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": "Study the concavity and find the inflection points of the function \\( f(x) = \\frac{x^4}{4} - 2x^3 + \\frac{9x^2}{2} \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Compute the second derivative**\n\n\\[\nf(x)=\\frac{x^{4}}{4}-2x^{3}+... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\( a \\) be a fixed positive number (\\( a > 0 \\)) and \\( z \\) be a complex number such that \\( \\left| z + \\frac{1}{z} \\right| = a \\). Find the maximum and minimum values of \\( |z| \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Determine all positive integers $n$ for which there exists a polynomial $P(x)$ with integer coefficients such that for each integer $m$, the number $P(m)$ is divisible by $n$ if and only if $m$ is not divisible by $n$.",
"tool_calls": null
},
{
"role": "assistant",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "The numbers from $1$ to $100$ are placed in a circle. A number is underlined if it divides its left neighboring number. What is the highest possible quantity of underlined numbers?",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\na_{1},a_... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the equation $f(x) = \\sin(F(x))$ where $F(x) = \\int f(x) \\, dx$ and $f: \\mathbb{R} \\to \\mathbb{R}$, $F: \\mathbb{R} \\to \\mathbb{R}$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "We are looking for real‑valued functions \\(f\\) and the... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all \\( t \\) such that for all \\( x, y, z \\in \\mathbb{R}^* \\):\n\\[ 2(x^3 + y^3 + z^3) + 3(1 + 3t)xyz \\geq (1 + t)(x + y + z)(xy + yz + zx) \\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n2(x^{3}+y^{3}+z^{3})+3(1+3t)xyz\\ge (1+t)(x... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all integers \\( x \\) such that \\( 2^{182} + 4^x + 8^{700} \\) is a perfect square.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nS(x)=2^{182}+4^{x}+8^{700}=2^{182}+2^{2x}+2^{2100}\n\\]\n\nAll three summands are powers of \\(2\\). \nIf ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the integral $\\int\\frac{1}{(a \\sin^2 x + b \\cos^2 x)^2} \\, dx$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\n\\[\nI=\\int\\frac{dx}{\\bigl(a\\sin ^2x+b\\cos ^2x\\bigr)^{2}}\\qquad \n(a\\neq0,\\;b\\neq0)\n\\]\n\n---\n\n... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "A checkerboard with size \\( 109 \\times 21 \\) will be tiled with some tiles with size \\( 3 \\times 1 \\). What is the maximum number of tiles which can be placed on the board so that no two tiles overlap, or touch (touching on each tile's vertices are also not allowed)?",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find positive distinct integers \\( x_1, x_2, \\ldots, x_{2002} \\) such that \\( x_1^2 + x_2^2 + \\cdots + x_{2002}^2 = 1335(x_1 + x_2 + \\cdots + x_{2002}) \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\nS=\\sum_{i=1}^{2002}x_i ,\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Simplify the expression $$\\dfrac{a^5+(a-1)^4}{(a-1)^5-a^4}.$$",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\frac{a^5+(a-1)^4}{(a-1)^5-a^4}\n\\]\n\n---\n\n### 1. Expand the powers \n\n\\[\n\\begin{aligned}\n(a-1)^4 &= a^4-4a^3+6a^2-4a+1,\\\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Simplify the expression\n$$\\cos^7{\\frac{\\pi}{9}}+\\cos^7{\\frac{5\\pi}{9}}+\\cos^7{\\frac{7\\pi}{9}}.$$",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nS=\\cos ^7\\frac{\\pi}{9}+\\cos ^7\\frac{5\\pi}{9}+\\cos ^7\\frac{7\\pi}{9}\n\\]\n\n---\n\n... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Three people sold their chickens at the market. The first person had 10 chickens, the second had 16, and the third had 26. They sold some chickens at a higher price in the morning and the rest at a lower price in the afternoon. Each person collected a total of 35 monies by the ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "How can we study the changes of the function $f(x) = x^2 - \\sqrt{x}$ without using the derivative?",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step‑by‑step study of** \n\n\\[\nf(x)=x^{2}-\\sqrt{x},\\qquad x\\ge 0\n\\]\n\n---\n\n### 1. Domain ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given a plane $P$, find the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 onto $P$, and determine the maximum area of $T'$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\nT=\\{V_{1},V_{2},V_{3},... