Any book on major (recent) math discovery (results) in an easy understanding way?

You might enjoy Piergiorgio Odifreddi, The Mathematical Century: The 30 Greatest Problems of the Last 100 Years, published by Princeton University Press in 2006. Also, Ben Yandell, The Honors Class: Hilbert's Problems and Their Solvers, published by AKPeters in 2001. Here's a link to a review.

$\displaystyle\lim_{n\to\infty}|\sin n|^{\frac{1}{n}}$

Random 3D points uniformly distributed on an ellipse shaped window of a sphere

Non-surjective but injective real polynomial functions $\mathbb{R}^n\to \mathbb{R}^n$

Evaluating the double limit $\lim_{m \to \infty} \lim_{n \to \infty} \cos^{2m}(n! \pi x)$

Is there a simple expression for $\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$

Constructing an $\epsilon$-net of $l_2$ unit ball

Prove that $\frac{(p^{n}-1)(p^{n}-p).....(p^{n}-p^{n-1})}{n!} \in \mathbb{N}$ with $p$ a prime number and $n \in \mathbb{N}$

Show $1+x+(x^2/2!)+ \cdots + (x^n/n!)=0$ has no rational solutions for all $n>1$.

Did Landau prove that there is a prime on $\bigl(x,\frac65x\bigr)$?

Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

Every tree has two leaves. Is my proof ok?

In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?