MathDB

Problem 2

Part of 2006 Federal Math Competition of S&M

Problems(3)

inequality, sum xy>=4sum x^2y^2+5xyz with x+y+z=1 (Serbia MO 2006 1st Grade P2)

Source:

4/10/2021
Let x,y,zx,y,z be positive numbers with x+y+z=1x+y+z=1. Show that yz+zx+xy≥4(y2z2+z2x2+x2y2)+5xyz.yz+zx+xy\ge4\left(y^2z^2+z^2x^2+x^2y^2\right)+5xyz.When does equality hold?
inequalities
locus of points in square, triangle centers

Source: Serbia MO 2006 2nd Grade P2

4/10/2021
For an arbitrary point MM inside a given square ABCDABCD, let T1,T2,T3T_1,T_2,T_3 be the centroids of triangles ABM,BCMABM,BCM, and DAMDAM, respectively. Let OMOM be the circumcenter of triangle T1T2T3T_1T_2T_3. Find the locus of points OMOM when MM takes all positions within the interior of the square.
geometry
1/x+1/y=1/p-1/q

Source: Serbia MO 2006 3&4th Grades P2

4/10/2021
Given prime numbers pp and qq with p<qp<q, determine all pairs (x,y)(x,y) of positive integers such that 1x+1y=1p−1q.\frac1x+\frac1y=\frac1p-\frac1q.
Diophantine equationnumber theory