"Graphs and Trees":
Basic concepts from graph theory: node degree, compute adjacency matrix of a given graph and the other way round, path/cycle, find (weak/strong) component, (rooted) tree, compute height/depth of a tree, are two graphs isomorphic?

"Order Relation":
Prove that a given relation on a given set is a partial (linear, well, dense) order, provide all the definitions, make Hasse diagram, denote min/max, largest/smallest elementsfind (maximal) chains/antichains

Find prime factorisation of the following numbers, find their gcd/lcm (also using prime factorisation)

"Mathematical Induction":
prove the statement by math induction (describe all the steps)

"Counting":
product and sum rule
representation method
inclusion-exclusion principle
pigeon principle 
compute the number of (all/k-element) subsets of a given set
count the sequences of numbers/letters with given constraints
count combinations/permutations (also with repetitions) of a given set
count functions/injections between given sets
compute the number of ways of putting (in/)distinguishable balls in distinguishable boxes/ number of composing b balls of ice cream in f flavours, etc.

"Basic Discrete Probability":
(can be mixed with counting problems)
check from the definition that two given events independent
compute conditional probability of A conditioned by B (where A and B are described and given)
compute total probability
use Bayes' formula for inverting conditional probability
compute distribution/expected value of given random variable (e.g. dice, coins, cards)

"Equipollence Relation":
prove from the definition that given set is countable (show a bijection)
prove that the two sets are equipollent (show a bijection)