MathDB

7

Part of 2011 IMO Shortlist

Problems(4)

IMO Shortlist 2011, Algebra 7

Source: IMO Shortlist 2011, Algebra 7

7/11/2012
Let a,ba,b and cc be positive real numbers satisfying min⁡(a+b,b+c,c+a)>2\min(a+b,b+c,c+a) > \sqrt{2} and a2+b2+c2=3.a^2+b^2+c^2=3. Prove that
a(b+c−a)2+b(c+a−b)2+c(a+b−c)2≥3(abc)2.\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.
Proposed by Titu Andreescu, Saudi Arabia
inequalitiesalgebraIMO Shortlist
IMO Shortlist 2011, Combinatorics 7

Source: IMO Shortlist 2011, Combinatorics 7

7/12/2012
On a square table of 20112011 by 20112011 cells we place a finite number of napkins that each cover a square of 5252 by 5252 cells. In each cell we write the number of napkins covering it, and we record the maximal number kk of cells that all contain the same nonzero number. Considering all possible napkin configurations, what is the largest value of kk?
Proposed by Ilya Bogdanov and Rustem Zhenodarov, Russia
inequalitiescombinatoricsIMO ShortlistExtremal combinatorics
IMO Shortlist 2011, G7

Source: IMO Shortlist 2011, G7

7/13/2012
Let ABCDEFABCDEF be a convex hexagon all of whose sides are tangent to a circle ω\omega with centre OO. Suppose that the circumcircle of triangle ACEACE is concentric with ω\omega. Let JJ be the foot of the perpendicular from BB to CDCD. Suppose that the perpendicular from BB to DFDF intersects the line EOEO at a point KK. Let LL be the foot of the perpendicular from KK to DEDE. Prove that DJ=DLDJ=DL.
Proposed by Japan
geometrycircumcircleIMO Shortlist
IMO Shortlist 2011, Number Theory 7

Source: IMO Shortlist 2011, Number Theory 7

7/11/2012
Let pp be an odd prime number. For every integer a,a, define the number Sa=∑j=1p−1ajj.S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}. Let m,n∈Z,m,n \in \mathbb{Z}, such that S3+S4−3S2=mn.S_3 + S_4 - 3S_2 = \frac{m}{n}. Prove that pp divides m.m.
Proposed by Romeo Meštrović, Montenegro
modular arithmeticnumber theoryIMO Shortlist