MathDB

2

Part of 2018 VJIMC

Problems(2)

Determinant divisible by p^3

Source: VJIMC 2018, Category I, Problem 2

4/14/2018
Find all prime numbers pp such that p3p^3 divides the determinant ∣2211…11321…111421⋮⋮⋱111(p+7)2∣.\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.
number theoryprime numbers
Inequality with integer parts

Source: VJIMC 2018, Category II, Problem 2

4/14/2018
Let nn be a positive integer and let a1≤a2≤⋯≤ana_1\le a_2 \le \dots \le a_n be real numbers such that a1+2a2+⋯+nan=0.a_1+2a_2+\dots+na_n=0. Prove that a1[x]+a2[2x]+⋯+an[nx]≥0a_1[x]+a_2[2x]+\dots+a_n[nx] \ge 0 for every real number xx. (Here [t][t] denotes the integer satisfying [t]≤t<[t]+1[t] \le t<[t]+1.)
inequalitiesfloor functionInteger Partalgebra