STEMvibe
Integral Cup 2026: Grand Prix
Sample Problems
Integration and Analysis
Problem 1. Let {an }∞
n=1 be a sequence defined as a1 = 1 and an =
limit
lim n − an
n→∞
exist? If it does, find it.
Problem 2. Let {un }n≥3 be a sequence defined by
un = un−1 + un−2 + un−3
with u0 = 1, u1 = 1 and u2 = 2. Consider the sum
∞
X
un
a
n n =
b
n=1 3
with gcd(a, b) = 1. Find a + b.
Problem 3. Calculate
Z ∞
0
Problem 4. Calculate
x2/3 (1 +
1
√
2x + x2 )
dx.
x4 + x 2
dx.
0 (x3 + 3x2 + 1)2
Z 1
1
q
n2 − an−1 . Does the