The construction "(as) + adjective/adverb + as subject (may)+ verb" is similar to "no matter how" or "however" + adjective/adverb + subject + (may) + verb. As can be replaced with though.
Note: Although the first as should be suppressed, it is increasingly used.
1) (As) smart as they may be , they're still kids
No matter how smart they are, they're still kids
However smart they are, they're still kids
|— Por muy listos que sean, siguen siendo niños|
2) (As) attractive as the offer may be, I wouldn't touch it with a barge pole
No matter how attractive the offer is, I wouldn't touch it with a barge pole
However attractive the offer is, I wouldn't touch it with a barge pole
Attractive though the offer may be
|— Por muy atractiva que sea la oferta, yo no la tocaría ni con un palo|
3) (As) strange as it may sound, it's the truth
No matter how strange it sounds, it's the truth
However strange it may sound, it's the truth
|—Por muy raro que parezca, es la verdad|
A similar construction (as) much as /no matter how much + verb phrase is used with similar meaning. Adding may increases the concession and reduces the speaker's commitment to the truth or reality of the information presented.
4) (As) much as we loved it, we felt like no one was listening to our music
(As) much as we may have loved it, we felt like no one was listening
—Por más que nos encantaba, teníamos la sensación de que nadie escuchaba nuestra música
—Por mucho que nos encantara, teníamos la sensación de que nadie escuchaba nuestra música
|5) No matter how much you say you are innocent, nobody is going to believe you||—Por mucho que digas que eres inocente, nadie te va a creer|
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