MathDB

3

Part of 2006 IMC

Problems(2)

IMC 2006, problem 3, day 1

Source:

7/22/2006
Let AA be an nnxnn matrix with integer entries and b1,b2,...,bkb_{1},b_{2},...,b_{k} be integers satisfying detA=b1⋅b2⋅...⋅bkdetA=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}. Prove that there exist nnxnn-matrices B1,B2,...,BkB_{1},B_{2},...,B_{k} with integers entries such that A=B1⋅B2⋅...⋅BkA=B_{1}\cdot B_{2}\cdot ...\cdot B_{k} and detBi=bidetB_{i}=b_{i} for all i=1,...,ki=1,...,k.
linear algebramatrixRing TheoryIMCcollege contests
IMC 2006, problem 3, day 2

Source: Horrible

7/25/2006
Compare tan⁡(sin⁡x)\tan(\sin x) with sin⁡(tan⁡x)\sin(\tan x), for x∈]0,π2[x\in \left]0,\frac{\pi}{2}\right[.
trigonometrylogarithmsIMCcollege contests