MathDB

7

Part of 2008 All-Russian Olympiad

Problems(3)

2008 appears on blackboard

Source: All Russian 2008, Grade 9, Problem 7

6/13/2008
A natural number is written on the blackboard. Whenever number x x is written, one can write any of the numbers 2x \plus{} 1 and \frac {x}{x \plus{} 2}. At some moment the number 2008 2008 appears on the blackboard. Show that it was there from the very beginning.
number theorygreatest common divisorrelatively primecombinatorics
product of sum becoming exponents

Source: ARO 2008, Grade 10

6/12/2008
For which integers n>1 n>1 do there exist natural numbers b1,b2,…,bn b_1,b_2,\ldots,b_n not all equal such that the number (b_1\plus{}k)(b_2\plus{}k)\cdots(b_n\plus{}k) is a power of an integer for each natural number k k? (The exponents may depend on k k, but must be greater than 1 1)
number theory unsolvednumber theory
harrrrrd quadrilateral property

Source: ARO 2008

6/11/2008
In convex quadrilateral ABCD ABCD, the rays BA,CD BA,CD meet at P P, and the rays BC,AD BC,AD meet at Q Q. H H is the projection of D D on PQ PQ. Prove that there is a circle inscribed in ABCD ABCD if and only if the incircles of triangles ADP,CDQ ADP,CDQ are visible from H H under the same angle.
geometrygeometric transformationdilationangle bisector