
0.035038888454437256,
-0.03498004749417305,
-0.0521552711725235,
0.042148757725954056,
-0.01934893988072872,
-0.005956651177257299,
0.060657430440187454,
0.04106106236577034,
-0.007797840982675552,
0.02876119315624237,
0.03266021981835365,
-0.06654992699623108,
-0.0149... | 0.833423 | 0.826913 | 0.006509 |
Suppose you are running a small food business and need to decide how much of two ingredients, A and B, to use in your recipe. The cost of each unit of A is $3, and the cost of each unit of B is $4. You want to create at least 50 servings of your dish, but want to ensure that the recipe remains affordable despite any po... | tiger_lab_math | [
0.013448771089315414,
0.024770215153694153,
-0.03920717164874077,
-0.014752520248293877,
0.028669485822319984,
-0.004777213092893362,
-0.02411852590739727,
0.0656798928976059,
0.043630070984363556,
-0.020209351554512978,
0.016315365210175514,
0.006747696548700333,
-0.02750900760293007,
-0.... | 0.81849 | 0.799332 | 0.019158 |
A right triangle has a hypotenuse of length 10 cm and one of its acute angles measures 30 degrees. Find the area of the triangle using trigonometry. | tiger_lab_math | [
0.008208942599594593,
0.03026876598596573,
-0.035238806158304214,
-0.036141812801361084,
0.023360060527920723,
-0.027323829010128975,
-0.014559291303157806,
0.05124417692422867,
0.039182376116514206,
-0.0268624909222126,
0.02021973393857479,
0.035620227456092834,
-0.03511225804686546,
-0.0... | 0.800051 | 0.79543 | 0.004621 |
Sure, here's a precise numerical problem for a math student to solve, related to Calculus and the subtopic of Solving Differential Equations:
Given a differential equation of the form y' + 3y = 6 with initial condition y(0) = 2, find the solution to the differential equation.
Good luck! | tiger_lab_math | [
0.023040305823087692,
0.01114229392260313,
-0.05112403258681297,
-0.03841329738497734,
0.04283721372485161,
-0.004996803589165211,
-0.02299751527607441,
0.046384263783693314,
0.03654806688427925,
-0.021203454583883286,
0.03156081214547157,
0.029948094859719276,
-0.030186545103788376,
-0.03... | 0.829009 | 0.827427 | 0.001582 |
An isosceles, obtuse triangle has one angle with a degree measure that is 50$\%$ larger than the measure of a right angle. What is the measure, in degrees, of one of the two smallest angles in the triangle? Express your answer as a decimal to the nearest tenth. | tiger_lab_math | [
0.015047935768961906,
0.015789611265063286,
-0.02878846414387226,
-0.04082602635025978,
0.020145000889897346,
-0.016954325139522552,
-0.018376506865024567,
0.05200859531760216,
0.03779340162873268,
-0.014235218055546284,
0.0351591631770134,
0.03104850836098194,
-0.031706660985946655,
-0.01... | 0.808642 | 0.80489 | 0.003752 |
Let X be a compact topological space and suppose the Betti numbers of X are given by b_0 = 4, b_1 = 2, b_2 = 3, b_3 = 1. Compute the Euler characteristic of X and verify Poincaré duality by showing the homology groups of X satisfy the necessary conditions. | tiger_lab_math | [
0.009814242832362652,
0.02199813537299633,
-0.027034487575292587,
-0.037969786673784256,
0.03090216964483261,
0.0029823845252394676,
-0.004189836326986551,
0.05004880949854851,
0.048463862389326096,
0.008707051165401936,
0.0018386549782007933,
0.04228931665420532,
-0.03065001405775547,
-0.... | 0.822476 | 0.793625 | 0.028851 |
In triangle $ABC$, $AB = 11$, $AC = 13$, and $BC = 20$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let $P$ be the foot of the altitude from $G$ to $BC$. Find $GP$.
