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10.1

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(2)

x^2+bx+c=0, arithm. progression (I Soros Olympiad 1994-95 R2 10.1)

Source:

5/25/2024
The equation x2+bx+c=0x^2 + bx + c = 0 has two different roots x1x_1 and x2x_2. It is also known that the numbers bb, x1x_1, cc, x2x_2 in the indicated order form an arithmetic progression. Find the difference of this progression.
algebraArithmetic Progression
f(f(n))=n=f(f(n+2)+2) (I Soros Olympiad 1994-95 Ukraine R2 10.1)

Source:

6/6/2024
The function f:Z→Zf: Z \to Z satisfies the following conditions: 1) f(f(n))=nf(f(n))=n for all integers nn 2) f(f(n+2)+2)=nf(f(n+2)+2) = n for all integers nn 3) f(0)=1f(0)=1. Find the value of f(1995)f(1995) and f(−1994)f(-1994).
algebranumber theoryfunctional