「ねえ、この[記事見]{きじみ}て。[昨日]{きのう}のスケート[大会]{たいかい}、[青木選手]{あおきせんしゅ}が___。」
a)[優勝]{ゆうしょう}したとか b)優勝したんだって c)優勝したという
Only (b) can be the correct answer because you have the newspaper right in front of you and therefore, you know exactly who won. You are NOT relying on hearsay or rumors, which is the source of information when you use 「とか」 or 「という」.
「~~したんだって」 can actually be used both ways --- quoting a hearsay or rumor AND quoting a direct source, but 「とか」 and 「という」 can only be used when quoting an indirect source. Tricky, I know but only (b) is correct.
「[今]{いま}、テレビの[天気予報]{てんきよほう}で[見]{み}たんだけど、[明日]{あした}は[全国的]{ぜんこくてき}に[雨]{あめ}だ___よ。」
a)と[聞]{き}いている b)と[言]{い}われている c)そうだ
Only (c) is correct and this question is more complicated than the first, I have to admit. It is more complicated because it is about the next day's weather, which is something no one can be 100% sure about. The first question was about a fact --- who won the race the previous day.
「と聞いている」 does not fit because it suggests that there has been some time between the present and the time you heard what you heard. The sentence clearly says 「今見た」. You saw the forecast a moment ago.
「と言われている」 does not fit, either, because it means "Many people say ~~." You just saw the forecast now. You did not have the time to hear what many people say about tomorrow's weather.
「そうだ」 might look like the wrong answer to some because it sounds like you are quoting a rumor, but it is correct because even though you have just seen the forecast on TV, there is a possibility that it might not actually rain tomorrow. That it might rain tomorow is not a fact at this point.