[asy]
unitsize(0.3 cm);
pair A, B, C, D, E, F, G, P;
A = (44/5,33/5);
B = (0,0);
C = (20,0);
D = (B + C)/2;
E =... | tiger_lab_math | [
0.046552129089832306,
0.0013850699178874493,
-0.047392308712005615,
-0.05428522452712059,
0.03191988915205002,
-0.047898367047309875,
-0.03142790123820305,
0.04405437409877777,
0.03477209806442261,
0.007669183425605297,
0.041442014276981354,
0.03238806501030922,
-0.031256284564733505,
-0.0... | 0.837581 | 0.819558 | 0.018023 |
Consider the surface defined by the equation x^2 + y^2 + z^2 + 6z = 0. Calculate the Gaussian curvature of this surface at the point (0,0,0). | tiger_lab_math | [
-0.0009949069935828447,
0.03557484969496727,
-0.031116368249058723,
-0.04417745769023895,
0.040981367230415344,
-0.005237707868218422,
-0.01195327565073967,
0.05842911824584007,
0.019185738638043404,
-0.034246038645505905,
0.01763964630663395,
0.029252877458930016,
-0.016548078507184982,
-... | 0.823222 | 0.820004 | 0.003218 |
Given a complex-valued function $f(z) = z^2 + 2z +2,$ find the minimum and maximum values of $|f(z)|$ on the closed unit disc $|z| \leq 1.$ | tiger_lab_math | [
0.028128961101174355,
0.029454130679368973,
-0.042982716113328934,
-0.0359802320599556,
0.04682521894574165,
-0.008220896124839783,
-0.021684709936380386,
0.04365788400173187,
0.03729696199297905,
0.006985324900597334,
0.03522012010216713,
0.02238425239920616,
-0.016539456322789192,
0.0013... | 0.825507 | 0.812171 | 0.013336 |
Consider the metric given by $g = dx^2 + dy^2 + dz^2$ in $\mathbb{R}^3$. Find the Levi-Civita connection of this metric at the point $(1,2,3)$. | tiger_lab_math | [
0.0007598567754030228,
0.03568664565682411,
-0.03593085706233978,
-0.04663208872079849,
0.03041777014732361,
-0.002681576646864414,
-0.014603552408516407,
0.03862936422228813,
0.02126968279480934,
-0.00924550462514162,
0.014291001483798027,
0.035939596593379974,
-0.031153198331594467,
-0.0... | 0.829132 | 0.815744 | 0.013389 |
Suppose we have a symplectic matrix A, which satisfies A^TJA = J, where J is the standard symplectic matrix. If A has eigenvalues 2-i, 3, 5, and 2+i, find the determinant of A. | tiger_lab_math | [
0.025059683248400688,
0.014812970533967018,
-0.04055096581578255,
-0.03745179995894432,
0.03352949768304825,
-0.027488842606544495,
-0.026615001261234283,
0.06486893445253372,
0.026008643209934235,
-0.01515787374228239,
-0.0002673221752047539,
0.017026390880346298,
-0.05559308081865311,
-0... | 0.831393 | 0.827665 | 0.003729 |
Let M be a 3-dimensional manifold equipped with a Riemannian metric g whose components in local coordinates (x,y,z) are given by:
g11 = x^2 + z^2
g22 = y^2 + z^2
g33 = x^2 + y^2
and zeros everywhere else. Calculate the Cartan connection of M. | tiger_lab_math | [
0.02358011156320572,
0.035723548382520676,
-0.028930580243468285,
-0.06553752720355988,
0.039781082421541214,
-0.014025506563484669,
-0.015759868547320366,
0.03694649040699005,
0.024868017062544823,
0.007043274585157633,
0.009250623174011707,
0.0590532161295414,
-0.055394187569618225,
-0.0... | 0.822166 | 0.814991 | 0.007175 |
Let G be an Abelian group and a, b, c be elements in G. Statement 1 | If a and b commute (ab=ba), then a, b, and c commute. Statement 2 | The inverse of a group element 'a' in G, denoted by a^-1, also belongs to G. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.03696482628583908,
0.038196809589862823,
-0.02020450308918953,
-0.037839099764823914,
0.037487685680389404,
-0.05276397243142128,
0.001665217918343842,
0.08389800786972046,
0.030754948034882545,
-0.0120834456756711,
0.00488101365044713,
0.023244209587574005,
-0.02421901933848858,
-0.0378... | 0.813381 | 0.811886 | 0.001495 |
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