Problem 1: Lattice points
We have five lattice points (points with integers as coordinates) in two-dimensional space. Show that the midpoint of one of the line segments that connect these points is also a lattice point. Hint: Each point has two coordinates each of which can be either even or odd.

Problem 2: Divisibility by n
Consider a list of n + 1 positive integers: a₁, a₂,...,aₙ, aₙ₊₁

(a) Prove that it is always possible to choose a pair of these whose difference is divisible by n. Hint: For each ni, consider the remainder in the division by n.
(b) Suppose aₙ₊₁ is now dropped from the list and n > 2. Prove that it is always possible to choose a pair whose sum or difference is divisible by n. Hint What happens if you put a₁,...,aₙ, in n boxes based on their remainder in the division by n?

Problem 3: Languages
In a certain class there are 25 students: 14 speak Spanish, 12 speak French, 6 speak French and Spanish, 5 speak German and Spanish, and 2 speak all three. The 6 that speak German all speak another language. How many speak no foreign language?

Problem 4: Card game
You are playing cards with 3 other players, call them Player 1, Player 2, and Player 3. You draw a 10. You will loose if any player gets J, Q, K, or A. What is the probability that you will loose? Hint: Say that a hand is good for Player i if he gets J, Q, K, or A. Let Si be the set of hands that are good for Player i.
(a) Using Inclusion-Exclusion, find the number of good hands |S₁ U S₂ U S₃|. Dividing this by the total number of hands gives you the probability.
(b) Find the number of good hands in another way: First find the number of bad hands, then subtract it from the total number of hands.

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Discrete Math and Probability Problems
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