I was trying out some problems from a Russian mathematics book when this question came up. Though it seems pretty obvious that this question is to be solved through triangle inequality, yet I am unable to find sufficient conditions to get the conditions right.
As per the problem we need to prove that it is impossible to construct a star (like the one given in the picture) which satisfies: $$BC>AB,\ DE>CD,\ FG>EF,\ HI>GH,\ KA>IK.$$
P.S.: I tried to work out with some triangles like $\triangle BIF, \triangle BEH$ but unfortunately the equations seemed inconclusive.
Note that for any $△UVW$, if $VW > UW$, then $∠U > ∠V$.
Now suppose such star exists, then$$ ∠BAC > ∠BCA = ∠DCE > ∠DEC = ∠FEG > ∠FGE\\ = ∠HGI > ∠HIG = ∠KIA > ∠KAI = ∠BAC, $$ a contradiction.