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nemotron-math-v2-10k-10
Given a discrete probability density $\{p(i)\}_{i=0}^\infty$, let $G(z)$ be the probability generating function defined as \[ G(z) = \sum_{i=0}^\infty p(i) \cdot z^i. \] Compute the $n$-th derivative of the ratio $\frac{G^{(1)}(z)}{G(z)}$, i.e., find the explicit expression of \[ \zeta_n(z) = \left(\frac{G^{(1)}(z)}{G(...
\; \begin{aligned} \zeta_n(z)&= \sum_{k=1}^{\,n+1}(-1)^{\,k-1}(k-1)!\, \frac{ B_{\,n+1,k}\!\Big( \displaystyle\sum_{i=1}^{\infty} i\,p(i)\,z^{i-1},\, \displaystyle\sum_{i=2}^{\infty} i(i-1)p(i)\,z^{i-2},\, \dots,\, \displaystyle\sum_{i=n+2-k}^{\infty}\frac{i!}{(i-n-2+k)!}\,p(i)\, z^{\,i-n-...
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4039846/general-quotient-rule-for-a-couple-of-special-functions
Gost91
https://math.stackexchange.com/users/266419/gost91
nemotron-math-v2-10k-100
Given the equation \( T = x_1 + (Ax_2) + (A^2x_3) + (A^3x_4) + (x_5 \cdot e^{0.1984A}) \), determine the value of \( A \) in terms of \( T, x_1, x_2, x_3, x_4, \) and \( x_5 \).
A=f^{-1}(T)
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1879386/variable-of-polynomial-equation
David
https://math.stackexchange.com/users/357946/david
nemotron-math-v2-10k-1016
Given the equation \( b = a_{1} x_{1} + a_{2} x_{2} + \sqrt{(a_{1} x_{11} + a_{2} x_{22})^2} \), isolate \( x_{11} \) on one side of the equation.
\,x_{11}= \dfrac{\pm\bigl(b-a_{1}x_{1}-a_{2}x_{2}\bigr)-a_{2}x_{22}}{a_{1}}\,
4
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4377014/how-to-define-unknown-parameter
A.F.R.S2022
https://math.stackexchange.com/users/1023319/a-f-r-s2022
nemotron-math-v2-10k-1021
Among 500 students, 55% passed and 45% failed. What is the probability that at least 7 out of 10 randomly selected students will be failed?
\(\frac{\binom {225}{7} \times \binom{275}{3}}{\binom{500}{10}}+\frac{\binom {225}{8} \times \binom{275}{2}}{\binom{500}{10}}+\frac{\binom {225}{9} \times \binom{275}{1}}{\binom{500}{10}}+\frac{\binom {225}{10} }{\binom{500}{10}}\)
5
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/477867/what-is-the-probability
sue
https://math.stackexchange.com/users/91252/sue
nemotron-math-v2-10k-1030
Solve the PDE $\Delta u = 0$ with the following boundary conditions: - $u(0,y) = c1$ - $u(a,y) = c1$ - $u(x,0) = g(x)$ - $\frac{\delta u}{\delta v}(x,b) = c2$ where $c1$ and $c2$ are real numbers, and $x \in [0,a]$, $y \in [0,b]$.
\begin{aligned} u(x,y)&=c_{1} +\sum_{n=1}^{\infty} \sin\!\Bigl(\frac{n\pi x}{a}\Bigr) \Bigl[ A_{n}\cosh\!\Bigl(\frac{n\pi y}{a}\Bigr) +B_{n}\sinh\!\Bigl(\frac{n\pi y}{a}\Bigr)\Bigr], \end{aligned}
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/2961240/to-solve-a-pde-using-separation-of-variables
Shaun
https://math.stackexchange.com/users/597346/shaun
nemotron-math-v2-10k-1037
Let $S$ be a closed convex set and $x$ be an extreme point of $S$. Determine whether $S-\{x\}$ is convex, not convex, or may or may not be convex. Consider both finite and infinite dimensional spaces.
Convex
5
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/833533/let-s-be-a-closed-convex-set-x-be-an-extreme-pt-of-s-then-s-x-is
David
https://math.stackexchange.com/users/117084/david
nemotron-math-v2-10k-1045
Define an explicit isotopy \( g: D \times I \rightarrow \mathbb{R}^3 \) from the closed disk \( D = \{(x,y,0) : x^2 + y^2 \leq 1\} \) in the \( xy \)-plane to the lower hemisphere \( L = \{(x,y,z) : x^2 + y^2 + z^2 = 1, z \leq 0\} \).
\( g((x,y,0),t) = (x,y,-t\sqrt{1-x^2-y^2}) \)
8
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4308276/how-to-define-an-isotopy-from-the-lower-hemisphere-in-mathbb-r-3-to-a-closed-dis
pyridoxal_trigeminus
https://math.stackexchange.com/users/922398/pyridoxal-trigeminus
nemotron-math-v2-10k-1050
Given the vector space \( V = \mathbb{R}^2 \), bases \( B = \{(1,2), (3,4)\} \) and \( C = \{(7,3), (4,2)\} \), and vector \( v = (1,0) \), find the coordinates of the vectors \( (1,2) \) and \( (3,4) \) in the basis \( C \).
\([-3, 5.5]\) and \([-5, 9.5]\)
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3821080/change-of-basis-calculation-step-2
Katie Melosto
https://math.stackexchange.com/users/684991/katie-melosto
nemotron-math-v2-10k-1053
Evaluate the integral $\int \cos^5(x) \, dx$ using the reduction formula.
\(\frac{1}{5}\sin^5(x) - \frac{2}{3}\sin^3(x) + \sin(x)\)
4
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1736511/int-cos-5-x-dx-with-reduction-formula
mavavilj
https://math.stackexchange.com/users/248602/mavavilj
nemotron-math-v2-10k-1066
Let \( f(x) = \ln\left(1 + \frac{x}{2}\right) \) and let \( P_6(x) \) be the 6th order interpolating polynomial. Given points \( x_0, \ldots, x_6 \), estimate the error \( \left| \int_{x_1}^{x_6} f(x) \, \mathrm{d}x - \int_{x_1}^{x_6} P_6(x) \, \mathrm{d}x \right| \).
\displaystyle |\,\int_{x_{1}}^{x_{6}}f(x)\,dx-\int_{x_{1}}^{x_{6}}P_{6}(x)\,dx| \;\le\; \frac{2459}{168}\,\frac{h^{8}}{(x_{1}+2)^{7}} \;\;(\approx 6.7\times10^{-3}\text{ for }x_i=i)
0
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4300530/estimate-error-in-integration-with-interpolation-polynomial
narek
https://math.stackexchange.com/users/978664/narek
nemotron-math-v2-10k-1078
Given the function \( f \) defined by \( f(0) = 3 \), \( f(1) = -1 \), and \( f(n) = f(n-2) - f(n-1) \), find \( f(100) \) in terms of the Fibonacci numbers \( (F_0, F_1, F_2, \ldots, F_{101}) \).
\,f(100)=3F_{99}+F_{100}\,
3
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1104
Given a set \( A \subseteq \{1, 2, 3, \ldots\} \), under what necessary or sufficient conditions on \( A \) will every member of the multiset of products of members of \( A \) have a multiplicity of only 1?
\begin{aligned} &\text{All products of distinct elements of }A\text{ occur with multiplicity }1\\[2pt] &\iff\ \forall\;B\neq C\subseteq A,\;\prod_{b\in B}b\neq\prod_{c\in C}c\\[2pt] &\iff\ \text{the exponent vectors } \mathbf v(a) \text{ are zero‑sum‑free, i.e. }\\ &\qquad\bigl(\sum_{a\in A}\varepsilon_a\mathbf v(a)=\...
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4342014/multiplicities-of-members-of-the-multiset-of-products-of-members-of-a-set-a-subs
Michael Hardy
https://math.stackexchange.com/users/11667/michael-hardy
nemotron-math-v2-10k-1115
Construct a $99\%$ confidence interval for the standard deviation of the annual growth of cacti, given that a sample of $64$ cacti had a mean annual growth of $52.80$ mm and a standard deviation of $4.5$ mm.
\;3.65\ \text{mm}\;<\;\sigma\;<\;5.81\ \text{mm}\;
8
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/340286/estimation-of-standard-deviation
