{
  "index": "1959-B-6",
  "type": "NT",
  "tag": [
    "NT",
    "COMB"
  ],
  "difficulty": "",
  "question": "6. Prove that, if \\( x \\) and \\( y \\) are positive irrationals such that \\( 1 / x+1 / y=1 \\), then the sequences \\( [x],[2 x], \\ldots,[n x], \\ldots \\) and \\( [y],[2 y], \\ldots,[n y], \\ldots \\) together include every positive integer exactly once. (The notation \\( [x] \\) means the largest integer not exceeding \\( x \\).)",
  "solution": "Solution. Evidently \\( x>1 \\), and therefore the numbers \\( 0, x, 2 x, 3 x, \\ldots \\) all differ by more than one. Hence the integers\n\\[\n[x],[2 x],[3 x], \\ldots\n\\]\nare all positive and different. Similarly the integers\n\\[\n[y],[2 y],[3 y], \\ldots\n\\]\nare all positive and different.\nSuppose some integer \\( p \\) appears in both (1) and (2). Say \\( p=[a x]= \\) \\( [b y] \\) where \\( a \\) and \\( b \\) are positive integers. Then \\( p<a x<p+1 \\) and \\( p<b y<p+1 \\). (The possibilities \\( p=a x \\) and \\( p=b y \\) are excluded here, and in similar places elsewhere, because \\( x \\) and \\( y \\) are irrational). Then\n\\[\np=\\frac{p}{x}+\\frac{p}{y}<a+b<\\frac{p+1}{x}+\\frac{p+1}{y}=p+1\n\\]\nand we have found an integer, \\( a+b \\), between the integers \\( p \\) and \\( p+1 \\). This is impossible. Hence no integers appear in both (1) and (2).\n\nSuppose some positive integer \\( \\boldsymbol{p} \\) is not included in either (1) or (2). Then we can find positive integers \\( a \\) and \\( b \\) such that\n\\[\na x<p<p+1<(a+1) x\n\\]\nand\n\\[\nb y<p<p+1<(b+1) y\n\\]\n\nThen\n\\[\n\\frac{a}{p}+\\frac{b}{p}<\\frac{1}{x}+\\frac{1}{y}=1\n\\]\nso \\( a+b<p \\). Also\n\\[\n\\frac{a+1}{p+1}+\\frac{b+1}{p+1}>\\frac{1}{x}+\\frac{1}{y}=1\n\\]\nso \\( a+b+2>p+1 \\). Thus we have found two integers, \\( p \\) and \\( p+1 \\), between the integers \\( a+b \\) and \\( a+b+2 \\), which is impossible. We conclude that every integer appears either in (1) or in (2).\n\nRemark. This is sometimes called Beatty's problem, after Samuel Beatty (1881-1970). In a slightly different form it appeared as Problem 3117, American Mathematical Monthly, vol. 34 (1927), pages 158-159. Howard Grossman, \"A Set Containing All Integers,\" American Mathematical Monthly, vol. 69 (1962), pages 532-533, gives a proof by analyzing lattice points. A. S. Fraenkel, \"The Bracket Function and Complementary Sets of Integers,\" Canadian Journal of Mathematics, vol. 21 (Jan. 1969), pages 6-27, gives a history, a bibliography, and a generalization of the problem.",
  "vars": [
    "n",
    "p",
    "a",
    "b"
  ],
  "params": [
    "x",
    "y"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "indexer",
        "p": "targetint",
        "a": "coefone",
        "b": "coeftwo",
        "x": "firstirrat",
        "y": "secondirrat"
      },
      "question": "6. Prove that, if \\( firstirrat \\) and \\( secondirrat \\) are positive irrationals such that \\( 1 / firstirrat+1 / secondirrat=1 \\), then the sequences \\( [firstirrat],[2 firstirrat], \\ldots,[indexer firstirrat], \\ldots \\) and \\( [secondirrat],[2 secondirrat], \\ldots,[indexer secondirrat], \\ldots \\) together include every positive integer exactly once. (The notation \\( [firstirrat] \\) means the largest integer not exceeding \\( firstirrat \\).)",
