Grade 7 Math Olympiad Problems

Problem 1: a sequence built from its own past terms

A sequence starts $3, 5$ , and every term after that is the sum of the previous two. Find the 8th term.

Solution: Just build it term by term:

3,\; 5,\; 8,\; 13,\; 21,\; 34,\; 55,\; 89

Each term is the sum of the two before it (e.g. $21+34=55$ ), so the 8th term is 89.

If $2^n \times 3^n = 216$ , find $n$ .

Solution: Combine the two powers first, since they share an exponent:

2^n \times 3^n = (2\times3)^n = 6^n

So $6^n = 216 = 6^3$ , giving $n=3$ .

In a class, 60% of students play basketball, 50% play soccer, and 30% play both. What percentage play neither?

Solution:Add the two sports, then subtract the overlap once so it isn't double-counted:

\text{Play at least one} = 60\% + 50\% - 30\% = 80\%

So $100\% - 80\% = 20\%$ play neither sport.

Three consecutive integers add up to 333. Find the largest one.

Solution: Call the smallest $n$ , so the three are $n, n+1, n+2$ :

n + (n+1) + (n+2) = 333 \;\Rightarrow\; 3n+3=333 \;\Rightarrow\; n=110

The three integers are 110, 111, 112. The largest is 112.

A 12-hour clock shows 3:00. What will it show 100 hours later?

Solution: Only the remainder after dividing by 12 matters, since every full 12 hours brings the clock back to the same position:

100 = 8 \times 12 + 4 \quad \Rightarrow \quad \text{remainder } 4

Adding just the remaining 4 hours to 3:00 gives 7:00.