MathDB

2

Part of 2004 IMC

Problems(2)

Problem 2 IMC 2004 Macedonia

Source:

7/25/2004
Let f1(x)=x2−1f_1(x)=x^2-1, and for each positive integer n≥2n \geq 2 define fn(x)=fn−1(f1(x))f_n(x) = f_{n-1}(f_1(x)). How many distinct real roots does the polynomial f2004f_{2004} have?
algebrapolynomialinductionIMCcollege contests
Problem 8 IMC 2004 Macedonia

Source:

7/26/2004
Let f,g:[a,b]→[0,∞)f,g:[a,b]\to [0,\infty) be two continuous and non-decreasing functions such that each x∈[a,b]x\in [a,b] we have ∫axf(t) dt≤∫axg(t) dt  and ∫abf(t) dt=∫abg(t) dt. \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. Prove that ∫ab1+f(t) dt≥∫ab1+g(t) dt. \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt.
functionintegrationgeometryperimeterinequalitiesIMCcollege contests