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the $2$-adic valuation $v_2$ of the sum $1 + \\frac{1}{3} + \\frac{1}{5} + \\cdots + \\frac{1}{2k-1}$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nS_k:=1+\\frac13+\\frac15+\\cdots+\\frac1{2k-1}\\qquad(k\\ge 1)\n\\]\n\nLet \n\n\\[\nD:=\\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\( p \\) and \\( q \\) be real numbers. Define a sequence \\( \\{a_n\\} \\) by \\( a_1 = p \\), \\( a_2 = q \\), and \\( a_{n+2} = 2a_{n+1} + a_n \\) for \\( n = 1, 2, \\ldots \\). Find the necessary and sufficient condition on \\( p \\) and \\( q \\) for which the series ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given a tree $T$, starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for a... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "A frog sitting at the point (1, 2) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length 1, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given a twentieth degree polynomial with integer coefficients, what is the largest number of marked points with integer coordinates and ordinates between 0 and 10 (inclusive) that can lie on the graph of this polynomial?",
"tool_calls": null
},
{
"role": "assistant"... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "How many ways are there to color an $n \\times n$ board with red and black such that every $2 \\times 2$ subsquare contains exactly two red and two black squares, for $n \\geq 2$?",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\na_{i,j}= ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve for \\( x \\) in the equation \\((1-\\tan{x})(1+\\tan{x})(\\tan^2{x}+1)+2^{\\tan^2{x}}=0\\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n(1-\\tan x)(1+\\tan x)(\\tan ^2x+1)+2^{\\tan ^2x}=0\n\\Longrightarrow (1-\\tan ^2x)(\\tan ^2x+1)+2^... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the real numbers \\( a \\) that satisfy the equation\n\\[ \\lfloor nx \\rfloor = \\sum_{k=1}^{n} \\lfloor x + (k-1)a \\rfloor, \\]\nfor any real numbers \\( x \\) and for a natural number \\( n \\ge 2. \\)",
"tool_calls": null
},
{
"role": "assistant",
"con... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\( D \\) be a point on side \\( [AB] \\) of triangle \\( ABC \\) with \\( |AB| = |AC| \\) such that \\( [CD] \\) is an angle bisector and \\( m(\\widehat{ABC}) = 40^\\circ \\). Let \\( F \\) be a point on the extension of \\( [AB] \\) after \\( B \\) such that \\( |BC| = |... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the system of equations:\n\\[\n\\begin{cases}\nx^2(y+3) = 4(2-y) \\\\\ny^2(z+3) = 4(2-z) \\\\\nz^2(x+3) = 4(2-x)\n\\end{cases}\n\\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\begin{cases}\nx^2(y+3)=4(2-y)\\\\[2mm]\ny^2(z+3)=4(2-z)\\\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Two ants start at points $(-1,1)$ and $(1,1)$ and walk to the right along the parabola $y=x^2$. Their midpoint moves along the line $y=1$ with constant speed $1$. What is the speed of the left ant when it first hits the line $y=1/2$?",
"tool_calls": null
},
{
"role"... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find a positive integer \\( n \\) for which\n\\[ \\bigg\\lfloor 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\frac{1}{\\sqrt{4}} + \\cdots + \\frac{1}{\\sqrt{n}} \\bigg\\rfloor = 12. \\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "First note... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "What is the maximum value of \n\\[\n\\sqrt{|x_1 - x_2|} + \\sqrt{|x_2 - x_3|} + \\sqrt{|x_3 - x_4|} + \\cdots + \\sqrt{|x_{n-1} - x_n|} + \\sqrt{|x_n - x_1|}\n\\]\nwhere \\(0 \\leq x_i \\leq 1\\) for \\(i = 1, 2, 3, \\ldots, n\\) and \\(n\\) is an odd integer greater than or eq... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\( f, g : \\mathbb{R} \\to \\mathbb{R} \\) such that \\( f \\) is Lebesgue measurable on \\( \\mathbb{R} \\) and \\( g \\) is continuous on \\( \\mathbb{R} \\). Determine whether \\( f \\circ g \\) is measurable. Provide a counterexample if it is not always measurable.",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "A sequence of numbers is generated as follows: Let \\( a_1 = 1 \\) and for \\( n \\ge 1 \\), choose \\( a_{n+1} \\) at random from the positive integers from 1 to \\( 10 - a_n \\), inclusive. On average, what fraction of the terms in the sequence will be equal to 1?",
