AMC 8, 2024 Problem Marathon

[Reyaansh Agrawal]

5. Here we can do trial and error.

Any number n greater than 6 and less than or equal to 12 is not going to be an answer.

So we have 5 and 6 as options.

5 can be divided into 2 and 3, whose product is divisible by 6.

But 6 cannot be divided into two numbers such that their product is divisible by 6.

So answer is B) 6.

[Reyaansh Agrawal]

6. Here we can do some guesswork.

The perimeter would be the smallest path, So P is the smallest.

Q contains the most detours from the perimeter, so it should be the longest.

S would be longer than R, because it has longer detours.

So the answer would be B) P, R, S, Q

[Reyaansh Agrawal]

7. The number of squares in a 3 $\times$ 7 board is 21.

So the number of 1 $\times$ 1 squares is going to be congruent to 21 mod 4 (since 2$\times$2 blocks and 1$\times$4 blocks have 4 blocks).

In the options, this condition reduces it to 1 and 5.

It is obvious that 1 cannot be the answer.

Hence E) 5 is the answer.

 

 

[Reyaansh Agrawal]

8. Monday: $2

Tuesday: $5, $4

Wednesday: $8, $10, $7

Thursday: $11, $16, $13, $20, $10, $14

Answer: D) 6

[Reyaansh Agrawal]

9. Let us say Maria has n blue marbles.

Then, she would have $\frac{n}{2}$ green marbles and $\frac{n}{4}$ red marbles.

Therefore, the total sum of marbles is $\frac{7n}{4}$.

Since gcd(7, 4) = 1, $\frac{7n}{4}$ would be a multiple of 7.

Hence the answer would be E) 28.

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