id stringlengths 23 36 | problem stringlengths 39 1.82k | reference_answer stringlengths 1 498 | difficulty_bin int64 0 8 | source stringclasses 2
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values | source_url stringlengths 58 129 ⌀ | author_name stringlengths 3 30 ⌀ | author_url stringlengths 40 75 ⌀ |
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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 | CC-BY-SA-4.0 | 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 | CC-BY-SA-4.0 | 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 | CC-BY-SA-4.0 | 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 | CC-BY-SA-4.0 | 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 | CC-BY-SA-4.0 | 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 | CC-BY-SA-4.0 | 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 | CC-BY-SA-4.0 | 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 | CC-BY-SA-4.0 | 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 | π314 | 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 |
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