HCMUS Olympic Team Selection Test

Problem 1.

Find $a,b$ such that the following system of equations has a solution $$ \begin{cases} (a+1)x_1+x_2+x_3+x_4=1\\ x_1+(a+1)x_2+x_3+x_4=1\\ x_1+x_2+(a+1)x_3+x_4=1\\ x_1+x_2+x_3+(a+1)x_4=b \end{cases} $$

Problem 2.

Consider the equation

$$ x^3-2026x+1=0 $$

1. Prove that the equation has three real roots $ a < b < c$ .
2. Let $$ \mathcal A= \begin{pmatrix} a & b & c\\ c & a & b\\ b & c & a \end{pmatrix} $$ Compute $r(\mathcal A)$.
3. Is the matrix $\mathcal A$ diagonalizable over $\mathbb R$? What about over $\mathbb C$? Justify your answer.

Problem 3.

Let $n$ be a positive integer. Consider tuples $(b_1,b_2,\ldots,b_n)\in\mathbb R^n$. Denote by $V_n$ the set of tuples $(b_1,b_2,\ldots,b_n)$ such that there exist pairwise distinct real numbers $a_1,a_2,\ldots,a_n$ satisfying $$ \sum_{i=1}^n b_iP(x+a_i)=0, \qquad \deg P < n $$ Find $\dim V_n$.

Problem 4.

Let $A\in M_n(\mathbb R)$. Prove that there exists $B\in M_n(\mathbb R)$ such that $$ (AB)^2=AB $$ and $$ r(AB)=r(A)=r(B). $$

Problem 5.

Let $$ S=\{1,2,\ldots,100\} $$ and let $\mathcal F$ be the set of functions $f:S\to S$ satisfying $$ f(k)\le f(k+1)\le f(k)+1, \qquad k=1,\ldots,99. $$

1. Determine the number of elements in $\mathcal F$.
2. For each $f\in\mathcal F$, let $n(f)$ be the number of solutions of $f(x)=x$. Compute $$ \sum_{f\in\mathcal F} n(f). $$

Problem 6.

Let $M\in M_n(\mathbb R)$ be a square matrix whose entries all have absolute value $1$. Compute $$ \sum \det(M) \qquad\text{and}\qquad \sum \det(M^2). $$