MathDB

4

Part of 2007 IMC

Problems(2)

Finite group, N_{ABC} = N_{CBA}

Source: IMC 2007, Day 1, Problem 4

8/6/2007
Let G G be a finite group. For arbitrary sets U,V,W⊂G U, V, W \subset G, denote by NUVW N_{UVW} the number of triples (x,y,z)∈U×V×W (x, y, z) \in U \times V \times W for which xyz xyz is the unity . Suppose that G G is partitioned into three sets A,B A, B and C C (i.e. sets A,B,C A, B, C are pairwise disjoint and G=A∪B∪C G = A \cup B \cup C). Prove that NABC=NCBA. N_{ABC}= N_{CBA}.
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Determinant of a certain kind of matrix

Source: IMC 2007 Day 2 Problem 4

8/7/2007
Let n>1 n > 1 be an odd positive integer and A=(aij)i,j=1..n A = (a_{ij})_{i, j = 1..n} be the n×n n \times n matrix with aij={2if i=j1if i−j≡±2(modn)0otherwise. a_{ij}= \begin{cases}2 & \text{if }i = j \\ 1 & \text{if }i-j \equiv \pm 2 \pmod n \\ 0 & \text{otherwise}\end{cases}. Find det⁡A \det A.
linear algebramatrixmodular arithmeticinductiontrigonometryinequalitiesalgebra