The cosmic inflation did resolve the flatness problem by the theory which states that the universe appears to have a flat geometry.

Several problems that were found with the Big Bang theory were solved by cosmic inflation. I suggest you read about cosmic inflation. One of these problems was the fact that the Universe is fairly homogeneous. Without inflation, it wouldn't have had time to "spread out" any irregularities.

observations of the cosmic microwave background that indicate a flat geometry for the universe

I'm not an expert in astrophysics, but I expect it to be relatively soon. You may have seen the news articles which stated that a research team at the Harvard-Smithsonian Center for Astrophysics has discovered "cosmic ripples" from when the universe was 1*10 power -36 seconds old or so. Even if these are proven to not be decisive, as far as I know most astrophysicists agree that the Big Bang and cosmic inflation are the likeliest theory for universal origin and expansion. And with the recent advances in optical and computing technology, even more conclusive results are surely on the horizon. (Unless, that is, the theory of cosmic inflation is proven false, in which case we will probably start from scratch in all astrophysical fields.)

Alan Guth did not really discover anything - maybe a bit of fluff in his navel or something.

He did, however, develop the idea of cosmic inflation.