Roger
https://math.stackexchange.com/users/67182/roger
nemotron-math-v2-10k-1116
How many ways are there to sample $x$ beads from a set of 3 red, 3 green, and 3 blue beads (where beads of the same color are indistinguishable)?
N(x)=\binom{x+2}{2}-3\binom{x-2}{2}+3\binom{x-6}{2}
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4217482/number-of-ways-to-sample-x-beads-from-3-red-beads-3-green-beads-and-3-blue-beads
BeepBoop
https://math.stackexchange.com/users/615169/beepboop
nemotron-math-v2-10k-1125
Find a formula or counting approach to determine the number of permutations where x dice are greater than 5 when rolling m fair 6-sided dice, with a constant c distributed across the dice.
\displaystyle N(m,x)=\binom{m}{x}\,5^{\,m-x}
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/2506002/multiple-dice-permutations-for-different-number-ranges-each-die
MikeX
https://math.stackexchange.com/users/499353/mikex
nemotron-math-v2-10k-1127
Given a matrix \( X \) of size \( n \times n \), how can one find all matrices \( Y \) with real entries such that \( YY = X \)?
\;Y = P\,\operatorname{diag}(\varepsilon_{1}\sqrt{\lambda_{1}},\dots ,\varepsilon_{n}\sqrt{\lambda_{n}})\,P^{-1}, \qquad \varepsilon_{i}\in\{+1,-1\} \;
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/347952/how-to-find-a-matrix-square-root-with-all-real-entries-if-it-exists
Rein
https://math.stackexchange.com/users/21209/rein
nemotron-math-v2-10k-1134
Construct a circle that is tangential to a given line \( p \) and orthogonal to two given circles \( c \) and \( d \) that intersect at points \( A \) and \( B \). The line \( p \) passes through point \( A \) and intersects circles \( c \) and \( d \) at points \( P_1 \) and \( P_2 \).
\;\omega(O,R)\;
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4746689/construction-of-a-circle-orthogonal-to-two-given-circles-and-tangential-to-a-giv
Katarina
https://math.stackexchange.com/users/1062464/katarina
nemotron-math-v2-10k-1153
Given the number of sets and the number of elements in each set, what mathematical equation efficiently computes the number of unique combinations in the Cartesian product? For example, if there are 3 sets with 3, 1, and 2 elements respectively, the Cartesian product would have 6 unique combinations. Find the count wit...
\( \text{Product of the sizes of all sets} \) or for the specific example, \( 6 \). Since the problem asks for a general equation, the final answer is:
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/471179/cartesian-product
Jacob
https://math.stackexchange.com/users/825/jacob
nemotron-math-v2-10k-1175
Given three polynomials \( y_1 = a_0 + a_1 x_1 \), \( y_2 = b_0 + b_1 x_2 \), and \( y_3 = c_0 + c_1 x_3 \), and the weighted averages \( x_{\rm avg} \) and \( y_{\rm avg} \) defined as: \[ x_{\rm avg} = \frac{x_1(x_2 - x_3)^2 + x_2(x_1 - x_3)^2 + x_3(x_1 - x_2)^2}{(x_1 - x_2)^2 + (x_1 - x_3)^2 + (x_2 - x_3)^2}, \] \[ ...
d_{0}=a_{0}=b_{0}=c_{0},\qquad d_{1}=a_{1}=b_{1}=c_{1}
6
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3433308/weighted-averaging-of-polynomial-coefficients
Thomas Kirven
https://math.stackexchange.com/users/367112/thomas-kirven
nemotron-math-v2-10k-1177
Given matrices \( A \) and \( B \) and column vectors \( x \) and \( y \), under what conditions can we deduce \( A \cdot x = y \) from the equation \( B \cdot A \cdot x = B \cdot y \)?
\( B \) must be invertible (full rank).
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/484669/from-b-cdot-a-cdot-x-b-cdot-y-to-a-cdot-x-y
realmatrix
https://math.stackexchange.com/users/92746/realmatrix
nemotron-math-v2-10k-1186
Given Stirling's approximation for \(\ln N!\) as \(N \ln N - N + \frac{1}{2} \ln N\), calculate the fractional error that comes from neglecting the term \(\frac{1}{2} \ln N\) for \(N = 10\) and \(N = 100\). The fractional error is given by \(\frac{\frac{1}{2} \ln N}{N \ln N - N + \frac{1}{2} \ln N}\). Verify the result...
\varepsilon(100)\;\approx\;0.00635\;(0.635\%)
4
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4012428/stirling-s-approximation-fractional-error
Agnese
https://math.stackexchange.com/users/882793/agnese
nemotron-math-v2-10k-12
Determine whether the sum $\sum_{n=1}^{\infty} \dfrac{1}{\binom{2n}{n}} = \sum_{n=1}^{\infty} \dfrac{(n!)^2}{(2n)!}$ has a closed form.
\(\frac{1}{3} + \frac{2\pi\sqrt{3}}{27}\)
8
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-120
Given a matrix \( P \in \mathbb{R}^{m \times n} \) with \( n \) vectors in \( \mathbb{R}^m \) as its columns, how can we determine the set \( S^* \) of \( r \) columns such that the Frobenius norm \( \|P - P_S R\|_F^2 \) is minimized, without performing an exhaustive search over all combinations? Here, \( P_S \) is the...
\begin{aligned} \text{1. Compute a rank‑revealing QR factorisation with column pivoting: } & P\Pi = QR .\\[4pt] \text{2. Let } \Pi=[\pi_{1},\dots,\pi_{n}] \text{ be the induced permutation.}\\[2pt] \text{3. Choose } S^{*}= \{\pi_{1},\pi_{2},\dots,\pi_{r}\}.\\[4pt] \text{4. Set } R^{*}=P_{S^{*}}^{\dagger}P =(P_{S^{*...
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/242902/finding-the-best-subset-in-terms-of-frobenius-norm
Helen
https://math.stackexchange.com/users/49578/helen
nemotron-math-v2-10k-1204
Calculate the sum $\sum_{k=1}^{33}\binom{33}{k} k$.
\( 33 \times 2^{32} \)
4
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1645828/how-do-i-calculate-sum-k-1-33-binom-33-k-k
Ilan Aizelman WS
https://math.stackexchange.com/users/145510/ilan-aizelman-ws
nemotron-math-v2-10k-1206
Given $\overline{AB} = \overline{AC} = 20$, $\overline{AD} = \overline{AE} = 12$, and $[ADFE] = 24$, find the area of $\triangle BCF$.
\(\frac{40}{3}\)
6
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3911250/triangle-area-problem-with-intersecting-lines
Snacc
https://math.stackexchange.com/users/617984/snacc
nemotron-math-v2-10k-121
Four married couples are sitting on a circular table with eight indistinguishable chairs. A husband and his wife always sit either adjacent to each other or directly across from each other. In addition, a man and a woman may not sit on adjacent seats unless they are married. In how many distinguishable ways can this be...
36
6
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1231
Find all complex numbers $z$ that satisfy the equation $\sin z = i \alpha \cos z$, where $\alpha \in [-1, 1]$ is a real number.
\( z = k\pi + \tanh^{-1}(\alpha) \)
5
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/249620/complex-solutions-of-sin-z-i-alpha-cos-z
methuselah
