Olympiad problems

2011 N.N. Mihăileanu Individual, 2

Let be a natural number $ k, $ and a matrix $ M\in\mathcal{M}_k(\mathbb{R}) $ having the property that $$ \det\left( I-\frac{1}{n^2}\cdot A^2 \right) +1\ge\det \left( I -\frac{1}{n}\cdot A \right) +\det \left( I +\frac{1}{n}\cdot A \right) , $$ for all natural numbers $ n. $ Prove that the trace of $ A $ is $ 0. $ [i]Nelu Chichirim[/i]

1985 ITAMO, 12

Tags: geometry , 3d geometry , tetrahedron , probability , linear algebra

Let $A$, $B$, $C$, and $D$ be the vertices of a regular tetrahedron, each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to the vertex at its opposite end. Let $p = n/729$ be the probability that the bug is at vertex $A$ when it has crawled exactly 7 meters. Find the value of $n$.

2012 District Olympiad, 3

Tags: linear algebra , matrix

Let be a natural number $ n, $ and two matrices $ A,B\in\mathcal{M}_n\left(\mathbb{C}\right) $ with the property that $$ AB^2=A-B. $$ [b]a)[/b] Show that the matrix $ I_n+B $ is inversable. [b]b)[/b] Show that $ AB=BA. $

2011 Iran MO (2nd Round), 2

Tags: linear algebra , matrix , trigonometry , angle bisector , geometry

In triangle $ABC$, we have $\angle ABC=60$. The line through $B$ perpendicular to side $AB$ intersects angle bisector of $\angle BAC$ in $D$ and the line through $C$ perpendicular $BC$ intersects angle bisector of $\angle ABC$ in $E$. prove that $\angle BED\le 30$.

2024 SEEMOUS, P4

Tags: linear algebra , matrix

Let $n\in\mathbb{N}$, $n\geq 2$. Find all values of $k\in\mathbb{N}$, $k\geq 1$, for which the following statement holds: $$\text{"If }A\in\mathcal{M}_n(\mathbb{C})\text{ is such that }A^kA^*=A\text{, then }A=A^*\text{."}$$ (here, $A^*$ denotes the conjugate transpose of $A$).

1977 IMO, 2

Tags: linear algebra , algebra , sequence , maximization , extremal combinatorics

In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.

2001 IMC, 1

Tags: matrix , linear algebra

Let $ n$ be a positive integer. Consider an $ n\times n$ matrix with entries $ 1,2,...,n^2$ written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for $ n \equal{} 3$ we get $ \left[\begin{array}{ccc}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$) We choose $ n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?

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 $j$.)

1972 Spain Mathematical Olympiad, 1

Tags: algebra , linear algebra , matrices , group theory , group

Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$. An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to: a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if $K$ is a commutative field, the elements of$ M$ that have inverse they are characterized by the condition $ad - bc \ne 0$. c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.

1950 Miklós Schweitzer, 6

Tags: linear algebra

Prove the following identity for determinants: $ |c_{ik} \plus{} a_i \plus{} b_k \plus{} 1|_{i,k \equal{} 1,...,n} \plus{} |c_{ik}|_{i,k \equal{} 1,...,n} \equal{} |c_{ik} \plus{} a_i \plus{} b_k|_{i,k \equal{} 1,...,n} \plus{} |c_{ik} \plus{} 1|_{i,k \equal{} 1,...,n}$

1989 IMO Longlists, 98

Tags: linear algebra , matrix , algebra

Let $ A$ be an $ n \times n$ matrix whose elements are non-negative real numbers. Assume that $ A$ is a non-singular matrix and all elements of $ A^{\minus{}1}$ are non-negative real numbers. Prove that every row and every column of $ A$ has exactly one non-zero element.

2007 All-Russian Olympiad, 7

Tags: linear algebra , matrix , geometry , rectangle , combinatorics

Given a matrix $\{a_{ij}\}_{i,j=0}^{9}$, $a_{ij}=10i+j+1$. Andrei is going to cover its entries by $50$ rectangles $1\times 2$ (each such rectangle contains two adjacent entries) so that the sum of $50$ products in these rectangles is minimal possible. Help him. [i]A. Badzyan[/i]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Tags: linear algebra , matrix , algebra , polynomial

Justify your answer whether $A=\left( \begin{array}{ccc} -4 & -1& -1 \\ 1 & -2& 1 \\ 0 & 0& -3 \end{array} \right)$ is similar to $B=\left( \begin{array}{ccc} -2 & 1& 0 \\ -1 & -4& 1 \\ 0 & 0& -3 \end{array} \right),\ A,\ B\in{M(\mathbb{C})}$ or not.

