MathDB

9

Part of 2015 Harvard-MIT Mathematics Tournament

Problems(2)

2015 Algebra #9: Divisible by Large Power

Source:

3/28/2015
Let N=302015N=30^{2015}. Find the number of ordered 4-tuples of integers (A,B,C,D)∈{1,2,…,N}4(A,B,C,D)\in\{1,2,\ldots,N\}^4 (not necessarily distinct) such that for every integer nn, An3+Bn2+2Cn+DAn^3+Bn^2+2Cn+D is divisible by NN.
Divisibilityalgebranumber theory
2015 Geometry #9

Source:

12/23/2016
Let ABCDABCD be a regular tetrahedron with side length 11. Let XX be the point in the triangle BCDBCD such that [XBC]=2[XBD]=4[XCD][XBC]=2[XBD]=4[XCD], where [ω‾][\overline{\omega}] denotes the area of figure ω‾\overline{\omega}. Let YY lie on segment AXAX such that 2AY=YX2AY=YX. Let MM be the midpoint of BDBD. Let ZZ be a point on segment AMAM such that the lines YZYZ and BCBC intersect at some point. Find AZZM\frac{AZ}{ZM}.