Books in the spirit of Problems and Theorems in Analysis by George Pólya and Gábor Szegő In the Preface of the first German Edition of the book Problems and Theorems in Analysis by George Pólya and Gábor Szegő, one can read  [emphasis mine] :

The chief aim of this book, which we trust is not unrealistic, is to 
  accustom advanced students of mathematics, through systematically 
  arranged problems in some important fields of analysis, to the ways 
  and means of independent thought and research. It is intended to serve
  the need for individual active study on the part of both the student
  and  the teacher. The book may be used by the student to extend his
  own  reading or lecture material, or he may work quite independently
  through  selected portions of the book in detail. The instructor may
  use it as an  aid in organizing tutorials or seminars.  
This book is no mere collection of problems. Its most important feature is the
  systematic arrangement of the material which aims to  stimulate the
  reader to independent work and to suggest to him useful  lines of
  thought.
  [...] 
The origin of the material is highly varied. We have made selections 
  from the classical body of knowledge of mathematics and also from 
  treatises of more recent date. We collected problems which had in part
  already been published in various periodicals and in part communicated
  to us verbally by their authors. We have adapted the material to our 
  purpose, completed, reformulated and substantially expanded it. In 
  addition we have published here for the first time in the form of
  problems  a number of our own original results. We thus hope to be
  able to offer  something new even to the expert.

To put it briefly, I think that the description of the key features that distinguish this book which is given in the preface is perfectly appropriate. With respect to these aspects, I regard this book as a sort of introduction to the methods, the ideas, and the results of mathematical research. 
It is actually the first book of this kind which I have ever come across, and I've really enjoyed going through it. So, the natural question that I pose is:

$\color{#c00}{\text{Question:}}$ 

Are there other similar books (in analysis or also different topics)
    that are in the spirit of Pólya and Szegő's work? That is, are there
    other books that are actually introductions to mathematical research
in the sense which is implied by the emphasized parts of the passage
    quoted above?


 A: There is a fine book by Laszlo Lovasz "Combinatorial Problems and Exercises". See (Wikipedia) http://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz or (Amazon) http://www.amazon.co.uk/Combinatorial-Problems-Exercises-Chelsea-Publishing/dp/0821842625/ref=sr_1_1?ie=UTF8&qid=1424531944&sr=8-1&keywords=Combinatorial+Problems+and+Exercises . Since you like George Polya check out his two volume "Mathematics and Plausible Reasoning" (ISBN-10: 1614275572).
A: I like Larson's "Problem Solving Through Problems." 
http://www.amazon.com/Problem-Solving-Through-Problems-Problem-Mathematics/dp/0387961712
Also here is a link to a list of a multitude of problem books.
http://link.springer.com/search/page/1?facet-content-type=%22Book%22&facet-series=%22714%22
Also here is a link to Prof. Noam Elkies website that has problems that might be of interest:
http://www.math.harvard.edu/~elkies/Misc/index.html
A: Note: Your question is really a challenge, cause the book you're pointing to as reference is a first-class evergreen of highest rank.
So, I was thinking: Which books give me a similar feeling when I am going through them as the classic by Pólya and Szegő and which of them could also play in the same leaque? One other criteria was, that they should provide a reasonable thorough survey through a part of mathematics.

The first two books which came into my mind:
Enumerative Combinatorics by Richard P. Stanley
An outstanding classic to study combinatorics with an enormous wealth of examples and solutions from easy to really hard. E.g. example 6.19  of Volume $2$ provides you with $66$ different combinatorial structures related with the ubiquitous Catalan Numbers.
Applied and Computational Complex Analysis by P. Henrici
is a classic on Complex Analysis from 1977. The keyword in this 3 volume set is applied. You will by guided through lots of enlightning examples, which help you to study complex analysis and become familiar with this part of mathematics. In fact I found this book only a few years ago. I was curious that so many papers had referenced P. Henrici's book. But since I've bought it and read parts of it with great pleasure I know the reason! :-)

A: I don't think any book can be deemed "an intro to research" but attempts have been done to present it opposite to something less elusive like math competitions. (Less elusive does not mean that competing is easier.) One such book, written by people strong in both is disciplines is
An Invitation to Mathematics: From Competitions to Research