      "solution": "Solution. Evidently \\( firstirrat>1 \\), and therefore the numbers \\( 0, firstirrat, 2 firstirrat, 3 firstirrat, \\ldots \\) all differ by more than one. Hence the integers\n\\[\n[firstirrat],[2 firstirrat],[3 firstirrat], \\ldots\n\\]\nare all positive and different. Similarly the integers\n\\[\n[secondirrat],[2 secondirrat],[3 secondirrat], \\ldots\n\\]\nare all positive and different.\nSuppose some integer \\( targetint \\) appears in both (1) and (2). Say \\( targetint=[coefone firstirrat]= \\) \\( [coeftwo secondirrat] \\) where \\( coefone \\) and \\( coeftwo \\) are positive integers. Then \\( targetint<coefone firstirrat<targetint+1 \\) and \\( targetint<coeftwo secondirrat<targetint+1 \\). (The possibilities \\( targetint=coefone firstirrat \\) and \\( targetint=coeftwo secondirrat \\) are excluded here, and in similar places elsewhere, because \\( firstirrat \\) and \\( secondirrat \\) are irrational). Then\n\\[\n targetint=\\frac{targetint}{firstirrat}+\\frac{targetint}{secondirrat}<coefone+coeftwo<\\frac{targetint+1}{firstirrat}+\\frac{targetint+1}{secondirrat}=targetint+1\n\\]\nand we have found an integer, \\( coefone+coeftwo \\), between the integers \\( targetint \\) and \\( targetint+1 \\). This is impossible. Hence no integers appear in both (1) and (2).\n\nSuppose some positive integer \\( \\boldsymbol{targetint} \\) is not included in either (1) or (2). Then we can find positive integers \\( coefone \\) and \\( coeftwo \\) such that\n\\[\ncoefone firstirrat<targetint<targetint+1<(coefone+1) firstirrat\n\\]\nand\n\\[\ncoeftwo secondirrat<targetint<targetint+1<(coeftwo+1) secondirrat\n\\]\n\nThen\n\\[\n\\frac{coefone}{targetint}+\\frac{coeftwo}{targetint}<\\frac{1}{firstirrat}+\\frac{1}{secondirrat}=1\n\\]\nso \\( coefone+coeftwo<targetint \\). Also\n\\[\n\\frac{coefone+1}{targetint+1}+\\frac{coeftwo+1}{targetint+1}>\\frac{1}{firstirrat}+\\frac{1}{secondirrat}=1\n\\]\nso \\( coefone+coeftwo+2>targetint+1 \\). Thus we have found two integers, \\( targetint \\) and \\( targetint+1 \\), between the integers \\( coefone+coeftwo \\) and \\( coefone+coeftwo+2 \\), which is impossible. We conclude that every integer appears either in (1) or in (2).\n\nRemark. This is sometimes called Beatty's problem, after Samuel Beatty (1881-1970). In a slightly different form it appeared as Problem 3117, American Mathematical Monthly, vol. 34 (1927), pages 158-159. Howard Grossman, \"A Set Containing All Integers,\" American Mathematical Monthly, vol. 69 (1962), pages 532-533, gives a proof by analyzing lattice points. A. S. Fraenkel, \"The Bracket Function and Complementary Sets of Integers,\" Canadian Journal of Mathematics, vol. 21 (Jan. 1969), pages 6-27, gives a history, a bibliography, and a generalization of the problem."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "satellite",
        "p": "pineapple",
        "a": "elephant",
        "b": "butterfly",
        "x": "triangle",
        "y": "hydrogen"
      },
