Problem List

These are the problems posted on the year 2018. Click to view!

01: Irrational Sum (Jan 2, 2018)
An infinite sequence \(a_n\) of natural numbers satisfy the following conditions for all \(n \in \mathbb{N}\).

(1) \(a_n < a_{n+1} \)

(2) \(a_{n+2} + a_{n} > 2a_{n+1}\)

Show that the sum
\[\Large \sum_{n=1}^{\infty} \frac{1}{10^{a_n}}\]
is irrational.

02: Periodic Function with Integral (Jan 23, 2018)
Given a continuous function \(f(x)\) with period \(T\), show that
\[\Large \lim_{n \rightarrow \infty} \int_a^b f(nx) dx = \frac{b-a}{T}\int_0^T f(x)dx\]
for any \(a < b\).

03: Cauchy-Schwarz ? (Mar 1, 2018)
For \(x, y \in \mathbb{R}\) such that \(x^2+y^2=1\), find the minimum and maximum values of
\[\large (3x+2y)^2 + (x+2y)^2\]

04: Differentiable Decimal (Apr 26, 2018)
For a real number \(x\in \left[0, 1\right)\), its decimal representation is
\[\large x=\sum_{n=1}^{\infty} \frac{a_n}{10^n} = 0.a_1a_2a_3\cdots\]
and there doesn't exist \(k\in\mathbb{N}\) such that for all \(n\geq k \left(n \in\mathbb{N}\right)\), \(a_n=9\).

Now define \(f: \left[0, 1\right) \rightarrow \mathbb{R}\) as
\[\large f(x)=\sum_{n=1}^{\infty} \frac{a_n}{10^{2n}}\]

Find a value \(a\) in the domain of \(f\) such that \(f'(a)\) does not exist, and show that \(f\) is not differentiable.

05: Who wants coffee? (Oct 10, 2018)
This problem is a little story.

When three professors are seated in a restaurant, the host asks them: "Does everyone want coffee?".

The first professor says: "I do not know".

The second professor says: "I do not know".

Finally, the third professor says: "No, not everyone wants coffee."

The host comes back and gives coffee to the professors who want it.

How did she figure out who wanted coffee?