Mathcounts National Sprint Round Problems And Solutions -

Answer: ( \boxed10 )

Key Takeaway: Systematic casework by counts, not sequences, avoids overcounting paths.

The first term of a sequence is 3. Each term after the first is 4 more than twice the previous term. What is the 5th term?

Solution:
Let ( a_1 = 3 ).
( a_2 = 2(3) + 4 = 10 )
( a_3 = 2(10) + 4 = 24 )
( a_4 = 2(24) + 4 = 52 )
( a_5 = 2(52) + 4 = 108 ) Mathcounts National Sprint Round Problems And Solutions

✅ Answer: (108)

What is the sum of the distinct prime factors of 210?

Solution:
First, factor 210:
(210 = 21 \times 10 = (3 \times 7) \times (2 \times 5) = 2 \times 3 \times 5 \times 7).
All factors are prime and distinct. Sum = (2 + 3 + 5 + 7 = 17). Answer: ( \boxed10 ) Key Takeaway: Systematic casework

Answer: (\boxed17)

Problem (based on 2018 Sprint #25):
How many three-digit integers ( \overlineabc ) (with ( a \neq 0 )) are such that ( \overlineab + \overlinebc ) is a perfect square?

Note: ( \overlineab = 10a + b ), ( \overlinebc = 10b + c ). The first term of a sequence is 3

The "Death Spiral" of the Sprint Round occurs when a student spends 5 minutes on Problem #15 because they refuse to skip it.

Sometimes the fastest solution is eliminating impossibilities. Problem: The square root of a number is between 15 and 16. Which digit is in the units place of the number? Since $15^2 = 225$ and $16^2 = 256$, the number is in the 200s. However, the question asks for the units digit. Squaring a number ending in 5 ends in 5; squaring a number ending in 6 ends in 6. Logic can narrow the options before any calculation is done.