      "question": "6. Prove that, if \\( triangle \\) and \\( hydrogen \\) are positive irrationals such that \\( 1 / triangle+1 / hydrogen=1 \\), then the sequences \\( [triangle],[2 triangle], \\ldots,[satellite triangle], \\ldots \\) and \\( [hydrogen],[2 hydrogen], \\ldots,[satellite hydrogen], \\ldots \\) together include every positive integer exactly once. (The notation \\( [triangle] \\) means the largest integer not exceeding \\( triangle \\).)",
      "solution": "Solution. Evidently \\( triangle>1 \\), and therefore the numbers \\( 0, triangle, 2 triangle, 3 triangle, \\ldots \\) all differ by more than one. Hence the integers\n\\[\n[triangle],[2 triangle],[3 triangle], \\ldots\n\\]\nare all positive and different. Similarly the integers\n\\[\n[hydrogen],[2 hydrogen],[3 hydrogen], \\ldots\n\\]\nare all positive and different.\nSuppose some integer \\( pineapple \\) appears in both (1) and (2). Say \\( pineapple=[elephant triangle]= \\) \\( [butterfly hydrogen] \\) where \\( elephant \\) and \\( butterfly \\) are positive integers. Then \\( pineapple<elephant triangle<pineapple+1 \\) and \\( pineapple<butterfly hydrogen<pineapple+1 \\). (The possibilities \\( pineapple=elephant triangle \\) and \\( pineapple=butterfly hydrogen \\) are excluded here, and in similar places elsewhere, because \\( triangle \\) and \\( hydrogen \\) are irrational). Then\n\\[\npineapple=\\frac{pineapple}{triangle}+\\frac{pineapple}{hydrogen}<elephant+butterfly<\\frac{pineapple+1}{triangle}+\\frac{pineapple+1}{hydrogen}=pineapple+1\n\\]\nand we have found an integer, \\( elephant+butterfly \\), between the integers \\( pineapple \\) and \\( pineapple+1 \\). This is impossible. Hence no integers appear in both (1) and (2).\n\nSuppose some positive integer \\( \\boldsymbol{pineapple} \\) is not included in either (1) or (2). Then we can find positive integers \\( elephant \\) and \\( butterfly \\) such that\n\\[\nelephant triangle<pineapple<pineapple+1<(elephant+1) triangle\n\\]\nand\n\\[\nbutterfly hydrogen<pineapple<pineapple+1<(butterfly+1) hydrogen\n\\]\n\nThen\n\\[\n\\frac{elephant}{pineapple}+\\frac{butterfly}{pineapple}<\\frac{1}{triangle}+\\frac{1}{hydrogen}=1\n\\]\nso \\( elephant+butterfly<pineapple \\). Also\n\\[\n\\frac{elephant+1}{pineapple+1}+\\frac{butterfly+1}{pineapple+1}>\\frac{1}{triangle}+\\frac{1}{hydrogen}=1\n\\]\nso \\( elephant+butterfly+2>pineapple+1 \\). Thus we have found two integers, \\( pineapple \\) and \\( pineapple+1 \\), between the integers \\( elephant+butterfly \\) and \\( elephant+butterfly+2 \\), which is impossible. We conclude that every integer appears either in (1) or in (2).\n\nRemark. This is sometimes called Beatty's problem, after Samuel Beatty (1881-1970). In a slightly different form it appeared as Problem 3117, American Mathematical Monthly, vol. 34 (1927), pages 158-159. Howard Grossman, \"A Set Containing All Integers,\" American Mathematical Monthly, vol. 69 (1962), pages 532-533, gives a proof by analyzing lattice points. A. S. Fraenkel, \"The Bracket Function and Complementary Sets of Integers,\" Canadian Journal of Mathematics, vol. 21 (Jan. 1969), pages 6-27, gives a history, a bibliography, and a generalization of the problem."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "fractional",
        "p": "noninteger",
        "a": "negative",
        "b": "nullified",
        "x": "rational",
        "y": "integral"
      },
