MathDB

5

Part of 2011 JBMO Shortlist

Problems(4)

Really easy problem

Source: JBMO 2011 Shortlist A5

5/15/2016
A5\boxed{\text{A5}} Determine all positive integers a,ba,b such that a2b2+208=4([a,b]+(a,b))2a^{2}b^{2}+208=4([a,b]+(a,b))^2 where [a,b][a,b]-lcm of a,ba,b and (a,b)(a,b)-gcd of a,ba,b.
number theory
sets where each element does not divides other's sum

Source: JBMO 2011 Shortlist C5

10/14/2017
A set SS of natural numbers is called good, if for each element x∈S,xx \in S, x does not divide the sum of the remaining numbers in SS. Find the maximal possible number of elements of a good set which is a subset of the set A={1,2,3,...,63}A = \{1,2, 3, ...,63\}.
JBMOcombinatoricsSets
2011 JBMO Shortlist G5

Source: 2011 JBMO Shortlist G5

10/8/2017
Inside the square ABCD{ABCD}, the equilateral triangle △ABE\vartriangle ABE is constructed. Let M{M} be an interior point of the triangle △ABE\vartriangle ABE such that MB=2MB=\sqrt{2}, MC=6MC=\sqrt{6}, MD=5MD=\sqrt{5} and ME=3{ME=\sqrt{3}}. Find the area of the square ABCD{ABCD}.
geometryJBMO
sum of digits 2011 and 6 / product of digits

Source: JBMO 2011 Shortlist N5

10/14/2017
Find the least positive integer such that the sum of its digits is 20112011 and the product of its digits is a power of 66.
JBMOnumber theorysum of digits