Is it possible to give an a priori answer to whether or not an infinite past is possible?
From my epistemic position, there is nothing contradictory about the possibility of an infinite past. So, from my epistemic position, it is possible the past is infinite. If by "give a correct answer using logic" you'll accept answers restricted to epistemic modality, then the correct answer is "yes, it is possible the past is infinite."
If, however, by "give a correct answer using logic" you mean determine - metaphysically speaking - that an infinite past is possible, then the answer seems "no" since this outstrips my epistemic position, and it's neither logically impossible that the past is finite nor does the possibility of an infinite past follow from any logical truths.
This is one of the antimonies in Kant’s Critique of Pure Reason. It is possible to give reasonable arguments in both directions, so pure reason isn’t able to resolve it “alone”... Kant famously leaves “room” for faith, but empirically there is data (measurements of the cosmic microwave background) that would suggest it is in fact the case that the universe began as a hot dense singularity at a particular point in the past, billions of years ago.
Is it possible to give a correct answer using logic to whether infinite past is possible or not?
There are articles on Wikipedia about Temporal Finitism, the philosophical view of Time and even a statistical mechanics and classical thermodynamics based explanation; which shows that choice of definition can certainly play a part in what the answer would be, and whether such an answer is logical and correct.
My offering is a physics based logical explanation:
References and Definitions:
"In special relativity, the relativity of simultaneity shows that there is no unique present, and that each point in the universe can have a different set of events that are in its present moment.
Many of special relativity's now-proven counter-intuitive predictions, such as length contraction and time dilation, are a result of this. Relativity of simultaneity is often taken to imply eternalism (and hence a B-theory of time), where the present for different observers is a time slice of the four dimensional universe. This is demonstrated in the Rietdijk–Putnam argument and additionally in an advanced form of this argument called the Andromeda paradox, created by mathematical physicist Roger Penrose.
It is therefore common (though not universal), for B-theorists to be four-dimensionalists, that is, to believe that objects are extended in time as well as in space and therefore have temporal as well as spatial parts. This is sometimes called a time-slice ontology.".
"That no inherent meaning can be assigned to the simultaneity of distant events is the single most important lesson to be learned from relativity.
— David Mermin, It’s About Time".
Fortunately, at the instance of our creation everything existed at the same point (and a moment later, at more than one point); prior to that instance, and arguably for some "duration" (not "time") after, "time" (the flow of existence forward, in such a manner that is not revertable) did not exist.
Put another way, for a period of unknown duration "time" did not advance and therefore the concept of time, (the movement of events forward; that is irreversible) did not exist. That is as far back as one can go in "time", to its creation.
Questions on Philosophy.SE:
Questions on Physics.SE
Past: "The past is the set of all events that occurred before a given point in time.".
Time Crystal: "A time crystal or space-time crystal is a structure that repeats in time, as well as in space.".
Prior to our existence we (our universe) had neither dimension, mass, nor time, anything - a state of nothingness. Elsewhere (if it existed) could have those properties, beings in another universe could say we did not exist (that we had no mass, were not located anywhere at any particular distance, up until a particular time). Another's existence is not "our past"; that they had dimensions and time prior to our existence does not impart the property upon us - we could not have had "time" prior to our existence.
Supported by: Wikipedia: The Very Early Universe:
Prior to 10^−43 seconds after creation:
"During the earliest moments of cosmic time, the energies and conditions were so extreme that current knowledge can only suggest possibilities, which may turn out to be incorrect. To give one example, eternal inflation theories propose that inflation lasts forever throughout most of the universe, making the notion of "N seconds since Big Bang" ill-defined. Therefore the earliest stages are an active area of research and based on ideas which are still speculative and subject to modification as scientific knowledge improves.
Although a specific "inflationary epoch" is highlighted at around 10^−32 seconds, observations and theories both suggest that distances between objects in space have been increasing at all times since the moment of the Big Bang, and are still increasing (with the exception of gravitationally bound objects such as galaxies and most clusters, once the rate of expansion had greatly slowed).".