id stringlengths 35 40 | final_answer stringlengths 0 305 | difficulty float64 -1 10 | topic stringclasses 82
values | messages listlengths 2 2 |
|---|---|---|---|---|
HuggingFaceTB/DeepMath-103K/00000/18 | 0 | 8 | Mathematics -> Number Theory -> Prime Numbers | [
{
"content": "Determine the limit of \\( \\frac{\\pi(x)}{x} \\) as \\( x \\to \\infty \\), where \\( \\pi(x) \\) is the prime counting function.",
"role": "user"
},
{
"content": "We are asked to determine the limit:\n\n$$\n\\lim_{x \\to \\infty} \\frac{\\pi(x)}{x}\n$$\n\nwhere $ \\pi(x) $ is the **p... |
HuggingFaceTB/DeepMath-103K/00000/13 | \mathbb{C}(t) | 4 | Mathematics -> Algebra -> Abstract Algebra -> Field Theory | [
{
"content": "Provide an example of a field that properly contains the field of complex numbers \\( \\mathbb{C} \\).",
"role": "user"
},
{
"content": "There is **no proper field** that strictly contains the field of complex numbers \\( \\mathbb{C} \\) **in the usual sense of field extensions**.\n\nH... |
HuggingFaceTB/DeepMath-103K/00000/9 | 0 | 8 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | [
{
"content": "Find the expectation \\( \\mathbb{E}[X_t] \\) where \\( X_t = \\sin(B_t) \\) and \\( B_t \\) is a standard Brownian motion for \\( t \\geq 0 \\).",
"role": "user"
},
{
"content": "We are asked to find the expectation:\n\n$$\n\\mathbb{E}[X_t] = \\mathbb{E}[\\sin(B_t)]\n$$\n\nwhere $ B_t... |
HuggingFaceTB/DeepMath-103K/00000/22 | 1 | 6.5 | Mathematics -> Algebra -> Intermediate Algebra -> Other | [
{
"content": "Using the generating function \\( \\sum_{n=0}^{\\infty}P_n(x)r^n=(1-2rx+r^2)^{-\\frac{1}{2}} \\), find the value of \\( P_n(1) \\).",
"role": "user"
},
{
"content": "We are given the generating function:\n\n$$\n\\sum_{n=0}^{\\infty} P_n(x) r^n = \\frac{1}{\\sqrt{1 - 2rx + r^2}}\n$$\n\n... |
HuggingFaceTB/DeepMath-103K/00000/37 | 1 | 5.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0}\\left(\\log\\frac{1}{x}\\right)^x \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to 0} \\left( \\log \\frac{1}{x} \\right)^x\n$$\n\n---\n\n### Step 1: Simplify the expression\n\nNote that:\n\n$$\n\\log... |
HuggingFaceTB/DeepMath-103K/00000/6 | |A| | 3 | Mathematics -> Algebra -> Intermediate Algebra -> Other | [
{
"content": "Let $A$ be a proper infinite subset of a set $X$. If $x$ and $y$ are two distinct elements of $X$ that are not in $A$, and we define $B = \\{x, y\\} \\cup A$, what is the cardinality of $B$ in terms of the cardinality of $A$?",
"role": "user"
},
{
"content": "We are given:\n\n- $ A $ i... |
HuggingFaceTB/DeepMath-103K/00000/10 | \emptyset | 5.5 | Mathematics -> Algebra -> Algebra -> Polynomial Operations | [
{
"content": "Determine the values of \\( k \\) for which the equation \\( x^3 - 3x^2 + 6x + k = 0 \\) has three real roots.",
"role": "user"
},
{
"content": "We are given the cubic equation:\n\n$$\nx^3 - 3x^2 + 6x + k = 0\n$$\n\nand we are to **determine the values of $ k $** for which this equatio... |
HuggingFaceTB/DeepMath-103K/00000/7 | \phi^4 | 5.5 | Mathematics -> Geometry -> Plane Geometry -> Polygons | [
{
"content": "Consider a regular pentagon circumscribed in a circle. Connect each vertex of this pentagon to every other non-adjacent vertex with a straight line segment to form a pentagram, which contains a smaller pentagon. What is the ratio of the area of the original (large) pentagon to the smaller one in t... |
HuggingFaceTB/DeepMath-103K/00000/4 | \infty | 5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Find the limit: \\[ \\lim_{x \\to \\infty} (x!)^{1/x} \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to \\infty} (x!)^{1/x}\n$$\n\nNote: Here, $ x $ is going to infinity, and $ x! $ is the factorial function. However, **factorials are only d... |
HuggingFaceTB/DeepMath-103K/00000/36 | \dfrac{1}{2e} | 5.5 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | [
{
"content": "Given the joint probability distribution of $X$ and $Y$:\n\\[ f(x,y) = \\begin{cases} \n e^{-(x+y)}, & x > 0, y > 0 \\\\\n 0, & \\text{otherwise} \n\\end{cases} \\]\ncompute the probability $P(Y > X+1)$.",
"role": "user"
},
{
"content": "We are given the joint probability density fun... |
HuggingFaceTB/DeepMath-103K/00000/12 | \dfrac{1}{p+1} | 4.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}} \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}}\n$$\n\nwhere $ p $ is a fixed real number (we'll assume $ p > 0 $... |
HuggingFaceTB/DeepMath-103K/00000/25 | \{\mathbf{0}\} | 6 | Mathematics -> Algebra -> Linear Algebra -> Matrices | [
{
"content": "Find the kernel of the Vandermonde matrix \\( A \\) given by\n\\[\nA = \\begin{pmatrix} x_1^0 & x_1^1 & \\ldots & x_1^n \\\\ x_2^0 & x_2^1 & \\ldots & x_2^n \\\\ \\vdots & \\vdots & \\ldots & \\vdots \\\\ x_m^0 & x_m^1 & \\ldots & x_m^n \\end{pmatrix}\n\\]\nwhere \\( n < m - 1 \\) and the \\( x_i ... |
HuggingFaceTB/DeepMath-103K/00000/46 | 3.241 | 5.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Calculate the integral: \\[ \\int_{0}^{2} \\sqrt{1+x^3} \\, dx \\]",
"role": "user"
},
{
