Learn Hardcore Russian: Я люблю решать задачи по математике.

this would sound wrong or at least very odd, because решить (perfective) refers to one completed act: “to solve (and finish solving)”. You don’t “love to complete once”; you love the process/activity, so Russian uses решать.

What is the difference between я люблю решать and мне нравится решать?

Both are often translated as I like to solve / I like solving, but there is a nuance:

Я люблю решать задачи по математике.
Literally: I love to solve math problems.
– люблю is a bit stronger, more emotional, closer to “I love / I really like”.
Мне нравится решать задачи по математике.
Literally: To me it is pleasing to solve math problems.
– нравится is more neutral, like “I like”.

In everyday speech, they overlap a lot. Context and tone decide whether люблю sounds like “really like” or “love”.

Why is it задачи, not задача?

Задачи is the plural of задача.

задача = a problem, a task
задачи = problems, tasks

In English, when we say I like solving math problems, we normally mean “problems in general”, not just one. Russian does the same: the plural expresses the idea of the activity in general.

You could say:

Я люблю решать задачу по математике.

but that would usually mean one specific problem (maybe one problem that is especially interesting or difficult), which is not what is meant here.

What exactly does задачи mean here: “tasks” or “problems”?

In a math/school context:

задача most often = a math problem / word problem / exercise.

So задачи по математике is best understood as math problems or math exercises, not “tasks” in a general work sense.

What does по математике mean, and why is по used instead of another preposition?

По математике literally means “in math / in mathematics”.

по
- dative (математике) is very common to indicate a field, subject, or area:
  - книга по истории – a book on history
  - урок по русскому языку – a lesson in Russian (language)
  - конференция по физике – a conference on physics

So:

задачи по математике = problems in (the area of) mathematics, i.e. math problems.

Other prepositions like о математике (about math) would mean something different; they wouldn’t be used for school exercises.

What case is математике, and why?

Математике is in the dative case, singular.

The preposition по here requires the dative to express “in the area of / concerning (a subject)”:

по математике – in math / math-related
по биологии – in biology
по литературе – in literature

So математика (nominative) → математике (dative).

Could we just say Я люблю решать задачи and omit по математике?

Yes, grammatically you can:

Я люблю решать задачи.
= I like solving problems / tasks.

But then it becomes general: any kind of tasks/problems (logical puzzles, work tasks, etc.).

Adding по математике specifies what kind of problems: math problems.

Is the word order fixed? Could I say Я люблю задачи по математике решать?

Russian word order is flexible, but not all variants sound equally natural.

Most natural here:

Я люблю решать задачи по математике. ✅

Other possibilities:

Решать задачи по математике я люблю.
– Possible, but sounds a bit expressive or poetic; it emphasizes решать задачи по математике.
Я люблю задачи по математике решать.
– Understandable, but feels less natural in standard speech. Moving решать to the end can sound colloquial or stylistically marked.

For a learner, stick with the original order; it’s the standard, neutral pattern:
Subject – verb – infinitive – object – modifier.

How is the verb люблю formed and why does it end in -ю?

The infinitive is любить (to love).

It’s a second-conjugation verb. Present tense forms:

я люблю
ты любишь
он/она любит
мы любим
вы любите
они любят

So я люблю is 1st person singular present: I love / I like.
The ending -ю is the standard 1st person singular ending for many -ить verbs.

Why is it я люблю решать, not something like я есть люблю решать?

Russian does not use the verb “to be” in the present tense in this kind of sentence.

English: I am happy.
Russian: Я счастлив. (literally: I happy.)

Similarly with liking/loving:

Я люблю решать задачи по математике.
→ You never add есть here. Есть is not used as “am/is/are” in the present.

Есть is mainly used as “to eat” or sometimes to emphasize existence (“there is / there are”).

How do you pronounce this sentence, especially the stress?

Stresses:

Я люблЮ решАть задАчи по математИке.

Rough guide:

Я – ya
люблЮ – lyoo-BLYU (stress on -лю)
решАть – ree-SHAT’ (stress on -шать, final syllable)
задАчи – za-DA-chee (stress on -да)
по – po (short “po”)
математИке – ma-tee-ma-TEE-ke (stress on -ти)

Try to make consonants soft where marked in Russian spelling (e.g. люблю, математике).

How would I negate this: “I don’t like solving math problems”?

You simply add не before люблю:

Я не люблю решать задачи по математике.
= I don’t like solving math problems.

The structure stays the same; only не is added before the verb.

Could I replace задачи по математике with математические задачи? Is there a difference?

Yes, you can:

Я люблю решать математические задачи.

Both are correct and natural, but there is a small stylistic difference:

задачи по математике – very common in school / everyday speech.
математические задачи – a bit more formal or technical, literally “mathematical problems”.

In many contexts they’re interchangeable; for a typical school context, задачи по математике is extremely common.