I have a test where I am given the following article, what is an answer for the question: What is the name of the piece Tchaikovsky dedicated to Balakirev? ?

In 1867, Rubinstein handed over the directorship of the Conservatory to Zaremba. Later that year he resigned his conductorship of the Russian Music Society orchestra, to be replaced by Balakirev. Tchaikovsky had already promised his Characteristic Dances (then called Dances of the Hay Maidens) from his opera The Voyevoda to the society. In submitting the manuscript (and perhaps mindful of Cui's review of the cantata), Tchaikovsky included a note to Balakirev that ended with a request for a word of encouragement should the Dances not be performed.At this point The Five as a unit was dispersing. Mussorgsky and Rimsky-Korsakov wanted to remove themselves from Balakirev's influence, which they now found stifling, and go in their individual directions as composers. Balakirev might have sensed a potential new disciple in Tchaikovsky. He explained in his reply from Saint Petersburg that while he preferred to give his opinions in person and at length to press his points home, he was couching his reply "with complete frankness", adding, with a deft touch of flattery, that he felt that Tchaikovsky was "a fully fledged artist" and that he looked forward to discussing the piece with him on an upcoming trip to Moscow.These letters set the tone for Tchaikovsky's relationship with Balakirev over the next two years. At the end of this period, in 1869, Tchaikovsky was a 28-year-old professor at the Moscow Conservatory. Having written his first symphony and an opera, he next composed a symphonic poem entitled Fatum. Initially pleased with the piece when Nikolai Rubinstein conducted it in Moscow, Tchaikovsky dedicated it to Balakirev and sent it to him to conduct in Saint Petersburg. Fatum received only a lukewarm reception there. Balakirev wrote a detailed letter to Tchaikovsky in which he explained what he felt were defects in Fatum but also gave some encouragement. He added that he considered the dedication of the music to him as "precious to me as a sign of your sympathy towards me—and I feel a great weakness for you"....

Fatum