"content": "The integral \n$$\n\\int_{0}^{2} \\sqrt{1+x^3} \\, dx\n$$ \ndoes **not** have an elementary antiderivative. That is, it **cannot** be expressed in terms of basic functions (polynomial... |
HuggingFaceTB/DeepMath-103K/00000/5 | \dfrac{\pi}{2} | 5 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Find the length of the polar curve given by \\( r = \\sqrt{1 + \\cos(2\\theta)} \\) for \\( 0 \\leq \\theta \\leq \\frac{\\pi\\sqrt{2}}{4} \\).",
"role": "user"
},
{
"content": "We are given a polar curve:\n\n$$\nr = \\sqrt{1 + \\cos(2\\theta)}\n$$\n\nand we are to find the **length** ... |
HuggingFaceTB/DeepMath-103K/00000/47 | \cosh x \sin y | 5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "For the function \\( u(x, y) = \\sinh x \\cos y \\), find the conjugate harmonic function \\( v(x, y) \\).",
"role": "user"
},
{
"content": "We are given a function:\n\n$$\nu(x, y) = \\sinh x \\cos y\n$$\n\nand we are to find the **conjugate harmonic function** $ v(x, y) $, such that $... |
HuggingFaceTB/DeepMath-103K/00000/1 | m^2 + 1 = 0 | 5 | Mathematics -> Differential Equations -> Ordinary Differential Equations (ODEs) | [
{
"content": "Find the auxiliary equation for the ordinary differential equation with constant coefficients: \\((x^2D^2 + xD + 1)y = \\sin(2\\log x)\\sin(\\log x)\\).",
"role": "user"
},
{
"content": "We are given the ordinary differential equation (ODE):\n\n$$\n(x^2 D^2 + x D + 1)y = \\sin(2\\log x... |
HuggingFaceTB/DeepMath-103K/00000/21 | \ln(2\pi + 1) | 5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "Find parametric equations for a unit circle with a speed of \\( e^t \\), starting from \\( x=1 \\), \\( y=0 \\). Determine when the circle is completed.",
"role": "user"
},
{
"content": "We are asked to find **parametric equations** for a **unit circle** that moves with **speed $ e^t $... |
HuggingFaceTB/DeepMath-103K/00000/29 | 1 | 6 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Evaluate the infinite series \\( \\sum_{n=1}^{\\infty} \\frac{2n+1}{(n^{2}+n)^{2}}. \\)",
"role": "user"
},
{
"content": "We are asked to evaluate the infinite series:\n\n$$\n\\sum_{n=1}^{\\infty} \\frac{2n+1}{(n^2 + n)^2}\n$$\n\n---\n\n### Step 1: Simplify the denominator\n\nNote that... |
HuggingFaceTB/DeepMath-103K/00000/45 | \dfrac{1}{e} | 6 | Mathematics -> Precalculus -> Limits | [
{
"content": "Compute the limit \\( \\lim_{n\\to\\infty}{(\\sqrt[n]{e}-\\frac{2}{n})^n} \\).",
"role": "user"
},
{
"content": "We are asked to compute the limit:\n\n$$\n\\lim_{n\\to\\infty} \\left(\\sqrt[n]{e} - \\frac{2}{n}\\right)^n\n$$\n\n---\n\n### Step 1: Understand the expression\n\nWe have:\n... |
HuggingFaceTB/DeepMath-103K/00000/49 | e^{-2} | 7 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: $$\\lim_{x\\to\\infty}\\left(\\frac1{x^2\\sin^2\\frac 1x}\\right)^\\frac 1{x\\sin\\frac 1x-1}$$ without using L'Hospital's Rule or Series expansion.",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x\\to\\infty}\\left(\\frac{1}{x^2... |
HuggingFaceTB/DeepMath-103K/00000/0 | 0 | 4.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\sqrt{x} \\left( \\sqrt[3]{x+1} - \\sqrt[3]{x-1} \\right) \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to \\infty} \\sqrt{x} \\left( \\sqrt[3]{x+1} - \\sqrt[3]{x-1} \\right)\n$$\n\n---\n\n###... |
HuggingFaceTB/DeepMath-103K/00000/41 | No | 8.5 | Mathematics -> Geometry -> Differential Geometry -> Other | [
{
"content": "Is the space \\(X = \\prod_{t \\in \\mathbb{Z}} \\mathbb{R}\\), with the product topology \\(T\\), completely metrizable by a metric \\(d\\) such that \\(d(\\tau x, \\tau y) = d(x, y)\\) for the shift map \\(\\tau\\) defined by \\(\\tau x = (\\ldots, x_0, x_1, x_2, \\ldots)\\)?",
"role": "user... |
HuggingFaceTB/DeepMath-103K/00000/3 | 34 | 4.5 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations | [
{
"content": "Determine the minimum sample size required such that the probability of at least two people being a match at all six genetic markers exceeds 0.001, given that the probability of a match at each individual marker is 1/9.",
"role": "user"
},
{
"content": "We are asked to determine the **... |
HuggingFaceTB/DeepMath-103K/00000/19 | 1 | 8 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | [
{
"content": "Consider a sequence $(r_n)$ resulting from infinite coin flips, where $R_n=1$ if $r_n$ is a head and $R_n=-1$ if $r_n$ is a tail. Determine if $P\\left(\\sum \\frac{R_n}{n} < \\infty\\right) = 1$. Provide a justification for your answer.",
"role": "user"
},
{
"content": "We are given a... |
HuggingFaceTB/DeepMath-103K/00000/2 | -\dfrac{2}{3} | 4 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left(\\dfrac{1}{\\tan^2 x}-\\dfrac{1}{x^2} \\right) \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to 0} \\left(\\dfrac{1}{\\tan^2 x}-\\dfrac{1}{x^2} \\right)\n$$\n\n---\n\n### Step 1: Use the ident... |
HuggingFaceTB/DeepMath-103K/00000/17 | 1 | 3.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Evaluate the limit: \\[ \\lim_{t\\to 0}\\left(\\frac{1}{\\ln(1 + t)}+\\frac{1}{\\ln(1-t)}\\right). \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{t\\to 0}\\left(\\frac{1}{\\ln(1 + t)}+\\frac{1}{\\ln(1-t)}\\right)\n$$\n\n---\n\n### Step 1: Unders... |
