I heard Magnus Carlsen's mother saying that he _____ in the chess competition…

Concept

Present Perfect Continuous (has/have + been + verb-ing) is used for an action or situation that began in the past and has continued up to the present moment. It is typically signalled by duration markers such as "for" or "since" (for example, "for the last 3 years", "since 2010").

Application

The blank carries the marker "for the last 3 years", showing that Magnus Carlsen's winning has continued without a fixed end point up to now:

"I heard Magnus Carlsen's mother saying that he has been winning in the chess competition for the last 3 years."

Contrast with the other forms

• "has won" is the Present Perfect Simple — it reports a completed action or result, not one still continuing over the marked duration.

• "is winning" is the Present Continuous — it describes an action happening right now, not one stretching back over a multi-year duration.

• "wins" is the Simple Present — it states a habitual fact or general truth, not an ongoing action tied to a specific duration.

The Present Perfect Continuous form that matches the ongoing multi-year duration is the correct completion.