"tool... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the equation \\( x^2 + 7x = (2x + 1) \\sqrt{x^2 + x + 6} \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nx^{2}+7x=(2x+1)\\sqrt{x^{2}+x+6}\\qquad (1)\n\\]\n\n---\n\n### 1. Isolate the square‑root \n\n\\(2x+1\\neq0\\) because for \\(x=... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\(\\beta \\in (0,1]\\) be fixed. Let \\(x, y \\in \\left[0, \\frac{\\pi}{2}\\right)\\) such that \\(\\tan x + \\tan y = \\sqrt{2\\beta}\\). Find the minimum and the maximum possible values of \\(\\cos x + \\cos y\\).",
"tool_calls": null
},
{
"role": "assistant... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the integral \\(\\int \\frac{x^4-1}{x^2\\sqrt{x^4+x^2+1}} \\, dx\\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nI=\\int\\frac{x^{4}-1}{x^{2}\\sqrt{x^{4}+x^{2}+1}}\\;dx\n\\]\n\n---\n\n### 1. Simplify the integrand \n\n\\[\n\\frac{x... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Determine the smallest number of squares that need to be colored on a \\( 1001 \\times 1001 \\) board such that:\n(i) If two squares share a side, then at least one of them must be colored;\n(ii) Among any six successive squares in a row or in a column, some two adjacent ones m... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the least positive integer $n$ such that $\\cos\\frac{\\pi}{n}$ cannot be written in the form $p+\\sqrt{q}+\\sqrt[3]{r}$ with $p,q,r\\in\\mathbb{Q}$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\text{Find the smallest }n\\text{ such tha... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let \\(a\\), \\(b\\), \\(c\\), and \\(d\\) be nonnegative numbers. Determine the least positive \\(\\alpha\\) and the best positive \\(k\\) for which the following inequality holds:\n\\[\n\\sqrt{1+\\alpha \\sum (a-b)^2} + k \\left(\\frac{abc + abd + acd + bcd}{4}\\right) \\ge (... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the equation: $\\tan\\left(\\frac{3\\pi}{4} \\cdot \\frac{x+6}{x+3}\\right) \\cdot \\tan\\left(\\frac{3\\pi}{4} \\cdot \\frac{x-1}{x+1}\\right) = 1$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\tan\\Bigl(\\frac{3\\pi }4\\,\\frac{x+6}{... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the integral $\\int \\frac{\\arcsin\\frac{x}{2}}{\\sqrt{2-x}} \\, dx$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nI=\\int \\frac{\\arcsin\\frac{x}{2}}{\\sqrt{2-x}}\\;dx\n\\]\n\n---\n\n### 1. Integration by parts \n\nTake \n\n\\[\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all \\( x, y \\in \\mathbb{N} \\) satisfying \\( \\frac{x^2 + y}{xy + 1} \\in \\mathbb{Z} \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\begin{aligned}\n&\\text{Let }d=xy+1.\\quad d\\mid x^{2}+y\\Longrightarrow \n\\frac{x^{2}+y}{xy+1... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let $\\ell$ be a line in the plane and let $90^\\circ < \\theta < 180^\\circ$. Consider distinct points $P, Q, R$ such that:\n- $P$ lies on $\\ell$ and $PQ$ is perpendicular to $\\ell$.\n- $R$ lies on the same side of $\\ell$ as $Q$, and $R$ does not lie on $\\ell$.\n- For any ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all primes \\( p \\) and \\( q \\) such that \\( q \\mid p^q + 1 \\) and \\( q + p \\mid q^{q+p} - p^p \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\text{Find all prime pairs }(p,q)\\text{ such that } \n\\begin{cases}\nq\\mid p^{\\,... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all convex polygons for which the following property holds: for any interior point of the given polygon and for any side, the foot of the perpendicular drawn from this point on the selected side lies inside that side.",
"tool_calls": null
},
{
