No. It gives established as a hint. That means that there are situations where established might be a good translation. It often is not for the current sentence. This is one of the many reasons for NOT being too free with the "my answer should have been correct" button. In order to allow some correct answers in certain situations Duo has to build in more possible situational synonyms that are for only a few special cases. It is a difficult enough job since many words already have diverse standard meanings. If anyone has used a Thesaurus much you will appreciate this. You are using the same word too often and want to vary it some. But you may well find that of the five or ten synonyms listed there is only one or two, if that, that can be substituted in your particular case.
So why is 'decided' not accepted? "Yes, we have decided something" was wrong, even though this dictionary would provide "decide" as one of the two main translations of "determinar":
The spanish audio doesn't seem to match the written spanish. To me, the audio sounds does not sound like determinado, the first two syllables may be closer to reco or something similar, rather than deter... I may be wrong, but Duolingo might want to check to verify. Has anyone else noticed that?
That should be accepted. It may be due to one of two things. Duo sometimes doesn't go for additional synonyms when a cognate is a perfect fit like this one. Alternatively sometimes when you have a separatable verb phrase, Duo prefers the separated version of the sentence. We have found something out. But either way, it should be accepted.
It isn't. That would be Vamos a determinar algo. This is the present perfect, just like the English. The present perfect uses the present tense of the verb Haber and the past participle. Haber is actually the verb that is related to the Latin verb to have, avere. This became somewhat mutated in Spanish. It is only used as the auxiliary for the perfect tenses except for the third person singular which is used in the various tenses and moods to mean There is/are (There were, There will be, etc). Of course the present tense form has further changed from ha to hay for that meaning only. The verb tener comes from the Latin tenere which means to hold, and it retains that meaning in modern Italian and French (tenere and tenir respectively)