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3 problems match “extremal reasoning”. Clear

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1. Minimum red balloons in a string ●●●●○ 100 ptsDavid is making a string of red, white, and blue balloons by these rules: - no red balloon is adjacent to another red balloon; - white balloons appear in groups of exactly two, and groups of white balloons are separated by at least two non-white balloons; - blue balloons appear in groups of exactly three, and groups of blue balloons are separated by at least three non-blue balloons. If David uses more than 600 balloons, find the smallest number of red balloons he can use.
2. Arm wrestling tournament rounds ●●●●○ 100 ptsIn an arm wrestling tournament, 510 athletes participate. A win gives 1 point and a loss gives 0 points. If the winner initially had fewer points than the opponent, the winner additionally receives one point from the loser. In each round, participants whose scores differ by at most 1 point compete. The tournament ends as soon as a sole leader is determined. What is the minimum number of rounds that need to be played?
3. Maximum smallest ball in bowl Jia ●●●●○ 100 ptsPlace the 27 balls numbered $1$ to $27$ into three bowls, Jia, Yi and Bing, so that the average ball number in bowl Jia is $15$, in bowl Yi is $3$, and in bowl Bing is $18$, and each bowl contains at least $4$ balls. Find the maximum possible value of the smallest ball number in bowl Jia.