Hash functions and proofs of work: Hash functions are designed to achieve collision resistance (no collisions can be found) and proof-of-work security (there is no more efficient algorithm for finding outputs starting with many zeros besides trial and error). Show (by providing two counterexamples) that these two properties are not equivalent. That is, construct a hash function that is collision-resistant, but not proof-of-work secure, and construct a hash function that is proof-of-work secure but not collision-resistant. You can construct your counterexamples using a standard hash function H that is collision-resistant and proof-of-work secure as a building block. Note: Both of your counter-examples will be existential constructions that nobody would ever design or use in practice.