HuggingFaceTB/DeepMath-103K/00000/43 | 0 | 7.5 | Mathematics -> Algebra -> Linear Algebra -> Matrices | [
{
"content": "Find a matrix $X \\in M_n(\\mathbb{C})$ such that the linear functional $f(A) = \\text{tr}(XA)$ on $M_n(\\mathbb{C})$ preserves matrix multiplication.",
"role": "user"
},
{
"content": "We are asked to find a matrix $ X \\in M_n(\\mathbb{C}) $ such that the linear functional \n$$\nf(A)... |
HuggingFaceTB/DeepMath-103K/00000/34 | 23 | 6.5 | Mathematics -> Algebra -> Intermediate Algebra -> Other | [
{
"content": "Let $a$ and $b$ be positive integers such that the range of the function \\( y = \\frac{x^2 + ax + b}{x^2 + 2x + 3} \\) is the interval \\(-5 \\leq y \\leq 4\\) for all real numbers $x$. Find the value of $a + b$. \\( \\boxed{} \\)",
"role": "user"
},
{
"content": "We are given a ratio... |
HuggingFaceTB/DeepMath-103K/00000/30 | True | 6.5 | Mathematics -> Calculus -> Integral Calculus -> Integrals | [
{
"content": "Determine whether the following inequality is true for all real numbers $0 < r < 1$ and $t \\geq 0$:\n\\[ \\int_t^{t+r} \\sin(x)\\, dx \\leq \\int_{\\frac{\\pi}{2}-\\frac{r}{2}}^{\\frac{\\pi}{2}+\\frac{r}{2}}\\sin(x)\\, dx. \\]",
"role": "user"
},
{
"content": "We are given the inequal... |
HuggingFaceTB/DeepMath-103K/00000/28 | B | 8 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Determine whether the following statement is true or false: \"Any bounded sequence in $L^4[0,1]$ has a convergent subsequence in $L^2[0,1]$.\"",
"role": "user"
},
{
"content": "We are asked to determine whether the following statement is **true or false**:\n\n> \"Any bounded sequence i... |
HuggingFaceTB/DeepMath-103K/00000/24 | 0 | 7.5 | Mathematics -> Number Theory -> Prime Numbers | [
{
"content": "Verify that for \\(n = 2^kN\\), where \\(N\\) is odd, the following identity holds:\n\\[ \\sum_{d\\mid n}(-1)^{n/d}\\phi(d) = \\sum_{d\\mid 2^{k-1}N}\\phi(d) - \\sum_{d\\mid N}\\phi(2^kd) = 0. \\]",
"role": "user"
},
{
"content": "We are given the identity to verify for $ n = 2^k N $, ... |
HuggingFaceTB/DeepMath-103K/00000/20 | Yes | 8.5 | Mathematics -> Other | [
{
"content": "Consider a locally path connected space $X$. If every open subset $U \\subseteq X$ is semi-locally simply connected, does it follow that $X$ is locally simply connected?",
"role": "user"
},
{
"content": "We are given a **locally path-connected space** $ X $, and the condition that **ev... |
HuggingFaceTB/DeepMath-103K/00000/31 | Yes | 8 | Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers | [
{
"content": "Let $E \\subset \\mathbb{C}$ be a set such that for any sequence of distinct elements $(e_n)_{n \\in \\mathbb{N}}$ from $E$, $e_n \\to 0$ in norm. Is $E$ necessarily countable? Provide a justification for your answer.",
"role": "user"
},
{
"content": "We are given a set $ E \\subset \\... |
HuggingFaceTB/DeepMath-103K/00000/35 | \dfrac{1}{2} | 7.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Evaluate the limit: \\[ \\lim _{n\\to \\infty }n \\int_{-1}^0(x + e^x)^{n}dx. \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{n \\to \\infty} n \\int_{-1}^0 (x + e^x)^n \\, dx.\n$$\n\n---\n\n### Step 1: Understand the integrand\n\nLet’s define $ ... |
HuggingFaceTB/DeepMath-103K/00000/33 | 292 | 6 | Mathematics -> Discrete Mathematics -> Combinatorics | [
{
"content": "Find the number of ways to make change for a dollar using generating functions. Specifically, determine the coefficient of the \\(x^{100}\\) term in the expansion of the generating function \\(\\frac{1}{(x-1)(x^5-1)\\cdots(x^{50}-1)}\\).",
"role": "user"
},
{
"content": "We are asked t... |
HuggingFaceTB/DeepMath-103K/00000/27 | \dfrac{5}{2} | 5.5 | Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions | [
{
"content": "Find the maximum value of $x^2 + y^2$ given that $(x, y)$ satisfy the following equations:\n\\[ 2x^2 + 5xy + 3y^2 = 2 \\]\n\\[ 6x^2 + 8xy + 4y^2 = 3 \\]\nNote: Calculus is not allowed.",
"role": "user"
},
{
"content": "We are given two equations:\n\n$$\n(1)\\quad 2x^2 + 5xy + 3y^2 = 2\... |
HuggingFaceTB/DeepMath-103K/00000/23 | \log 2 | 5 | Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers | [
{
"content": "Evaluate the expression \\( \\log \\left| 1 + \\alpha + \\alpha^2 + \\alpha^3 - \\frac{1}{\\alpha} \\right| \\), where \\( \\alpha \\) is a fifth root of unity.",
"role": "user"
},
{
"content": "We are asked to evaluate the expression:\n\n$$\n\\log \\left| 1 + \\alpha + \\alpha^2 + \\a... |
HuggingFaceTB/DeepMath-103K/00000/32 | Yes | 9 | Mathematics -> Algebra -> Abstract Algebra -> Field Theory | [
{
"content": "Let $S$ be an algebraic smooth surface over $\\\\mathbb{C}\\\\). Suppose there is a fibration $p: S \\rightarrow C$ onto a smooth curve, and let $f$ be a fiber of this fibration. Let $K$ be a canonical divisor on $S$ such that $K \\cdot f = 0$. Determine whether each $m$-canonical map $\\varphi_{m... |
HuggingFaceTB/DeepMath-103K/00000/42 | 9 | 5 | Mathematics -> Geometry -> Plane Geometry -> Triangulations | [
{
"content": "A triangle is inscribed in a circle with radius 1. What is the maximum value of the sum of the squares of the sides of the triangle?",
