Mathcounts National Sprint Round Problems And Solutions

In Mathcounts, answers to Sprint Round problems are almost always positive integers. If you solve a problem and get a fraction like $15/4$, double-check your work. While not impossible, non-integer answers are rare and often signal an arithmetic error.

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 )

✅ Answer: (108)

Because calculators are banned, all arithmetic must be done mentally or on paper. This round tests computational fluency, number sense, algebraic manipulation, and problem-solving agility. Mathcounts National Sprint Round Problems And Solutions

Problem (based on 2019 Sprint #12):
What is the sum of all positive integers ( n ) such that ( \frac36n ) is an integer and ( n ) is a multiple of 4?

Answer: ( \boxed\frac32 )

Key Takeaway: Coordinate geometry is your friend when no calculator is allowed. Shoelace is fast and accurate.

Let’s consolidate five representative problems with concise solutions: In Mathcounts, answers to Sprint Round problems are

Problem (Modeled after 2017 National Sprint #27):
If (x + y = 8) and (x^2 + y^2 = 34), find the value of (x^3 + y^3).

Solution:
We use identities:
((x+y)^2 = x^2 + 2xy + y^2 \Rightarrow 64 = 34 + 2xy \Rightarrow 2xy = 30 \Rightarrow xy = 15).

Then (x^3 + y^3 = (x+y)(x^2 - xy + y^2) = 8 \cdot (34 - 15) = 8 \cdot 19 = 152).

Answer: (\boxed152)

Variation: A harder version asks for (x^4 + y^4). You’d use (x^4 + y^4 = (x^2+y^2)^2 - 2(xy)^2 = 34^2 - 2(15)^2 = 1156 - 450 = 706).

Key takeaway: Memorize symmetric polynomial identities. They save precious seconds.

Without a calculator, practice: