MathDB

3

Part of 2014 Bosnia Herzegovina Team Selection Test

Problems(2)

Nice exponential

Source: Bosnia and Herzegovina TST 2014 day 1 problem 3

5/10/2014
Find all nonnegative integer numbers such that 7x−2⋅5y=−17^x- 2 \cdot 5^y = -1
modular arithmeticnumber theory unsolvedZsigmondy
Circumcenter and orhocenter

Source: Bosnia and Herzegovina TST 2014 day 2 problem 3

5/11/2014
Let DD and EE be foots of altitudes from AA and BB of triangle ABCABC, FF be intersection point of angle bisector from CC with side ABAB, and OO, II and HH be circumcenter, center of inscribed circle and orthocenter of triangle ABCABC, respectively. If CFAD+CFBE=2\frac{CF}{AD}+ \frac{CF}{BE}=2, prove that OI=IHOI = IH.
geometrycircumcircleangle bisectorperpendicular bisectorgeometry unsolved