"role": "assistan... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the integral \\(\\int_0^1 \\frac{\\tanh^{-1} x \\cdot \\ln x}{x(1-x)(1+x)} \\, dx\\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\n\\[\nI=\\int_{0}^{1}\\frac{\\operatorname{artanh}x\\,\\ln x}{x\\,(1-x)(1+x)}dx\n =\\int_... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the system of equations:\n\\[\n\\left\\{\\begin{matrix}\n(x+y)\\sqrt{x^2-4x+5}+(x-2)\\sqrt{(x+y)^2+1}=0\n\\\\\nx-y(x^2-y^2)=2(1-x^2+y^2)\n\\end{matrix}\\right.\n\\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\begin{cases}\n(x+y)\\sqrt... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the sum $\\sum_{n=1}^{\\infty} \\frac{1}{2^n (1+\\sqrt[2^n]{2})}$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nS=\\sum_{n=1}^{\\infty}\\frac{1}{2^{n}\\bigl(1+\\sqrt[2^{n}]{2}\\bigr)}\n\\]\n\n---\n\n**1. Rewrite each term**\n\n\\[\n\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all values of positive integers $a$ and $b$ such that it is possible to place $a$ ones and $b$ zeros at the vertices of a polygon with $a+b$ sides. After rotating the numbers at the vertices, exactly one pair of neighboring $0$ and $1$ should switch places, while all other... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "How many integer tuples $(x, y, z)$ satisfy $0 \\leq x, y, z < 2011$, $xy + yz + zx \\equiv 0 \\pmod{2011}$, and $x + y + z \\equiv 0 \\pmod{2011}$?",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\np=2011\\qquad(\\text{prime}).\n\\]\n\nAl... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the sum $\\tan(1) + \\tan(5) + \\tan(9) + \\cdots + \\tan(173) + \\tan(177)$. Note: $\\tan$ is the tangent function.",
"tool_calls": null
},
{
"role": "assistant",
"content": "The angles are equally spaced by \\(4^{\\circ}\\).\nSince \n\n\\[\n4^{\\circ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all real numbers \\( a \\) such that the equation \\( x(x+1)^3 = (2x + a + 1)(x + a) \\) has exactly four real roots.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nx(x+1)^{3}=(2x+a+1)(x+a)\n\\qquad (x\\in\\mathbb R)\n\\]\n\n---\n\n### 1. ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all harmonic functions \\( u: \\mathbb{R}^n \\to \\mathbb{R} \\) such that \\( |u(x)| \\le C|x|^m \\) for all \\( |x| \\ge 1 \\), where \\( C \\) is a constant and \\( m \\in (0,2) \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Answer.** ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the following system of equations:\n\\[\n\\left\\{\n\\begin{matrix}\nx^5 = 2x + 7y + 27 \\\\\ny^5 = 5x + 4y + 27\n\\end{matrix}\n\\right.\n\\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "First observe that the two equations give a simple rel... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "A piece of paper in the shape of a right triangle is folded such that the smaller leg rests on the hypotenuse. If the segment $a = 9$ and segment $b = \\frac{24}{5}$, calculate the lengths of the sides of the paper.",
"tool_calls": null
},
{
"role": "assistant",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "There are a number of towns on a circular road. They are served by five buses. Each bus travels the entire road but only stops at five towns. For every pair of towns, there is a bus that stops at both. Find the largest possible number of towns.",
"tool_calls": null
},
{... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the system of equations:\n\\[ \\begin{cases} x^3 + y^3 = 68 \\\\ x^2 - y^2 = 7 \\sqrt{5} \\end{cases} \\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\begin{cases}\nx^3+y^3=68\\\\[2mm]\nx^2-y^2=7\\sqrt5\n\\end{cases}\n\\]\n\n---\n\n**1... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Bob and Tom each have a cube and some red and blue paint. They paint each side of their cubes either red or blue. What is the probability that the two painted cubes are indistinguishable?",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Coun... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all triples \\((k, m, n)\\) of positive integers such that \\(m\\) is a prime and the following conditions are satisfied:\n1. \\(kn\\) is a perfect square.\n2. \\(\\frac{k(k-1)}{2} + n\\) is a fourth power of a prime.\n3. \\(k - m^2 = p\\) where \\(p\\) is a prime.\n4. \\(... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all polynomials \\( P(x) \\) with odd degree such that \\( P(x^2 - 2) = P^2(x) - 2 \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\boxed{P(x)=2\\,T_{2k+1}\\!\\Bigl(\\frac{x}{2}\\Bigr)\\qquad(k=0,1,2,\\dots)}\n\\]\n\nwhere \\(T_n\\) de... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all polynomials $P$ with real coefficients such that $$(x+1)P(x-1)-(x-1)P(x)=c \\text{ for all } x \\in \\mathbb{R},$$ where $c$ is a given real number.