      "question": "6. Prove that, if \\( rational \\) and \\( integral \\) are positive irrationals such that \\( 1 / rational+1 / integral=1 \\), then the sequences \\( [rational],[2 rational], \\ldots,[fractional rational], \\ldots \\) and \\( [integral],[2 integral], \\ldots,[fractional integral], \\ldots \\) together include every positive integer exactly once. (The notation \\( [rational] \\) means the largest integer not exceeding \\( rational \\).)",
      "solution": "Solution. Evidently \\( rational>1 \\), and therefore the numbers \\( 0, rational, 2 rational, 3 rational, \\ldots \\) all differ by more than one. Hence the integers\n\\[\n[rational],[2 rational],[3 rational], \\ldots\n\\]\nare all positive and different. Similarly the integers\n\\[\n[integral],[2 integral],[3 integral], \\ldots\n\\]\nare all positive and different.\nSuppose some integer \\( noninteger \\) appears in both (1) and (2). Say \\( noninteger=[negative rational]=[nullified integral] \\) where \\( negative \\) and \\( nullified \\) are positive integers. Then \\( noninteger<negative rational<noninteger+1 \\) and \\( noninteger<nullified integral<noninteger+1 \\). (The possibilities \\( noninteger=negative rational \\) and \\( noninteger=nullified integral \\) are excluded here, and in similar places elsewhere, because \\( rational \\) and \\( integral \\) are irrational). Then\n\\[\nnoninteger=\\frac{noninteger}{rational}+\\frac{noninteger}{integral}<negative+nullified<\\frac{noninteger+1}{rational}+\\frac{noninteger+1}{integral}=noninteger+1\n\\]\nand we have found an integer, \\( negative+nullified \\), between the integers \\( noninteger \\) and \\( noninteger+1 \\). This is impossible. Hence no integers appear in both (1) and (2).\n\nSuppose some positive integer \\( \\boldsymbol{noninteger} \\) is not included in either (1) or (2). Then we can find positive integers \\( negative \\) and \\( nullified \\) such that\n\\[\nnegative rational<noninteger<noninteger+1<(negative+1) rational\n\\]\nand\n\\[\nnullified integral<noninteger<noninteger+1<(nullified+1) integral\n\\]\n\nThen\n\\[\n\\frac{negative}{noninteger}+\\frac{nullified}{noninteger}<\\frac{1}{rational}+\\frac{1}{integral}=1\n\\]\nso \\( negative+nullified<noninteger \\). Also\n\\[\n\\frac{negative+1}{noninteger+1}+\\frac{nullified+1}{noninteger+1}>\\frac{1}{rational}+\\frac{1}{integral}=1\n\\]\nso \\( negative+nullified+2>noninteger+1 \\). Thus we have found two integers, \\( noninteger \\) and \\( noninteger+1 \\), between the integers \\( negative+nullified \\) and \\( negative+nullified+2 \\), which is impossible. We conclude that every integer appears either in (1) or in (2).\n\nRemark. This is sometimes called Beatty's problem, after Samuel Beatty (1881-1970). In a slightly different form it appeared as Problem 3117, American Mathematical Monthly, vol. 34 (1927), pages 158-159. Howard Grossman, \"A Set Containing All Integers,\" American Mathematical Monthly, vol. 69 (1962), pages 532-533, gives a proof by analyzing lattice points. A. S. Fraenkel, \"The Bracket Function and Complementary Sets of Integers,\" Canadian Journal of Mathematics, vol. 21 (Jan. 1969), pages 6-27, gives a history, a bibliography, and a generalization of the problem."
