MathDB

5

Part of 2002 IMO Shortlist

Problems(4)

IMO ShortList 2002, geometry problem 5

Source: IMO ShortList 2002, geometry problem 5

9/28/2004
For any set SS of five points in the plane, no three of which are collinear, let M(S)M(S) and m(S)m(S) denote the greatest and smallest areas, respectively, of triangles determined by three points from SS. What is the minimum possible value of M(S)/m(S)M(S)/m(S) ?
geometryratioareaIMO ShortlistTrianglepentagon
IMO ShortList 2002, number theory problem 5

Source: IMO ShortList 2002, number theory problem 5

9/28/2004
Let m,n≥2m,n\geq2 be positive integers, and let a1,a2,…,ana_1,a_2,\ldots ,a_n be integers, none of which is a multiple of mn−1m^{n-1}. Show that there exist integers e1,e2,…,ene_1,e_2,\ldots,e_n, not all zero, with ∣ ei ∣<m\left|{\,e}_i\,\right|<m for all ii, such that e1a1+e2a2+ … +enane_1a_1+e_2a_2+\,\ldots\,+e_na_n is a multiple of mnm^n.
modular arithmeticnumber theoryIMO Shortlistgenerating functionsroots of unitycomplex numbersHi
IMO ShortList 2002, algebra problem 5

Source: IMO ShortList 2002, algebra problem 5

9/28/2004
Let nn be a positive integer that is not a perfect cube. Define real numbers a,b,ca,b,c by a=\root3\of n\kern1.5pt,\qquad b={1\over a-[a]}\kern1pt,\qquad c={1\over b-}\kern1.5pt, where [x][x] denotes the integer part of xx. Prove that there are infinitely many such integers nn with the property that there exist integers r,s,tr,s,t, not all zero, such that ra+sb+tc=0ra+sb+tc=0.
linear algebraalgebrasystem of equationsIMO Shortlist
IMO ShortList 2002, combinatorics problem 5

Source: IMO ShortList 2002, combinatorics problem 5

9/28/2004
Let r≥2r\geq2 be a fixed positive integer, and let FF be an infinite family of sets, each of size rr, no two of which are disjoint. Prove that there exists a set of size r−1r-1 that meets each set in FF.
combinatoricsIMO ShortlistSet systems