domain stringclasses 7
values | sub_source stringclasses 15
values | coding_subclass stringclasses 6
values | messages listlengths 2 24 | tools listlengths 1 94 ⌀ |
|---|---|---|---|---|
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Calculate the determinant of the matrix \\( A \\):\n\n\\[ A = \\left | \\begin{array}{ccccc} \n2 & 3 & 4 & \\cdots & n \\\\\n3 & 6 & 10 & \\cdots & \\frac{n(n+1)}{2!} \\\\\n4 & 10 & 20 & \\cdots & \\frac{n(n+1)(n+2)}{3!} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Three cevians are drawn in a triangle that do not intersect at one point, forming 4 triangles and 3 quadrilaterals. If the area of each of the four triangles is 8, find the sum of the areas of the three quadrilaterals.",
"tool_calls": null
},
{
"role": "assistant",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the maximum number of regions in the plane that can be determined by $n$ triangles.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let the \\(n\\) triangles be drawn in the plane in *general position* :\n\n* no three sides meet at a common point... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let there be an $n$-sided polygon and $m$ numbers $1, 2, 3, \\ldots, m$ corresponding to each variable $x_1, x_2, x_3, \\ldots, x_m$ such that each number $j$ is placed on the vertices of the polygon $x_j$ and $\\sum_{j=1}^{m} x_j = n$. How many ways can the numbers be assigned... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the inequality \\[2\\cos x \\le |\\sqrt{1+\\sin 2x} - \\sqrt{1-\\sin 2x}|\\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\begin{aligned}\n2\\cos x &\\le \\bigl|\\sqrt{1+\\sin 2x}-\\sqrt{1-\\sin 2x}\\bigr| \\\\\n\\bigl(1\\pm\\sin 2x\\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\( A \\) be the set of all positive integers greater than 104 and less than 211. Find the subset \\( B \\) of \\( A \\) with the maximal cardinality such that every two elements in \\( B \\) are relatively prime.",
"tool_calls": null
},
{
"role": "assistant",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the sum $\\sum_{n=0}^{t-q} (-1)^n \\binom{q+n}{q}\\binom{t}{q+n}$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "First rewrite the sum in a more convenient form. \nLet \n\n\\[\nj=q+n\\qquad\\Longrightarrow\\qquad n=j-q,\\;\\;j=q,q+1,\\dot... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "The number \\( a = \\frac{p}{q} \\), where \\( p \\) and \\( q \\) are relatively prime positive integers, has the property that the sum of all real numbers \\( x \\) satisfying \n\\[ \\lfloor x \\rfloor \\cdot \\{x\\} = a \\cdot x^2 \\]\nis 420, where \\( \\lfloor x \\rfloor \... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given a large circle with radius \\(2r\\) and a smaller circle inside it, where the radius of the smaller circle bisects a radius of the larger circle (creating a 30-60-90 triangle), find the ratio of the area of the smaller circle to the area of the larger circle.",
"tool_... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the trigonometric equation $8\\sin x\\cos x\\cos 4x + \\cos 3x = 0$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n8\\sin x\\cos x\\cos 4x+\\cos 3x=0\\qquad (x\\in\\mathbb R)\n\\]\n\nUsing \\(8\\sin x\\cos x=4\\sin 2x\\) the equation beco... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Three circles with radii \\( r \\), \\( r_1 \\), and \\( R \\) are tangent to each other externally. The internal common tangent of the first two circles intersects the circle with radius \\( R \\) at points \\( A \\) and \\( B \\). Calculate the length of the chord \\( \\overl... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "For the function \\( f(z) = \\frac{1}{(z + \\frac{\\pi}{2})\\cos(z)} \\), find the isolated singularities, determine their types (if it’s a pole, give the order of the pole), and compute the corresponding residues.",
"tool_calls": null
},
{
