{
  "index": "2011-B-1",
  "type": "NT",
  "tag": [
    "NT",
    "ANA"
  ],
  "difficulty": "",
  "question": "$\\epsilon > 0$, there are positive integers $m$ and $n$ such that\n\\[ \\epsilon < |h \\sqrt{m} - k \\sqrt{n}| < 2\\epsilon.  \\]",
  "solution": "Since the rational numbers are dense in the reals, we can find positive integers $a,b$\nsuch that\n\\[\n\\frac{3\\epsilon}{hk} < \\frac{b}{a}\n< \\frac{4\\epsilon}{hk}.\n\\]\nBy multiplying $a$ and $b$ by a suitably large positive integer, we can also ensure that $3a^2 > b$. We then have\n\\[\n\\frac{\\epsilon}{hk} < \\frac{b}{3a} < \\frac{b}{\\sqrt{a^2+b} + a} = \\sqrt{a^2+b} - a\n\\]\nand\n\\[\n\\sqrt{a^2+b} - a = \\frac{b}{\\sqrt{a^2+b} + a} \\leq \\frac{b}{2a} < 2 \\frac{\\epsilon}{hk}.\n\\]\nWe may then take $m = k^2 (a^2+b), n = h^2 a^2$.",
  "vars": [
    "m",
    "n",
    "a",
    "b"
  ],
  "params": [
    "\\\\epsilon",
    "h",
    "k"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "m": "measure",
        "n": "numberr",
        "a": "anchor",
        "b": "ballast",
        "\\epsilon": "tolerance",
        "h": "height",
        "k": "keyvar"
      },
      "question": "For every $tolerance > 0$, there are positive integers $measure$ and $numberr$ such that\n\\[ tolerance < |height \\sqrt{measure} - keyvar \\sqrt{numberr}| < 2tolerance.  \\]",
      "solution": "Since the rational numbers are dense in the reals, we can find positive integers $anchor,ballast$\nsuch that\n\\[\n\\frac{3tolerance}{heightkeyvar} < \\frac{ballast}{anchor}\n< \\frac{4tolerance}{heightkeyvar}.\n\\]\nBy multiplying $anchor$ and $ballast$ by a suitably large positive integer, we can also ensure that $3anchor^2 > ballast$. We then have\n\\[\n\\frac{tolerance}{heightkeyvar} < \\frac{ballast}{3anchor} < \\frac{ballast}{\\sqrt{anchor^2+ballast} + anchor} = \\sqrt{anchor^2+ballast} - anchor\n\\]\nand\n\\[\n\\sqrt{anchor^2+ballast} - anchor = \\frac{ballast}{\\sqrt{anchor^2+ballast} + anchor} \\leq \\frac{ballast}{2anchor} < 2 \\frac{tolerance}{heightkeyvar}.\n\\]\nWe may then take $measure = keyvar^2 (anchor^2+ballast), numberr = height^2 anchor^2$. "
    },
    "descriptive_long_confusing": {
      "map": {
        "m": "cloudshade",
        "n": "emberlight",
        "a": "harvestor",
        "b": "dragonfly",
        "\\epsilon": "\\porcupine",
        "h": "sandcastle",
        "k": "lighthouse"
      },
      "question": "$\\porcupine > 0$, there are positive integers $cloudshade$ and $emberlight$ such that\n\\[ \\porcupine < |sandcastle \\sqrt{cloudshade} - lighthouse \\sqrt{emberlight}| < 2\\porcupine.  \\]",
      "solution": "Since the rational numbers are dense in the reals, we can find positive integers $harvestor,dragonfly$\nsuch that\n\\[\n\\frac{3\\porcupine}{sandcastle lighthouse} < \\frac{dragonfly}{harvestor}\n< \\frac{4\\porcupine}{sandcastle lighthouse}.\n\\]\nBy multiplying $harvestor$ and $dragonfly$ by a suitably large positive integer, we can also ensure that $3harvestor^2 > dragonfly$. We then have\n\\[\n\\frac{\\porcupine}{sandcastle lighthouse} < \\frac{dragonfly}{3harvestor} < \\frac{dragonfly}{\\sqrt{harvestor^2+dragonfly} + harvestor} = \\sqrt{harvestor^2+dragonfly} - harvestor\n\\]\nand\n\\[\n\\sqrt{harvestor^2+dragonfly} - harvestor = \\frac{dragonfly}{\\sqrt{harvestor^2+dragonfly} + harvestor} \\leq \\frac{dragonfly}{2harvestor} < 2 \\frac{\\porcupine}{sandcastle lighthouse}.\n\\]\nWe may then take $cloudshade = lighthouse^2 (harvestor^2+dragonfly), emberlight = sandcastle^2 harvestor^2$. "
    },
    "descriptive_long_misleading": {
      "map": {
        "m": "negativeinteger",
        "n": "zerointeger",
        "a": "noninteger",
        "b": "fractional",
        "\\epsilon": "\\giganticvalue",
        "h": "depthscalar",
        "k": "horizontalvalue"
      },
      "question": "$\\giganticvalue > 0$, there are positive integers $negativeinteger$ and $zerointeger$ such that\n\\[ \\giganticvalue < |depthscalar \\sqrt{negativeinteger} - horizontalvalue \\sqrt{zerointeger}| < 2\\giganticvalue.  \\]",