https://math.stackexchange.com/users/18349/methuselah
nemotron-math-v2-10k-126
Factor the number $5^{1985} - 1$ into a product of three integers, each greater than $5^{100}$.
\;5^{1985}-1=(5^{397}-1)\,\;d\;\,\frac{5^{1588}+5^{1191}+5^{794}+5^{397}+1}{d}\;
2
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1262
Show how to place 9 queens on a \( 9 \times 9 \) chessboard such that no two queens can attack each other.
The configuration provided in Post 18 is a valid solution for placing 9 queens on a \(9 \times 9\) chessboard such that no two queens can attack each other.
1
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1279
Find the probability that when a person ties two bundles of 8 strings each (with no knowledge of how the strings are crossed), a single large ring can be formed.
\dfrac{1}{8}
2
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-13
Find a Dedekind domain \( D \) and a prime ideal \( \mathfrak{p} \) in \( D \) such that the quotient \( D/\mathfrak{p} \) is an infinite field.
\( D = \mathbb{C}[T] \) and \( \mathfrak{p} = (T-a) \) for any \( a \in \mathbb{C} \) (or other examples provided).
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1053986/counterexample-in-dedekind-domains
user126033
https://math.stackexchange.com/users/126033/user126033
nemotron-math-v2-10k-1316
Given a point \( T(1,2,3) \) in space, a plane passes through \( T \) and intersects all three coordinate axes. Determine the minimal possible area of the tetrahedron formed by the coordinate axes, the plane through \( T \), and the point \( T \).
27
8
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1325
Find the number of non-trivial solutions of the equation $${a \choose 3} - {b \choose 3} = c^3$$, where \(a, b, c \in \mathbb{Z}\) and \(a, b \ge 3\).
\text{Infinitely many}
8
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1328
Find an example of a non-quadratic analytic function \( f: \mathbb{C} \to \mathbb{C} \) (a power series with infinite radius of convergence) that satisfies the following properties: 1. \( f(0) = 0 \). 2. \( f'(\mathbb{R}) = \mathbb{R} \). 3. \( f''(x) \neq 0 \) for all \( x \in \mathbb{C} \).
\( f(x) = \sum_{n} \frac{x^{2n +2}}{(2n+2)(2n+1)n!} \)
4
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/647070/example-of-a-certain-locally-univalent-function
Rabee Tourky
https://math.stackexchange.com/users/39780/rabee-tourky
nemotron-math-v2-10k-1341
Given a polynomial $Q(x) = x^k + a_{k-1}x^{k-1} + \ldots + a_0$, under what conditions on $p$ and the coefficients of the polynomial does there exist a $p \times p$ matrix $A$ such that $A^k + a_{k-1}A^{k-1} + \ldots + a_0I = 0$?
\text{There exists a }p\times p\text{ matrix }A\text{ over }F\text{ with }Q(A)=0 \iff p\in M(Q) \iff\exists\;f_{1},\dots ,f_{t}\mid Q\text{ such that }\sum_{j=1}^{t}\deg f_{j}=p.
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4099769/existence-of-matrix-which-satisfies-the-equation
NeoFanatic
https://math.stackexchange.com/users/842645/neofanatic
nemotron-math-v2-10k-1349
Determine the intersection points of the functions $f(x) = \sqrt{5 - x}$ and $g(x) = 5 - x^2$ using the concept of inverse functions.
\displaystyle \left(\frac{1-\sqrt{17}}{2},\;\frac{1+\sqrt{17}}{2}\right), \quad \left(\frac{\sqrt{21}-1}{2},\;\frac{\sqrt{21}-1}{2}\right)
6
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3718590/how-to-make-inverse-to-work-here
Alma Do
https://math.stackexchange.com/users/360748/alma-do
nemotron-math-v2-10k-1356
For the ring extension $\mathbb{Z}\subset\mathbb{Z}[x]/(x^2-3)$, find a nonzero prime ideal $P$ in $\mathbb{Z}$ such that there are more than one prime ideal in $\mathbb{Z}[x]/(x^2-3)$ lying over $P$.
P=(11)
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1191961/ideals-lying-over-a-given-prime-ideal
user181158
null
nemotron-math-v2-10k-1358
Simplify the sum \( \sum_{i=0}^k \cos^2\left(\frac{\pi}{2}+\left\lfloor\frac{n}{b^i}\right\rfloor \frac{2 \pi}{b}\right) \) in terms of \( n \), \( k \), and \( b \).
\; \sum_{i=0}^{k}\cos ^2\!\Bigl(\frac{\pi}{2} +\Bigl\lfloor\frac{n}{b^{i}}\Bigr\rfloor\frac{2\pi}{b}\Bigr) = \sum_{i=0}^{\min\{k,\;\lfloor\log_{b}n\rfloor\}} \sin ^2\!\Bigl(\frac{2\pi}{b}\,d_{i}\Bigr), \quad d_{i}= \Bigl\lfloor\frac{n}{b^{i}}\Bigr\rfloor\bmod b \;
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4163472/is-there-a-way-to-simplify-sum-i-0-k-cos-2-left-frac-pi-2-left-lfloor-frac-n-b-i
Isaac Victor Silva Rodrigues
https://math.stackexchange.com/users/573123/isaac-victor-silva-rodrigues
nemotron-math-v2-10k-137
Find any 158 consecutive integers such that the sum of the digits of any of these numbers is not divisible by 17.
999999999999921 through 1000000000000078
4
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1391
Evaluate the integral $$\int\frac{\cos^2x\,\mathrm dx}{\sin^2x+4\sin x\cos x}.$$
\displaystyle \frac{17\ln|\sin x|-\ln|\sin x+4\cos x|-4x}{68}+C
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/2666075/what-is-the-best-way-to-solve-int-frac-cos-2x-mathrm-dx-sin-2x-4-sin-x-cos-x
M.Mass
https://math.stackexchange.com/users/418323/m-mass
nemotron-math-v2-10k-1406
Evaluate the series \(\sum_{n=1}^{\infty}\ln^2 \!\left(1+\frac1{2n}\right) \!\ln^2\!\left(1+\frac1{2n+1}\right)\).
% \displaystyle S= \frac{-353\,\pi^{4} +4410\,\pi^{2}\ln^{2}2 +4275\,\ln^{4}2 +7470\,\zeta(3)\ln2 +9630} {2160\,\pi^{4}}
0
stackexchange
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https://math.stackexchange.com/questions/1394843/calculating-sum-n-1-infty-ln-2-left-1-frac1-2n-right-ln-2-left-1-frac1-2n-1-righ
user 1591719
https://math.stackexchange.com/users/32016/user-1591719
nemotron-math-v2-10k-1443
Given the supply function \( X_S(p) = \ln\left(\frac{e^2}{4}p\right) \) and the demand function \( X_D(p) = \sqrt{20 - 4p} \), with equilibrium price \( EP = 4 \) and equilibrium quantity \( EQ = 2 \), calculate the consumer surplus and producer surplus.
\displaystyle CS=\frac{4}{3}\;\approx\;1.33,\qquad PS=4+\frac{4}{e^{2}}\;\approx\;4.54
8
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3285130/calculate-consumer-and-producer-surplus-area-between-curves
QWERTYZ
https://math.stackexchange.com/users/686965/qwertyz
nemotron-math-v2-10k-1469
Given $N$ sets, each containing any number of elements from $1$ to $100$, count the number of arrangements where one element is selected from each set such that no two selected elements are the same. Two arrangements are considered different if at least one element differs. Example: For sets $\{1, 2, 3\}$, $\{4\}$, an...
\displaystyle \sum_{\substack{J\subseteq\{1,\dots ,100\}\\ |J|=N}} \operatorname{perm}\!\bigl(A_{[N],\,J}\bigr)