"role": "user"
},
{
"content": "We are given a triangle inscribed in a circle of **radius 1**, and we are to find the **maximum value of the sum of the... |
HuggingFaceTB/DeepMath-103K/00000/16 | Yes | 9 | Mathematics -> Differential Equations -> Partial Differential Equations (PDEs) | [
{
"content": "Let \\( A \\) be a C* algebra of operators on a Hilbert space \\( H \\). Determine if there exists an \\( x \\) in \\( H \\) such that the set \\( Ax \\) is dense in \\( H \\) but not equal to the whole \\( H \\).",
"role": "user"
},
{
"content": "We are given a C*-algebra $ A $ of ope... |
HuggingFaceTB/DeepMath-103K/00000/38 | 4603 | 5.5 | Mathematics -> Number Theory -> Congruences | [
{
"content": "Evaluate \\(1234^{1234} \\pmod{5379}\\). Note that \\(5379 = 3 \\times 11 \\times 163\\).",
"role": "user"
},
{
"content": "We are asked to evaluate:\n\n$$\n1234^{1234} \\mod 5379\n$$\n\nGiven that:\n\n$$\n5379 = 3 \\times 11 \\times 163\n$$\n\nWe will use the **Chinese Remainder Theor... |
HuggingFaceTB/DeepMath-103K/00000/48 | Yes | 8 | Mathematics -> Geometry -> Differential Geometry -> Curvature | [
{
"content": "In a compact metric space $X$, consider a finite subset $F \\subset X$ that is $\\epsilon$-equidistant, meaning that the distance $d(x,y) = \\epsilon$ for all distinct $x,y \\in F$. For a given $\\epsilon > 0$, does there exist a maximum number $N$, depending on both $\\epsilon$ and $X$, such that... |
HuggingFaceTB/DeepMath-103K/00000/40 | \dfrac{1}{3} | 6 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Find the maximum value of the function \\( f(x) = \\int^{x}_{0} \\sqrt{(x^2-x)^2+y^4}~dy \\) for \\( 0 \\leq x \\leq 1 \\).",
"role": "user"
},
{
"content": "We are given the function:\n\n$$\nf(x) = \\int_0^x \\sqrt{(x^2 - x)^2 + y^4} \\, dy\n$$\n\nand we are to **find the maximum valu... |
HuggingFaceTB/DeepMath-103K/00000/26 | 25 | 5.5 | Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions | [
{
"content": "Find the minimum value of the product \\(abc\\) if the quadratic equation \\(ax^2 - bx + c = 0\\) has two distinct roots in the interval \\((0, 1)\\), where \\(a\\), \\(b\\), and \\(c\\) are natural numbers.",
"role": "user"
},
{
"content": "We are given a quadratic equation:\n\n$$\nax... |
HuggingFaceTB/DeepMath-103K/00000/44 | 6156 | 5 | Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions | [
{
"content": "Find the minimum natural number \\( n \\) such that the expression \\( f(n) = \\sqrt{100+\\sqrt{n}} + \\sqrt{100-\\sqrt{n}} \\) is an integer.",
"role": "user"
},
{
"content": "We are given the function:\n\n$$\nf(n) = \\sqrt{100 + \\sqrt{n}} + \\sqrt{100 - \\sqrt{n}}\n$$\n\nand we are ... |
HuggingFaceTB/DeepMath-103K/00000/15 | \dfrac{5}{6} | 7 | Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions | [
{
"content": "Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8, 4$. Define $n = (10a + b)^{10c + d}$. Calculate the probability that $1984!$ is divisible by $n$. Use Fermat's Little Theorem to assist in your calculations.",
"role": "user"
},
{
"content": "We are given that $ a, b, c, d $ is... |
HuggingFaceTB/DeepMath-103K/00000/8 | 6 | 4 | Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions | [
{
"content": "Find the longest geometric progression with a common ratio greater than 1 that can be formed from the set \\( \\{100, 101, 102, \\ldots, 1000\\} \\). What is the length of this progression?",
"role": "user"
},
{
"content": "We are asked to find the **longest geometric progression (GP)*... |
HuggingFaceTB/DeepMath-103K/00000/11 | 0.5 | 6.5 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Verify that the distance from the function \\( f = x \\) to the set \\( Y = \\{ f \\in C[0,1] : \\int_0^1 f = 0 \\} \\) is 0.5, given that \\( Y \\) is a closed subset of \\( X = \\{ f \\in C[0,1] : f(0) = 0 \\} \\).",
"role": "user"
},
{
"content": "We are asked to **verify that the d... |
HuggingFaceTB/DeepMath-103K/00000/39 | \mathbf{c}^\top \otimes \mathbf{A} | 6 | Mathematics -> Algebra -> Linear Algebra -> Matrices | [
{
"content": "Find the derivative of the matrix product $\\mathbf{A}\\mathbf{B}\\mathbf{c}$ with respect to the matrix $\\mathbf{B}$, where $\\mathbf{A}$ is an $n\\times m$ matrix, $\\mathbf{B}$ is an $m\\times k$ matrix, and $\\mathbf{c}$ is a $k\\times 1$ vector.",
"role": "user"
},
{
"content": "... |
HuggingFaceTB/DeepMath-103K/00000/14 | No | 8.5 | Mathematics -> Algebra -> Linear Algebra -> Linear Transformations | [
{
"content": "Let $T$ be a continuous linear operator. Suppose $(u_n)$ is a sequence that converges weakly to $u$, denoted as $(u_n) \\rightharpoonup u$. Additionally, assume $T(u_n) \\rightharpoonup T(u)$ and there exists a subsequence $(u_{n_k})$ such that $T(u_{n_k}) \\rightarrow T(u)$. Does it follow that $... |
HuggingFaceTB/DeepMath-103K/00000/73 | 0 | 6 | Mathematics -> Calculus -> Differential Calculus -> Derivatives | [
{
"content": "Calculate the first partial derivative with respect to $x$ of the function \\( f(x,y) \\) at the point \\((0,0)\\), where\n\\[\nf(x,y) =\n\\begin{cases}\n0 & (x,y)=(0,0)\\\\\n\\frac{xy}{|x|+|y|} & (x,y) \\neq (0,0)\n\\end{cases}\n\\]",
