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Omni-MATH / For positive integers a and b, let M(a,b)=(lcm(a,b))/(gcd(a,b)), and for each positive integer n≥2, define x_(n)=M(1,M(2,M(3,…,M(n−2,M(n−1,n))…))) Compute the number of positive in…

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problem

For positive integers aa and bb, let M(a,b)=lcm(a,b)gcd(a,b)M(a, b)=\frac{\operatorname{lcm}(a, b)}{\operatorname{gcd}(a, b)}, and for each positive integer n≥2n \geq 2, define xn=M(1,M(2,M(3,…,M(n−2,M(n−1,n))…)))x_{n}=M(1, M(2, M(3, \ldots, M(n-2, M(n-1, n)) \ldots))) Compute the number of positive integers nn such that 2≤n≤20212 \leq n \leq 2021 and 5xn2+5xn+12=26xnxn+15 x_{n}^{2}+5 x_{n+1}^{2}=26 x_{n} x_{n+1}.
Plain-text mathematical notation (without MathML)
For positive integers a and b, let M(a,b)=(lcm(a,b))/(gcd(a,b)), and for each positive integer n≥2, define x_(n)=M(1,M(2,M(3,…,M(n−2,M(n−1,n))…))) Compute the number of positive integers n such that 2≤n≤2021 and 5x_(n)²+5x_(n+1)²=26x_(n)x_(n+1).
Original LaTeX notation
For positive integers $a$ and $b$, let $M(a, b)=\frac{\operatorname{lcm}(a, b)}{\operatorname{gcd}(a, b)}$, and for each positive integer $n \geq 2$, define $$x_{n}=M(1, M(2, M(3, \ldots, M(n-2, M(n-1, n)) \ldots)))$$ Compute the number of positive integers $n$ such that $2 \leq n \leq 2021$ and $5 x_{n}^{2}+5 x_{n+1}^{2}=26 x_{n} x_{n+1}$.

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task_id
cc3b790e-e518-5df8-993c-e6e3d1857b94
task_key
test--cc3b790e-e518-5df8-993c-e6e3d1857b94
task_revision_id
1
config / split
/ test

Official source

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