"role": "assistant",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "The probability of player $A$ winning a particular game against player $B$ is $\\frac{1}{3}$, and the probability of player $B$ winning the game is $\\frac{2}{3}$. They play a series of games where $A$ wins the series if he wins 2 consecutive games, and $B$ wins the series if h... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given 20 points in space, where 7 points are on one line and 5 points are on another line, find the largest number of planes that can pass through these points if the two lines are parallel.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – C... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the integral \\(\\int \\left(\\frac{1}{\\ln x}\\right)^n \\, dx\\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nI_n=\\int\\frac{dx}{(\\ln x)^n}\\qquad (x>0,\\;x\\neq1)\n\\]\n\n---\n\n**Step 1 – logarithmic substitution** \n\nLet \n... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the integral \\( \\int \\frac{x^{2}-1}{x^{4}+1}\\ dx \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\n\\[\nI=\\int\\frac{x^{2}-1}{x^{4}+1}\\,dx .\n\\]\n\n---\n\n### 1. Factor the denominator \n\n\\[\nx^{4}+1=(x^{2}+\\sqr... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "The base of a first right cylinder is a circle of radius $y$, the base of a second right cylinder is a circle of radius $x$, the radius of the first circle being greater than that of the second. The altitude of the first cylinder is 1/8, the altitude of the second, 1/2. Find th... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Mad scientist Kyouma writes $N$ positive integers on a board. Each second, he chooses two numbers $x, y$ with $x > y$, and writes the number $x^2 - y^2$ on the board. After some time, all numbers from 1 to 1000 are present on the board. Find the minimum possible value of $N$.",... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "In a right triangle \\(ABC\\), the radius of the inscribed circle is \\(r\\), and \\(AH\\) is the altitude drawn from the vertex of the right angle. Circles are inscribed in each of the resulting right triangles. Find the distance between the centers of these inscribed circles.... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be a triangle with $\\angle BAC > 90^{\\circ}$, $AB = 5$, and $AC = 7$. Points $D$ and $E$ lie on segment $BC$ such that $BD = DE = EC$. If $\\angle BAC + \\angle DAE = 180^{\\circ}$, compute the length of $BC$.",
"tool_calls": null
},
{
"role":... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "How many ways can 4 knights (2 white and 2 black) be placed on an 8x8 chessboard such that knights of the same color cannot attack each other?",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Total placements without the “same‑colour” restri... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Two circles are externally tangent. Lines $\\overline{PAB}$ and $\\overline{PA'B'}$ are common tangents with $A$ and $A'$ on the smaller circle and $B$ and $B'$ on the larger circle. If $PA = AB = 4$, then what is the area of the smaller circle?",
"tool_calls": null
},
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all triples of prime numbers $(p, q, r)$ such that $p^{q} + p^{r}$ is a perfect square.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\np,q,r\\ \\text{be primes},\\qquad p^{q}+p^{r}=m^{2}\n\\]\n\nand, without loss of generality, ass... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the integral $$\\int_0^{\\infty} \\frac{\\ln (1+\\cosh x)}{1+\\cosh x} \\cdot x \\mathrm{~d} x.$$",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nI=\\int_{0}^{\\infty}\\frac{x\\,\\ln (1+\\cosh x)}{1+\\cosh x}\\,dx\n\\]\n\n---\n\n### 1. ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "How many permutations $\\pi$ of the set $\\{1,2,...,n\\}$ are there such that $|\\pi(i+1)-\\pi(i)| \\geq 2$ for all $i \\in \\{1,2,...,n-1\\}$?",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution.** \nFor a permutation \\(\\pi\\) of \\(\\{1,\\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "In triangle ABC, the measure of angle A minus the measure of angle B is equal to 120 degrees, and the length of the circumradius is 8 times the length of the inradius. Find the cosine of angle C.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "For each positive integer $n$, let $f(n)$ be the fewest number of terms needed to write $n$ as a sum of factorials. For example, $f(28) = 3$ because $4! + 2! + 2! = 28$ and 28 cannot be written as the sum of fewer than $3$ factorials. Evaluate $f(1) + f(2) + \\ldots + f(720)$."... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the integral\n$$\\iint_A \\frac{2x+y}{x^2+y^2} \\, dxdy$$\nwhere $A$ is the domain delimited by the curves $x=0$, $y=0$, $y=\\sqrt{1-x^2}$, and $y=2-x$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**1. Description of the domain**\n\nThe ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the last geometric progression (GP) in the set \\(A = \\{1, 2, 3, \\ldots, 999, 1000\\}\\) such that \\(a_5\\) is the maximum possible value.