2022 AIME I Problem 6

Count the bad pairs among the $\binom{24}{2}$ choices with $5 \lt a \lt b \lt 30:$ every four-term progression must use $a$ or $b$

Beyond $a = 6$ and $20 \in \{a, b\},$ exactly three pairs $(a, b)$ complete a progression with two of the fixed terms. Watch for double counting.

The sequence is increasing exactly when $5 \lt a \lt b \lt 30,$ giving $\binom{24}{2} = 276$ pairs. The six fixed terms contain no four-term arithmetic progression, so every progression must involve $a$ or $b.$ If only one of them is involved, three fixed terms must already be in progression: $3, 4, 5$ extends only by $6,$ and $30, 40, 50$ extends only by $20.$ So the single-variable violations are $a = 6$ ($23$ pairs) and $20 \in \{a, b\}$ ($23$ pairs), which overlap in the pair $(6, 20).$

If both $a$ and $b$ are involved, two fixed terms complete the progression. Checking the possible positions: $(4, 5, a, b)$ gives $(6, 7);$ $(3, 5, a, b)$ gives $(7, 9);$ $(3, a, b, 30)$ gives $(12, 21);$ $(4, a, b, 40)$ gives $(16, 28);$ $(5, a, b, 50)$ gives $(20, 35),$ out of range; and $(a, b, 30, 40)$ gives $(10, 20).$ Of these, $(6, 7)$ and $(10, 20)$ are already counted, so $(7, 9),$ $(12, 21),$ and $(16, 28)$ are the only new bad pairs.

The number of valid pairs is $276 - (23 + 23 - 1)$ $- 3 = 276 - 48 = 228.$