MathDB

3

Part of 2024 AIME

Problems(2)

How to Win the Game

Source: 2024 AIME I #3

2/2/2024
Alice and Bob play the following game. A stack of nn tokens lies before them. The players take turns with Alice going first. On each turn, the player removes 11 token or 44 tokens from the stack. The player who removes the last token wins. Find the number of positive integers nn less than or equal to 20242024 such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.
AIME I
Beautiful grid problem

Source: 2024 AIME II Problem 3

2/8/2024
Find the number of ways to place a digit in each cell of a 2×32 \times 3 grid so that the sum of the two numbers formed by reading left to right is 999999, and the sum of the three numbers formed by reading top to bottom is 9999. The grid below is an example of such an arrangement because 8+991=9998+991 = 999 and 9+9+81=999+9+81 = 99. [asy] unitsize(0.7cm); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); label("99", (0.5,0.5)); label("99", (1.5,0.5)); label("11", (2.5,0.5)); label("00", (0.5,1.5)); label("00", (1.5,1.5)); label("88", (2.5,1.5)); [/asy]