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\n(a_1,a_2,a_3,a_4,a_5)\n\\]\n\nbe a geometr... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "The radius of the base circle of a cone is equal to $R$ and the angle at the vertex of its axial section is $\\alpha $. Find the volume of a reqular triangular pyramid circumscribed about the cone.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "I take 500mL of a liquid and dilute it with 100mL of water. Then I take 500mL of the mixture and again dilute it with 100mL of water. I repeat this process 20 times in all. What percentage of the liquid remains in the mixture at the end?",
"tool_calls": null
},
{
"r... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle such that \\(\\angle A = 60^\\circ\\). Let \\(P\\) and \\(Q\\) be the feet of the perpendiculars drawn from \\(A\\) to the internal bisectors of \\(\\angle B\\) and \\(\\angle C\\) respectively. Given that \\(BP = 104\\) and \\(CQ = 105\\), find the ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Calculate the integral:\n\\[\n\\int_0^{\\frac{\\pi}{2}} \\left(\\frac{2304 \\cos x}{(\\cos 4x - 8 \\cos 2x + 15)^2}\\right) \\, dx\n\\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Simplify the denominator**\n\n\\[\n\\begin{aligned}\n\\c... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\( A \\in \\mathcal{M}_n(\\mathbb{R}) \\). Define \\( C_A \\) as the subset of \\( M \\in \\mathcal{M}_n(\\mathbb{R}) \\) such that \\( AM = MA \\) and \\( M \\) is similar to \\( A \\) (i.e., there exists \\( P \\in \\mathrm{GL}_n(\\mathbb{R}) \\) such that \\( M = PAP^{-... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "A circle \\( \\Omega \\) of unit radius is inscribed in quadrilateral \\( ABCD \\). Let circle \\( \\omega_A \\) be the unique circle of radius \\( r_A \\) externally tangent to \\( \\Omega \\) and also tangent to segments \\( AB \\) and \\( DA \\). Similarly define circles \\(... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "For how many real numbers $x$ with $|x| \\le 3$ is $x(x - 1)$ an integer?",
"tool_calls": null
},
{
"role": "assistant",
"content": "For \n\n\\[\nx\\in[-3,3],\\qquad x(x-1)\\in\\mathbb Z .\n\\]\n\nLet \n\n\\[\nx(x-1)=n\\qquad (n\\in\\mathbb Z).\n\\]\n\nThen \... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Factorize the polynomial \\( x^4 - 2x + 3 \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nx^4-2x+3\n\\]\nhas no rational linear factor (the only possible rational zeros are \\(\\pm1,\\pm3\\) and none of them make the polynomial zero), so it ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the equation $\\tan x = x + 1$ where $x \\in \\mathbb{R}$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\text{Solve}\\qquad \\tan x = x+1 ,\\qquad x\\in\\mathbb R .\n\\]\n\n---------------------------------------------------------------... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the smallest positive real \\( k \\) such that for any positive integer \\( n \\ge 2 \\) and positive reals \\( a_0, a_1, \\ldots, a_n \\),\n\\[\n\\frac{1}{a_0 + a_1} + \\frac{1}{a_0 + a_1 + a_2} + \\ldots + \\frac{1}{a_0 + a_1 + \\ldots + a_n} < k \\left( \\frac{1}{a_0} +... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Determine all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the equation \\( f(xy + f(y)) + f(x) = f(x + y) + yf(x) \\) for all real numbers \\( x \\) and \\( y \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\text{Find all... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "In a game where two contestants A and B each have ten chips numbered from 1 to 10, A places a chip numbered $n$ on the first row and B places a chip numbered $n$ on the second row at the $n$-th turn. B wins if, after the 10th turn, both rows show the numbers of the chips in the... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "If \\( S \\) is a set of positive integers less than or equal to 100 with no two distinct elements of \\( S \\) summing to an element of \\{7, 12, 33, 45, 69, 81\\}, then what is the maximum number of elements \\( S \\) can have?",
"tool_calls": null
},
{
"role": "a... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find a continuous bounded function \\( f:(0,1] \\to \\mathbb{R} \\) that is not uniformly continuous. Extend \\( f \\) with continuity in such a way that \\( f(0) = 0 \\) and find the oscillation \\( \\omega_f(0) \\) of \\( f \\) at \\( 0 \\).",