2008 Irish Math Olympiad, 4

Tags: linear algebra , matrix , number theory

Given $ k \in [0,1,2,3]$ and a positive integer $ n$, let $ f_k(n)$ be the number of sequences $ x_1,...,x_n,$ where $ x_i \in [\minus{}1,0,1]$ for $ i\equal{}1,...,n,$ and $ x_1\plus{}...\plus{}x_n \equiv k$ mod 4 a) Prove that $ f_1(n) \equal{} f_3(n)$ for all positive integers $ n$. (b) Prove that $ f_0(n) \equal{} [{3^n \plus{} 2 \plus{} [\minus{}1]^n}] / 4$ for all positive integers $ n$.

2008 Alexandru Myller, 1

Tags: linear algebra , matrix

Let be a real $ 4\times 4 $ real matrix with $ \text{det} \left( A^2-I\right) <0. $ Prove that there is a number $ \alpha\in (-1,1) $ so that $ A+\alpha I $ is singular. [i]Mihai Haivas[/i]

2004 Italy TST, 3

Tags: linear algebra , matrix , combinatorics

Given real numbers $x_i,y_i (i=1,2,\ldots ,n)$, let $A$ be the $n\times n$ matrix given by $a_{ij}=1$ if $x_i\ge y_j$ and $a_{ij}=0$ otherwise. Suppose $B$ is a $n\times n$ matrix whose entries are $0$ and $1$ such that the sum of entries in any row or column of $B$ equals the sum of entries in the corresponding row or column of $A$. Prove that $B=A$.

2014 Regional Olympiad of Mexico Center Zone, 6

Tags: linear algebra , combinatorics , graph theory

In a school there are $n$ classes and $n$ students. The students are enrolled in classes, such that no two of them have exactly the same classes. Prove that we can close a class in a such way that there still are no two of them which have exactly the same classes.

2018 ISI Entrance Examination, 8

Tags: linear algebra

Let $n\geqslant 3$. Let $A=((a_{ij}))_{1\leqslant i,j\leqslant n}$ be an $n\times n$ matrix such that $a_{ij}\in\{-1,1\}$ for all $1\leqslant i,j\leqslant n$. Suppose that $$a_{k1}=1~~\text{for all}~1\leqslant k\leqslant n$$ and $~~\sum_{k=1}^n a_{ki}a_{kj}=0~~\text{for all}~i\neq j$. Show that $n$ is a multiple of $4$.

2012 Tuymaada Olympiad, 3

Tags: induction , linear algebra , matrix , combinatorics

Prove that $N^2$ arbitrary distinct positive integers ($N>10$) can be arranged in a $N\times N$ table, so that all $2N$ sums in rows and columns are distinct. [i]Proposed by S. Volchenkov[/i]

2011 SEEMOUS, Problem 3

Tags: vector , inequalities , linear algebra

Given vectors $\overline a,\overline b,\overline c\in\mathbb R^n$, show that $$(\lVert\overline a\rVert\langle\overline b,\overline c\rangle)^2+(\lVert\overline b\rVert\langle\overline a,\overline c\rangle)^2\le\lVert\overline a\rVert\lVert\overline b\rVert(\lVert\overline a\rVert\lVert\overline b\rVert+|\langle\overline a,\overline b\rangle|)\lVert\overline c\rVert^2$$where $\langle\overline x,\overline y\rangle$ denotes the scalar (inner) product of the vectors $\overline x$ and $\overline y$ and $\lVert\overline x\rVert^2=\langle\overline x,\overline x\rangle$.

2021 SEEMOUS, Problem 2

Tags: linear algebra

Let $n \ge 2$ be a positive integer and let $A \in \mathcal{M}_n(\mathbb{R})$ be a matrix such that $A^2=-I_n$. If $B \in \mathcal{M}_n(\mathbb{R})$ and $AB = BA$, prove that $\det B \ge 0$.

1994 Irish Math Olympiad, 4

Tags: vector , matrix , modular arithmetic , search , linear algebra

Consider all $ m \times n$ matrices whose all entries are $ 0$ or $ 1$. Find the number of such matrices for which the number of $ 1$-s in each row and in each column is even.

2012 Putnam, 5

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

2017 Miklós Schweitzer, 2

Tags: linear algebra , matrix , abstract algebra , field

Prove that a field $K$ can be ordered if and only if every $A\in M_n(K)$ symmetric matrix can be diagonalized over the algebraic closure of $K$. (In other words, for all $n\in\mathbb{N}$ and all $A\in M_n(K)$, there exists an $S\in GL_n(\overline{K})$ for which $S^{-1}AS$ is diagonal.)

2004 District Olympiad, 4

Tags: matrix , calculus , derivative , algebra , polynomial , linear algebra

Let $A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})$ such that $a_{12}=a_{23}=\ldots=a_{p-1,p}=1$ and $a_{ij}=0$ for any other entry. a)Prove that $A^{p-1}\neq O_p$ and $A^p=O_p$. b)If $X\in \mathcal{M}_{p}(\mathbb{C})$ and $AX=XA$, prove that there exist $a_1,a_2,\ldots,a_p\in \mathbb{C}$ such that: \[X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)\] c)If there exist $B,C\in \mathcal{M}_p(\mathbb{C})$ such that $(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*$, prove that $B=O_p$ or $C=O_p$.