MathDB

4

Part of 2009 IMC

Problems(2)

IMC 2009 Day 1 P4

Source:

7/15/2014
Let p(z)=a0+a1z+a2z2+⋯+anznp(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n be a complex polynomial. Suppose that 1=c0≥c1≥⋯≥cn≥01=c_0\ge c_1\ge \cdots \ge c_n\ge 0 is a sequence of real numbers which form a convex sequence. (That is 2ck≤ck−1+ck+12c_k\le c_{k-1}+c_{k+1} for every k=1,2,⋯ ,n−1k=1,2,\cdots ,n-1 ) and consider the polynomial q(z)=c0a0+c1a1z+c2a2z2+⋯+cnanzn q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n Prove that : max⁡∣z∣≤1q(z)≤max⁡∣z∣≤1p(z) \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z)
algebrapolynomialIMCcollege contests
IMC 2009 Day 2 P4

Source:

7/17/2014
Let pp be a prime number and W⊆Fp[x]\mathbf{W}\subseteq \mathbb{F}_p[x] be the smallest set satisfying the following :
(a) x+1∈Wx+1\in \mathbf{W} and xp−2+xp−3+⋯+x2+2x+1∈Wx^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W} (b) For γ1,γ2\gamma_1,\gamma_2 in W\mathbf{W}, we also have γ(x)∈W\gamma(x)\in \mathbf{W}, where γ(x)\gamma(x) is the remainder (γ1∘γ2)(x)(modxp−x)(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}. How many polynomials are in W?\mathbf{W}?
modular arithmeticalgebrapolynomialIMCcollege contests