"tool_calls": null
},
{
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\(a, b, c\\) be positive numbers such that \\(a + b + c = 1\\). Find the minimum value of \n\\[ P = \\frac{a}{1 + b^2 + c^2} + \\frac{b}{1 + c^2 + a^2} + \\frac{c}{1 + a^2 + b^2}. \\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\n... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all functions $f$ such that for any function $g$ that has primitives on an interval $I$, the function $g(f(x))$ also has primitives.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Answer**\n\n\\[\n\\boxed{\\text{All continuous functions }f:I\\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "$x+\\sqrt x=a$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nx+\\sqrt{x}=a ,\\qquad x\\ge 0 .\n\\]\n\nSet \\(t=\\sqrt{x}\\;(t\\ge0)\\). Then \\(x=t^{\\,2}\\) and \n\n\\[\nt^{2}+t=a\\qquad\\Longrightarrow\\qquad t^{2}+t-a=0 .\n\\]\n\nSolve th... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the x- and y- intercepts of the straight line that is tangent to the curve \\( y = f(x) \\) at the point (1,3).",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\n1. **Equation of the tangent line** \n\n At the point \\((x_0,y_0)=... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Calculate the area covered by the straight line \\(x - y = 3\\) and the curve \\(y^2 = 4x\\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "The region bounded by the line \n\n\\[\nx-y=3\\qquad\\Longleftrightarrow\\qquad x=y+3\n\\]\n\nand the parabo... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve for $(x, y)$ in the equation $\\sin^4(x) + 4\\cos^4(y) - 4\\sqrt{2}\\sin(x)\\cos(y) + 2 = 0$, where $x, y \\in \\left(0, \\frac{\\pi}{2}\\right)$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\sin ^4x+4\\cos ^4y-4\\sqrt 2\\,\\sin x\\cos... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Determine the value of \\[\\sum_{k=0}^{1009}(-1)^k\\frac1{2019-k}\\binom{2019-k}k=\\frac1{2019}\\binom{2019}0-\\frac1{2018}\\binom{2018}1+\\frac1{2017}\\binom{2017}2-\\frac1{2016}\\binom{2016}3+\\dots-\\frac1{1010}\\binom{1010}{1009}.\\]",
"tool_calls": null
},
{
"r... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "If the binomial $(hx^n - ky^m)$ is raised to the fifth power, one of the terms in the expansion is $1250x^9 y^{12},$ and another term is $-6750x^6 y^{18}.$ Find the sum of $h, k, n,$ and $m.$",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the value of \\[ \\tan^6\\left( \\frac{\\pi}{18} \\right) + \\tan^6\\left( \\frac{5\\pi}{18} \\right) + \\tan^6\\left( \\frac{7\\pi}{18} \\right). \\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nS=\\tan^{6}\\frac{\\pi}{18}+\\tan^{6}\\fra... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given the rational function \\( R(z) = \\frac{p(z)}{q(z)} \\) where \\( p(z) \\) and \\( q(z) \\) are polynomials and \\( z \\in \\mathbb{C} \\), with \\( \\deg(R(z)) = \\max(\\deg(p), \\deg(q)) \\), find \\( \\deg(R'(z)) \\).",
"tool_calls": null
},
{
"role": "assi... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{Q^+} \\rightarrow \\mathbb{Q} \\) satisfying the equation\n\\[ f(x) + f(y) = \\left(f(x+y) + \\frac{1}{x+y}\\right) \\left(1 - xy + f(xy)\\right) \\]\nfor all \\( x, y \\in \\mathbb{Q^+} \\).",
"tool_calls": null
},
{
"role": "assi... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given 21 gold coins to be divided among 5 people (Alex, Billy, Calxin, David, and Edward) according to the following rules:\n- Alex will have half of the remaining gold coins.\n- Billy will have 2/3 of the remaining gold coins.\n- Calxin will have half of the remaining gold coi... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find a matrix $W_{k\\times n}$ such that $I_{m\\times k} W_{k\\times n} = O_{m\\times n} + \\delta_{m\\times n}$, where $I$ and $O$ are known matrices, and $\\delta_{ij} \\le \\epsilon$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Write... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given \\(a, b, c > 0\\), solve the equation\n\\[\n\\frac{bc + a^2 - ax + 1}{a} + \\frac{1}{1 - ax} = 4\\sqrt[4]{\\frac{bc}{a}}\n\\]\nfor \\(x\\), and find \\(x^a\\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\begin{aligned}\n\\frac{bc+a^{2... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Calculate the area of intersection between the parabola \\( y^2 = -(x-2) \\) and the absolute value function \\( y = |x| \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Write the curves in a convenient form**\n\n\\[\ny^{2}=-(x-2)\\;\\L... