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1976 IMO Problems/Problem 6

Revision as of 19:57, 28 May 2025 by Ddk001 (talk | contribs) (→Solution)
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Problem

A sequence $(u_{n})$ is defined by

\[u_{0} = 2 \quad u_{1} = \frac {5}{2}, u_{n + 1} = u_{n}(u_{n - 1}^{2} - 2) - u_{1} \ \text{ for } n = 1,\cdots\]

Prove that for any positive integer $n$ we have

\[\lfloor u_{n} \rfloor = 2^{\frac {(2^{n} - ( - 1)^{n})}{3}}\]

(where $\lfloor x\rfloor$ denotes the smallest integer $\leq$ $x$)$.$

Solution

Let the sequence $(x_n)_{n \geq 0}$ be defined as \[x_{0}=0,x_{1}=1, x_{n}=x_{n-1}+2x_{n-2}\] We notice \[x_n=\frac{2^n-(-1)^n}{3}\] Because the roots of the characteristic polynomial $x_{n}=x_{n-1}+2x_{n-2}$ are $-1$ and $2$.

We also see $\frac{2^1-(-1)^1}{3}=1=x_1$, $\frac{2^2-(-1)^2}{3}=1=x_2$ We want to prove \[2^{x_{n}-2x_{n-1}}+2^{-x_{n}+2x_{n-1}}=2^{x_1}+2^{-x_1}\] This is done by induction

Base Case: For $n=1$ ses det $2^{1-0}+2^{0-1}=2^{1}+2^{-1}$

Inductive step: Assume $2^{x_{n-1}-2x_{n-2}}+2^{-x_{n-1}+2x_{n-2}}=2^{x_1}+2^{-x_1}$ We notice \begin{align*} 2^{x_{n}-2x_{n-1}}+2^{-x_{n}+2x_{n-1}} &=2^{x_{n-1}+2x_{n-2}-2x_{n-1}}+2^{-(x_{n-1}+2x_{n-2})+2x_{n-1}}\\ &=2^{-x_{n-1}+2x_{n-2}}+2^{x_{n-1}-2x_{n-2}}\\ &=2^{x_1}+2^{-x_1} \end{align*} We then want to show \[a_n=2^{x_n}+2^{-x_n}\] This can be done using induction

Base Case

For $n=1$, it is clear that \[a_1=\frac{5}{2}\] and \[2^{x_1}+2^{-x_1}=2^1+2^{-1}=2+\frac{1}{2}=\frac{5}{2}\] Therefore, the base case is proved.

Inductive Step

Assume for all natural $k<n$ at $a_k=2^{x_k}+2^{-x_k}$\newline Then we have that: \begin{align*} a_n &= a_{n}(a_{n-1}^{2}-2)-a_{1} \\ &= (2^{x_{n-1}}+2^{-x_{n-1}})((2^{x_{n-2}}+2^{-x_{n-2}})^2-2)-(2^{x_{1}}+2^{-x_{0}}) \\ &= (2^{x_{n-1}}+2^{-x_{n-1}})((2^{x_{n-2}})^2+(2^{-x_{n-2}})^2+2*2^{x_{n-2}}*2^{-x_{n-2}}-2)-(2^{x_{1}}+2^{-x_{0}}) \\ &=(2^{x_{n-1}}+2^{-x_{n-1}})((2^{2x_{n-2}})+(2^{-2x_{n-2}})+2*2^{x_{n-2}-x_{n-2}}-2)-(2^{x_{1}}+2^{-x_{0}}) \\ &=(2^{x_{n-1}}+2^{-x_{n-1}})((2^{2x_{n-2}})+(2^{-2x_{n-2}})+2-2)-(2^{x_{1}}+2^{-x_{0}}) \\ &=2^{x_{n-1}}*2^{2x_{n-2}}+2^{-x_{n-1}}2^{-2x_{n-2}}+2^{x_{n-1}}*2^{-2x_{n-2}}+2^{-x_{n-1}}*2^{2x_{n-2}}-2^{x_{1}}-2^{-x_{0}}\\ &=2^{x_{n-1}+2x_{n-2}}+2^{-(x_{n-1}+2x_{n-2}})+2^{x_{n-1}-2x_{n-2}}+2^{-x_{n-1}+2x_{n-2}}-2^{x_{1}}-2^{-x_{0}}\\ &=2^{x_{n}}+2^{-x_{n}}+2^{x_{n-1}-2x_{n-2}}+2^{-x_{n-1}+2x_{n-2}}-2^{x_{1}}-2^{-x_{0}} \end{align*} From our first induction proof we have that: \[2^{x_{n-1}-2x_{n-2}}+2^{-x_{n-1}+2x_{n-2}}=2^{x_1}+2^{-x_1}\] Then: \[a_n=2^{x_n}+2^{-x_n}+2^{x_1}+2^{-x_1}-(2^{x_1}+2^{-x_1})=2^{x_n}+2^{-x_n}\] We notice $\left[a_n\right]=\left[2^{x_n}+2^{-x_n}\right]=2^{x_n}$, Because $2^{x_n} \in \mathbb{N}$ and $2^{-x_n}<1$, for all $n>0$ Finally we conclude \[\left[a_n\right]=2^{\frac{2^n-(-1)^n}{3}}\]