    },
    "garbled_string": {
      "map": {
        "n": "qzwvxrta",
        "p": "mdlkhsqe",
        "a": "fbmqratn",
        "b": "txhpswle",
        "x": "rjdkeovc",
        "y": "kmptsaxy"
      },
      "question": "6. Prove that, if \\( rjdkeovc \\) and \\( kmptsaxy \\) are positive irrationals such that \\( 1 / rjdkeovc+1 / kmptsaxy=1 \\), then the sequences \\( [rjdkeovc],[2 rjdkeovc], \\ldots,[qzwvxrta rjdkeovc], \\ldots \\) and \\( [kmptsaxy],[2 kmptsaxy], \\ldots,[qzwvxrta kmptsaxy], \\ldots \\) together include every positive integer exactly once. (The notation \\( [rjdkeovc] \\) means the largest integer not exceeding \\( rjdkeovc \\).)",
      "solution": "Solution. Evidently \\( rjdkeovc>1 \\), and therefore the numbers \\( 0, rjdkeovc, 2 rjdkeovc, 3 rjdkeovc, \\ldots \\) all differ by more than one. Hence the integers\n\\[\n[rjdkeovc],[2 rjdkeovc],[3 rjdkeovc], \\ldots\n\\]\nare all positive and different. Similarly the integers\n\\[\n[kmptsaxy],[2 kmptsaxy],[3 kmptsaxy], \\ldots\n\\]\nare all positive and different.\nSuppose some integer \\( mdlkhsqe \\) appears in both (1) and (2). Say \\( mdlkhsqe=[fbmqratn rjdkeovc]= \\) \\( [txhpswle kmptsaxy] \\) where \\( fbmqratn \\) and \\( txhpswle \\) are positive integers. Then \\( mdlkhsqe<fbmqratn rjdkeovc<mdlkhsqe+1 \\) and \\( mdlkhsqe<txhpswle kmptsaxy<mdlkhsqe+1 \\). (The possibilities \\( mdlkhsqe=fbmqratn rjdkeovc \\) and \\( mdlkhsqe=txhpswle kmptsaxy \\) are excluded here, and in similar places elsewhere, because \\( rjdkeovc \\) and \\( kmptsaxy \\) are irrational). Then\n\\[\nmdlkhsqe=\\frac{mdlkhsqe}{rjdkeovc}+\\frac{mdlkhsqe}{kmptsaxy}<fbmqratn+txhpswle<\\frac{mdlkhsqe+1}{rjdkeovc}+\\frac{mdlkhsqe+1}{kmptsaxy}=mdlkhsqe+1\n\\]\nand we have found an integer, \\( fbmqratn+txhpswle \\), between the integers \\( mdlkhsqe \\) and \\( mdlkhsqe+1 \\). This is impossible. Hence no integers appear in both (1) and (2).\n\nSuppose some positive integer \\( \\boldsymbol{mdlkhsqe} \\) is not included in either (1) or (2). Then we can find positive integers \\( fbmqratn \\) and \\( txhpswle \\) such that\n\\[\nfbmqratn rjdkeovc<mdlkhsqe<mdlkhsqe+1<(fbmqratn+1) rjdkeovc\n\\]\nand\n\\[\ntxhpswle kmptsaxy<mdlkhsqe<mdlkhsqe+1<(txhpswle+1) kmptsaxy\n\\]\n\nThen\n\\[\n\\frac{fbmqratn}{mdlkhsqe}+\\frac{txhpswle}{mdlkhsqe}<\\frac{1}{rjdkeovc}+\\frac{1}{kmptsaxy}=1\n\\]\nso \\( fbmqratn+txhpswle<mdlkhsqe \\). Also\n\\[\n\\frac{fbmqratn+1}{mdlkhsqe+1}+\\frac{txhpswle+1}{mdlkhsqe+1}>\\frac{1}{rjdkeovc}+\\frac{1}{kmptsaxy}=1\n\\]\nso \\( fbmqratn+txhpswle+2>mdlkhsqe+1 \\). Thus we have found two integers, \\( mdlkhsqe \\) and \\( mdlkhsqe+1 \\), between the integers \\( fbmqratn+txhpswle \\) and \\( fbmqratn+txhpswle+2 \\), which is impossible. We conclude that every integer appears either in (1) or in (2).\n\nRemark. This is sometimes called Beatty's problem, after Samuel Beatty (1881-1970). In a slightly different form it appeared as Problem 3117, American Mathematical Monthly, vol. 34 (1927), pages 158-159. Howard Grossman, \"A Set Containing All Integers,\" American Mathematical Monthly, vol. 69 (1962), pages 532-533, gives a proof by analyzing lattice points. A. S. Fraenkel, \"The Bracket Function and Complementary Sets of Integers,\" Canadian Journal of Mathematics, vol. 21 (Jan. 1969), pages 6-27, gives a history, a bibliography, and a generalization of the problem."