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all values of \\( x \\) which satisfy the inequality \\( (2a-1)x^2 < (a+1)x + 3a \\) for any \\( a \\) in the interval \\( (1, 2) \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n(2a-1)x^{2}<(a+1)x+3a ,\\qquad a\\in(1,2)\n\\]\n\nWe have ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Compute the function \\( f(t) = \\sum_{n \\in \\mathbb{Z}} \\frac{1}{a^2 + (t - 2n\\pi)^2} \\) where \\( a > 0 \\) and \\( t \\) is a real number.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\nLet \n\n\\[\nf(t)=\\sum_{n\\in\\mathbb ... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "How many positive integers \\( n \\) satisfy the following conditions?\n(1) The integer \\( n \\) has exactly 2017 digits when written in base 2017.\n(2) The sum of the 2017 2016-digit numbers obtained by deleting exactly one digit of \\( n \\) when \\( n \\) is written in base... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Let \\( S \\subset [0, 1] \\) be a set of 5 points with \\(\\{0, 1\\} \\subset S\\). The graph of a real function \\( f : [0, 1] \\to [0, 1] \\) is continuous and increasing, and it is linear on every subinterval \\( I \\) in \\([0, 1]\\) such that the endpoints but no interior... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "In triangle \\(ABC\\), \\(AB = 20\\) and \\(AC = 18\\). The angle bisector of \\(\\angle BAC\\) intersects the circumcircle of \\(\\triangle ABC\\) at \\(D\\). If the area of \\(\\triangle ABC\\) is five times the area of \\(\\triangle BCD\\), find the area of quadrilateral \\(... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "In a rectangle of dimensions 11 x 13, a cell is called \"peculiar\" if after removing it, the remaining figure can be divided into squares of dimensions 2 x 2 and 3 x 3. How many of the 143 cells are \"peculiar\"?",
"tool_calls": null
},
{
"role": "assistant",
"... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Find the largest $c$ such that for any $\\lambda \\ge 1$, there exists an $a$ satisfying the inequality\n\n$$\\sin a + \\sin (a\\lambda) \\ge c.$$",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution Overview**\n\nFor a fixed \\(\\lambda\\ge 1\\... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Determine $n$ such that $n = d_6^2 + d_7^2 - 1$, where $d_1, d_2, \\ldots, d_k$ are all the divisors of $n$ with $1 = d_1 < d_2 < \\ldots < d_k = n$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\n1=d_{1}<d_{2}<\\dots <d_{k}=n ,\n\\qqua... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "How many corner turns are at least necessary to move a rectangular outdoor enclosure, which is 2 meters long and 1 meter wide, by exactly its width (1 meter) in the same direction, such that each corner of the enclosure is ultimately exactly 1 meter away from its starting posit... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given a $2n$-vertices equilateral polygon and its circumcircle $(O)$, find the number of pentagons whose vertices are five of the $2n$ vertices and contain $O$ inside (with $O$ not lying on the edges).",
"tool_calls": null
},
{
"role": "assistant",
"content": "*... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "How many guesses are needed for Banana to figure out the ten-esque quadruple $(a_1, a_2, a_3, a_4)$ chosen by Ana, given that at each stage Banana offers a ten-esque quadruple $(x_1, x_2, x_3, x_4)$ and Ana tells her the value of \n\\[|a_1 - x_1| + |a_2 - x_2| + |a_3 - x_3| + |... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Determine the least $m$ such that every distribution of stickers among 33 friends, where for any two friends there are at least $m$ stickers that both don't have, results in an incomplete album (i.e., there is at least one sticker that no friend has).",
"tool_calls": null
... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "In a country with 2018 cities, each city is connected by roads to at least 3 other cities, and it is possible to travel from any city to any other city using these roads. Determine the maximum number of roads that can be included in the shortest route between any two cities.",
... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "An invisible tank is on a $100 \\times 100$ table. A cannon can fire at any $k$ cells of the board. After each shot, the tank moves to one of the adjacent cells (by side). Determine the smallest value of $k$ such that the cannon can definitely hit the tank after some time.",
... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given a cyclic quadrilateral with three known sides \\(a\\), \\(b\\), and \\(c\\), compute the smallest possible value for the fourth side in terms of \\(a\\), \\(b\\), \\(c\\), and the diagonals.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "In the middle of a vast prairie, a firetruck is stationed at the intersection of two perpendicular straight highways. The truck travels at $50$ miles per hour along the highways and at $14$ miles per hour across the prairie. Consider the set of points that can be reached by the... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given a positive integer \\( m \\), find the smallest real number \\( \\lambda \\) (possibly depending on \\( m \\)) such that for any positive integer \\( n \\) and any reals \\( x_1, \\cdots, x_n \\in [0,1] \\), there exists an \\( m \\)-element set \\( B \\) of real numbers ... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Find the number of pairwise coprime triples of positive integers $(a, b, c)$ with $a < b < c$ such that $a \\mid (bc - 31)$, $b \\mid (ca - 31)$, and $c \\mid (ab - 31)$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\nLet \n\n\\[\na\... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "How many ways are there to split a $4$ by $4$ unit square into four rectangles with integer side lengths by cutting along the unit lines?",
"tool_calls": null
},
{
"role": "assistant",
"content": "We work on the \\(4\\times 4\\) grid whose interior grid‑lines ar... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "How many ways are there to fill a $3 \\times 3$ tic-tac-toe board such that it results in a win for either player, with no ties?",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution Overview**\n\nWe consider a completely filled \\(3\\times3\\) b... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Let \\( A_1, A_2, A_3, A_4 \\) be four points in the plane, with \\( A_4 \\) being the centroid of \\( \\triangle A_1A_2A_3 \\). Find a point \\( A_5 \\) on the plane that maximizes the ratio\n\\[\n\\frac{\\min\\limits_{1\\le i<j<k\\le 5}(A_iA_jA_k)}{\\max\\limits_{1\\le i<j<k\... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given two positive integers \\( p \\) and \\( q \\), there is a blackboard with \\( n \\) positive integers written on it. An operation consists of choosing two identical numbers \\( a \\) and \\( a \\) on the blackboard and replacing them with \\( a + p \\) and \\( a + q \\). ... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given the sequences \\( x_{n+1} = x_n^2 - 2 \\) and \\( y_{n+1} = (y_n^2 - 2)^2 - 2 \\) with initial values \\( x_0 = 4 \\) and \\( y_0 = 14 \\), find the smallest integer \\( k \\) such that \\( \\gcd(y_k - x_k, 100!) = 100! \\).",
"tool_calls": null
},
{
"role": "... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Alison has compiled a list of 20 hockey teams, ordered by how good she thinks they are. Benjamin can ask her about any three teams and she will either tell him which she thinks is the weakest or which she thinks is the strongest. Determine the largest $N$ such that Benjamin can... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution Overview**\n\nLet \n\n\\[\nL=\\operatorna... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "In a game where the set \\(\\{1, 2, \\ldots, 10\\}\\) is partitioned into equally-sized sets \\(A\\) and \\(B\\), Alice and Bob each choose an element from \\(A\\) or \\(B\\) respectively in five consecutive rounds. The player who chooses the larger number receives a point, and... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given a 5x5 table filled with the numbers 1 through 25 (each number used exactly once), determine the smallest value of \\( k \\) such that the sum of the numbers in any 3x3 subtable does not exceed \\( k \\).",
"tool_calls": null
},
{
"role": "assistant",
"cont... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given that one root of the equation \\(x^4 - 5x^3 + ax^2 + bx + c = 0\\) is \\(3 + \\sqrt{2}\\), and all roots are real, find the extreme values of \\(a\\), \\(b\\), and \\(c\\) (where \\(a\\), \\(b\\), and \\(c\\) are rational numbers). Also, determine the roots of the equatio... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "In an isosceles triangle \\(ABC\\) with \\(BC = 80\\) and \\(AB = 45\\), point \\(M\\) is on \\(BC\\) such that \\(BM = 50\\). Points \\(P\\) and \\(N\\) are on sides \\(AB\\) and \\(AC\\) respectively, such that triangle \\(MPN\\) is isosceles with base \\(PN\\) and similar to... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Among all the planes that are tangent to the surface $x(yz)^2 = 1$, find the one that is farthest from the origin.",
