Examples of seemingly elementary problems that are hard to solve?

Singmaster's conjecture. Any 15-year-old can understand what it says. Singmaster wrote that Paul Erdős told him it's probably true but probably very hard to prove.

It says there's a finite upper bound on the number of times a number other than 1 appears in Pascal's triangle. For all anyone knows for sure, the upper bound could be 8. And only one number is known to appear that many times: $$ \binom{3003}{1}=\binom{78}{2}=\binom{15}{5}=\binom{14}{6}. $$ It is known that infinitely many numbers appear 6 times; infinitely many appear 4 times; infinitely many appear 3 times, and infinitely many appear 2 times. (And only one appears just once.)

Whether any number appears an odd number of times where the odd number is more than 3 is unknown.

Is the sequence of fractional parts of $(3/2)^n$ dense in $[0,1]$?

This problem has a completeley elementary statement, yet noone has any idea how to prove it.

It is known that for almost every $t$, $t (3/2)^n \bmod 1$ is equidistributed in $[0,1]$ (this follows from a much more general result of Weyl), and also that for almost every $\beta>1$, the sequence $\beta^n\bmod 1$ is equidistributed in $[0,1]$ (This was proved by Koksma in the 30s).

It also follows from results of Pisot that $(3/2)^n \bmod 1$ has infinitely many accumulation points.

This question is somewhat related to the question (already mentioned in an answer) of whether famous irrational numbers are normal, but in a sense more elementary and frustrating because it is only about multiplication/division by $2$ and $3$!

Here are five problems which you might like:

(1) Are there 44 unit vectors in $\mathbb{R}^5$ where the dot product between each pair is less than $\frac{1}{2}$?

I originally saw this formulation on this MO post. It is open, and related to the kissing number in 5 dimensions.

(2) A Hadamard Matrix is a square matrix all of whose entries are $\pm1$, and whose rows are mutually orthogonal. Prove that there exists a Hadamard Matrix of size $4k$ for every $k\geq 1$.

This is known as the Hadamard Conjecture. While it was considered by Hadamard in the 19th century, it is still open today.

(3) Given $n$ distinct points in the plane, what is the minimum number of distinct distances between those points?

This is a famous problem of Erdos, and while it has not completely been resolved, a near optimal bound belongs to Guth and Katz. (See Terence Tao's blog post)

(4) Prove that $$\sigma(n)\leq H_n +e^{H_n}\log H_n$$ where $\sigma(n)$ is the sum of divisors function and $H_n=1+\frac{1}{2}+\cdots+\frac{1}{n}$ is the $n^{th}$ Harmonic number.

This problem is equivalent to the Riemann Hypothesis. This reformulation was done by Jeffrey Lagarias. (Since the reformulation was more recent, I think it qualifies for this list)

(5) If $\alpha\neq 0,1$ is algebraic, and $\beta$ is irrational algebraic, will $\alpha^\beta$ be transcendental?

This is the well known Hilbert's 7th Problem. This was resolved by Gelfond and Schneider in 1934. Related to this, there is the famous story of Hilbert giving a lecture in 1919 where he said that he might see the proof of the Riemann hypothesis in his life time, that the youngest members of the audience might live to see Fermat’s Last Theorem proved, but no one present in the hall would live to see a proof of transcendence of $2^{\sqrt{2}}$. Of course the exact opposite happened.