6 Subset Card Game

6 Subset Card Game Jonathan and Yiming are playing a card game. Jonathan has k > 2 cards, and each card has a real number written on it. Jonathan tells Yiming (truthfully), that the sum of the card values is 0, and that the sum of squares of the values on the cards is 1. Specically, if the card values are c1,cz,...,ck, then we have 2L1 c,- = 0 and Efe? = 1. Jonathan and Yiming also agree on a positive target value of a. The cards are then going to be dealt randomly in the following fashion: for each card in the deck, a fair coin is ipped. If the coin lands heads, then the card goes to Yiming, and if the coin lands tails, the card goes to Jonathan. Note that it is possible for either player to end up with no cards/all the cards. A player wins the game if the sum of the card values in their hand is at least a, otherwise it is a tie. Prove that the probability that Yiming wins is at most #