2
stackexchange
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https://math.stackexchange.com/questions/885412/select-elements-from-n-sets
user3878046
https://math.stackexchange.com/users/167397/user3878046
nemotron-math-v2-10k-147
Calculate the limit $\lim_{x \to 0} \frac{\cos(x)}{\sin(x)}$.
The limit does not exist.
8
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4481652/calculate-lim-x-to-0-frac-cos-x-sin-x
Hector Auvinen
https://math.stackexchange.com/users/982896/hector-auvinen
nemotron-math-v2-10k-1470
Determine the area of the region given in polar coordinates: \( D=\{(r, \theta):\, 1+\cos \theta \leq r \leq 3\cos \theta \} \).
\(\pi\)
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3283453/area-of-a-region-given-in-polar-coordinates
AaronTBM
https://math.stackexchange.com/users/657519/aarontbm
nemotron-math-v2-10k-1471
Given a multivariable polynomial \( p(x_1, x_2, \ldots, x_n) \), determine if there exists a real number \( C \) such that \( p(x_1, x_2, \ldots, x_n) \ge C \). In other words, determine if the polynomial is bounded from below.
\; p(x)\text{ is bounded from below } \Longleftrightarrow \forall\,v\neq0,\; \text{let }k(v)=\max\{j\mid p_{j}(v)\neq0\}; \text{ then }p_{k(v)}(v)\ge0. \;
3
stackexchange
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https://math.stackexchange.com/questions/2102408/how-to-determine-if-a-polynomial-is-bounded-from-below
Max Radin
https://math.stackexchange.com/users/68324/max-radin
nemotron-math-v2-10k-1475
Let \( A \) be an \( n \times m \) random matrix with entries from the finite field \( \mathbb{F}_q \). What is the probability that the rank of \( A \) is full?
\displaystyle \Pr\big(\operatorname{rank}(A)=r\big)= \prod_{i=0}^{\,r-1}\frac{q^{N}-q^{i}}{q^{N}} =\frac{\displaystyle\prod_{i=0}^{\,r-1}\bigl(q^{N}-q^{i}\bigr)} {q^{nm}}
5
stackexchange
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https://math.stackexchange.com/questions/4040934/let-a-be-a-n-times-m-random-matrix-with-entries-from-finite-field-mathbb-f-q-wha
Shweta Aggrawal
https://math.stackexchange.com/users/581242/shweta-aggrawal
nemotron-math-v2-10k-1477
Determine a general method to find the number of solutions to the equation $z = x^2 + y^2$ where $x$ and $y$ are integers, or to bound the number of solutions without explicitly computing them.
\displaystyle r_{2}(z)=\begin{cases} 4\!\displaystyle\prod_{p\equiv1(4)}(a_p+1), & \text{if every }p\equiv3(4)\text{ has even exponent},\\[6pt] 0, & \text{otherwise}. \end{cases}
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/2536097/representing-a-given-number-as-the-sum-of-two-squares
Faust
https://math.stackexchange.com/users/60353/faust
nemotron-math-v2-10k-1478
For $n=4$, determine the number of non-equivalent (distinct) cycles of length 4.
6
5
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4603470/number-of-distinct-n-cycles-of-n-objects
jiten
https://math.stackexchange.com/users/424260/jiten
nemotron-math-v2-10k-1481
How can the product of the first \( n \) integers starting from 140 be expressed in a way that also holds for negative numbers? Specifically, find a general expression for \( 140 \times 139 \times \cdots \times (140 - n + 1) \) that works even when \( n \) is such that the sequence includes negative numbers.
\( 140^{\underline{n}} \)
5
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/326540/method-of-expressing-the-product-of-first-n-integers
fosho
https://math.stackexchange.com/users/33691/fosho
nemotron-math-v2-10k-1486
Define a set $S$ of nonnegative integers such that: 1) $0 \in S$ 2) For all $n > 0$, $n \in S$ if and only if there exist $a, b \in S$ (not necessarily distinct) such that $n = (2^a)(3^b)$. Determine whether a given integer $n$ is in the set $S$.
\;n\in S\iff F(n)\;
3
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1496
Find two disjoint sets \(\{a, b, c\}\) and \(\{m, n, p\}\) such that \(a, b, c, m, n, p \in \mathbb{N}\), \(a, b, c, m, n, p \geq 2002\), and \(a + b + c = m + n + p\) and \(a^2 + b^2 + c^2 = m^2 + n^2 + p^2\).
\{a,b,c\}=\{2002,\;2006,\;2007\},\qquad \{m,n,p\}=\{2003,\;2004,\;2008\}
5
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1500
Transform the sum of min functions into one min function: $$\sum_{i=1}^{s} \pi_i \text{min}(x, \omega_i)$$ into the form: $$\text{min}_{k=1,\dots, K} \{ a_k \ x \}.$$
\displaystyle \sum_{i=1}^{s}\pi_i\min (x,\omega_i)= \min_{k=0,\dots ,s}\; \tilde a_k^{\!\top}\tilde x
1
stackexchange
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https://math.stackexchange.com/questions/1677918/transforming-the-sum-of-min-functions-into-one-min-function-with-the-aim-of-tran
Milan
https://math.stackexchange.com/users/78672/milan
nemotron-math-v2-10k-1506
How many ways can $n$ people (where $n$ is even) sitting around a circular table be assigned one of $d$ colors such that no two adjacent people have the same color?
\((d-1)[(d-1)^{n-1}+1]\)
5
stackexchange
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https://math.stackexchange.com/questions/4465680/an-even-number-n-of-people-sit-around-a-table-with-d-choices-for-colors
Cantor
https://math.stackexchange.com/users/564470/cantor
nemotron-math-v2-10k-1514
For which values of \( n \) is \( a_n = \lfloor \pi \times 10^{n-1} \rfloor \) a prime number?
\,n=1,\;2,\;6,\;38\,
0
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3362902/finding-primes-in-pi
sam wolfe
https://math.stackexchange.com/users/487230/sam-wolfe
nemotron-math-v2-10k-1531
Solve the differential equation \(\frac{\mathrm{d^2} y}{\mathrm{d} x^2} = \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^2\).
\( y = -\ln(x + C) + K \)
3
stackexchange
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https://math.stackexchange.com/questions/2754153/problem-in-the-differential-equation
userNoOne
https://math.stackexchange.com/users/518028/usernoone
nemotron-math-v2-10k-1539
Given the equation \( y = \frac{\ln(p + qe^x)}{x} \), where \( p \) and \( q \) are constants, express \( x \) in terms of \( y \).
\;x=\ln u\;
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1009880/y-ln-p-qe-x-x-solve-for-x
Najmie
https://math.stackexchange.com/users/190531/najmie
nemotron-math-v2-10k-1542
Find the $n$th term of the sequence $2, 3, 6, 7, 14, 15, 30, \ldots$, where the pattern is to add one to the first term to get the second term, then double the second term to get the third term, and repeat.
\,a_n= \begin{cases} 2^{\frac{n+3}{2}}-2, & n\ \text{odd},\\[6pt] 2^{\frac{n+2}{2}}-1, & n\ \text{even}. \end{cases}
7
