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IMO Shortlist 1982 problem 3

Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$ .

a) Prove that for every such sequence there is an $n\ge1$ such that: ${x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999.$

b) Find such a sequence such that for all $n$ : ${x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4.$

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IMO Shortlist 1959 problem 3

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ $a \cos^2{x}+b \cos{x}+c=0.$ Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$ , $b=2$ , $c=-1$ .

IMO Shortlist 1976 problem 4

A sequence $(u_{n})$ is defined by $u_{0}=2 \quad u_{1}=\frac{5}{2}, u_{n+1}=u_{n}(u_{n-1}^{2}-2)-u_{1} \quad \textnormal{for } n=1,\ldots$ Prove that for any positive integer $n$ we have $[u_{n}]=2^{\frac{(2^{n}-(-1)^{n})}{3}}$ (where {{ Nevaljan tag "x" }} denotes the smallest integer $\leq$ x) $.$

IMO Shortlist 1977 problem 15

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

IMO Shortlist 1978 problem 9

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive $.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

IMO Shortlist 1988 problem 24

Let $\{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:
$a_k - 2 \cdot a_{k + 1} + a_{k + 2} \geq 0$
and $\sum^k_{j = 1} a_j \leq 1$ for all $k = 1,2, \ldots$ . Prove that:
$0 \leq (a_{k} - a_{k + 1}) < \frac {2}{k^2}$
for all $k = 1,2, \ldots$ .

IMO Shortlist 1989 problem 11

Define sequence $(a_n)$ by $\sum_{d|n} a_d = 2^n.$ Show that $n|a_n.$