IMC 2019 – Problems from Day 1

July 30, 2019 Leave a comment Go to comments

Problem 1. Evaluate the product

\displaystyle \prod_{n=3}^\infty \frac{(n^3+3n)^2}{n^6-64}.

Problem 2. A four-digit {YEAR} is called very good if the system

\displaystyle \left\{\begin{array}{rcl} Yx+Ey+Az+Rw & = & Y \\ Rx+Yy+Ez+Aw & = & E \\ Ax+Ry+Yz+Ew & = & A \\ Ex+Ay+Rz+Yw & = & R \end{array} \right.

of linear equations in the variables {x,y,z,w} has at least two solutions. Find all very good {YEAR}s in the {21}st century (between {2001} and {2100}).

Problem 3. Let {f : (-1,1) \rightarrow \Bbb{R}} be a twice differentiable function such that

\displaystyle 2f'(x)+xf''(x)\geq 1 \text{ for } x \in (-1,1).

Prove that

\displaystyle \int_{-1}^1 xf(x) dx \geq \frac{1}{3}.

Problem 4. Define the sequence {a_0,a_1,...} of numbers by the following recurrence:

\displaystyle a_0 = 1, \ \ a_1 = 2, \ \ (n+3)a_{n+2} = (6n+9)a_{n+1}-na_n \text{ for } n \geq 0.

Prove that all terms of this sequence are integers.

Problem 5. Determine whether there exist an odd positive integer {n} and {n \times n} matrices {A} and {B} with integer entries that satisfy the following conditions:

  1. {\det B = 1}.
  2. {AB = BA}.
  3. {A^4+4A^2B^2+16B^4 = 2019 I}

As usual {I} denotes the {n\times n} identity matrix.

Source: imc-math.org.uk

Categories: Algebra, Analysis, Higher Algebra, IMO, Linear Algebra, Number theory, Olympiad, Problem Solving Tags: , , , ,
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