    },
    "kernel_variant": {
      "question": "Let α, β>1 be positive real numbers and define their Beatty sequences  \n\nA = (⌊kα⌋)_{k≥1}, B = (⌊kβ⌋)_{k≥1}.  \n\n(a) Assume that every positive integer belongs to exactly one of A or B.  Prove that  \n (i) α and β are both irrational, and (ii) 1/α + 1/β = 1.  \n\n(b) Conversely, assume α, β are positive irrationals with 1/α + 1/β = 1.  Prove that A and B then partition ℕ.  \n\nThus obtain a necessary-and-sufficient condition for two Beatty sequences to give a perfect two-way partition of the positive integers.",
      "solution": "We treat necessity and sufficiency separately, keeping the same style and level of detail.\n\nStep 1 – Strict growth.  \nNote that α, β>1, so  \n ⌊(k+1)α⌋ − ⌊kα⌋ > α−1>0,  \nand similarly for β.  Hence A and B are strictly increasing; no term may repeat within a single sequence.\n\nStep 2 – Irrationality is forced.  \nSuppose, for a contradiction, α = p/q with p,q coprime.  \nThen qα = p is integral, so ⌊qα⌋ = p.  But  \n (q+q)α = 2p is again integral, so ⌊2qα⌋ = 2p.  \nThus A takes the value p twice (at k=q and k=2q), contradicting Step 1.  Therefore α is irrational; an identical argument forces β to be irrational.\n\nStep 3 – Counting yields 1/α + 1/β = 1.  \nLet A(n) (resp. B(n)) be the number of terms of A (resp. B) not exceeding n.  Since A∪B = {1,…,n}, we have A(n)+B(n)=n.  \nObserve  \n n/α−1 < A(n) ≤ n/α and n/β−1 < B(n) ≤ n/β,  \nbecause ⌊kα⌋≤n⇔k≤n/α, with possible ±1 error.  Combining and dividing by n gives  \n 1/α+1/β − 2/n < 1 < 1/α+1/β + 2/n.  \nLetting n→∞ we obtain 1/α+1/β=1, establishing part (a).\n\nStep 4 – Sufficiency (Beatty’s theorem).  \nNow assume α, β are positive irrationals with reciprocals summing to 1.  \n\nDisjointness.  If some p were in both sequences, write p=⌊aα⌋=⌊bβ⌋.  Since α,β are irrational we have  \n p<aα,p<bβ<p+1,  \nwhence p/α+p/β<a+b<(p+1)/α+(p+1)/β, i.e. p<a+b<p+1, impossible for integral a+b.\n\nExhaustion.  Suppose p escaped both lists.  Choose a,b such that aα<p<(a+1)α and bβ<p<(b+1)β.  Then  \n a/p+b/p<1< (a+1)/(p+1)+(b+1)/(p+1),  \nyielding a+b<p<a+b+2, so the two consecutive integers p,p+1 lie strictly between consecutive integers a+b,a+b+2, contradiction.\n\nThus every positive integer appears exactly once in A∪B, completing part (b).\n\nStep 5 – Conclusion.  \nParts (a) and (b) together show that two Beatty sequences partition ℕ precisely when their moduli are irrational and their reciprocals add to 1.",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.104573",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}