Let r be the region consisting of the points x y of the

Question

1: Let f be a real-valued function on the plane such that for every square ABCD in the plane, f(A) + f(B) + f(C) + f(D) = 0. Does it follow that f(P) = 0 for all points P in the plane?

2: Let f: R² → R be a function such that f(x, y) + f(y, z) + f(z, x) = 0 for all real numbers x, y, and z. Prove that there exists a function g : R → R such that f(x, y) = g(x)-g(y) for all real numbers x and y.

3: Let R be the region consisting of the points (x, y) of the cartesian plane satisfying both |x| - |y| ≤ 1 and |y| ≤ 1. Sketch the region R and find its area.

4: Let f be a polynomial with positive integer coefficients. Prove that if n is a positive integer, then f(n) divides f(f(n) + 1) if and only if n = 1.

5: Let S be a class of functions from [0,∞) to [0,∞) that satisfies:

(i) The functions f₁(x) = e^x - 1 and f₂(x) = ln(x + 1) are in S;

(ii) If f(x) and g(x) are in S, the functions f(x) + g(x) and f(g(x)) are in S;

(iii) If f(x) and g(x) are in S and f(x) ≥ g(x) for all x ≥ 0, then the function f(x) - g(x) is in S.

Prove that if f(x) and g(x) are in S, then the function f(x)g(x) is also in S.

6: Given a positive integer n, what is the largest k such that the numbers 1, 2, . . . , n can be put into k boxes so that the sum of the numbers in each box is the same? [When n = 8, the example {1, 2, 3, 6}, {4, 8}, {5, 7} shows that the largest k is at least 3.]

7: Inscribe a rectangle of base b and height h and an isosceles triangle of base b in a circle of radius one as shown. For what value of h do the rectangle and triangle have the same area?

8: A composite (positive integer) is a product ab with a and b not necessarily distinct integers in 2, 3, 4, . . .. Show that every composite is expressible as xy + xz + yz + 1, with x, y, z positive integers.

9: Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all 20 integers is 39. Show that there are two faces that share a vertex and have the same integer written on them.

10: For each integer n ≥ 0, let S(n) = n - m², where m is the greatest integer with m² ≤ n. Define a  sequence (a_k)_k=0^∞ by a₀ = A and a_k+1 = a_k + S(a_k) for k ≥ 0. For what positive integers A is this sequence eventually constant?