noun
competitive spirit; aversion to losing
Describes a personality trait of hating to lose and being determined to win; often used positively for someone with a strong fighting spirit, but can also imply stubbornness.
彼女は負けん気が強いから、どんな試合でも最後まで諦めない。
She has a strong competitive spirit, so she never gives up until the end of any match.
負けん気を出して、もう一度挑
My aversion to losing kicked in, and I tried again.
From 負け (losing) + ん (contracted old negative ぬ) + 気 (spirit), literally 'spirit of not losing'. The ん is a remnant of the classical negative form.