Art of Problem Solving: 2018 AMC 12 A #24

Art of Problem Solving's Richard Rusczyk solves the 2018 AMC 12 A #24.

[@derekd.4156]

I would just look that Alice will on average choose 1/2 and on average bob will choose 7/12. For carol to have the best chance she needs to select half way between them. (6/12 + 7/12)/2 = 13/24

[@prathikkannan3324]

Great think about it solution!

[@aadidalia9679]

For the final step you could have just used -b/2a to find the vertex

[@זאבגלברד]

Or equate the derivative to zero

[@armanbolouri9440]

Which is how -b/2a is derived (c term is gone after taking the derivative, b term is negated as you need to move it to the other side of the equation, and a term is doubled (because quadratics have a power of 2 then divides into b to find the remaining variable that was formerly connecting to be leaving x= -b/2a just thought of this real quickly right now but it’s pretty cool lmao)

[@armanbolouri9440]

^at derivative guy

[@tzakl5556]

Arman Bolouri or you can see it from the quadratic formula. It’s just -b/2a plus or minus the square root junk for the roots so the axis of symmetry is the midline from which u add and subtract from therefore it’s -b/2a

[@wontpower]

I love the AMC problems so much, I only wish I got into competition mathematics sooner! Is there anything like AMC for college students?

[@FT029]

www.maa.org/math-competitions/putnam-competition The Putnam competition is also a math competition, but it's proof-based and has calculus.

[@legendariersgaming]

www.amatyc.org/ is for 2-year colleges.

[@theevilmathematician]

i do not know. The only college math contest ik is Putnam

[@jerryli1995]

Fantastic video! :) One thing I think should be mentioned is that the probability of winning by choosing c = 13/24 is greater than 1/2. Plugging this into the expression it is obviously true, but I think it should be mentioned somehow.

[@aryanshshrivastava2650]

Expected value is much faster.

[@artix2468]

expected value of who?

[@rootsofdisunity3414]

@@artix2468 Expected value for Alice is (1+0)/2, or 1/2. Expected value for BOB is (1/2+2/3)/2, or 7/12. In order to maximize success, Carol should be right in between here, (1/2+7/12)/2 is 13/24 and we're done.

[@theevilmathematician]

I headsolved this question. So there is a number X between 1/2 & 2/3. Clearly X>1/2 & X

[@outmanewac9311]

Can you solve some imo problem

[@helo3827]

This is too easy for an AMC 12 Problem 24 imo.