MathDB

4

Part of 2014 Moldova Team Selection Test

Problems(3)

Moldova TST, greatest prime divisor of the sum

Source: Moldova TST 2014, First Day, Problem 4

3/4/2014
Define p(n)p(n) to be th product of all non-zero digits of nn. For instance p(5)=5p(5)=5, p(27)=14p(27)=14, p(101)=1p(101)=1 and so on. Find the greatest prime divisor of the following expression: p(1)+p(2)+p(3)+...+p(999).p(1)+p(2)+p(3)+...+p(999).
AMCAIMEnumber theory proposednumber theory
How many distinct midpoints

Source: Moldova TST 2014, Second Day, Problem 4

3/30/2014
Consider n≥2n \geq 2 distinct points in the plane A1,A2,...,AnA_1,A_2,...,A_n . Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points?
combinatorics proposedcombinatorics
Cyclic sums

Source: Moldova TST 2014, Third Day, Problem 4

3/31/2014
On a circle n≥1n \geq 1 real numbers are written, their sum is n−1n-1. Prove that one can denote these numbers as x1,x2,...,xnx_1, x_2, ..., x_n consecutively, starting from a number and moving clockwise, such that for any kk (1≤k≤n1\leq k \leq n) x1+x2+...+xk≥k−1 x_1 + x_2+...+x_k \geq k-1.
combinatorics proposedcombinatorics