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In this note, it is shown that the differential polynomial of the form 
$Q(f)^{(k)}-p$ has infinitely many zeros and particularly 
$Q(f)^{(k)}$ has infinitely many fixed points for any positive integer k, where f is a transcendental meromorphic function, p is a nonzero polynomial and Q is a polynomial with coefficients in the field of small functions of f. The results are traced back to Problems 1.19 and 1.20 in the book of research problems by Hayman and Lingham [Research Problems in Function Theory, Springer, 2019]. As a consequence, we give an affirmative answer to an extended problem on the zero distribution of 
$(f^n)'-p$, proposed by Chiang and considered by Bergweiler [Bull. Hong Kong Math. Soc. 1(1997), p. 97–101].
