MathDB

2

Part of 2019 CMIMC

Problems(4)

2019 A/NT2: Fraction is Not an Integer

Source:

1/27/2019
For all positive integers nn, let f(n)f(n) return the smallest positive integer kk for which nk\tfrac{n}{k} is not an integer. For example, f(6)=4f(6) = 4 because 11, 22, and 33 all divide 66 but 44 does not. Determine the largest possible value of f(n)f(n) as nn ranges over the set {1,2,…,3000}\{1,2,\ldots, 3000\}.
2019number theory
2019 C/CS2: Coloring the Vertices of a Cube

Source:

1/27/2019
How many ways are there to color the vertices of a cube red, blue, or green such that no edge connects two vertices of the same color? Rotations and reflections are considered distinct colorings.
2019combinatoricsgeometry3D geometry
2019 G2: Minimum Distance Between Circumcenters

Source:

1/27/2019
Suppose X,Y,ZX, Y, Z are collinear points in that order such that XY=1XY = 1 and YZ=3YZ = 3. Let WW be a point such that YW=5YW = 5, and define O1O_1 and O2O_2 as the circumcenters of triangles △WXY\triangle WXY and △WYZ\triangle WYZ, respectively. What is the minimum possible length of segment O1O2‾\overline{O_1O_2}?
2019geometrycircumcircle
2019 T2: Ordered Pairs Satisfying GCD Property

Source:

1/27/2019
Determine the number of ordered pairs of positive integers (m,n)(m,n) with 1≤m≤1001\leq m\leq 100 and 1≤n≤1001\leq n\leq 100 such that gcd⁡(m+1,n+1)=10gcd⁡(m,n). \gcd(m+1,n+1) = 10\gcd(m,n).
number theorygreatest common divisor2019team