MathDB

Problem 4

Part of 1999 VJIMC

Problems(2)

x^n+y^n in Z for n in [4], then n=5 holds for complex x,y

Source: VJIMC 1999 1.4

8/1/2021
Show that the following implication holds for any two complex numbers xx and yy: if x+yx+y, x2+y2x^2+y^2, x3+y3x^3+y^3, x4+y4∈Zx^4+y^4\in\mathbb Z, then xn+yn∈Zx^n+y^n\in\mathbb Z for all natural n.
algebranumber theorycomplex numbers
integral inequality in [0,1]^n

Source: VJIMC 1999 2.4

7/29/2021
Let u1,u2,…,un∈C([0,1]n)u_1,u_2,\ldots,u_n\in C([0,1]^n) be nonnegative and continuous functions, and let uju_j do not depend on the jj-th variable for j=1,…,nj=1,\ldots,n. Show that (∫[0,1]n∏j=1nuj)n−1≤∏j=1n∫[0,1]nujn−1.\left(\int_{[0,1]^n}\prod_{j=1}^nu_j\right)^{n-1}\le\prod_{j=1}^n\int_{[0,1]^n}u_j^{n-1}.
calculusintegration