MathDB

3

Part of 2014 JBMO TST - Turkey

Problems(2)

m^6+5n^2=m+n^3

Source: Turkey JBMO TST 2014 P3

6/21/2014
Find all pairs (m,n)(m, n) of positive integers satsifying m6+5n2=m+n3m^6+5n^2=m+n^3.
number theory proposednumber theory
Equivalence of identities in a triangle

Source: Turkey JBMO TST 2014 P7

6/21/2014
Let a line ℓ\ell intersect the line ABAB at FF, the sides ACAC and BCBC of a triangle ABCABC at DD and EE, respectively and the internal bisector of the angle BACBAC at PP. Suppose that FF is at the opposite side of AA with respect to the line BCBC, CD=CECD = CE and PP is in the interior the triangle ABCABC. Prove that FB⋅FA+CP2=CF2  ⟺  AD⋅BE=PD2.FB \cdot FA+CP^2 = CF^2 \iff AD \cdot BE = PD^2.
geometryincenterangle bisectorgeometry proposed