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Omni-MATH / The Fibonacci numbers are defined by F₀=0,F₁=1, and F_(n)=F_(n−1)+F_(n−2) for n≥2. There exist unique positive integers n₁,n₂,n₃,n₄,n₅,n₆ such that …

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The Fibonacci numbers are defined by F0=0,F1=1F_{0}=0, F_{1}=1, and Fn=Fn−1+Fn−2F_{n}=F_{n-1}+F_{n-2} for n≥2n \geq 2. There exist unique positive integers n1,n2,n3,n4,n5,n6n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6} such that ∑i1=0100∑i2=0100∑i3=0100∑i4=0100∑i5=0100Fi1+i2+i3+i4+i5=Fn1−5Fn2+10Fn3−10Fn4+5Fn5−Fn6\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}. Find n1+n2+n3+n4+n5+n6n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}.
Plain-text mathematical notation (without MathML)
The Fibonacci numbers are defined by F₀=0,F₁=1, and F_(n)=F_(n−1)+F_(n−2) for n≥2. There exist unique positive integers n₁,n₂,n₃,n₄,n₅,n₆ such that ∑_(i₁=0)¹⁰⁰∑_(i₂=0)¹⁰⁰∑_(i₃=0)¹⁰⁰∑_(i₄=0)¹⁰⁰∑_(i₅=0)¹⁰⁰F_(i₁+i₂+i₃+i₄+i₅)=F_(n₁)−5F_(n₂)+10F_(n₃)−10F_(n₄)+5F_(n₅)−F_(n₆). Find n₁+n₂+n₃+n₄+n₅+n₆.
Original LaTeX notation
The Fibonacci numbers are defined by $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exist unique positive integers $n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}$ such that $\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}$. Find $n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}$.

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task_id
880311b6-92f5-5732-a2db-194ab0e44696
task_key
test--880311b6-92f5-5732-a2db-194ab0e44696
task_revision_id
1
config / split
/ test

Official source

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