stackexchange
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https://math.stackexchange.com/questions/872485/find-the-n-th-term-of-1-2-5-10-13-26-29
Guest
https://math.stackexchange.com/users/165334/guest
nemotron-math-v2-10k-1544
Given a continuous function \( g: B \rightarrow \mathbb{R} \) where \( B \) is the closed unit disk, how can this function be extended to the entire plane \( \mathbb{R}^2 \)?
\widetilde g(x)= \begin{cases} g(x), & \|x\|\le 1,\\[6pt] (2-\|x\|)\;g\!\Big(\dfrac{x}{\|x\|}\Big), & 1<\|x\|\le 2,\\[10pt] 0, & \|x\|> 2 . \end{cases}
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/271435/continuous-extension-from-closed-ball-to-whole-plane
Myshkin
https://math.stackexchange.com/users/24690/myshkin
nemotron-math-v2-10k-1548
Determine if there exists any prime number generated by the polynomial \( x^{n-1} + 2x^{n-2} + 3x^{n-3} + \ldots + (n-1)x + n \) for \( x = 3k + 1 \), where \( k \in \mathbb{Z}^+ \).
P_{13}(4)=29826157\ \text{(prime)}
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4251840/smarandache-number-and-some-generator-polynomial
Juan Moreno
https://math.stackexchange.com/users/567304/juan-moreno
nemotron-math-v2-10k-1552
In general, how many maps \( f: [n] \to [6] \) are there such that at least \( a_i \) distinct elements must map to \( i \) for \( i = 1, \ldots, 6 \)?
N(n;a_1,\ldots ,a_6)= \sum_{\substack{x_1,\ldots ,x_6\ge a_1,\ldots ,a_6\\ x_1+\cdots +x_6=n}} \frac{n!}{x_1!\,x_2!\cdots x_6!}
4
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4370662/number-of-maps-from-n-to-6-if-there-are-conditions-on-at-least-how-many-elements
Gina
https://math.stackexchange.com/users/963265/gina
nemotron-math-v2-10k-1553
Find the minimal value of $n$ such that there exist 4 distinct positive integers $a, b, c, d$ among $n$ distinct positive integers for which $2003 \mid (a+b-c-d)$.
48
4
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1587
Construct a scalene triangle such that \[a(\tan B - \tan C) = b(\tan A - \tan C)\]
a(\tan B-\tan C)=b(\tan A-\tan C)
0
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1590
Find the smallest positive integer \( n \) such that for every integer \( m \) with \( 0 < m < 1993 \), there exists an integer \( k \) for which \[ \frac{m}{1993} < \frac{k}{n} < \frac{m + 1}{1994}. \]
3987
8
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1602
Solve the equation \[\dfrac{1}{\tan^{2}x+(1+\tan x)^{2}}+\dfrac{1}{\cot^{2}x+(1+\cot x)^{2}}+\dfrac{(\sin x+\cos x)^{4}}{4}=\dfrac{5+\sin^{3}2x}{4}\] for \(x \in [0, \frac{\pi}{2}]\).
\,x=0\quad\text{or}\quad x=\dfrac{\pi}{2}\,
7
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1607
A cardboard windmill with three blades is made from four equilateral triangles of side length 6 cm. Two triangles sharing a common vertex have the corresponding sides lying on the same straight line. The area of the circle swept out by the blades of the windmill is \( x \) cm\(^2\). What is the greatest integer less th...
263
7
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1625
Given an undirected graph \( G \) with \( n \) vertices and \( m \) edges, how many distinct trees \( G' \) can be formed by removing some edges from \( G \) such that \( G' \) contains all \( n \) vertices?
\;\tau(G)=\frac{1}{n}\,\prod_{i=2}^{n}\lambda_{i}\;
1
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1636
Evaluate the expression \( \left(\frac{n-2k-1}{k+1}\right) C^n_k \) and determine if it is an integer for all \( k \) and \( n \), for all even values of \( k \) and \( n \), for all odd values of \( k \) and \( n \), if \( k = 1 \) or \( n-1 \), or if \( n \) is divisible by \( k \).
\displaystyle\Bigl(\frac{n-2k-1}{k+1}\Bigr)\binom{n}{k} =\binom{n}{k+1}-\binom{n}{k}
6
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1653
Let \( S = \{1, 2, \ldots, n\} \). Let \( A_i \subset S \) for \( i \in \{1, 2, \ldots, m\} \). Impose the following conditions: - \( |A_i| = r \) with \( r < n \) for all \( i \). - \( |A_i \cap A_j| = t \) for all \( i \neq j \), with \( t < r \). Given \( n, r, t \), what is the maximum number \( m \) of subsets \(...
m_{\max}=n
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3062304/subsets-of-a-set-with-common-elements-between-themselves
Vladimir Vargas
https://math.stackexchange.com/users/187578/vladimir-vargas
nemotron-math-v2-10k-1667
Solve the second-order differential equation for \( x(t) \): \[ 2x''(t) x(t) - 3 (x'(t))^2 + 4x^4(t) = 0 \]
\,x(t)\equiv0\,
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3853407/solving-two-coupled-odes-without-use-of-complex-numbers
Daniels Krimans
https://math.stackexchange.com/users/417573/daniels-krimans
nemotron-math-v2-10k-168
Express the variable \( k \) in a closed form given the equation \( k^4 = 2^k \).
\( k = -\frac{4}{\ln(2)}W_n\left(-\frac{1}{4}\ln(2)\right),\ -\frac{4}{\ln(2)}W_n\left(\frac{1}{4}\ln(2)i\right),\ -\frac{4}{\ln(2)}W_n\left(\frac{1}{4}\ln(2)\right),\ -\frac{4}{\ln(2)}W_n\left(-\frac{1}{4}\ln(2)i\right) \) for \( n \in \mathbb{Z} \).
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4549744/how-to-get-a-variable-in-a-closed-form-given-an-equation
user996159
https://math.stackexchange.com/users/996159/user996159
nemotron-math-v2-10k-1692
Find the $\sigma$-algebra $\mathbf{X} = \sigma(S)$ generated by the collection of subsets $S$ where $X = \mathbb{R}$ and $S = \{(-\infty, n): n \in \mathbb{Z}\}$ (i.e., all intervals in the form $(-\infty, n)$).
\(\sigma(S)= \left \{\bigcup_{i\in \mathbb{N}} A_i: \textrm{for all } i \in \mathbb{N}, A_i \in S_f \right \}\) where \(S_f = \{[m, n): m,n \in \mathbb{Z} \textrm{ and } m\leqslant n \}\)
5
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3907161/finding-the-sigma-algebra-mathbf-x-sigma-s-generated-by-the-collection-of-subset
Shmulley Boteach
https://math.stackexchange.com/users/626529/shmulley-boteach
nemotron-math-v2-10k-1706
Let \( X \) be a finite set. Define a bijection \( T: X^2 \rightarrow X^2 \) such that the binary operations \( \ast_{l,+}, \ast_{l,-}, \ast_{r,+}, \ast_{r,-} \) derived from \( T \) and \( T^{-1} \) are quasigroup operations. Determine the number of such well-mixing algebras \( (X, T) \) up to isotopy for small values...
t_{1}=1,\;t_{2}=0,\;t_{3}=1,\;t_{4}=2,\;t_{5}=2,\;t_{6}=0,\; t_{7}=3,\;t_{8}=4,\;t_{9}=5,\;t_{10}=7,\;t_{11}=5,\;t_{12}=10,\dots
1
stackexchange
CC-BY-SA-4.0
https://mathoverflow.net/questions/454973/counting-permutations-of-x-2-that-induce-4-quasigroup-operations-up-to-isotopy
Joseph Van Name
https://mathoverflow.net/users/22277/joseph-van-name
nemotron-math-v2-10k-1716
