MathDB

4

Part of 1988 Greece National Olympiad

Problems(3)

k^2+2m, m^2+2k never both perfect squares 1988 Greece MO Grade XI p4

Source:

9/7/2024
Prove that there are do not exist natural numbers k,mk, m such that numbers k2+2mk^2+2m, m2+2km^2+2k to be squares of integers.
number theoryPerfect Squares
\sqrt[ 2^{1453}]{2^{1821}} \in A, if 1,2 \in A and \sqrt{ab} \in A for a,b \in A

Source: 1988 Greece MO Grade XI p4

9/7/2024
Let A⊆RA\subseteq \mathbb{R} such that:
i) If a,b∈Aa,b\in A then ab∈A\sqrt{ab} \in A ii) 1∈A1\in A and 2∈A2\in A
Prove that 2182121453∈A\sqrt[\displaystyle 2^{1453}]{2^{1821}}\in A.
algebra
lim sum 1/ (a_i ... a_n) when a_{n+1}= a^2_{n}-2

Source: 1988 Greece MO Grade XII p4

9/7/2024
Let a1=5a_1=5 and an+1=an2−2a_{n+1}= a^2_{n}-2 for any n=1,2,...n=1,2,....
a) Find lim⁡n→∞an+1a1a2...an\lim_{n \rightarrow \infty}\frac{a_{n+1}}{a_1a_2 ...a_{n}}
b) Find lim⁡ν→∞(1a1+1a1a2+...+1a1a2...aν)\lim_{\nu \rightarrow \infty}\left(\frac{1}{a_1}+\frac{1}{a_1a_2}+...+\frac{1}{a_1a_2 ...a_{\nu}}\right)
Sequenceslimitreal analysis