## Euclid 2024 Question 2, CEMC UWaterloo

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#### Difficulty: 5

#### This problem is tagged with euclid, euclid24, highschool.

**(Euclid 2024, Question 2, CEMC - UWaterloo)**

(a) In a sequence with six terms, each term after the second is the sum of the previous two terms. If the fourth term is $13$ and the sixth term is $36$, what is the first term?

(b) For some real number $r \neq 0$, the sequence $5r$, $5r^2$, $5r^3$ has the property that the second term plus the third term equals the square of the first term. What is the value of $r$?

(c) Jimmy wrote four tests last week. The average of his marks on the first, second and third tests was $65$. The average of his marks on the second, third and fourth tests was $80$. His mark on the fourth test was $2$ times his mark on the first test. Determine his mark on the fourth test.

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