In this exercise you must use the contractions with the preposition à. The preposition à (to, at, in) contracts ONLY with the masculine singular: LE and plural for both genders: LES
À + LE = AU
À + LES = AUX
There is no contraction with LA (f.sg.) nor with L’.

So here we have À + LE parc (m.sg.) = AU parc

Florence n’aime pas aller au parc.

bibliothèque… is feminine singular, so LA bibliothèque. There is no contraction with the article LA, so in this case we have : À LA bibliothèque

…à la bibliothèque…

école… is feminine singular that starts with a vowel and has the article LA. BUT, le and la elide with nouns that begin with a vowel, so if we add the article to it, that will be: l’école.
There is no contraction with the preposition À and the elided definite article L’.
Therefore, in this case we have : À L’ÉCOLE:

à l’école

MJC stands for: Maison des Jeunes et de la Culture.
Maison is feminine singular, LA MAISON.
As we know, there’s no contraction with LA, so in this case we have: à la MJC.

à la MJC

À contracts with the definite plural article LES, and there is only one plural definite article for both genders. So, even if in this case CARTES is feminine plural (UNE CARTE, LES CARTES) and LES ÉCHECS (*chess*) is masculine plural, the result is the same for both:
À + LES = AUX

In French, unlike English, LES ÉCHECS is a **plurale-tantum**: a noun that appears only in plural form and doesn’t have its singular variant referring to a single object. So, LES ÉCHECS means simply “chess” (usually uncountable in English), not “chesses”.
L’ÉCHEC in singular has nothing to do with the game of chess and it means: failure. Therefore, LES ÉCHECS can be both “chess” and “failures”

aux cartes – aux échecs

So, PISCINE (pool) is feminine singular: LA PISCINE, and there is no contraction with LA. In this case we’ll have: à la piscine

Inès nage à la piscine.

UN LYCÉE, LE LYCÉE, m.sg.
So, we’ll have the contraction with À and LE (AU).

au lycée.