MathDB

3

Part of 2007 JBMO Shortlist

Problems(4)

(m+1)/(m+1,n+1) \in A => A \equiv N^*

Source: JBMO Shortlist 2007 A3

10/14/2017
Let AA be a set of positive integers containing the number 11 and at least one more element. Given that for any two different elements m,nm, n of A the number m+1(m+1,n+1) \frac{m+1 }{(m+1,n+1) } is also an element of AA, prove that AA coincides with the set of positive integers.
JBMOalgebra
n | (p - 1) & p | (n^6 - 1) prove p-n or p+n square

Source: JBMO Shortlist 2007 N3

10/14/2017
Let n>1n > 1 be a positive integer and pp a prime number such that n∣(p−1)n | (p - 1) and p∣(n6−1)p | (n^6 - 1). Prove that at least one of the numbers p−np- n and p+np + n is a perfect square.
Perfect SquareJBMOnumber theory
squares in a bw chessboard

Source: JBMO Shortlist 2007 C3

10/14/2017
The nonnegative integer nn and (2n+1)×(2n+1) (2n + 1) \times (2n + 1) chessboard with squares colored alternatively black and white are given. For every natural number mm with 1<m<2n+11 < m < 2n+1, an m×mm \times m square of the given chessboard that has more than half of its area colored in black, is called a BB-square. If the given chessboard is a BB-square, fi nd in terms of nn the total number of BB-squares of this chessboard.
JBMOcombinatoricsChessboard
2007 JBMO Shortlist G3

Source: 2007 JBMO Shortlist G3

10/10/2017
Let the inscribed circle of the triangle △ABC\vartriangle ABC touch side BCBC at MM , side CACA at NN and side ABAB at PP . Let DD be a point from [NP]\left[ NP \right] such that DPDN=BDCD\frac{DP}{DN}=\frac{BD}{CD} . Show that DM⊥PNDM \perp PN .
geometryJBMO