MathDB

2

Part of 2011 South East Mathematical Olympiad

Problems(2)

a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3)

Source: SMO 2011 Problem 2

8/18/2011
If positive integers, a,b,ca,b,c are pair-wise co-prime, and,  a2∣(b3+c3),b2∣(a3+c3),c2∣(a3+b3)\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3) find a,b,a,b, and cc
number theorygreatest common divisornumber theory unsolved
Geometric inequality

Source: China South East Mathematical Olympiad2011

8/18/2011
Let PiP_i i=1,2,......ni=1,2,......n be nn points on the plane , MM is a point on segment ABAB in the same plane , prove : ∑i=1n∣PiM∣≤max⁡(∑i=1n∣PiA∣,∑i=1n∣PiB∣)\sum_{i=1}^{n} |P_iM| \le \max( \sum_{i=1}^{n} |P_iA| , \sum_{i=1}^{n} |P_iB| ). (Here ∣AB∣|AB| means the length of segment ABAB) .
inequalitiesvectorinequalities proposed