I’ve been looking for this exact book for such a long time (I very distinctly remember the penguin test you mentioned being a part of it). I’m 90% sure that “The Ability” by M.M. Vaughan is the book. I’m going to try to find it at my local library to be sure but I hope this was the book you were looking for!!
A r/whatsthebook question where this was identified as the correct answer gave the description:
world where some kids are “gifted”, but the ones who are only retain their powers until they are 13 (i think) years old. the main character learns when he is 12ish that he has powers, and the way they test them is by hiding some image and asking him what image he thinks it is — i remember the image having been a penguin image. kid was very surprised that he got it right, and they brought him to a place to meet other kids.
fast forward to the end of the first book, he’s about to turn 13 and lose his powers. he stares at himself in the mirror and reads his own mind as the minutes tick down, observing his mind space (which he views as a room with drawers) filling up with cement. he scoops out the cement as time goes on — there is a sequel, i think he retains his powers.