Problem 1: We call a 5-tuple of integers arrangeable if its elements can be labeled a, b, c, d, e in some order so that a – b + c – d + e = 29. Determine all 2017-tuples of integers n_{1}, n_{2}, …, n_{2017}such that if we place them in a circle in clockwise order, then any 5-tuple of numbers in consecutive positions on the circle is arrangeable.

**Problem 2. **Let ABC be a triangle with AB < AC. Let D be the intersection point of the internal bisector of angle BAC and the circumcircle of ABC. Let Z be the intersection point of the perpendicular bisector of AC with the external bisector of angle ∆BAC. Prove that the midpoint of the segment AB lies on the circumcircle of triangle ADZ.

**Problem 3: **Let A(n) denote the number of sequences a_{1} ≥ a_{2} ≥ … ≥ a_{k} of positive integers for which a_{1} +· · · + a_{k} = n and each a_{i} + 1 is a power of two (i = 1, 2, …, k).

Let B(n) denote the number of sequences b_{1} ≥ b_{2} ≥ … ≥ b_{m} of positive integers for which b_{1} + · · · + b_{m} = n and each inequality b_{j} ≥ 2b_{j+1 }holds (j = 1, 2, …, m – 1).

Prove that A(n) = B(n) for every positive integer n.

**Problem 4. **Call a rational number r powerful if r can be expressed in the form p^{k}/q for some relatively prime positive integers p, q and some integer k > 1. Let a, b, c be positive rational numbers such that abc = 1. Suppose there exist positive integers x, y, z such that a^{x} + b^{y} + c^{z} is an integer. Prove that a, b, c is all powerful.

**Problem 5. **Let *n *be a positive integer. A pair of *n*-tuples (*a*_{1}*, … a _{n}*) and (

*b*

_{1}

*, …, b*) with integer entries is called an

_{n}*exquisite pair*if |a

_{1}b

_{1}+ … + a

_{n}b

_{n}| ≤ 1.

*Determine the maximum number of distinct*

*n*-tuples with integer entries such that any two of them form an exquisite pair.