This is my first post in this forum (so please be nice!). Cyber-shihan Ubaldo Alcantara first asked me for information on this topic and it took me some time to realise that it was in connection with Jun Akiyama's web site. I sent Ubaldo a reply, but then realised it was better to contribute to the forum myself. So here is what I have discovered in O Sensei's writings about competition.
1. The word which O Sensei uses for 'competition' is 競争 (きょうそう: KYOU-SOU). The first character is composed of the Radials 117 (standing) and the character (344 in Nelson's dictionary) for 'ani' (brother). The second character is also read as 'araso' and the meaning of the compound is basically 'rivalry'.
The word 試合 (しあい SHI-ai) has the sense of two teams meeting for a game or match and is not used by O Sensei.
2. Where does O Sensei discuss competition in Aikido? I have seen no evidence for any general declaration made by O Sensei against competition. There is a reference to sports understood by O Sensei in a western sense on Page 50 of Hideo Takahashi's book, "Takemusu Aiki", which records lectures given by the Founder. A translation of O Sensei's ideas is given on Page 21 of Issue 117 of Aikido Journal. There O Sensei does talk about competition as applied to aikido, always using the Chinese character I explained above. His views are clearly old-fashioned and he makes statements about Japan and western sports which are no longer true.
3. The reference to Tomiki Sensei and competition appears on pp.184-188 of "Aikido Ichiro", by Kisshomaru Ueshiba. Doshu explains that Tomiki Sensei became a professor at Waseda University in 1954 but often came to visit the Founder in Iwama and Tokyo. Tomiki Sensei was a POW in Siberia and developed a system of aiki-taiso, probably to stay alive, and explained his system to O Sensei. In Kisshomaru Doshu's words,
"On seeing this (sc. Tomiki Sensei's system), my father said,
そのよおうなものを 合気 と称されて困る
"If you call this sort of thing "Aiki", it will cause problems."
Kisshomaru Doshu then goes on to discuss what happened afterwards. I muself have had lengthy conversations with Kisshomaru Doshu and with Okumura Shigenobu Sensei (9th dan), who first learned aikido at the hands of Tomiki Sensei and was later deputed to negotiate over whether Tomiki Sensei should use 'aikido' for his art. Okumura Sensei was clearly torn between loyalty to his sensei and loyalty to the Founder.
Best regrds to all,