MathDB

3

Part of 2010 Bulgaria National Olympiad

Problems(2)

Find all possible values of n - [Bulgaria NMO 2010]

Source:

12/29/2010
Let a0,a1,…,a9a_0, a_1, \ldots, a_9 and b1,b2,…,b9b_1 , b_2, \ldots,b_9 be positive integers such that a9<b9a_9<b_9 and ak≠bk,1≤k≤8.a_k \neq b_k, 1 \leq k \leq 8. In a cash dispenser/automated teller machine/ATM there are n≥a9n\geq a_9 levs (Bulgarian national currency) and for each 1≤i≤91 \leq i \leq 9 we can take aia_i levs from the ATM (if in the bank there are at least aia_i levs). Immediately after that action the bank puts bib_i levs in the ATM or we take a0a_0 levs. If we take a0a_0 levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of nn such that after finite number of takings money from the ATM there will be no money in it.
modular arithmeticcombinatorics unsolvedcombinatorics
Prove that M ≡ D - [Bulgaria NMO 2010]

Source:

5/19/2010
Let kk be the circumference of the triangle ABC.ABC. The point DD is an arbitrary point on the segment AB.AB. Let II and JJ be the centers of the circles which are tangent to the side AB,AB, the segment CDCD and the circle k.k. We know that the points A,B,IA, B, I and JJ are concyclic. The excircle of the triangle ABCABC is tangent to the side ABAB in the point M.M. Prove that M≡D.M \equiv D.
geometrytrapezoidgeometric transformationincenterinradiusgeometry unsolved