Learn Hardcore Esperanto: Foje la biciklo havas malgrandan problemon, sed mia fratino solvas la problemon rapide. - Sometimes the bicycle has a small problem, but my sister solves the problem quickly.

Foje is often used for a general now and then, occasionally sense.
Kelkfoje literally means some times (several times), and often suggests on several occasions / quite a few times.

In this sentence, Foje la biciklo havas malgrandan problemon..., foje sounds like every now and then the bike has a small problem.
If you said Kelkfoje, it could sound a bit more like on a number of occasions the bike has a small problem — still correct, just a slightly different nuance.

Does Foje have to be at the beginning of the sentence?

No. Esperanto word order is flexible. You could say:

Foje la biciklo havas malgrandan problemon...
La biciklo foje havas malgrandan problemon...
La biciklo havas foje malgrandan problemon...

All are grammatically correct.
Placing Foje at the very beginning gives it a bit more emphasis, similar to English: Sometimes, the bicycle has a small problem...

Why do we say la biciklo but only mia fratino (without la)?

In Esperanto:

The definite article la is used much like the in English: la biciklo = the bicycle.
Possessive pronouns like mia, via, lia, ŝia, nia, ilia (my, your, his, her, our, their) already make the noun specific/definite.

So:

la biciklo – the bicycle (we’re talking about a particular bike)
mia fratino – my sister (already specific because she’s my sister)

You could say la mia fratino, but that’s unusual and usually unnecessary. Normally, you omit la when you use mia.

Why is it biciklo and not biciklon?

Because biciklo is the subject of the verb havas.

la biciklo havas malgrandan problemon
- la biciklo – subject (the doer)
- malgrandan problemon – direct object (the thing that is had)

In Esperanto, only the direct object gets the -n ending (accusative).
Subjects stay in the base form, so it is biciklo, not biciklon.

Why do malgrandan and problemon both end in -n?

Because malgrandan problemon is the direct object of the verbs havas and solvas.

The noun problemo takes -n: problemon, marking the problem as the object.
The adjective malgranda must agree with the noun in:
- number (singular/plural) and
- case (nominative/accusative).

So:

base forms: malgranda problemo – a small problem (no object marking)
as direct object: malgrandan problemon – a small problem (object, so both get -n).

That agreement rule is general in Esperanto: adjectives match the noun in -j (plural) and -n (accusative).

Could I say problemo malgranda instead of malgranda problemo?

Yes, Esperanto allows both orders:

malgranda problemo
problemo malgranda

Both mean a small problem.
The most common neutral order is adjective + noun (malgranda problemo), but putting the adjective after the noun (problemo malgranda) can give a bit of stylistic or emphatic flavor, or fit verse/rhythm in poetry. Grammatically, both are correct, and both would still take -n in the object role:

malgrandan problemon
problemon malgrandan

Why is the object repeated as la problemon instead of using a pronoun like ĝin?

Both are possible; the choice is stylistic.

Current sentence:

...sed mia fratino solvas la problemon rapide.

Possible variant:

...sed mia fratino solvas ĝin rapide. – but my sister solves it quickly.

Using la problemon:

repeats the noun and keeps it very clear what she solves.
can sound a bit more formal or explicit.

Using ĝin (it):

avoids repetition and is more compact.
is perfectly natural once the referent (the problem) is clear.

In real texts, you’ll see both patterns; pronouns are very common when the context is clear.

Why do we say havas problemon? Is that natural in Esperanto?

Yes, havi problemon is a normal, natural expression in Esperanto:

La biciklo havas problemon. – The bicycle has a problem.

Other possibilities exist:

La biciklo estas difekta. – The bicycle is damaged.
Estas problemo kun la biciklo. – There is a problem with the bicycle.

But havi problemon is completely standard and used a lot, especially in everyday speech.

Why is it solvas with -as at the end?

Solvas is the present tense form of the verb solvi (to solve):

solvi – infinitive (to solve)
solvas – present (solves / is solving)
solvis – past (solved / was solving)
solvos – future (will solve).

So:

mia fratino solvas la problemon rapide – my sister solves the problem quickly (present, general statement of habit/ability).

Why is it rapide and not rapida?

Because rapide modifies the verb solvas, not a noun.

In Esperanto:

Adjectives (modifying nouns) end in -a:
- rapida biciklo – a fast bicycle
- rapida solvo – a quick solution.
Adverbs (modifying verbs, adjectives, or whole sentences) end in -e:
- Ŝi solvas rapide. – She solves (it) quickly.
- Tre rapide – very quickly.

Here, rapide tells us how she solves the problem, so we need the adverb form rapide, not the adjective rapida.

Could I move rapide to another position, like mia fratino rapide solvas la problemon?

Yes. Common options:

...mia fratino solvas la problemon rapide.
...mia fratino rapide solvas la problemon.
...mia fratino solvas rapide la problemon.

All are grammatically correct.
The most typical are:

mia fratino rapide solvas la problemon
mia fratino solvas la problemon rapide

The meaning stays the same (my sister solves the problem quickly); small differences are more about rhythm/emphasis than grammar.

Why is the conjunction sed used, and not kaj?

kaj means and and simply joins things together.
sed means but and introduces a contrast.

The sentence says, roughly:

Sometimes the bicycle has a small problem, *but my sister solves the problem quickly.*

So the contrast is:

something negative or inconvenient (the bike has problems)
versus something positive (my sister quickly fixes them).

Because of that contrast, sed is the natural choice, not kaj.