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\boxed{\\,P(x)=\\frac{c}{2}+A\\,x(x+1)\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Consider a hemisphere centered at the origin with radius 1 and $z > 0$. Consider also a cone whose base is the circle of radius 1 in the $x$-$y$ plane centered at the origin and whose vertex is the point $(0,0,2)$. These two surfaces intersect along some curve with $z > 0$. Wha... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the range of the function \\( f(x) = \\frac{2x}{\\sqrt{x^2 + x + 1}} \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1. Domain**\n\n\\[\nf(x)=\\frac{2x}{\\sqrt{x^{2}+x+1}},\\qquad \nx^{2}+x+1=\\Bigl(x+\\tfrac12\\Bigr)^{2}+\\tfrac34>0... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the locus of the point \\( P \\) in the plane of an equilateral triangle \\( ABC \\) that satisfies the condition:\n\n\\[ \\text{max} \\{\\overline{PA}, \\overline{PB}, \\overline{PC} \\} = \\frac{1}{2} \\cdot \\left(\\overline{PA} + \\overline{PB} + \\overline{PC} \\right... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find positive integers $k$ such that there exist 2014 distinct positive integers whose sum is divisible by exactly $k$ distinct positive integers, with each of the 2014 integers being distinct.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Suppose $\\triangle ABC$ is an equilateral triangle with side length $3$. Points $D$, $E$, and $F$ are taken on sides $BC$, $CA$, and $AB$ respectively, such that $BD=1$ and $\\angle ADE = \\angle DEF = 60^\\circ$. Find the length of $AF$.",
"tool_calls": null
},
{
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the maximum value of \\( n \\) such that the sum \\(\\left\\lfloor \\sqrt{1} \\right\\rfloor + \\left\\lfloor \\sqrt{2} \\right\\rfloor + \\cdots + \\left\\lfloor \\sqrt{n} \\right\\rfloor\\) is a prime number.",
"tool_calls": null
},
{
"role": "assistant",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given that $q = \\frac{9}{10}p$, find the values of $p$ and $q$ such that the population at the end of the second year is the same as at the beginning of the first year.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\nLet the initial p... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve the cryptarithm:\n\\[ A2B7 \\]\n\\[ +9C8D \\]\n\\[ \\text{------} \\]\n\\[ 1E80F \\]\n\nWhere each letter represents a unique digit.",
"tool_calls": null
},
{
"role": "assistant",
"content": "The addition forces \n\n\\[\n7+D=F+10c_1,\\qquad B+8+c_1=10,\\q... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find \\( a_n \\) in terms of \\( a_1 \\) if \n\\[ a_{n-1}^2 = 2 \\left( a_n^2 + a_n \\left( \\sum_{k=1}^{n-1} a_k - \\frac{a_1}{2} \\right) \\right). \\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution.** \nLet \n\n\\[\nS_{n}=\\sum_{k=1}^... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Determine if the element $y=5+3\\sqrt[3]{2}+7\\sqrt[3]{4}$ is prime in the ring $R=\\{a+b\\sqrt[3]{2}+c\\sqrt[3]{4}\\mid a,b,c\\in\\mathbb{Z}\\}$. If not, factor it.",
"tool_calls": null
},
{
"role": "assistant",
"content": "The element \n\n\\[\ny=5+3\\sqrt[3]{... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the prime number \\( p \\) such that there exist positive integers \\( n, x, y \\) satisfying \\( p^n = x^3 + y^3 \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\np^n=x^{3}+y^{3}\\qquad (p\\text{ prime},\\;n,x,y>0)\n\\]\n\nFactor the sum... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "In triangle \\(ABC\\), let \\(M\\) be the midpoint of \\(AC\\) and \\(N\\) be the midpoint of \\(AB\\). Let point \\(F\\) lie on \\(MN\\) such that \\(\\angle ABF = \\angle CBF\\). Let \\(MN\\) meet the circumcircle of \\(ABF\\) again at a point \\(P \\neq F\\). If \\(MF=3\\), ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given the function \\( y = \\sin^{-1} \\left( \\frac{2 \\cdot 6^x}{1 + 36^x} \\right) \\), find \\(\\frac{dy}{dx}\\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\ny=\\sin ^{-1}\\!\\Bigl(\\frac{2\\;6^{x}}{1+36^{x}}\\Bigr),\\qquad \nu(x)=\\frac{... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "In $\\triangle ABC$ with side lengths $AB=13$, $BC=14$, and $CA=15$, let $M$ be the midpoint of $\\overline{BC}$. Let $P$ be the point on the circumcircle of $\\triangle ABC$ such that $M$ is on $\\overline{AP}$. There exists a unique point $Q$ on segment $\\overline{AM}$ such ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Suppose there is an ant at each vertex of a regular octahedron. Suddenly each ant randomly (with equal probability) picks a neighboring edge and crawls along it to the next vertex. What is the probability that no two ants collide, neither on a vertex, nor while traveling on an ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Evaluate the double integral \\(\\int_0^a \\int_0^{a-y} \\dfrac{x}{\\left(x^2 + y^2 + (a - x - y)^2\\right)^{\\frac{3}{2}}} \\, dx \\, dy\\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nI=\\int_{0}^{a}\\int_{0}^{\\,a-y}\\frac{x}{\\bigl(x^{2}+... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "On a square board made up of 9 unit squares with both diagonals of each unit square drawn, how many total rectangles can be formed using the line segments created by the original squares and the diagonals?",
"tool_calls": null
},
{
"role": "assistant",
"content"... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Consider the function \\( g(x,y) \\) defined as:\n\\[ g(x,y) = \\begin{cases} \n\\frac{x^3 y}{x - y^2} & \\text{if } x \\neq y^2 \\\\\n0 & \\text{if } y = x^2 \n\\end{cases} \\]\n\nDetermine if the limit \\(\\lim_{(x,y) \\to (0,0)} g(x,y)\\) exists.",
"tool_calls": null
}... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Compute $\\lim_{n\\to\\infty}\\sum_{k=1}^n(-1)^k\\binom{n}{k}^{-1/k}$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n\\[\nS_{n}:=\\sum_{k=1}^{n}(-1)^{k}\\,\\binom{n}{k}^{-1/k}\\qquad (n\\ge 1).\n\\]\n\nWrite \\(k=n p\\;(0<p\\le 1)\\). Usin... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Consider the sequence of integers $a_n$ defined by $a_1 = 1$, $a_p = p$ for prime $p$ and $a_{mn} = ma_n + na_m$ for $m, n > 1$. Find the smallest $n$ such that $\\frac{a_n^2}{2022}$ is a perfect power of $3$.",
"tool_calls": null
},
{
"role": "assistant",
"cont... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Four balls of radius $1$ are mutually tangent, with three resting on the floor and the fourth resting on the others. A tetrahedron, each of whose edges has length $s$, is circumscribed around the balls. Find the value of $s$.",
"tool_calls": null
},
{
"role": "assis... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy the functional equation \\( f(x + y^2 - f(y)) = f(x) \\) for all real numbers \\( x \\) and \\( y \\).",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\nLet ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all real numbers $x$ such that\n\\[\n\\left \\lfloor \\frac{x}{2} \\right \\rfloor + \\left \\lfloor \\frac{x}{3} \\right \\rfloor + \\left \\lfloor \\frac{x}{5} \\right \\rfloor = x\n\\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\beg... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Two cards are drawn from a well-shuffled pack of 52 cards. Find the probability that one of them is a red card and the other is a queen.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Count the relevant cards**\n\n- Total cards in a deck: ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "In a triangle \\(ABC\\), the altitude \\(AD\\) and the median \\(AE\\) divide \\(\\angle A\\) into three equal parts. If \\(BC = 28\\), then what is the nearest integer to \\(AB + AC\\)?",
"tool_calls": null
},
{
"role": "assistant",
"content": "Let \n\n* \\(D\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given three non-negative real numbers \\(a\\), \\(b\\), and \\(c\\) such that \\(|a - b| + |b - c| + |c - a| = 1\\), what can the sum \\(a + b + c\\) be equal to?",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\boxed{a+b+c\\ge\\frac12\\qquad\\t... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) with a continuous derivative that satisfy the following condition: On any closed interval \\([a, b]\\) \\((a < b)\\), there exist \\( m, n \\in [a, b] \\) such that\n\n\\[ f'(m) = \\min_{x \\in [a, b]} f(x) \\quad \\tex... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Solve for \\(x\\):\n\\[ \\sqrt{5-x} = 5-x^2 \\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "First note the domain restrictions.\n\n\\[\n\\sqrt{5-x}\\ \\text{is defined when }5-x\\ge0\\Longrightarrow x\\le5,\n\\]\n\nand the right–hand side must be ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "A box contains 18 green and 12 blue balls. Three balls are drawn without replacement, and then another three balls are drawn without replacement. What is the probability that at least 2 of the first 3 balls drawn are green and at least 2 of the second 3 balls drawn are blue?",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let $\\{a_n\\}$ be a sequence of integers such that for $n \\geq 1$, we have $(n-1) \\cdot a_{n+1} = (n+1) \\cdot a_n - 2(n-1)$. If 2000 divides $a_{1999}$, find the smallest $n \\geq 2$ such that 2000 divides $a_n$.",