Given two matrices \( Q_1 \) and \( Q_2 \): \[ Q_1 = \begin{bmatrix}1 & 0 & 0\\0 & -1 & 0\\0 & 0 & -2\end{bmatrix} \] \[ Q_2 = \begin{bmatrix}1/2 & 0 & 0\\0 & -1 & 0\\0 & 0 & -1\end{bmatrix} \] set \( \det(\lambda Q_1 + \mu Q_2) = 0 \) and solve for \(\lambda\) and \(\mu\) without using symbolic manipulation libraries.
\;\lambda=-\mu\;
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/320483/solving-symbolic-equation-from-determinant-numerically
David Doria
https://math.stackexchange.com/users/25749/david-doria
nemotron-math-v2-10k-1748
Solve the following linear programming problem graphically: Maximize \( Z = 10x + 15y \) Subject to: \[ x \ge 400 \] \[ x + y \le 1000 \] \[ 0.7x - 0.3y \ge 0 \] \[ x, y \ge 0 \] How do you graph the inequality \( 0.7x - 0.3y \ge 0 \)?
\,x^{*}=400,\;y^{*}=600,\;Z_{\max}=13000\,
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1049399/how-to-solve-this
fidrus
https://math.stackexchange.com/users/197227/fidrus
nemotron-math-v2-10k-1764
Given two water jugs with capacities \(a, b \in \mathbb{N}\), the goal is to measure exactly \(c\) units of water using the following operations: - Fill one of the jugs to its capacity limit, - Empty one of the jugs, - Fill water from one jug to the other until the first jug is empty or the second jug is full. Find an ...
6
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/2399425/jugs-of-water-puzzle-minimum-number-of-operations
neutron-byte
https://math.stackexchange.com/users/431114/neutron-byte
nemotron-math-v2-10k-177
What is the probability that a randomly selected graph from the family $G(n,m)$ (graphs with $n$ vertices and $m$ edges) has exactly $c$ connected components?
(9)
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/2206987/number-of-connected-components
maverick
https://math.stackexchange.com/users/313200/maverick
nemotron-math-v2-10k-1779
Solve the Laplace initial value problem given by $$ y'' + 36y = f(t), $$ where $$ f(t) = \begin{cases} 1, & \text{0 ≀ t < 8} \\ 0, & \text{8 ≀ t < ∞} \end{cases} $$ with initial conditions $$ y(0) = 0 \quad \text{and} \quad y'(0) = 1. $$
\( y(t) = \frac{1}{36}(-\cos(6t) + H(t-8)(\cos(6(8-t)) - 1) + 6\sin(6t) + 1) \)
5
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/983318/how-to-solve-laplace-initial-value-problem
Ally
https://math.stackexchange.com/users/180277/ally
nemotron-math-v2-10k-1784
Let \( g(z,t) \) be a continuous function in both variables \( z \) and \( t \) on \( \mathbb{C} \times [0,\pi] \), and analytic in \( z \). Consider the function \( f(z) = \int_{0}^{\pi} g(z,t) \, dt \). What can we say about the continuity and analyticity of the function \( f(z) \)?
\; f(z)=\displaystyle\int_{0}^{\pi}g(z,t)\,dt\; \text{ is continuous on the domain of }g\text{ and is holomorphic there (indeed entire if }g(z,t)\text{ is entire in }z).\;
6
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4839003/a-doubt-on-continuity-and-anlyticity-of-a-function-defined-via-a-integral
MathRookie2204
https://math.stackexchange.com/users/1191954/mathrookie2204
nemotron-math-v2-10k-1801
Evaluate the volume \( V \) given by the integral: \[ V = \pi \int_{-(\sqrt{5}+3)/2}^{(\sqrt{5}-3)/2} \left(1 - \left(\frac{1}{x^2 + 3x + 2}\right)^2\right) \, dx \]
\text{The integral diverges; the volume is not finite.}
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/2964926/find-the-volume
S.H.W
https://math.stackexchange.com/users/325808/s-h-w
nemotron-math-v2-10k-181
Transpose the formula for the Present Value of an Ordinary Annuity to solve for the interest rate \( i \). The formula is given by: \[ P = PMT \times \frac{1 - (1 + i)^{-n}}{i} \] Where: - \( P \) is the Present Value of an Ordinary Annuity, - \( PMT \) is the Payment, - \( i \) is the Interest Rate, - \( n \) is the...
\,i = \Bigl[1\!-\!\frac{P}{PMT}\,i\Bigr]^{-1/n} - 1\,
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1794181/transpose-present-value-of-an-ordinary-annuity-formula-for-interest-rate
BlueRhapsody
https://math.stackexchange.com/users/310898/bluerhapsody
nemotron-math-v2-10k-1821
Define a sequence that takes every value of the set of integers \( \mathbb{Z} \) exactly once and no other values.
\( a_n = \begin{cases} \frac{n}{2} & \text{if } n \text{ is even} \\ -\frac{n+1}{2} & \text{if } n \text{ is odd} \end{cases} \)
6
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/2007316/how-to-define-a-sequence-that-takes-every-value-of-z-once-and-no-other
Hitchens
https://math.stackexchange.com/users/366974/hitchens
nemotron-math-v2-10k-1857
Find the least-squares solution \( x^* \) of the system \( Ax = b \), where \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} \] and \[ b = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix}. \] Use the formula \( x^* = (A^TA)^{-1} A^Tb \).
\,x^{*}= \begin{pmatrix} -\dfrac{23}{36}\\[4pt] -\dfrac{1}{18}\\[4pt] \dfrac{19}{36} \end{pmatrix}\,
6
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1206941/what-am-i-doing-wrong-in-attempting-to-find-the-least-squares-solution-of-the-sy
120MinuteMan
https://math.stackexchange.com/users/226508/120minuteman
nemotron-math-v2-10k-1863
Consider a vibrating string occupying the interval $0 \leq x \leq l$. Suppose the string is struck in the middle so that its initial displacement is zero but its initial velocity $u_t(x,0)$ is $1$ for $|x - \frac{1}{2}l| < \delta$ and $0$ elsewhere. Find $u(x,t)$ for $t > 0$.
u(x,t)=\frac{4l}{c\pi^{2}} \sum_{n=1}^{\infty} \frac{\sin\!\big(\frac{n\pi}{2}\big)\, \sin\!\big(\frac{n\pi\delta}{l}\big)} {n^{2}}\, \sin\!\Big(\frac{n\pi c t}{l}\Big)\, \sin\!\Big(\frac{n\pi x}{l}\Big)
4
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3656260/using-fourier-series-to-model-a-vibrating-string
ferris
https://math.stackexchange.com/users/717480/ferris
nemotron-math-v2-10k-188
Consider $n$ unit vectors $v_i \in \mathbb{R}^d$ in $d$-dimensional Euclidean space. Assume that the angle between any pair of two vectors $v_i$ and $v_j$ is the same. What is the maximum possible value for this angle?
\displaystyle \theta_{\max}= \arccos\!\Bigl(-\frac{1}{\,n-1\,}\Bigr)
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4274173/maximum-equal-angle-separation-between-n-unit-vectors-in-d-dimensional-euclidean
le4m
https://math.stackexchange.com/users/41872/le4m
nemotron-math-v2-10k-1890
Determine if there is a closed-form solution for the probability $P(\theta_1 \le \theta_2 \le \theta_3)$ given by the integral \[ P(\theta_1 \le \theta_2 \le \theta_3) = \int_{{\theta_3}=0}^1 \int_{{\theta_2}=0}^{\theta_3}\int_{{\theta_1}=0}^{\theta_2} f(\theta_1, \theta_2, \theta_3)\ d{\theta_1}d{\theta_2}d{\theta_3}....