"role": "user"
},
{
"content": "We are given the f... |
HuggingFaceTB/DeepMath-103K/00000/61 | \sqrt{2} | 4 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "Determine the radius of convergence for the function \\( \\frac{1}{z^2 + 2z + 2} \\).",
"role": "user"
},
{
"content": "We are given the function:\n\n$$\nf(z) = \\frac{1}{z^2 + 2z + 2}\n$$\n\nand we are to **determine the radius of convergence** of this function.\n\n---\n\n### Step 1: ... |
HuggingFaceTB/DeepMath-103K/00000/99 | 6\sqrt[4]{3} | 5 | Mathematics -> Geometry -> Plane Geometry -> Triangulations | [
{
"content": "Find the minimum value of the perimeter of a triangle whose area is 3 cm².",
"role": "user"
},
{
"content": "We are asked to **find the minimum value of the perimeter** of a triangle whose **area is 3 cm²**.\n\n---\n\n### Step 1: Use known geometric inequality\n\nFor a given area, the ... |
HuggingFaceTB/DeepMath-103K/00000/98 | 0 | 4.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Calculate the limit: $$ \\lim_{n\\rightarrow\\infty}\\frac{2^n + n\\sin{n}}{\\log_2{n} + e^n} $$",
"role": "user"
},
{
"content": "We are asked to compute the limit:\n\n$$\n\\lim_{n\\rightarrow\\infty}\\frac{2^n + n\\sin{n}}{\\log_2{n} + e^n}\n$$\n\n---\n\n### Step 1: Understand the gr... |
HuggingFaceTB/DeepMath-103K/00000/66 | 30 | 4.5 | Mathematics -> Discrete Mathematics -> Combinatorics | [
{
"content": "A cube is to be colored using 6 distinct colors such that no two adjacent faces share the same color. How many distinct ways can this be done?",
"role": "user"
},
{
"content": "We are given a cube and asked to color its **6 faces** using **6 distinct colors**, such that **no two adjace... |
HuggingFaceTB/DeepMath-103K/00000/95 | \dfrac{1}{3} | 6.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Evaluate \\( \\lim_{n \\rightarrow \\infty} \\int_0^1 \\frac{nx^{n-1}}{2+x} \\, dx \\).",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{n \\to \\infty} \\int_0^1 \\frac{nx^{n-1}}{2+x} \\, dx\n$$\n\n---\n\n### Step 1: Understand the integrand\n\nWe ar... |
HuggingFaceTB/DeepMath-103K/00000/78 | 1 | 5.5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "Calculate the minimum distance from the curve \\( f(x) = \\begin{pmatrix} \\cos(\\pi x) \\\\ \\sin(\\pi x) \\\\ 1-x^2 \\end{pmatrix} \\) to the origin in \\( \\mathbb{R}^3 \\).",
"role": "user"
},
{
"content": "We are given a vector-valued function:\n\n$$\nf(x) = \\begin{pmatrix} \\cos... |
HuggingFaceTB/DeepMath-103K/00000/89 | 11 | 5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "Given the fixed point iteration \\( p_n = \\frac{p_{n - 1}^2 + 3}{5} \\), which converges for any initial \\( p_0 \\in [0, 1] \\), estimate an expression for the number of iterations \\( n \\) required to achieve an absolute error \\( \\left| p_n - p \\right| < 10^{-4} \\) when \\( p_0 = 1 \\). Us... |
HuggingFaceTB/DeepMath-103K/00000/50 | 2^n | 8 | Mathematics -> Geometry -> Differential Geometry -> Manifolds | [
{
"content": "Determine the number of orientations of a smooth manifold with \\(n\\) maximal connected components.",
"role": "user"
},
{
"content": "The question asks:\n\n> Determine the number of orientations of a smooth manifold with $n$ maximal connected components.\n\nLet’s carefully interpret a... |
HuggingFaceTB/DeepMath-103K/00000/79 | 0 | 4.5 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "Determine whether the sequence \\( \\frac{2^n - 3n^3}{1-4^n} \\) converges or diverges using the ratio test.",
"role": "user"
},
{
"content": "We are given the sequence:\n\n$$\na_n = \\frac{2^n - 3n^3}{1 - 4^n}\n$$\n\nand we are asked to **determine whether it converges or diverges usi... |
HuggingFaceTB/DeepMath-103K/00000/59 | 9e^4 - 1 | 5.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Multi-variable | [
{
"content": "Using the Fundamental Theorem of Line Integrals, evaluate \\( \\int_{C} e^x \\, dy + e^{x}y \\, dx, \\) where \\( C \\) is the parabola parameterized by \\( r(t)=\\langle t+1,t^2 \\rangle \\) for \\( t\\in[-1,3]. \\)",
"role": "user"
},
{
"content": "We are asked to evaluate the line i... |
HuggingFaceTB/DeepMath-103K/00000/91 | 0 | 8 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\frac{\\left(\\int_0^x e^{t^2} \\, dt\\right)^2}{\\int_0^x e^{2t^2} \\, dt} \\]",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{x \\to \\infty} \\frac{\\left(\\int_0^x e^{t^2} \\, dt\\right)^2}{\\int_0^... |
HuggingFaceTB/DeepMath-103K/00000/96 | 0.8 | 4 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | [
{
"content": "Calculate the unconditional variance of the random variable $N$, where $N|\\Lambda$ follows a binomial distribution with parameters $\\Lambda$ and $q = 0.4$, and $\\Lambda$ has a probability function defined by $p(1) = p(2) = p(3) = p(4) = 0.25$. Provide your answer as a numerical value.",
"ro... |
HuggingFaceTB/DeepMath-103K/00000/54 | 0 | 8 | Mathematics -> Differential Equations -> Partial Differential Equations (PDEs) | [
{
"content": "Find all real harmonic functions $u$ on the unit disk $D$ centered at $0$ in the complex plane such that $u(0) = 0$ and $u^2$ is also harmonic on $D$.",
