2010 AMC 12A Problem 5

The opponent can score at most $50\cdot10=500$ on the last $50$ shots. Since Chelsea leads by $50,$ she must score more than $500-50=450$ points on her remaining shots to guarantee victory. Her $n$ bullseyes give $10n$ points, and her other $50-n$ shots give at least $4(50-n)$ points, so $$10n+4(50-n)\gt450.$$ This simplifies to $6n\gt250,$ i.e. $n\gt41\tfrac23.$ Therefore Chelsea needs at least $42$ bullseyes. Thus, C is the correct answer.