"tool_calls": null
},
{
"role": "assistant",
"content": "The surface is given implicitly by \n\n\\[\nF(x,y,z)=x(yz)^{2}-1=0 .\n\\]\n\nFor a poin... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given a function \\( f(x) \\) such that \\( f(0) = 0 \\), \\( \\int_{0}^{\\frac{\\pi}{4}} [f'(x)]^2 \\, dx = 2 \\), and \\( \\int_{0}^{\\frac{\\pi}{4}} \\sin(2x) f(x) \\, dx = \\frac{1}{2} \\), find the value of \\( \\int_{0}^{\\frac{\\pi}{4}} f(x) \\, dx \\).",
"tool_calls... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "A sequence \\( x_1, x_2, \\ldots, x_n, \\ldots \\) consists of an initial block of \\( p \\) positive distinct integers that then repeat periodically. Given that \\( 1 \\le p \\le 10 \\), find the least \\( n \\) such that there is a strategy which allows to find \\( p \\) by r... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "If 10 different balls are to be placed in 4 distinct boxes at random, what is the probability that two of these boxes contain exactly 2 and 3 balls?",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Total number of placements**\n\nEach of the... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Choose a natural number (up to three digits) and write it. Then, add the squares of its digits to get a second number. Repeat this process, writing the result each time, until a number repeats. Which starting number will produce the longest list of unique numbers?",
"tool_c... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "For each permutation $\\sigma$ of the set $\\{1, 2, \\ldots, N\\}$, define its correctness as the number of triples $1 \\leqslant i < j < k \\leqslant N$ such that $\\sigma(j)$ lies between $\\sigma(i)$ and $\\sigma(k)$. Find the difference between the number of permutations wi... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given 2013 cards, each with a distinct number and initially face down, player $A$ can pick 10 arbitrary cards in a move. Player $B$ then announces one of the numbers on these 10 cards, but does not reveal which card it is on. Determine the maximum number of cards such that play... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Given a number \\( n \\) with divisors \\( 1 = d_1 < d_2 < \\cdots < d_k = n \\) arranged in ascending order, if \\( d_7^2 + d_{15}^2 = d_{16}^2 \\), find \\( d_{17} \\) (assuming \\( k \\leq 17 \\)).",
"tool_calls": null
},
{
"role": "assistant",
"content": "Fi... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Let \\( M \\) be a set of \\( n \\) elements. How many composition laws (operations) can be defined on \\( M \\) such that \\( M \\) must be a monoid?",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\nM=\\{x_{1},x_{2},\\ldots ,x_{n}\\}\n\\... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "A circle with diameter \\(AB\\) intersects a circle with center \\(A\\) at points \\(C\\) and \\(D\\). The point \\(E\\) is the intersection of \\(AB\\) and \\(CD\\). Given that \\(P\\) is a point on the circle such that \\(PC = 16\\) cm, \\(PD = 28\\) cm, and \\(PE = 14\\) cm,... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Consider an isosceles trapezoid with two circles inscribed such that each circle is tangent to two non-parallel sides of the trapezoid and to each other. Find the length of the segment $AB$ that connects the centers of the two circles.",
"tool_calls": null
},
{
"rol... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "Determine the smallest real constant \\( c \\) such that for any positive real numbers \\( a_1, a_2, a_3, a_4, a_5 \\), not necessarily distinct, one may choose pairwise distinct subscripts \\( i, j, k, l \\) such that \\(\\left|\\frac{a_i}{a_j} - \\frac{a_k}{a_l}\\right| \\leq... | null |
Math | nemotron-math-v2-high-aime-style | null | [
{
"role": "user",
"content": "In the coordinate plane, a point is called a lattice point if both of its coordinates are integers. Let \\( A \\) be the point \\((12, 84)\\). Find the number of right-angled triangles \\( ABC \\) where \\( B \\) and \\( C \\) are lattice points, having a right angle at vertex ... | null |
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