butt heads 激しく争う

butt heads は文字通り、頭をぶつけ合う、衝突するという意味です。 0:01 there was a time when people thought Donald Trump's run to the White House was a joke. now the night after he and Hillary Clinton butted heads in the final debate, the pair were back together to attend the Al Smith Dinner where those in attendance roast each other as they raise money for Catholic charities. 昨夜の最終討論会で火花を散らした二人は今夜カトリック教会主催の晩餐会で再び顔を合わせまたも舌戦を繰り広げた。

Trump and Clinton 'roast' themselves and each other at charity dinner

https://www.youtube.com/watch?v=IF3n6L3P_f0  2:43 Russia remains a powerful country on the world stage, often butting heads with the U.S. and other Western countries. ロシアはアメリカや西側諸国といつも対立問題を抱えている大国だ。
How Corrupt Is Russia?

https://www.youtube.com/watch?v=GVc8cENFu9o  4:47 she wasn't the type of person to back down. and he clearly wasn't the type of person to back down. so they butt heads more often than not. 彼女は負けることを知らないし、彼も一歩も引くタイプじゃなかったから、二人の喧嘩は絶えなかったわ。
A Daughter's Last Days

https://www.youtube.com/watch?v=R1QZQeEU_Cw  0:16 Perry and Romney really butted heads. 二人の候補は激しくやりあった。
Romney & Perry Trade Jabs at GOP Presidential Debate

参考リンク
http://bridge-english.blogspot.jp/2015/08/blog-post_21.html  激しい口論を英語でなんと言うか？
http://bridge-english.blogspot.jp/2009/05/altercation.html  altercation

練習問題解答
a-6741 butted heads with
b-0378 butting heads with