NOTE The OP didn't state "Preferably, suggest some open problems so that new results can be obtained." when this was answered.
I can provide you with Burton's Elementary Number Theory. It has a series of historical introductions and great examples you'll probably find worth of a research project. He has information and obivously theory about results from Fermat, Euler, Diophantus, Wilson, Möbius, and others. I can also provide you with the three volumes of the History of Number Theory, which might be a great source.
A few examples are
Fermat's Little Theorem If $p\not\mid a$ then$$a^{p-1} \equiv 1 \mod p$$
Wilson's Theorem If $p$ is a prime then
$$({p-1})! \equiv -1 \mod p$$
Möbius Inversion Formula If we have two arithmetical functions $f$ and $g$ such that
$$f(n) = \sum_{d \mid n} g(d)$$
Then
$$g(n) = \sum_{d \mid n} f(d)\mu\left(\frac{n}{d}\right)$$
Where $\mu$ is the Möbius function.
Maybe so interesting as the previous,
The $\tau$ and $\sigma$ functions
Let $\tau(n)$ be the number of divisors of $n$ and $\sigma(n)$ its sum. Then if $$n=p_1^{l_1}\cdots p_k^{l_k}$$
$$\tau(n)=\prod_{m=1 }^k(1+l_m)$$
$$\sigma(n)=\prod_{m=1 }^k \frac{p^{l_m+1}-1}{p-1}$$
Legendre's Identity
The multiplicty (i.e. number of times) with which $p$ divides $n!$ is
$$\nu(n)=\sum_{m=1}^\infty \left[\frac{n}{p^m} \right]$$
However odd that might look, the argument is somehow simple. The multiplicity with which $p$ divides $n$ is $\left[\dfrac{n}{p} \right]$, for $p^2$ it is $\left[\dfrac{n}{p^2} \right]$, and so forth. To get that of $n!$ we sum all these values to get the above, since each of $1,\dots,n$ is counted $l$ times as a multiple of $p^m$ for $m=1,2,\dots,l$, if $p$ divides it exactly $l$ times. Note the sum will terminate because the least integer function $[x]$ is zero when $p^m>n$.
Perfect numbers
A number is called a perfect number is the sum if its divisors equals the number, this means
$$\sigma(n) =2n$$
Euclid showed if $p=2^n-1$ is a prime, then $$\frac{p(p+1)}{2}$$ is always a perfect number
Euler showed that if a number is perfect, then it is of Euclid's kind.
$n$ - agonal or figurate numbers.
The greeks were very interested in numbers that could be decomposed into geometrical figures. The square numbers are well known to us, namely $m=n^2$. But what about triangular, or pentagonal numbers?
Explicit formulas have been found, namely
$$t_n=\frac{n(n+1)}{2}$$
$$p_n=\frac{n(3n-1)}{2}$$
You can try, as a good olympiadish excercise, to prove the following:
$${t_1} + {t_2} + {t_3} + \cdots + {t_n} = \frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{6}$$
We can arrange the numbers in a pentagon as a triangle and a square:
$${p_n} = {t_{n - 1}} + {n^2}$$
Largely (very largely, so please take everything here with a grain of salt), there are two types of mathematical research, commonly referred to as 'theorem proving/problem solving' vs. 'theory building'. Typical characteristics of theorem proving/problem solving type research is to try and tackle a famous open problem, usually stated in the form of a conjecture as to the validity of a statement or the specification of a problem. Quite often this will entail spending a lot of time learning the relevant material, analyzing particular attempts at solutions, trying to figure out why they don't work, and hopefully come up with some improvement to an existing attempt, or a whole new attempt, that has a good change of working. Very famous open-standing questions include: The Riemann hypothesis and $P\ne NP$ (which are examples of theorem proving) and the solution of the Navier-Stokes equations (an example of problem solving), all three are in the Clay's Institute millennium problems list.
Theory building is a somewhat different activity that involves the creation of new structures, or the extension of existing structures. Usually, the motivation behind the study of these new structures is coming from a desire to generalize (in order to gain better insight or be able to apply particular techniques of one area to a broader class of problems) or there might be a need to these new structures to exist, due to some application in mind. Typical activities would include a lot of reading on relevant structures, understanding their global role, figuring out what generalizations or new structures would make sense, what the aim of the new theory will be, and then a long process of proving basic structure theorems for the new structures that will necessitate tweaking the axioms. A striking example of this kind of research is Grothendieck's reformalization of modern algebraic geometry. Cantor's initial work on set theory can also be said to fall into this kind of research, and there are many other examples.
Of course, quite often a combination of the two approaches is required.
Today, research can be assisted by a computer (experimentally, computationally, and exploratory). Any mathematics research will require extensive amount of learning (both of results and of techniques) and will certainly include long hours of thinking. I find the entire process extremely creative.
I hope this helps. As should be clear, this is a rather subjective answer and I don't intend any of what I said to be taken to be said with any kind of mathematical rigor.
Best Answer
Training for competitions will help you solve competition problems - that's all. These are not the sort of problems that one typically struggles with later as a professional mathematician - for many different reasons. First, and foremost, the problems that one typically faces at research level are not problems carefully crafted so that they may be solved in certain time limits. Indeed, for problems encountered "in the wild", one often does not have any inkling whether or not they are true. So often one works simultaneously looking for counterexamples and proofs. Often solutions require discovering fundamentally new techniques - as opposed to competition problems - which typically may be solved by employing variations of methods from a standard toolbox of "tricks". Moreover, there is no artificial time limit constraint on solving problems in the wild. Some research level problems require years of work and immense persistence (e.g. Wiles proof of FLT). Those are typically not skills that can be measured by competitions. While competitions might be used to encourage students, they should never be used to discourage them.
There is a great diversity among mathematicians. Some are prolific problem solvers (e.g. Erdos) and others are grand theory builders (e.g. Grothendieck). Most are somewhere between these extremes. All can make significant, surprising contributions to mathematics. History is a good teacher here. One can learn from the masters not only from their mathematics, but also from the way that they learned their mathematics. You will find much interesting advice in the (auto-)biographies of eminent mathematicians. Time spent perusing such may prove much more rewarding later in your career than time spent learning yet another competition trick. Strive to aim for a proper balance of specialization and generalization in your studies.