MathDB

1

Part of 2012 JBMO TST - Turkey

Problems(2)

Find the greatest positive integer such that

Source: Turkey JBMO Team Selection Test Problem 1

5/30/2012
Find the greatest positive integer nn for which nn is divisible by all positive integers whose cube is not greater than n.n.
geometry3D geometrynumber theoryleast common multiplenumber theory proposedauyesl
Geometric Inequality

Source: Turkey JBMO Team Selection Test Problem 5

5/30/2012
Let a,b,ca, b, c be the side-lengths of a triangle, rr be the inradius and ra,rb,rcr_a, r_b, r_c be the corresponding exradius. Show that a+b+ca2+b2+c2≤2⋅ra2+rb2+rc2ra+rb+rc−3r \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r}
inequalitiesgeometryinradiusCauchy Inequalitygeometry proposed