Prove Vigenere cipher isn't perfectly secret
One way to approach this is to look at the decryption process; the plaintext is a function of the ciphertext and the key.  The conditional probability says you know the ciphertext (entropy 0), you don't know the key (entropy $n \log_2 26$); what's the entropy of the decrypted plaintext (which is a function of those two)?  How does that compare to the entropy of an arbitrary plaintext of length $2n$?