IMC 2015 for University Students: Day 1

Problem 1. For any integer n \ge 2 and two n \times n matrices with real entries A,B that satisfy the equation


prove that \det(A)=\det(B). Does the same conclusion follow for matrices with complex entries?

Problem 2. For a positive integer n, let f(n) be the number obtained by writing n in binary and replacing every 0 with 1 and vice versa. For example, n=23 is 10111  in binary, so f(n) is 1000 in binary, therefore f(23)=8. Prove that

\sum_{k=1}^n f(k) \le \frac{n^2}{4}.

When does the equality hold?

Problem 3. Let F(0)=0,F(1)=\frac{3}{2}, and F(n)=\frac{5}{2}F(n-1)-F(n-2) for n \ge 2.

Determine whether or not \displaystyle \sum_{n=0}^\infty \frac{1}{F(2^n)} is a rational number.

Problem 4. Determine whether or not there exist 15 integers m_1,m_2,\dots,m_{15} such that

\displaystyle \sum_{k=1}^{15} m_k \cdotp \text{arctan}(k) = \text{arctan}(16).

Problem 5. Let n \ge 2, and let A_1,A_2,\dots,A_{n+1} be n+1 points in the n-dimensional Euclidean space, not lying on the same hyperplane, and let B be a point strictly inside the convex hull of A_1,A_2,\dots,A_{n+1}. Prove that \angle A_iBA_j > 90^{\circ} holds for at least n pairs (i,j) with 1 \le i < j \le n+1.


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