Olympiad problems

2000 Irish Math Olympiad, 5

Tags: analytic geometry , algebra , parabola , linear algebra , matrix , conic

Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.

2009 AIME Problems, 14

Tags: modular arithmetic , linear algebra , matrix

For $ t \equal{} 1, 2, 3, 4$, define $ \displaystyle S_t \equal{} \sum_{i \equal{} 1}^{350}a_i^t$, where $ a_i \in \{1,2,3,4\}$. If $ S_1 \equal{} 513$ and $ S_4 \equal{} 4745$, find the minimum possible value for $ S_2$.

2017 India IMO Training Camp, 3

Tags: matrix , linear algebra , combinatorics

Let $n \ge 1$ be a positive integer. An $n \times n$ matrix is called [i]good[/i] if each entry is a non-negative integer, the sum of entries in each row and each column is equal. A [i]permutation[/i] matrix is an $n \times n$ matrix consisting of $n$ ones and $n(n-1)$ zeroes such that each row and each column has exactly one non-zero entry. Prove that any [i]good[/i] matrix is a sum of finitely many [i]permutation[/i] matrices.

2014 Putnam, 2

Tags: induction , linear algebra , matrix , putnam matrices

Let $A$ be the $n\times n$ matrix whose entry in the $i$-th row and $j$-th column is \[\frac1{\min(i,j)}\] for $1\le i,j\le n.$ Compute $\det(A).$

2005 District Olympiad, 1

Tags: matrix , linear algebra

Let $H$ denote the set of the matrices from $\mathcal{M}_n(\mathbb{N})$ and let $P$ the set of matrices from $H$ for which the sum of the entries from any row or any column is equal to $1$. a)If $A\in P$, prove that $\det A=\pm 1$. b)If $A_1,A_2,\ldots,A_p\in H$ and $A_1A_2\cdot \ldots\cdot A_p\in P$, prove that $A_1,A_2,\ldots,A_p\in P$.

2013 Romania National Olympiad, 2

Tags: matrix , linear algebra

Whether $m$ and $n$ natural numbers, $m,n\ge 2$. Consider matrices, ${{A}_{1}},{{A}_{2}},...,{{A}_{m}}\in {{M}_{n}}(R)$ not all nilpotent. Demonstrate that there is an integer number $k>0$ such that ${{A}^{k}}_{1}+{{A}^{k}}_{2}+.....+{{A}^{k}}_{m}\ne {{O}_{n}}$

2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 1

Tags: algebra , matrix , linear algebra , polynomial

Let $A=\left( \begin{array}{ccc} 1 & 1& 0 \\ 0 & 1& 0 \\ 0 &0 & 2 \end{array} \right),\ B=\left( \begin{array}{ccc} a & 1& 0 \\ b & 2& c \\ 0 &0 & a+1 \end{array} \right)\ (a,\ b,\ c\in{\mathbb{C}}).$ (1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$ (2) Under the condition of (1), find the condition for $a,\ b,\ c$ such that $B$ is diagonalizable.

2012 Putnam, 5

Tags: function , algebra , vector , linear algebra , matrix , domain

Let $\mathbb{F}_p$ denote the field of integers modulo a prime $p,$ and let $n$ be a positive integer. Let $v$ be a fixed vector in $\mathbb{F}_p^n,$ let $M$ be an $n\times n$ matrix with entries in $\mathbb{F}_p,$ and define $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$ by $G(x)=v+Mx.$ Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x)=G(x)$ and $G^{(k+1)}(x)=G(G^{(k)}(x)).$ Determine all pairs $p,n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0),$ $k=1,2,\dots,p^n$ are distinct.

1991 Arnold's Trivium, 87

Tags: derivative , function , calculus , quadratic , vector , linear algebra , matrix

Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.

2003 Gheorghe Vranceanu, 1

Tags: matrix , linear algebra , cayley-hamilton

Prove that if a $ 2\times 2 $ complex matrix has the property that there exists a natural number $ n $ such that $ \text{tr}\left( A^n\right) =\text{tr}\left( A^{n+1} \right) =0, $ then $ A^2=0. $

2007 Grigore Moisil Intercounty, 2

Tags: linear algebra , matrix

Le be a real number $ |a|<1, $ a natural number $ n\ge 2, $ and a $ 2\times 2 $ real matrix $ A $ that verifies $$ \det \left( A^{2n} -aA^{2n-1} -aA+I \right)=0. $$ Show that $ \det A=1. $

2003 VJIMC, Problem 4

Tags: matrix , linear algebra

Let $A$ and $B$ be complex Hermitian $2\times2$ matrices having the pairs of eigenvalues $(\alpha_1,\alpha_2)$ and $(\beta_1,\beta_2)$, respectively. Determine all possible pairs of eigenvalues $(\gamma_1,\gamma_2)$ of the matrix $C=A+B$. (We recall that a matrix $A=(a_{ij})$ is Hermitian if and only if $a_{ij}=\overline{a_{ji}}$ for all $i$ and $j$.)

2001 Romania National Olympiad, 2

Tags: algebra , polynomial , linear algebra , matrix , complex number

We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$. a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command. B=%Error. "rank" is a bad command. C = r$, such that $A=BC$. b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.