Is it permissible to hold a competition in an Islamic women's forum based on rivalry in increasing the number of times one says "Astaghfirullah" (I seek forgiveness from God)?

There is no harm in disclosing one's charity to motivate people to do good deeds. This is not considered the forbidden act of ostentation (tasmee'), which is prohibited. The forbidden ostentation is when a servant informs others of their acts of obedience out of hypocrisy and seeking people's praise. However, informing others of one's actions so that they may be emulated or compete in good deeds is permissible. Indeed, Abu Bakr and Umar, may Allah be pleased with them both, informed others of what they spent in equipping the army, and the Messenger of Allah, peace and blessings be upon him, did not object to their doing so.