ダービー is the transliteration of derby, referring to the horse race 東京優駿 (Tokyo Yushun), aka Japanese Derby, that took place on May 29.
クルトワ is Thibaut Courtois, the goalkeeper for Real Madrid that just won the UEFA Champions League.
冨樫先生 refers to 冨樫義博(TOGASHI Yoshihiro), the author of Yu Yu Hakusho, Level E, and Hunter X Hunter, who just tweeted on May 24 indicating that he had been working on new contents of Hunter X Hunter.
キスの日 means Kiss Day. May 23 1946 was the time when the first movie with kiss scene was released in cinemas.
三角関数 means trigonometric function. A member of the House of Representatives of Japan mentioned it on his Twitter account on May 17.