"role": "user"
},
{
"content": "We are asked to find all **real harmonic functions** $ u $ on the **unit disk** $ D = \\{ z \\in \\ma... |
HuggingFaceTB/DeepMath-103K/00000/81 | 1 | 5 | Mathematics -> Calculus -> Differential Calculus -> Derivatives | [
{
"content": "Use Logarithmic Differentiation to find \\(\\frac{d}{dx} (x^{{x}^{x}})\\) at \\(x=1\\).",
"role": "user"
},
{
"content": "We are asked to find the derivative of the function:\n\n$$\nf(x) = x^{x^x}\n$$\n\nusing **logarithmic differentiation**, and then evaluate it at $x = 1$.\n\n---\n\n... |
HuggingFaceTB/DeepMath-103K/00000/56 | 1 | 5.5 | Mathematics -> Precalculus -> Limits | [
{
"content": "Find the value of \\( \\lim_{x \\to 0} \\frac{4\\cos^2(f(x)) - 1}{1 - x^2} \\), given that the function \\( f(x) \\) satisfies \\( f(x) = f(2x) \\) for all \\( x \\in \\mathbb{R} \\) and \\( f(2017) = \\frac{\\pi}{4} \\).",
"role": "user"
},
{
"content": "We are given the limit:\n\n$$\... |
HuggingFaceTB/DeepMath-103K/00000/68 | \pi - 1 | 7.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Evaluate the limit \\( \\lim_{n\\to \\infty} \\int_{1}^{\\pi}\\frac{\\cos(\\frac{x}{n})}{1-e^{-xn}}dx \\).",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n$$\n\\lim_{n\\to \\infty} \\int_{1}^{\\pi}\\frac{\\cos\\left(\\frac{x}{n}\\right)}{1-e^{-xn}}\\,dx.\n$$\n\... |
HuggingFaceTB/DeepMath-103K/00000/65 | -4x^2 + 3x + 17 | 5.5 | Mathematics -> Algebra -> Linear Algebra -> Linear Transformations | [
{
"content": "Given a linear transformation \\( T: M^R_{2x2} \\rightarrow R_4[x] \\) defined by the following mappings: \\( T\\begin{pmatrix} 2 & 3 \\\\ 1 & 0 \\end{pmatrix} = x^2 \\), \\( T\\begin{pmatrix} 1 & 0 \\\\ 0 & 2 \\end{pmatrix} = 3x - 4 \\), and \\( T\\begin{pmatrix} 0 & 2 \\\\ 4 & 5 \\end{pmatrix} =... |
HuggingFaceTB/DeepMath-103K/00000/62 | 36\pi | 6 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Find the surface area of the solid of revolution obtained by rotating the function \\( x=\\frac{1}{15}(y^2+10)^{3/2} \\) from \\( y=2 \\) to \\( y=4 \\) about the \\( x \\)-axis.",
"role": "user"
},
{
"content": "We are given a function defined as:\n\n$$\nx = \\frac{1}{15}(y^2 + 10)^{3... |
HuggingFaceTB/DeepMath-103K/00000/86 | Yes | 8 | Mathematics -> Differential Equations -> Partial Differential Equations (PDEs) | [
{
"content": "Consider the upper half-plane \\( \\mathbb{H} = \\{ x \\in \\mathbb{R}^n : x_n > 0 \\} \\). Let \\( u \\in C^2(\\mathbb{H}) \\cap C(\\bar{\\mathbb{H}}) \\) be a bounded harmonic function such that \\( u \\leq 0 \\) on \\( \\partial\\mathbb{H} = \\{ x_n = 0 \\} \\). Determine if it is possible to c... |
HuggingFaceTB/DeepMath-103K/00000/55 | A | 8 | Mathematics -> Algebra -> Abstract Algebra -> Ring Theory | [
{
"content": "Let \\( A \\) be an integral domain. For every maximal ideal \\( m \\) in \\( A \\), consider \\( A_m \\) as a subring of the quotient field \\( K \\) of \\( A \\). Prove that \\( \\bigcap_{m \\text{: maximal ideal}} A_m = A \\), where the intersection is taken over all maximal ideals \\( m \\) of... |
HuggingFaceTB/DeepMath-103K/00000/58 | 300 | 5 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations | [
{
"content": "A stock market trader buys 100 shares of stock A and 200 shares of stock B. Let X and Y be the price changes of stock A and B, respectively, over a certain time period. Assume that the joint probability density function (PDF) of X and Y is uniform over the set of integers (x, y) satisfying −2 ≤ x ... |
HuggingFaceTB/DeepMath-103K/00000/70 | Yes | 6 | Mathematics -> Algebra -> Linear Algebra -> Vectors | [
{
"content": "If a vector space over the complex numbers has dimension $n$, is it possible to redefine the operations of addition and scalar multiplication such that the dimension of the vector space changes?",
"role": "user"
},
{
"content": "No, it is **not possible** to redefine the operations of ... |
HuggingFaceTB/DeepMath-103K/00000/52 | \dfrac{16}{9} | 5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Multi-variable | [
{
"content": "Evaluate the double integral \\( \\iint_\\Omega \\sqrt{x^2+y^2} \\,\\mathrm{d}x\\mathrm{d}y \\), where \\( \\Omega \\) is the region defined by \\((x-1)^2+y^2 \\leq 1\\) and \\(0 \\leq y\\).",
"role": "user"
},
{
"content": "We are asked to evaluate the double integral:\n\n$$\n\\iint_\... |
HuggingFaceTB/DeepMath-103K/00000/53 | ii | 5 | Mathematics -> Algebra -> Linear Algebra -> Matrices | [
{
"content": "Let \\( A \\) and \\( B \\) be \\( n \\times n \\) matrices. Which of the following is equal to \\( \\text{trace}(A^2 B^2) \\)?\n\n(i) \\( (\\text{trace}(AB))^2 \\)\n(ii) \\( \\text{trace}(AB^2 A) \\)\n(iii) \\( \\text{trace}((AB)^2) \\)\n(iv) \\( \\text{trace}(BABA) \\)",
"role": "user"
},
... |
HuggingFaceTB/DeepMath-103K/00000/85 | -7875000 | 6.5 | Mathematics -> Calculus -> Differential Calculus -> Derivatives | [
{
"content": "Find the 9th derivative of \\( \\frac{\\cos(5 x^2)-1}{x^3} \\) and evaluate it at \\( x=0 \\) using the Maclaurin Series.",
"role": "user"
},
{