\frac{1}{6}
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4163971/integral-of-joint-beta-distribution
Dominic Joseph
https://math.stackexchange.com/users/650540/dominic-joseph
nemotron-math-v2-10k-1893
Solve the 3D-Laplace equation on the domain \(\Omega = [0, x_{1}]\times[0, x_{2}]\times[0, x_{3}]\) with the following boundary conditions: \[ \begin{align} \Delta U(X_{1},X_{2},X_{3}) &= 0, \quad \text{on} \ \Omega, \\ U(X_{1},X_{2},X_{3}) \big|_{(X_{1},X_{2},X_{3}) = (x_{1},x_{2},x_{3})} &= 0, \\ (\nabla U \cdot \til...
\,U(X_1,X_2,X_3)= \tilde n_{1}\,(X_{1}-x_{1}) +\tilde n_{2}\,(X_{2}-x_{2}) +\tilde n_{3}\,(X_{3}-x_{3})\,
4
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4493277/solving-3d-laplace-equation-with-boundaries-at-one-single-point
Abdulhameed Qahtan Abbood Alta
https://math.stackexchange.com/users/1067032/abdulhameed-qahtan-abbood-alta
nemotron-math-v2-10k-1940
Calculate the integral $\int_{-\infty}^{+\infty} H_n(x)H_m(x)e^{-\alpha x^2} d x$ with $\alpha > 0$, where $(H_n,H_m)_{w = e^{-x^2}} = 2^n n! \sqrt{\pi}\delta_{nm}$. (Avoid using Gauss-Hermite quadrature.)
\; \int_{-\infty }^{\infty }H_{n}(x)H_{m}(x)e^{-\alpha x^{2}}dx = \sqrt{\frac{\pi}{\alpha}}\;n!\,m!\! \sum_{\substack{r=0\\ n-r\equiv m-r\equiv0\;(\mathrm{mod}\,2)}}^{\min (n,m)} \frac{\displaystyle \Bigl(\frac{2}{\alpha}\Bigr)^{r}\, \lambda^{\frac{n+m}{2}-r}} {r!\,\bigl(\frac{n-r}{2}\bigr)!\,\bigl(\frac{m-r}{2}\bigr)...
2
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3641312/integral-involving-hermite-polynomial
Nikita Gayanov
https://math.stackexchange.com/users/778260/nikita-gayanov
nemotron-math-v2-10k-1962
Find a linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) such that the image of \( T \) is spanned by the vectors \( (1,2,3) \) and \( (4,5,6) \).
\(\begin{pmatrix} 1 & 4 & 1\\ 2 & 5 & 2 \\ 3 & 6 & 3 \end{pmatrix}\)
1
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/1946605/find-a-linear-transformation
Anonymous
https://math.stackexchange.com/users/371304/anonymous
nemotron-math-v2-10k-1963
Given that \(\sum_{k=1}^n k^2 |a_k| = 1\), find the maximum value of \(\prod_{1 \leq i < j \leq n} |a_j - a_i|\).
\; \max_{\,\sum k^{2}|a_{k}|=1} \prod_{1\le i<j\le n}|a_{j}-a_{i}| =\frac{\displaystyle \prod_{1\le i<j\le n-1} \bigl(i^{2}u_{j}-j^{2}u_{i}\bigr)} {(n!)^{\,n-1}}, \;
0
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-1967
What are the smallest sets of $n$-permutations of $\{1, 2, \ldots, n\}$ where all combinations of $\{1, 2, \ldots, n\}$ are "prefix-permutation" of at least one element of the set? Definition: A permutation $a=(a_1,\ldots,a_n)$ is a "prefix-permutation" of another permutation $b=(b_1,\ldots,b_m)$ if there is a permuta...
\( \binom{n}{\lfloor \frac{n}{2} \rfloor} \)
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3146568/what-are-the-smallest-sets-of-n-permutations-of-1-2-ldots-n-where-all-n-combinat
amirouche
https://math.stackexchange.com/users/23663/amirouche
nemotron-math-v2-10k-1992
Estimate the value of the integral $\int_{-1}^1(e^{x^3}+e^{-x^3})\,dx$ and determine which of the following intervals it falls into: $(4, 2(e+\frac{1}{e}))$, $(\frac{1}{e}+2, 2e)$.
(4, 2(e + \frac{1}{e}))
6
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4515420/estimation-of-int-1-1-e-x-3-e-x-3-dx
Aspirant
https://math.stackexchange.com/users/910694/aspirant
nemotron-math-v2-10k-20
Point $B$ is in the exterior of the regular $n$ -sided polygon $A_1A_2\cdots A_n$, and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_1$, $A_n$, and $B$ are consecutive vertices of a regular polygon?
042
6
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-2007
Calculate the square roots of $-1-i$ in trigonometric form.
\(\sqrt{2} e^{-i\frac{3\pi}{8}}\)
3
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4362104/sqrt-1-i-in-exponential-form
Aureole
https://math.stackexchange.com/users/1017026/aureole
nemotron-math-v2-10k-2012
Solve the following system of coupled ordinary differential equations in MATLAB with the given initial conditions: \[ \ddot{\phi} + \dot{\phi} + g\phi = \ddot{x} + x + M \] \[ \ddot{x} + \dot{x}^2 + x = -\ddot{\phi} + \phi \] where \( M = 1 \) and the initial conditions are: \[ \dot{\phi}(0) = 0, \quad \phi(0) = 0, \qu...
\begin{aligned} \phi(t) &= t-2+2\,e^{-t/2},\\[4pt] x(t) &= t-2+\frac{4}{5}\cos t-\frac{2}{5}\sin t+\frac{6}{5}\,e^{-t/2}. \end{aligned}
0
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4197673/solving-the-coupled-system-of-ordinary-differential-equations-in-matlab
shyam prithvi
https://math.stackexchange.com/users/949811/shyam-prithvi
nemotron-math-v2-10k-202
Find the radius of convergence and the value of the power series $\sum_{n=0}^{\infty} a_nz^n$, given that $a_0=1$, $a_1=-1$, and $3a_n + 4a_{n-1} - a_{n-2} = 0$ for $n > 1$.
\displaystyle \sum_{n=0}^{\infty}a_nz^{\,n} =\frac{3+z}{3+4z-z^{2}}\qquad (|z|<\sqrt7-2)
7
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/284307/power-series-and-recurrence
Ran Kashtan
https://math.stackexchange.com/users/48874/ran-kashtan
nemotron-math-v2-10k-2022
Let \( X \) be a set, \( \mathcal{D} \) a distribution over \( X \), and \( F \) a set of functions from \( X \) to \( \{0,1\} \). Given \( \varepsilon, \delta \in (0,1) \) and i.i.d. samples \( x_1, x_2, \dots, x_m \) from \( \mathcal{D} \), find the minimal \( m \) such that with probability at least \( 1-\delta \) o...
\displaystyle m = \Theta\!\Bigg(\frac{\VC(F)\,\log\!\bigl(1/\varepsilon\bigr) \;+\;\log\!\bigl(1/\delta\bigr)}{\varepsilon^{2}}\Bigg)
6
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/3743043/what-is-known-about-convergence-of-empirical-extrema
Ο€314
https://math.stackexchange.com/users/128581/314
nemotron-math-v2-10k-2028
Find all pairs of integers $(a, b)$ such that the polynomial $ax^{17} + b^{16} + 1$ is divisible by $x^2 - x - 1$.
(a,b)=(987,\,-1597)
2
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-2062
Let $P_k(n)$ be the polynomial representing the sum of the first $n$ $k$-th powers. Evaluate $P_7(-3) + P_6(-4).$
-665
8
aops
CC-BY-4.0
null
null
null
nemotron-math-v2-10k-2077
Find an upper bound on $f(n)$, considering the pigeonhole principle applied to the exponents modulo 2 for any set of $2^m + 1$ positive integers all of whose prime factors belong to the set $\{p_1, ..., p_m\}$.
\,f(n)\le 2^{\,n}+1\,
5
stackexchange
CC-BY-SA-4.0
https://math.stackexchange.com/questions/4779266/largest-subset-of-1-2-n-containing-no-two-elements-whose-product-is-a-square
Prism
https://math.stackexchange.com/users/49227/prism
End of preview. Expand in Data Studio

