Square ABCD has side length 2. Points E and F are midpoints of AB and BC respectively. What is the area of triangle DEF?
Trying to calculate the number (impossible by hand). The National Solution: Look for a pattern in the powers of 2 modulo 7. $2^1 = 2$ $2^2 = 4$ $2^3 = 8 \equiv 1 \pmod7$ Since $2^3 \equiv 1 \pmod7$, the powers cycle every three: 2, 4, 1. We need to find where $2023$ falls in the cycle. $2023 \div 3$ leaves a remainder of $2$. Therefore, $2^2023$ has the same remainder as $2^2$, which is 4 . Mathcounts National Sprint Round Problems And Solutions
Because the National Competition is the highest level of the program, the problems are proprietary, but several sites host archives for practice: Official MATHCOUNTS Store Mathcounts Foundation Store is the only source for official, curated books like The All-Time Greatest MATHCOUNTS Problems The Most Challenging MATHCOUNTS Problems Solved . These include detailed, step-by-step solutions. Art of Problem Solving (AoPS) Wiki Square ABCD has side length 2
Final thought: The Mathcounts National Sprint Round isn’t about being a human calculator. It’s about being a strategic, resilient problem-solver who can execute clean mathematics on the fly. Trying to calculate the number (impossible by hand)