"tool_calls": null
},
{
"role": "assistant",
... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Consider a triangle $ABC$. Let $A_1$ be the symmetric point of $A$ with respect to the line $BC$, $B_1$ the symmetric point of $B$ with respect to the line $CA$, and $C_1$ the symmetric point of $C$ with respect to the line $AB$. Determine the possible set of angles of triangle... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all right triangles with integer sides where one of the legs is equal to 60.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Step 1 – Set up the Diophantine equation**\n\nLet the two legs be \\(a=60\\) and \\(b\\) and let the hypotenuse be \\(c... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the smallest two-digit number \\( n \\) such that \\( \\binom{n}{3} \\binom{n}{4} \\binom{n}{5} \\binom{n}{6} \\) is a perfect square.",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\nP(n)=\\binom{n}{3}\\binom{n}{4}\\binom{n}{5}\\binom{n}{6}\... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Let $\\mathcal{C}$ be the set of functions $f: [0,1] \\to \\mathbb{R}$ such that $f$ is twice differentiable on $[0,1]$, there exist $x_1, x_2 \\in [0,1]$ with $x_1 \\neq x_2$ satisfying either $f(x_1) = f(x_2) = 0$ or $f(x_1) = f'(x_1) = 0$, and $f''(x) < 1$ for all $x \\in [0... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Determine all positive integer tuples $(a,b,c,d)$ such that\n\\[ \\frac{ab + bc + cd + 1 + abcd}{2020} = \\frac{2( a + c + abd)}{3131} \\]",
"tool_calls": null
},
{
"role": "assistant",
"content": "\\[\n\\frac{ab+bc+cd+1+abcd}{2020}=\\frac{2(a+c+abd)}{3131}\n\\q... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find the ratio of the volume of a regular $n$-gonal pyramid to that of the sphere inscribed into it knowing that the circles circumscribed about the base and the lateral faces of the pyramid are equal. %",
"tool_calls": null
},
{
"role": "assistant",
"content": ... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Find all $2011$-tuples $(a_1, a_2, \\cdots, a_{2010}, a_{2011})$ that satisfy the conditions $\\sum_{i=1}^{2011} a_i = 2011$ and $|a_1 - a_2| = |a_2 - a_1| = |a_3 - a_1| = \\cdots = |a_{2011} - a_1|$.",
"tool_calls": null
},
{
"role": "assistant",
"content": "Le... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given 12 points \\(a_1, a_2, \\dots, a_{12}\\) in a plane with no three points collinear, and a binary value \\(I_{ij}\\) (0 or 1) assigned randomly for each pair \\(i < j\\), where \\(I_{ij} = 0\\) allows movement from \\(a_i\\) to \\(a_j\\) and \\(I_{ij} = 1\\) allows movemen... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "A positive integer \\( n \\) is called good if it can be uniquely written simultaneously as \\( a_1 + a_2 + \\cdots + a_k \\) and as \\( a_1 a_2 \\cdots a_k \\), where \\( a_i \\) are positive integers and \\( k \\ge 2 \\). Find, in terms of prime numbers, all good natural numb... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the angle bisectors of $\\angle B$ and $\\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\\triangle ABC$.",
"tool_calls": null
},
{
"role": "assistant",... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "Given a pyramid with a square base \\(ABCD\\) and vertex \\(S\\), find the shortest path starting and ending at \\(S\\) that passes through all the vertices of the base.",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution**\n\nLet \n\n* the sq... | null |
Math | nemotron-cascade2-math | null | [
{
"role": "user",
"content": "How many 2023-digit capicua numbers can be formed if they must have at least 2021 equal digits?",
"tool_calls": null
},
{
"role": "assistant",
"content": "**Solution Overview**\n\nA “capicua’’ number is a palindrome. \nLet the length be \\(n=2023\\) (odd). \nF... | null |
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