🧠 MBT Traces: teacher reasoning with inserted metacognitive behaviors

This dataset accompanies the paper "Distilling Metacognitive Behaviors through Agentic Editing" (anonymous submission). It contains 1,000 competition-style math problems from Nemotron-Math-v2, the original reasoning traces of the teacher model DeepSeek-V4.1-Flash on those problems, and the same traces after three editing strategies inserted seven metacognitive behaviors into them:

Strategy What the editor does
Unedited Nothing. The teacher's trace as generated.
MBT-R (rewrite) Rewrites the whole trace in one call, returning the full edited reasoning and response.
MBT-I (insert) Sees the reasoning with a numbered marker after each paragraph and returns, in one call, a list of passages to insert after chosen markers.
MBT-A (agentic) A coding agent edits the trace as a file in a git repository, inserting one passage per commit.

The seven target behaviors are subgoal setting, logical check, substantive verification, targeted backtracking, resuming abandoned paths, self-correction and sufficiency judgment. The companion GitHub repository contains the generation and editing code, the exact prompts, and both annotation schemes.

Models fine-tuned from Qwen3.5-4B-Base on each condition: Qwen3.5-4B-Unedited-SFT, Qwen3.5-4B-MBT-R, Qwen3.5-4B-MBT-I and Qwen3.5-4B-MBT-A.

πŸ“¦ Contents

Config Rows What a row is
problems 1,000 A problem, its reference answer, its difficulty bin and its source attribution
traces_unedited 1,000 The teacher's reasoning and response for one problem
traces_mbt_r 1,000 The MBT-R edit of that trace (776 edited, 224 unsuccessful)
traces_mbt_i 1,000 The MBT-I edit (1,000 edited, 6,472 insertions)
traces_mbt_a 1,000 The MBT-A edit (1,000 edited, 10,759 insertions)
sft_unedited 995 Chat-format training examples for Unedited SFT
sft_mbt_r 773 Chat-format training examples for MBT-R
sft_mbt_i 995 Chat-format training examples for MBT-I
sft_mbt_a 995 Chat-format training examples for MBT-A

sft_* configs

Each condition is trained separately on its own config. Each keeps the traces that were edited successfully and fit within the student's maximum length of 131,072 tokens. Token counts are under the Qwen3.5-4B chat template (paper Table D1).

Config Examples Tokens (Table D1)
sft_unedited 995 22,821,876
sft_mbt_r 773 7,557,835
sft_mbt_i 995 23,483,410
sft_mbt_a 995 24,432,120

πŸ—‚οΈ Fields

problems

Field Type Description
id string Problem id, shared by every file
problem string The problem statement as given to the teacher
reference_answer string The reference answer from Nemotron-Math-v2 (never shown to any editor)
difficulty_bin int 0 to 8: how many of eight gpt-oss-120b attempts were correct
source string stackexchange or aops
license string CC-BY-SA-4.0 (StackExchange) or CC-BY-4.0 (AoPS)
source_url, author_name, author_url string or null Attribution as given by Nemotron-Math-v2 (StackExchange rows)

traces_unedited: id, problem, reasoning (the text inside the thinking block), response (the text after it), finish_reason.

traces_mbt_r, traces_mbt_i, traces_mbt_a: id, method, edited (false when the strategy produced no acceptable edit), reasoning, response (null when not edited), response_changed (the editor changed the response section), and insertions, a list of {behavior, position, text} (MBT-I also gives marker). position is the share of the original reasoning, counted in non-whitespace characters, that comes before the insertion: 0 is the start and 1 the end. MBT-I positions follow from the markers; MBT-A positions come from replaying the agent's commits. MBT-R rewrites the whole trace, so it has no insertion list.

sft_unedited, sft_mbt_r, sft_mbt_i, sft_mbt_a: id and messages, a user message (the problem followed by "Please reason step by step, and put your final answer within \boxed{}.") and an assistant message <think>\n{reasoning}\n</think>\n\n{response}. Each config is one condition's training data (Table D1).

πŸš€ Usage

from datasets import load_dataset

problems = load_dataset("metacognitive-behavioral-tuning/mbt-traces", "problems", split="train")
mbt_a = load_dataset("metacognitive-behavioral-tuning/mbt-traces", "traces_mbt_a", split="train")
sft_mbt_a = load_dataset("metacognitive-behavioral-tuning/mbt-traces", "sft_mbt_a", split="train")

πŸ› οΈ How the data was made

  1. Problems. A pool of 10,000 Nemotron-Math-v2 problems after deduplication and decontamination against the five evaluation benchmarks (AIME 2025, AIME 2026, HMMT February 2025, HMMT February 2026 and HMMT November 2025) and seven more math benchmarks (MATH-500, AMC 2023, AIME 2024, HMMT February 2023, HMMT February 2024, OlympiadBench and GSM8K). The pool is divided into nine difficulty bins by how many of eight gpt-oss-120b attempts are correct, and 111 or 112 problems are drawn per bin.
  2. Teacher. DeepSeek-V4.1-Flash solves each problem once (reasoning effort max, temperature 1.0, top_p 0.95). Incorrect traces are kept.
  3. Editing. The editor is the teacher model itself. It sees the problem, the reasoning and the response, never the reference answer, and may not look the problem up. All three strategies share the same behavior instructions.

βš–οΈ License and attribution

The dataset is released under CC BY-SA 4.0 (see LICENSE.md), following the license of the StackExchange subsets of Nemotron-Math-v2. It contains only Nemotron-Math-v2 problems and outputs of DeepSeek-V4.1-Flash. See NOTICE.md for the upstream licenses and the per-problem attribution.

⚠️ Limitations

  • The traces are from one teacher on one domain (math). Incorrect teacher answers are retained; editing does not guarantee a correct answer.
  • Inserted passages are model-written. They are meant to demonstrate habits of checking and revising, and individual passages can themselves contain errors.
  • StackExchange problems are user-contributed and may contain ambiguities.

πŸ“š Citation

@misc{mbt2026,
  title  = {Distilling Metacognitive Behaviors through Agentic Editing},
  author = {Anonymous},
  year   = {2026},
  note   = {Under review}
}
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