Maths Olympiad Prep

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Algebra Difficulty 5.3 AIME, harder Find the answer

Task B-4.4. A sequence of real numbers is given by the formula xn+1=n+1xnx_{n+1}=\frac{n+1}{x_{n}}, for all n≥1n \geq 1, where x1=123456789x_{1}=123456789. What is x1⋅x2⋅x18⋅x19⋅x52⋅x53x_{1} \cdot x_{2} \cdot x_{18} \cdot x_{19} \cdot x_{52} \cdot x_{53}?

A number or a short expression. Spacing and $ signs are ignored.

Solution

## Solution.

Notice that from the given equality it follows that

xn⋅xn+1=n+1, for all n≥1 x_{n} \cdot x_{n+1}=n+1, \text { for all } n \geq 1

Then, x1⋅x2=2,x18⋅x19=19,x52⋅x53=53x_{1} \cdot x_{2}=2, x_{18} \cdot x_{19}=19, x_{52} \cdot x_{53}=53

so x1⋅x2⋅x18⋅x19⋅x52⋅x53=2⋅19⋅53=2014x_{1} \cdot x_{2} \cdot x_{18} \cdot x_{19} \cdot x_{52} \cdot x_{53}=2 \cdot 19 \cdot 53=2014.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.