Question 62
*Ali* has $100$ cards with numbers $1, 2, \ldots, 100$. Ali and *Amin* play a game together. In each step, Ali firstly chooses a card from the remaining cards and Amin decides to pick that card for himself or throw it away. In the case that he picks the card, he can't pick the next card chosen by Ali, and he has to throw it away. This procedure continues until there is no remaining card for Ali. Amin wants to pick cards in a way that the sum of the numbers of his cards is maximized and Ali wants to choose cards in a way that the sum of the numbers of Amin's cards is minimized. Find the largest value of $k$ such that Amin can play in a way that guarantees the sum of the numbers of his cards will be at least $k$.