MathDB

1

Part of 2022 VJIMC

Problems(2)

polynomial has no real roots

Source: VJIMC 2022 1.1

4/11/2022
Assume that a real polynomial P(x)P(x) has no real roots. Prove that the polynomial Q(x)=P(x)+P′′(x)2!+P(4)(x)4!+…Q(x)=P(x)+\frac{P''(x)}{2!}+\frac{P^{(4)}(x)}{4!}+\ldots also has no real roots.
algebrapolynomial
integral inequality, existence of function

Source: VJIMC 2022 2.1

4/11/2022
Determine whether there exists a differentiable function f:[0,1]→Rf:[0,1]\to\mathbb R such that f(0)=f(1)=1,∣f′(x)∣≤2 for all x∈[0,1]and∣∫01f(x)dx∣≤12.f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.
calculusintegrationinequalitiesfunction