Learn Hardcore Russian: Мой брат гордый, когда сам решает задачу по математике. - My brother is proud when he solves a math problem by himself.

гордый – a long-form adjective: Мой брат гордый = My brother is proud (as a quality / tendency).
горд – a short-form adjective: Мой брат горд = My brother is proud (right now / in this situation).
- Short forms are often a bit more formal or literary.
гордиться – a verb meaning to be proud (of something) and it needs an object in the instrumental case:
- Он гордится сестрой. – He is proud of his sister.
- Он гордится тем, что сам решает задачу по математике. – He is proud of the fact that he solves the math problem himself.

Your sentence uses гордый to give a description of the brother’s emotional reaction in such situations in a fairly neutral, spoken style.
You could also say:

Мой брат горд, когда сам решает задачу по математике.

This would sound a bit more formal or “bookish.”
But Мой брат гордится, когда сам решает задачу по математике sounds incomplete, because гордиться wants a specific object (proud of what?).

Where is the word “is” in Мой брат гордый? Why is there no verb?

Russian usually omits the present tense of быть (to be) in sentences like this.

Мой брат гордый. literally: My brother proud.
Understood meaning: My brother *is proud.*

The structure is:

Мой брат – subject (nominative case)
гордый – predicate adjective (also nominative, masculine singular to agree with брат)

In the present tense, есть (is/are) is normally left out:

Она учитель. – She is a teacher. (no есть)
Мой брат умный. – My brother is smart.

You only see есть in special emphatic or contrastive cases, or of course in other tenses (был, будет, etc.).

Why is there a comma before когда?

In Russian, когда introduces a subordinate clause of time, and such clauses are normally separated by a comma.

Your sentence has:

Main clause: Мой брат гордый – My brother is proud
Subordinate clause: когда сам решает задачу по математике – when he solves a math problem by himself

So you must write:

Мой брат гордый, когда сам решает задачу по математике.

The comma is obligatory in standard written Russian in this structure. You cannot drop it.

What exactly does сам mean here, and why not один or себя?

сам has two closely related meanings:

“himself” (emphasising who does the action)
“by himself / without help”

In your sentence, сам means something like by himself, without anyone’s help:

когда сам решает задачу – when he solves the problem himself / on his own.

Compare:

Он сам это сделал. – He did it himself (not someone else).
Он сделал это один. – He did it alone (nobody else was there / involved).

один focuses more on being alone, not having company.
сам focuses on doing the action without someone doing it for you or helping you.

себя is a reflexive object (whom? what?), not this kind of emphasis:

Он видит себя в зеркале. – He sees himself in the mirror.

So сам is the natural choice for “solves the problem himself (without help).”

Could we change the position of сам? For example, say когда решает сам задачу по математике?

Yes, Russian word order is flexible, and several versions are possible:

когда сам решает задачу по математике
когда он сам решает задачу по математике
когда решает задачу по математике сам

All are grammatically correct, but the nuance of emphasis changes slightly:

сам решает (before the verb) – often emphasizes who does it: he himself, not someone else.
решает … сам (after everything) – can highlight “on his own, without help,” as a kind of afterthought.

In everyday speech, когда он сам решает задачу по математике and когда сам решает задачу по математике are the most natural and common variants, with only very subtle differences in focus.

Why is it задачу and not задача?

задача is a feminine noun. Its forms:

Nominative singular: задача (dictionary form)
Accusative singular: задачу

The verb решать (to solve) is transitive and takes a direct object in the accusative case:

решать что? – задачу

So:

Он решает задачу. – He solves a (the) problem.

If you used задача (nominative), it would sound like “the problem solves” (as if the problem itself were the subject), which is ungrammatical here.

What does the phrase задачу по математике literally mean, and how is it different from математическую задачу?

Both are correct but feel slightly different:

задачу по математике
- literally: a problem *in/for mathematics*
- по
  - dative (математике) is very common with school subjects:
    - урок по математике – a math lesson
    - контрольная по математике – a math test
    - задача по физике – a physics problem
- This phrase strongly suggests a problem from a math class, textbook, test, etc.
математическую задачу
- adjective + noun: a mathematical problem
- sounds a bit more “technical” or more general: any mathematically-formulated problem, not necessarily connected to schoolwork.

In everyday school context, задачу по математике is more typical and natural.

Why is решает in the present tense and imperfective aspect here?

решает is:

Present tense
Imperfective aspect
3rd person singular of решать

In Russian, the present imperfective is often used for regular, repeated, or typical actions, not only for “right now.”

Your sentence means something like:

My brother is proud whenever he solves a math problem himself.

This “whenever/whenever that happens” idea is very naturally expressed with present imperfective + когда:

Он счастлив, когда играет в футбол. – He is happy when(ever) he plays football.

If you used решит (perfective future):

Мой брат будет гордый, когда сам решит задачу по математике.
– My brother will be proud when he solves the math problem (on that occasion).

That sounds like a one-time, specific future situation, not a general habit.

Is гордый positive, negative, or neutral here?

гордый can be:

Positive/neutral: proud in a good way, satisfied with your achievements.
- Я гордый отец. – I’m a proud father.
Negative: arrogant, too proud, haughty.
- Он слишком гордый, чтобы просить о помощи. – He is too proud to ask for help.

The meaning depends on context and sometimes tone.

In your sentence:

Мой брат гордый, когда сам решает задачу по математике.

this naturally sounds positive or neutral: he feels proud and satisfied with himself when he manages to solve math problems on his own.

Why is there no он before сам решает? Could we say когда он сам решает задачу по математике?

Yes, you can say:

Мой брат гордый, когда он сам решает задачу по математике.

That is perfectly correct.

However, in Russian you do not need to repeat a pronoun like он when the subject is already clear from the previous noun (мой брат). The default, more compact version is:

Мой брат гордый, когда сам решает задачу по математике.

Adding он often gives extra emphasis or contrast:

Мой брат гордый, когда он сам решает задачу, а не кто-то помогает.
– …when *he solves the problem himself, and not when someone helps.*

So in the simple version without contrast, omitting он is normal and natural.