Olympiad problems

2007 Stanford Mathematics Tournament, 22

Katie begins juggling five balls. After every second elapses, there is a chance she will drop a ball. If she is currently juggling $ k$ balls, this probability is $ \frac{k}{10}$. Find the expected number of seconds until she has dropped all the balls.

2018 Miklós Schweitzer, 3

Tags: linear algebra , matrix , combinatorics

We call an $n\times n$ matrix [i]well groomed[/i] if it only contains elements $0$ and $1$, and it does not contain the submatrix $\begin{pmatrix} 1& 0\\ 0 & 1 \end{pmatrix}.$ Show that there exists a constant $c>0$ such that every well groomed, $n\times n$ matrix contains a submatrix of size at least $cn\times cn$ such that all of the elements of the submatrix are equal. (A well groomed matrix may contain the submatrix $\begin{pmatrix} 0& 1\\ 1 & 0 \end{pmatrix}.$ )

1993 IMO Shortlist, 4

Tags: linear algebra , matrix , algebra , system of equations

Solve the following system of equations, in which $a$ is a given number satisfying $|a| > 1$: $\begin{matrix} x_{1}^2 = ax_2 + 1 \\ x_{2}^2 = ax_3 + 1 \\ \ldots \\ x_{999}^2 = ax_{1000} + 1 \\ x_{1000}^2 = ax_1 + 1 \\ \end{matrix}$

1967 IMO Shortlist, 4

Tags: linear algebra , matrix , algebra , system of equations , arithmetic sequence

In what case does the system of equations $\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$ have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression.

2023 China Team Selection Test, P18

Tags: matrix , combinatorics , square grid

Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$, the sum of entries in each row and each column is at least $\lambda$. Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.

2000 Iran MO (2nd round), 3

Tags: linear algebra , matrix , combinatorics

Let $M=\{1,2,3,\ldots, 10000\}.$ Prove that there are $16$ subsets of $M$ such that for every $a \in M,$ there exist $8$ of those subsets that intersection of the sets is exactly $\{a\}.$

2001 SNSB Admission, 1

Tags: linear algebra , matrix

Show that $ \det \left( I_n+A \right)\ge 1, $ for any $ n\times n $ antisymmetric real matrix $ A. $

2020 Brazil Undergrad MO, Problem 3

Tags: linear algebra , matrix , number theory

Let $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$ be the finite field with $13$ elements (with sum and product modulus $13$). Find how many matrix $A$ of size $5$ x $5$ with entries in $\mathbb{F}_{13}$ exist such that $$A^5 = I$$ where $I$ is the identity matrix of order $5$

2013 SEEMOUS, Problem 2

Tags: linear algebra , matrix

Let $M,N\in M_2(\mathbb C)$ be two nonzero matrices such that $$M^2=N^2=0_2\text{ and }MN+NM=I_2$$where $0_2$ is the $2\times2$ zero matrix and $I_2$ the $2\times2$ unit matrix. Prove that there is an invertible matrix $A\in M_2(\mathbb C)$ such that $$M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.$$

2011 Tournament of Towns, 7

Tags: matrix , combinatorics

In every cell of a square table is a number. The sum of the largest two numbers in each row is $a$ and the sum of the largest two numbers in each column is b. Prove that $a = b$.

2010 IberoAmerican Olympiad For University Students, 5

Tags: matrix , algebra , polynomial , inequalities , linear algebra

Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.

2005 Brazil Undergrad MO, 1

Tags: matrix , algebra , polynomial , function , linear algebra

Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.

2022 IMC, 2

Tags: linear algebra , matrix

For a positive integer $n$ determine all $n\times n$ real matrices $A$ which have only real eigenvalues and such that there exists an integer $k\geq n$ with $A + A^k = A^T$.

2012 SEEMOUS, Problem 3

Tags: matrix , linear algebra

a) Prove that if $k$ is an even positive integer and $A$ is a real symmetric $n\times n$ matrix such that $\operatorname{tr}(A^k)^{k+1}=\operatorname{tr}(A^{k+1})^k$, then $$A^n=\operatorname{tr}(A)A^{n-1}.$$ b) Does the assertion from a) also hold for odd positive integers $k$?

2008 IMC, 4

Tags: inequalities , linear algebra , matrix , group theory , abstract algebra , analytic geometry , geometry

We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$. For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

2011 Putnam, A4

Tags: linear algebra , matrix , modular arithmetic , vector , polynomial , putnam matrices

For which positive integers $n$ is there an $n\times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?

2004 IMC, 4

Tags: linear algebra , matrix , algebra , polynomial , vector

For $n\geq 1$ let $M$ be an $n\times n$ complex array with distinct eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_k$, with multiplicities $m_1,m_2,\ldots,m_k$ respectively. Consider the linear operator $L_M$ defined by $L_MX=MX+XM^T$, for any complex $n\times n$ array $X$. Find its eigenvalues and their multiplicities. ($M^T$ denotes the transpose matrix of $M$).

1969 Putnam, B6

Tags: matrix multiplication , matrix

Let $A$ and $B$ be matrices of size $3\times 2$ and $2\times 3$ respectively. Suppose that $$AB =\begin{pmatrix} 8 & 2 & -2\\ 2 & 5 &4 \\ -2 &4 &5 \end{pmatrix}.$$ Show that the product $BA$ is equal to $\begin{pmatrix} 9 &0\\ 0 &9 \end{pmatrix}.$

2006 IMC, 3

Tags: linear algebra , matrix , ring theory

Let $A$ be an $n$x$n$ matrix with integer entries and $b_{1},b_{2},...,b_{k}$ be integers satisfying $detA=b_{1}\cdot b_{2}\cdot ...\cdot b_{k}$. Prove that there exist $n$x$n$-matrices $B_{1},B_{2},...,B_{k}$ with integers entries such that $A=B_{1}\cdot B_{2}\cdot ...\cdot B_{k}$ and $detB_{i}=b_{i}$ for all $i=1,...,k$.

2009 AIME Problems, 14

Tags: linear algebra , matrix , modular arithmetic

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$.

1976 Spain Mathematical Olympiad, 6

Tags: matrices , linear algebra , matrix

Given a square matrix $M$ of order $n$ over the field of numbers real, find, as a function of $M$, two matrices, one symmetric and one antisymmetric, such that their sum is precisely $ M$.

1991 Vietnam Team Selection Test, 3

Tags: limit , linear algebra , matrix , vector , invariant , complex number , logarithm

Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have: \[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\] Show that this sequence has a finite limit. Determine this limit.

2011 SEEMOUS, Problem 2

Tags: eigenvalue , matrix , linear algebra

Let $A=(a_{ij})$ be a real $n\times n$ matrix such that $A^n\ne0$ and $a_{ij}a_{ji}\le0$ for all $i,j$. Prove that there exist two nonreal numbers among eigenvalues of $A$.

1995 Putnam, 6

Tags: linear algebra , matrix

Suppose that each of $n$ people writes down the numbers $1, 2, 3$ in random order in one column of a $3\times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a, b, c$ of the resulting matrix be rearranged (if necessary) so that $a \le b \le c$. Show that for some $n \ge 1995$ ,it is at least four times as likely that both $b = a+1$ and $c = a+2$ as that $a = b = c$.

2007 Nicolae Păun, 1

Tags: matrix , linear algebra

Prove that $ \exists X,Y,Z\in \mathcal{M}_n(\mathbb{C})$ such that a)$ X^2\plus{}Y^2\equal{}A$ b) $ X^3\plus{}Y^3\plus{}Z^3\equal{}A$ , where $ A\in \mathcal{M}_n(\mathbb{C})$