Mathcounts National Sprint Round Problems And Solutions __exclusive__

Mastering the Mathcounts National Sprint Round: Problems, Solutions, and Strategic Insights

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).

Triangle BEF: vertices B(8,0), E(3,15), F(24/11, 120/11). Use shoelace formula: Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 1/2 | 8(15 - 120/11) + 3(120/11 - 0) + (24/11)(0 - 15) | = 1/2 | 8( (165-120)/11 ) + 3(120/11) + (24/11)(-15) | = 1/2 | 8*(45/11) + 360/11 - 360/11 | = 1/2 | 360/11 | = 180/11. Mathcounts National Sprint Round Problems And Solutions

What is the Mathcounts National Sprint Round?

Answer:

( \boxed\frac32 )

Problem:

Find remainder when (2^100) is divided by 7. Solution: Cycle of 2^n mod 7: 2,4,1,2,4,1,... period 3. 100 mod 3 = 1, so 2^1 mod7 =2. Use shoelace formula: Area = 1/2 | x1(y2-y3)

Step-by-Step Solution

Key skills tested: