7th Wina math olympiad - series 1 questions (English)

This is a translation of the paper I got at the Wina math olympiad yesterday (21st of November 2017). It doesn't mention who I could attribute these questions to ... Let's just say they were almost certainly all written by people at Wina (https://www.wina.be/) and from what I know probably mostly (if not entirely) by Art Waeterschoot. Series 1 is the competition for "casuals", there was also a series 2 for "die-hards".



7th Wina math olympiad - series 1

Time available: 3h00. No calculator or text books. Each question counts equally towards your total score and the questions are roughly sorted in order of increasing difficulty. Have fun and good luck!

Questions

1. A number of people go to a restaurant and each order one dish that they can afford. Later, when they have to pay, it turns out they all forgot what they ordered. The waiter helps by proposing that the person with the least money pays for the cheapest dish, the person with the second-least amount of money pays for the second-cheapest dish, etc. Prove that the waiter's strategy guarantees that everyone can pay.

2. Consider the points A_n = (n, 1) in the plane and let O be the origin. Prove that sum(i = 0, infinity, tan(angle defined by A_i, O and A_i+1)) < 2.

3.
a) Prove that for all integers n >= 1 and real numbers x >= 0 that (1 + x/n)^n >= 1 + x.
b) Determine the unique n for which n <= (3/2)^(1/3) + (4/3)^(1/4) + ... + (2017/2016)^(1/2017) < n + 1.

4. Let ABCD be a trapezoid with AD || BC. Let E be the intersection of the diagonals AC and BD and say that the straight line through D and the centroid of the triangle ABC intersects the line segment BC in F. Prove that EF divides the triangle ABC into two parts of equal area.

5. Johan Q is organizing a party for 2017 people. At parties amongst mathematicians it's an unspoken rule that each conversation with 3 people must go smoothly and this happens only if these three people all know each other or if they all don't know each other, else the conversation is asymmetric. Johan checks this and proposes to, at a given moment, have as many smooth conversations in groups of 3 take place with all party guests. What's the maximal amount of conversations that can take place? Of course every person can only participate in at most one conversation.

[Some clarifications I remember:
- the "A knows B" relationship is symmetrical
- the challenge is to find the "lowest upper bound", so the amount of conversations that can always take place in any situation
- something else I forgot, I think]

6. Determine all primes p for which a natural number a exists so that 2^(p + 1) + 2^a = 2*a^2 + (a^2)*(2^a)