MathDB

3

Part of ICMC 4

Problems(2)

intergal of sin(nx)^n

Source: ICMC 2021 Round 1 P3

3/1/2022
Let sn=∫01sinn(nx) dx\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx.
(a) Prove that sn≤2ns_n \leq \dfrac 2n for all odd nn.
(b) Find all the limit points of the sequence s1,s2,s3,…s_1, s_2, s_3, \dots.
Proposed by Cristi Calin
calculusintegrationcollege contestsICMCsin
expected mess

Source: ICMC 2021 Round 2 P3

12/12/2021
Let f,g,h:R→Rf,g,h : \mathbb R \to \mathbb R be continuous functions and XX be a random variable such that E(g(X)h(X))=0E(g(X)h(X))=0 and E(g(X)2)≠0≠E(h(X)2)E(g(X)^2) \neq 0 \neq E(h(X)^2). Prove that E(f(X)2)≥E(f(X)g(X))2E(g(X)2)+E(f(X)h(X))2E(h(X)2).E(f(X)^2) \geq \frac{E(f(X)g(X))^2}{E(g(X)^2)} + \frac{E(f(X)h(X))^2}{E(h(X)^2)}. You may assume that all expected values exist.
Proposed by Cristi Calin
expectationICMCcollege contestsInequality