MathDB

1

Part of 2021 Francophone Mathematical Olympiad

Problems(2)

R < 1/15 < S, R=/2 x 3/4 x 5/6 x ...x 223/ 224

Source: 2021 Francophone MO Juniors p1

4/3/2021
Let RR and SS be the numbers defined by R=12×34×56×⋯×223224 and S=23×45×67×⋯×224225.R = \dfrac{1}{2} \times \dfrac{3}{4} \times \dfrac{5}{6} \times \cdots \times \dfrac{223}{224} \text{ and } S = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7} \times \cdots \times \dfrac{224}{225}.Prove that R<115<SR < \dfrac{1}{15} < S.
algebrainequalitiesFrancophone
exists c : b_n=c a_n. a_{n+2}=a_n+a_{n+1}, b_{n+2}=b_n+b_{n+1}, a_n / b_n

Source: 2021 Francophone MO Seniors p1

4/3/2021
Let a1,a2,a3,…a_1,a_2,a_3,\ldots and b1,b2,b3,…b_1,b_2,b_3,\ldots be positive integers such that an+2=an+an+1a_{n+2} = a_n + a_{n+1} and bn+2=bn+bn+1b_{n+2} = b_n + b_{n+1} for all n≥1n \ge 1. Assume that ana_n divides bnb_n for infinitely many values of nn. Prove that there exists an integer cc such that bn=canb_n = c a_n for all n≥1n \ge 1.
Sequencenumber theorydividesrecurrence relationFrancophone