Why did they need such short lived isotopes to find pear shaped nuclei? Recently, pear-shaped nuclei have been discovered. This discovery has interesting implications, as noted in this question: Do pear-shaped nuclei really have anything to do with time travel?
At the moment, the nuclei known to be pear-shaped are 220Rn (half-life of 55.6 s), 224Rn (half-life of 3.63 days) [1] and 144Ba (half-life of 11.5 s) [2].
Why are all pear-shaped nuclei so short lived? Is there any reason preventing pear-shaped nuclei to be stable?
[1] L. P. Gaffney, P. A. Butler et al.:Studies of pear-shaped nuclei using accelerated radioactive beams. Nature 497, 199–204 (09 May 2013) (E-print on L.P. Gaffney's L.U. page)
[2] Bucher, B. et al.: Direct Evidence of Octupole Deformation in Neutron-Rich 144Ba. Phys. Rev. Lett. 116, 112503 (2016) (link to arXiv preprint)
 A: The question is ill formed. That is because these are not short half-lives for nuclei in those regions. Exploring around this interactive table of nuclides shows that $^{220}$Rn's neighbors have half lives of 4.9 minutes ($^{221}$Fr), 25 minutes ($^{221}$Rn), 56 seconds ($^{219}$At), and 3.96 seconds ($^{219}$Rn) and similarly, $^{144}$Ba's neighbors vary from 1.791 seconds to 24.8 seconds. $^{224}$Rn does have an unusually long half life of 107 minutes (not 3.63 days, maybe you meant $^{222}$Rn?) when it's immediate neighbor $^{223}$Rn is only 24.3 minutes (even so, $^{226}$Ra with only two more protons has a half life of 1600 years).
In answer to the question why those elements have such short half lives compared to elements like Uranium or stable elements, that's because the strong nuclear force is a short ranged force. After about a proton-width or two, it is effectively zero. For nuclei near the valley of stability, this force is enough to keep the nuclei together. There are also effects from preserving a good ratio of protons to neutrons (equal near H, but roughly 3:2 in favor of neutrons at Pb) and whether you have an odd or even number of protons/neutrons.
As you add protons (move up on the table of nuclides) or neutrons (move right on the table of nuclides), you move away from that nice balance of protons to neutrons and the nucleus becomes more and more unstable. If you make the nucleus too large, you create too large of a surface area for the strong nuclear force to hold on to the outer layers of nucleons.
