Problems from an AoPS post (by Eduline, dated April 16, 2019)
1. Let S be the set of all integers k, 1 ≤ k ≤ n, such that gcd(k, n) = 1. What is the
arithmetic mean of the integers in S?
2. Let f (u) be a continuous function and, for any real number u, let [u] denote the greatest
integer less than or equal to u. Show that for any x > 1,
Z x [x] Z x
X
[u]([u] + 1)f (u)du = 2 i f (u)du.
1 i=1 i
3. Let I1 , I2 , I3 be three open intervals of R such that none is contained in another. If
I1 ∩ I2 ∩ I3 is non-empty, then show that at least one of these intervals is contained in
the union of the other two.
4. Given n positive integers, consider all possible sums formed by one or more of them.
Prove that these sums can be divided into n groups such that in each group the ratio
of the largest to the smallest does not exceed 2.
5. Let f : [0, 1] → [0, ∞) be a differentiable function, such that f 0 (x) is decreasing, f (0) = 0
and f 0 (1) > 0. Prove that the following inequality holds:
Z 1
1 f (1)
dx ≤ .
0 f (x)2 + 1 f 0 (1)
Can equality be achieved?
6. Let (xn ) be an integer sequence such that 0 ≤ x0 < x1 ≤ 100 and
xn+2 = 7xn+1 − xn + 280, for all n ≥ 0.
a) Prove that if x0 = 2, x1 = 3 then for each positive integer n, the sum of divisors of
the following number xn xn+1 + xn+1 xn+2 + xn+2 xn+3 + 2018 is divisible by 24.
b) Find all pairs of numbers (x0 , x1 ) such that xn xn+1 + 2019 is a perfect square for
infinitely many nonnegative integer numbers n.
7. Let f : R → (0, +∞) be a continuous function such that lim f (x) = lim f (x) = 0.
x→−∞ x→+∞
a) Prove that f (x) attains a maximum value on R.
b) Prove that there exist two sequeneces xn and yn with xn < yn , for all n = 1, 2, 3, ...
such that they have the same limit when n → ∞ and f (xn ) = f (yn ) for all n ≥ 1.
1
Z b
8. Let f : [a, b] → R be a continuous function such that f (x)dx 6= 0. Prove that there
a
are numbers a < α < β < b such that
Z α
f (x)dx = (b − α)f (β).
a
9. Let n > 1 be an integer and let f : [0, 1] → R be a continuous function such that
Z 1
1 1
f (x)dx = 1 + + ··· + .
0 2 n
1 − xn0
Prove that there is a real number x0 ∈ (0, 1), such that f (x0 ) = .
1 − x0
10. Let ψ : R → R be a continuous function such that
Z x+y Z x
ψ(t)dt = ψ(t)dt,
x x−y
for all x, y ∈ R. Prove that ψ is a constant function.
11. Let f, g : [a, b] → R be continuous functions. Prove that there is a real number c ∈ (a, b)
such that Z c Z b
f (x)dx + (c − a)g(c) = g(x)dx + (b − c)f (c).
a c
12. Let f : R → R be an injective and differentiable function. Prove that the function
F : (0, ∞) → R,
1 x
Z
F (x) = f (t)dt
x 0
is monotone.
Source: https://artofproblemsolving.com/community/c260h1551195p12175705
(You may go there for solutions and discussions regarding these problems.)