"content": "We are asked to find the **9th derivative** of the function \n$$\nf(x) = \\frac{\\cos(5x^2) - 1}{x^3}\n$$\nand evaluate it at $ x... |
HuggingFaceTB/DeepMath-103K/00000/92 | 9 | 5.5 | Mathematics -> Algebra -> Linear Algebra -> Matrices | [
{
"content": "Let \\( A \\in M_{5\\times 6}(\\mathbb{R}) \\) be a matrix with rank 4. Consider the block matrix \\( D = \\begin{pmatrix} I_5 & A \\\\ A^T & 0 \\end{pmatrix} \\). Determine the rank of \\( D \\).",
"role": "user"
},
{
"content": "We are given:\n\n- $ A \\in M_{5 \\times 6}(\\mathbb{R}... |
HuggingFaceTB/DeepMath-103K/00000/90 | -132x + 74 | 6.5 | Mathematics -> Algebra -> Linear Algebra -> Linear Transformations | [
{
"content": "Find the unique polynomial \\( r(x) \\) such that for all \\( p(x) \\in \\mathbb{P}^2 \\), the functional \\( f(p(x)) = p'(-15) + 8p(-1) \\) can be expressed as \\( \\langle p, r \\rangle = \\int_0^1 p(x)r(x)dx \\). Here, \\( \\mathbb{P}^2 \\) is the space of polynomials of degree less than two.",... |
HuggingFaceTB/DeepMath-103K/00000/57 | 4\pi^3 i | 7 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Multi-variable | [
{
"content": "Evaluate the integral \\( \\int_{\\gamma} \\frac{e^{2 \\pi z}}{(z+i)^3}dz \\) using the Cauchy Integration Formula, where \\( \\gamma(t)=2e^{it}, t \\in [0,2 \\pi] \\). Determine if the calculation \\( \\int_{\\gamma} \\frac{f(z)}{z-0}dz = 2 \\pi i f(0) = 0 \\) is correct.",
"role": "user"
}... |
HuggingFaceTB/DeepMath-103K/00000/82 | 999967355 | 5.5 | Mathematics -> Number Theory -> Factorization | [
{
"content": "Determine the number of integers in the range from 1 to \\(10^9\\) that are not perfect squares, perfect cubes, or perfect fifth powers.",
"role": "user"
},
{
"content": "We are asked to find the number of **integers from 1 to $10^9$** that are **not** perfect squares, perfect cubes, o... |
HuggingFaceTB/DeepMath-103K/00000/60 | \mathbb{Z} | 8 | Mathematics -> Algebra -> Abstract Algebra -> Group Theory | [
{
"content": "Determine the fundamental group of the space \\((S^1 \\times S^1)/(S^1 \\times \\{x\\})\\), where \\(x\\) is a point in \\(S^1\\).",
"role": "user"
},
{
"content": "We are asked to compute the **fundamental group** of the space \n$$\n(S^1 \\times S^1) / (S^1 \\times \\{x\\}),\n$$\nwhe... |
HuggingFaceTB/DeepMath-103K/00000/84 | Yes | 8 | Mathematics -> Algebra -> Abstract Algebra -> Field Theory | [
{
"content": "Let $F$ be a finite Galois extension of the rational function field $\\mathbb{Q}(x)$. Let $k$ be the field of constants of $F$, defined as the algebraic closure of $\\mathbb{Q}$ in $F$. Is $k$ necessarily a Galois extension of $\\mathbb{Q}$? Provide a justification for your answer.",
"role": "... |
HuggingFaceTB/DeepMath-103K/00000/88 | + | 8 | Mathematics -> Algebra -> Abstract Algebra -> Other | [
{
"content": "Determine the sign of the map induced by the covering map \\( p: S^n \\to \\mathbb{R}P^n \\) on homology, specifically \\( p_{*}: H_n(S^n, R) \\to H_n(\\mathbb{R}P^n, R) \\), where \\( R \\) is an arbitrary ring. Given that \\( H_n(S^n, R) \\cong R \\) is generated by the class \\([\\sigma_n^{(1)}... |
HuggingFaceTB/DeepMath-103K/00000/63 | 3 - 2\sqrt{2} | 6 | Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives | [
{
"content": "Find the minimum value of the function \\( f(x) = \\max_{t \\in [0,1]} |t^2 - tx| \\) for \\( x \\in (0,1) \\).",
"role": "user"
},
{
"content": "We are given the function:\n\n$$\nf(x) = \\max_{t \\in [0,1]} |t^2 - tx|\n$$\n\nfor $ x \\in (0,1) $, and we are to **find the minimum value... |
HuggingFaceTB/DeepMath-103K/00000/71 | 40 | 6 | Mathematics -> Discrete Mathematics -> Graph Theory | [
{
"content": "Consider a connected undirected simple non-planar graph $G$ with 15 vertices. If removing any edge from $G$ results in a planar graph, how many edges does $G$ have?",
"role": "user"
},
{
"content": "We are given a **connected undirected simple non-planar graph $ G $** with **15 vertice... |
HuggingFaceTB/DeepMath-103K/00000/64 | A | 6 | Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals | [
{
"content": "Assume that \\( f \\) is a non-negative real function, and let \\( a > 0 \\) be a real number. Define \\( I_a(f) \\) as follows:\n\\[ I_a(f) = \\frac{1}{a}\\int_{0}^{a} f(x) \\, dx \\]\nSuppose \\( \\lim_{x \\rightarrow \\infty} f(x) = A \\) exists. Determine whether \\( \\lim_{a \\rightarrow \\in... |
HuggingFaceTB/DeepMath-103K/00000/87 | 20 | 5 | Mathematics -> Geometry -> Plane Geometry -> Angles | [
{
"content": "A ladder is being moved through a corridor in the first quadrant of the $x$-$y$ plane. The ladder is represented by the line $y = mx + c$, where $m < 0$ and $c > 0$. The length of the ladder is $L$. The goal is to find the maximum length $L$ such that the ladder just touches the corner of the corr... |
HuggingFaceTB/DeepMath-103K/00000/72 | a | 5.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty} \\sum_{k=1}^{n} \\sin\\left(\\frac{(2k-1)a}{n^2}\\right) \\) by expressing it as a Riemann sum and finding the corresponding integral, where \\( a \\in \\mathbb{R} \\).",
"role": "user"
},
{
"content": "We are asked to evaluate the limit:\n\n... |