Solution 2

We note that there is a slightly easier claim that can be proved also by induction. By testing out smaller terms, one sees that

\[u_n= 2^{\frac{2^n-(-1)^n}{3}} + 2^{- \frac{2^n-(-1)^n}{3}}\]

for $n=1,2,3$ (note that since we are using strong induction to prove this result this suffices as the base case).

Suppose that the equation above holds for all $k \le n$. Then for $n+1$, one have

\begin{align*} u_{n+1} &= u_n (u_{n-1} ^2 -2) -u_1 \\ &= (2^{\frac{2^n-(-1)^n}{3}}+2^{- \frac{2^n-(-1)^n}{3}}) \cdot ( (2^{\frac{2^{n-1}-(-1)^{n-1}}{3}}+2^{- \frac{2^{n-1}-(-1)^{n-1}}{3}} )^2 -2) - \frac{5}{2} \\ &= (2^{\frac{2^n-(-1)^n}{3}}+2^{- \frac{2^n-(-1)^n}{3}}) \cdot ( (2^{\frac{2^{n-1}-(-1)^{n-1}}{3}})^2 + (2^{- \frac{2^{n-1}-(-1)^{n-1}}{3}})^2 )- \frac{5}{2} \\ &= (2^{\frac{2^n-(-1)^n}{3}}+2^{- \frac{2^n-(-1)^n}{3}}) \cdot (2^{\frac{2^{n}-2 \cdot (-1)^{n-1}}{3}} + 2^{- \frac{2^{n}-2 \cdot (-1)^{n-1}}{3}})- \frac{5}{2} \\ &= 2^{\frac{2^n-(-1)^n}{3} + \frac{2^{n}-2 \cdot (-1)^{n-1}}{3}} + 2^{- \frac{2^n-(-1)^n}{3} - \frac{2^{n}-2 \cdot (-1)^{n-1}}{3}} + 2^{(-1)^n} + 2^{-(-1)^n} -\frac{5}{2} \\ &= 2^{\frac{2^n-(-1)^n}{3} + \frac{2^{n}-2 \cdot (-1)^{n-1}}{3}} + 2^{- \frac{2^n-(-1)^n}{3} - \frac{2^{n}-2 \cdot (-1)^{n-1}}{3}} \\ &= 2^{\frac{2^{n+1}-(-1)^{n+1}}{3}}+2^{- \frac{2^{n+1}-(-1)^{n+1}}{3}} \end{align*}

as desired. The strong induction proves our claim. Now, it is rather obvious that $2^{- \frac{2^n-(-1)^n}{3}}$ is between 0 and 1, so indeed our claim proves the result. $\square$

~Ddk001

See also

1976 IMO (Problems) • Resources
Preceded by
Problem 5 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Final Question
All IMO Problems and Solutions
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