проявлять живой интерес(All our neighbours expressed a keen interest in the investigation and more than once offered their help in getting to the bottom of the mysterious sounds.ART Vancouver)
живо интересоваться(Himmler's diary, which he kept intermittently from the age of 10, shows that he took a keen interest in current events, dueling, and "the serious discussion of religion and sex".4uzhoj)
