Mathcounts National Sprint Round Problems And Solutions Jun 2026

. The formula is often remembered by the mnemonic man + dad = bmb + cnc :

Use the recursive formula to find $a_2$, $a_3$, and $a_4$. $a_2 = 3a_1 + 1 = 3(2) + 1 = 7$. $a_3 = 3a_2 + 1 = 3(7) + 1 = 22$. $a_4 = 3a_3 + 1 = 3(22) + 1 = 67$.

Use the Pythagorean Theorem: $a^2 + b^2 = c^2$, where $c$ is the length of the diagonal. Let $a = 8$ and $b = 5$. Then $8^2 + 5^2 = c^2$, so $64 + 25 = c^2$. Simplify: $89 = c^2$. Take the square root: $c = \sqrt89$.

How many three-digit integers ( \overlineabc ) (with ( a \neq 0 )) are such that ( \overlineab + \overlinebc ) is a perfect square? Mathcounts National Sprint Round Problems And Solutions

Reaching the level of speed required for the National Competition takes structured, deliberate practice over many months.

✅ (108)

Use inclusion-exclusion: Divisible by 3: ( \lfloor 99/3 \rfloor = 33 ) Divisible by 5: ( \lfloor 99/5 \rfloor = 19 ) Divisible by 15 (both): ( \lfloor 99/15 \rfloor = 6 ) So divisible by 3 or 5: ( 33 + 19 - 6 = 46 ) We want not both , so subtract the 6: ( 46 - 6 = 40 ) $a_3 = 3a_2 + 1 = 3(7) + 1 = 22$

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Area = ( \frac12 | x_Dy_E + x_Ey_F + x_Fy_D - (y_Dx_E + y_Ex_F + y_Fx_D) | )

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Number theory in the Sprint Round rewards knowledge of divisor function and prime factorization.

These official 2024 Chapter-level problems and their detailed solutions are an excellent starting point for preparation.

What is the value of ( 25^2 - 24^2 )?