MathDB

5

Part of 2006 All-Russian Olympiad

Problems(2)

Numbers increasing, greatest divisors decreasing

Source: All-Russian Olympiad 2006 finals, problem 10.5 = 9.5

5/7/2006
Let a1a_1, a2a_2, ..., a10a_{10} be positive integers such that a1<a2<...<a10a_1<a_2<...<a_{10}. For every kk, denote by bkb_k the greatest divisor of aka_k such that bk<akb_k<a_k. Assume that b1>b2>...>b10b_1>b_2>...>b_{10}. Show that a10>500a_{10}>500.
number theory proposednumber theory
X_k will win [Two sequences of positives: some x_k is &gt; y_k]

Source: All-Russian Olympiad 2006 finals, problem 11.5

5/6/2006
Two sequences of positive reals, (xn) \left(x_n\right) and (yn) \left(y_n\right), satisfy the relations x_{n \plus{} 2} \equal{} x_n \plus{} x_{n \plus{} 1}^2 and y_{n \plus{} 2} \equal{} y_n^2 \plus{} y_{n \plus{} 1} for all natural numbers n n. Prove that, if the numbers x1 x_1, x2 x_2, y1 y_1, y2 y_2 are all greater than 1 1, then there exists a natural number k k such that xk>yk x_k > y_k.
logarithmsalgebra unsolvedalgebra