MathDB

15

Part of 2016 Online Math Open Problems

Problems(2)

2015-2016 Spring OMO #15

Source:

3/29/2016
Let a,b,c,da,b,c,d be four real numbers such that a+b+c+d=20a+b+c+d=20 and ab+bc+cd+da=16ab+bc+cd+da=16. Find the maximum possible value of abc+bcd+cda+dababc+bcd+cda+dab.
Proposed by Yannick Yao
Online Math Open
2016-2017 Fall OMO Problem 15

Source:

11/16/2016
Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of $1,$2,$5\$ 1, \$ 2,\$ 5, and $10\$ 10 bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly $1,000,000\$1{,}000{,}000, and whoever makes the last move wins the $1,000,000\$1{,}000{,}000 in the pile (if the pile is worth more than $1,000,000\$1{,}000{,}000 after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?
Proposed by Yannick Yao
Online Math Open