      "solution": "Since the rational numbers are dense in the reals, we can find positive integers $noninteger,fractional$ such that\n\\[\n\\frac{3\\giganticvalue}{depthscalarhorizontalvalue} < \\frac{fractional}{noninteger}\n< \\frac{4\\giganticvalue}{depthscalarhorizontalvalue}.\n\\]\nBy multiplying $noninteger$ and $fractional$ by a suitably large positive integer, we can also ensure that $3noninteger^2 > fractional$. We then have\n\\[\n\\frac{\\giganticvalue}{depthscalarhorizontalvalue} < \\frac{fractional}{3noninteger} < \\frac{fractional}{\\sqrt{noninteger^2+fractional} + noninteger} = \\sqrt{noninteger^2+fractional} - noninteger\n\\]\nand\n\\[\n\\sqrt{noninteger^2+fractional} - noninteger = \\frac{fractional}{\\sqrt{noninteger^2+fractional} + noninteger} \\leq \\frac{fractional}{2noninteger} < 2 \\frac{\\giganticvalue}{depthscalarhorizontalvalue}.\n\\]\nWe may then take $negativeinteger = horizontalvalue^2 (noninteger^2+fractional), zerointeger = depthscalar^2 noninteger^2$.}  \n"
    },
    "garbled_string": {
      "map": {
        "m": "wplskqaz",
        "n": "xjcdarbe",
        "a": "vghtyuil",
        "b": "czoupqer",
        "\\epsilon": "qzxwvtnp",
        "h": "hjgrksla",
        "k": "bmncrdle"
      },
      "question": "$qzxwvtnp > 0$, there are positive integers $wplskqaz$ and $xjcdarbe$ such that\n\\[ qzxwvtnp < |hjgrksla \\sqrt{wplskqaz} - bmncrdle \\sqrt{xjcdarbe}| < 2qzxwvtnp.  \\]\n",
      "solution": "Since the rational numbers are dense in the reals, we can find positive integers $vghtyuil,czoupqer$\nsuch that\n\\[\n\\frac{3qzxwvtnp}{hjgrkslabmncrdle} < \\frac{czoupqer}{vghtyuil}\n< \\frac{4qzxwvtnp}{hjgrkslabmncrdle}.\n\\]\nBy multiplying $vghtyuil$ and $czoupqer$ by a suitably large positive integer, we can also ensure that $3vghtyuil^2 > czoupqer$. We then have\n\\[\n\\frac{qzxwvtnp}{hjgrkslabmncrdle} < \\frac{czoupqer}{3vghtyuil} < \\frac{czoupqer}{\\sqrt{vghtyuil^2+czoupqer} + vghtyuil} = \\sqrt{vghtyuil^2+czoupqer} - vghtyuil\n\\]\nand\n\\[\n\\sqrt{vghtyuil^2+czoupqer} - vghtyuil = \\frac{czoupqer}{\\sqrt{vghtyuil^2+czoupqer} + vghtyuil} \\leq \\frac{czoupqer}{2vghtyuil} < 2 \\frac{qzxwvtnp}{hjgrkslabmncrdle}.\n\\]\nWe may then take $wplskqaz = bmncrdle^2 (vghtyuil^2+czoupqer), xjcdarbe = hjgrksla^2 vghtyuil^2$. "
    },
    "kernel_variant": {
      "question": "Let $h,k$ be fixed positive integers and let $\\varepsilon>0$.  Prove that there exist positive integers $m$ and $n$ such that\n\\[\n\\varepsilon\\;<\\;\\bigl|\\,h\\sqrt m-k\\sqrt n\\,\\bigr|\\;<\\;3\\varepsilon.\n\\]",
      "solution": "Fix ε>0 and positive integers h,k.  By the density of the rationals we can choose positive integers a,b such that\n  (1)   3ε/(hk) < b/a < 6ε/(hk).  \nIf necessary, replace (a,b) by (t a, t b) for a large integer t so that also\n  (2)   b < 3a².\n\nDefine Δ = √(a²+b) − a.  Then algebraically\n  Δ = b/(√(a²+b) + a).  \nSince b<3a² we have\n  √(a²+b) ≤ √(a²+3a²)=2a,  so  √(a²+b)+a ≤3a,\nand plainly √(a²+b)>a gives  √(a²+b)+a>2a.  Hence\n  b/(3a) ≤ b/(√(a²+b)+a) = Δ < b/(2a).\n\nUsing (1) we get\n  lower:  Δ > b/(3a) > (3ε/(hk))/3 = ε/(hk),\n  upper:  Δ < b/(2a) < (6ε/(hk))/2 = 3ε/(hk).\n\nFinally set\n  m = k²(a²+b),   n = h²a².\nThen √m = k√(a²+b),  √n = ha,  so\n  |h√m − k√n| = hk·(√(a²+b)−a) = hk·Δ,\nand therefore\n  ε < hk·Δ < 3ε,\nas required.",
      "_meta": {
        "core_steps": [
          "Use density of the rationals to pick positive integers a,b with b/a lying in a prescribed ε–interval around ε/(hk).",
          "Dilate (a,b) so that b is smaller than a fixed multiple of a², guaranteeing a concrete upper bound on √(a²+b)+a.",
          "Convert the desired square-root difference with the identity √(a²+b)−a = b/(√(a²+b)+a).",
          "Apply the prepared upper / lower bounds on the denominator to force the expression between ε/(hk) and 2ε/(hk).",
          "Define m = k²(a²+b) and n = h²a² to obtain ε < |h√m − k√n| < 2ε."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Lower constant used when trapping b/a between two multiples of ε/(hk)",
            "original": "3"
          },
          "slot2": {
            "description": "Upper constant used when trapping b/a between two multiples of ε/(hk)",
            "original": "4"
          },
          "slot3": {
            "description": "Multiplier in the condition b < 3a² that provides the denominator upper bound ( < 3a )",
            "original": "3"
          },
          "slot4": {
            "description": "Coefficient 2 in the final target inequality ε < |…| < 2ε stated in the problem",
            "original": "2"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}