HuggingFaceTB/DeepMath-103K/00000/76 | Yes | 8 | Mathematics -> Number Theory -> Other | [
{
"content": "Is the statement \"If $\\gcd(y,z)=1$, then $yz$ is perfect if and only if $D(y)D(z)=2s(y)s(z)$\" always true for odd perfect numbers $q^k n^2$? Here, $D(x) = 2x - \\sigma(x)$ is the deficiency of $x$, $s(x) = \\sigma(x) - x$ is the sum of the aliquot divisors of $x$, and $\\sigma(x)$ is the sum of... |
HuggingFaceTB/DeepMath-103K/00000/97 | True | 8.5 | Mathematics -> Algebra -> Abstract Algebra -> Group Theory | [
{
"content": "Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$, the inequality $|P|^2 < |G|$ holds?",
"role": "user"
},
{
"content": "We are asked whether the following statement is true:\n\n> For all Sylow subgroups $ P $ of a nonabelian simple group $ G $, the inequalit... |
HuggingFaceTB/DeepMath-103K/00000/77 | \Delta | 6.5 | Mathematics -> Differential Equations -> Partial Differential Equations (PDEs) | [
{
"content": "Given that \\( f(\\phi) = \\Delta \\phi \\), where \\( \\Delta \\phi = \\nabla \\cdot \\nabla \\phi \\), find the expression for \\( \\frac{df(\\phi)}{d\\phi} \\).",
"role": "user"
},
{
"content": "We are given:\n\n$$\nf(\\phi) = \\Delta \\phi = \\nabla \\cdot \\nabla \\phi\n$$\n\nThis... |
HuggingFaceTB/DeepMath-103K/00000/75 | f(x) = x + 1 | 6.5 | Mathematics -> Algebra -> Abstract Algebra -> Group Theory | [
{
"content": "Find all strictly increasing functions \\( f : \\Bbb{Z} \\rightarrow \\Bbb{Z} \\) such that \\( f(f(x)) = x + 2 \\) for all integers \\( x \\).",
"role": "user"
},
{
"content": "We are given a functional equation:\n\n> Find all **strictly increasing** functions $ f : \\mathbb{Z} \\to \... |
HuggingFaceTB/DeepMath-103K/00000/80 | \dfrac{1}{4040} | 7.5 | Mathematics -> Calculus -> Integral Calculus -> Techniques of Integration -> Single-variable | [
{
"content": "Compute \\( \\lim\\limits_{n\\to \\infty} \\int\\limits_0^1 x^{2019} \\{nx\\} \\, dx \\), where \\( \\{a\\} \\) denotes the fractional part of the real number \\( a \\).",
"role": "user"
},
{
"content": "We are asked to compute the limit:\n\n$$\n\\lim_{n\\to \\infty} \\int_0^1 x^{2019}... |
HuggingFaceTB/DeepMath-103K/00000/69 | 36 | 6 | Mathematics -> Algebra -> Algebra -> Equations and Inequalities | [
{
"content": "Find the sum of all integer values of \\( a \\) such that \\( a(x^2+x-1) \\leq (x^2+x+1)^2 \\) for all real numbers \\( x \\).",
"role": "user"
},
{
"content": "We are given the inequality:\n\n$$\na(x^2 + x - 1) \\leq (x^2 + x + 1)^2\n$$\n\nfor **all real numbers** $ x $, and we are to... |
HuggingFaceTB/DeepMath-103K/00000/51 | Yes | 7.5 | Mathematics -> Algebra -> Linear Algebra -> Matrices | [
{
"content": "Let $X$ and $Y$ be orthogonal skew symmetric matrices. Determine if $X$ and $Y$ are orthogonally conjugate, i.e., if there exists an orthogonal matrix $U \\in O(n)$ such that $UX = YU$. Provide a justification for your answer.",
"role": "user"
},
{
"content": "We are given that $ X $ a... |
HuggingFaceTB/DeepMath-103K/00000/94 | (0, \infty) | 5 | Mathematics -> Calculus -> Differential Calculus -> Derivatives | [
{
"content": "Determine the intervals where the function \\(f(x) = (x+\\frac{1}{x})^{x}\\) is increasing and where it is decreasing.",
"role": "user"
},
{
"content": "We are given the function:\n\n$$\nf(x) = \\left(x + \\frac{1}{x}\\right)^x\n$$\n\nand we are to **determine the intervals where it is... |
HuggingFaceTB/DeepMath-103K/00000/93 | 2 | 5.5 | Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other | [
{
"content": "A consumer is searching for 1 unit of a product across stores with prices 1, 2, 3, ..., each occurring with probabilities 1/2, 1/4, 1/8, ..., 1/2^n respectively. The consumer incurs a search cost of 1 for each store visited. Determine the price at which the consumer should stop searching.",
"r... |
HuggingFaceTB/DeepMath-103K/00000/67 | Yes | 9 | Mathematics -> Algebra -> Abstract Algebra -> Field Theory | [
{
"content": "Let $k$ be a number field, and $F/k$ a finite extension. Is it possible to find a countable family of extensions $k_i/k$ of degree 2 and a place $v_i$ of $k_i$ such that if $v$ is the place of $k$ lying below $v_i$, then $[k_{v_i}:k_v] = 2$, where $k_{v_i}$ and $k_v$ are the completions of $k_i$ a... |
HuggingFaceTB/DeepMath-103K/00000/83 | 3 | 8.5 | Mathematics -> Algebra -> Abstract Algebra -> Group Theory | [
{
"content": "Determine whether the hyperoctahedral group, defined as the wreath product of $S_2$ and $S_n$ or equivalently $G = S_2^n \\rtimes S_n$, has only 3 maximal normal subgroups. Consider $G$ as a subgroup of $S_{2n}$ with its natural action on $[2] \\times [n]$. The group $G$ has two known normal subgr... |
HuggingFaceTB/DeepMath-103K/00000/74 | 6^n | 7.5 | Mathematics -> Algebra -> Abstract Algebra -> Group Theory | [
{
"content": "Determine the minimal possible order of a group $G$ that contains a subset $A \\subset G$ with $|A| = 2n$, such that for every $a \\in A$, there exists a unique $b \\in A$ with $[a, b] \\neq e$. Provide your answer as a single integer.",
"role": "user"
},
{
"content": "We are given a g... |
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