Olympiad problems

1985 Traian Lălescu, 1.3

Let be two matrices $ A,B\in M_2\left(\mathbb{R}\right) $ and two natural numbers $ m,n. $ Prove that: $$ \det\left( (AB)^m-(BA)^m\right)\cdot\det\left( (AB)^n-(BA)^n\right)\ge 0. $$

2022 IMC, 7

Tags: linear algebra , idempotence , rank

Let $A_1, \ldots, A_k$ be $n\times n$ idempotent complex matrices such that $A_iA_j = -A_iA_j$ for all $1 \leq i < j \leq k$. Prove that at least one of the matrices has rank not exceeding $\frac{n}{k}$.

2017 Korea USCM, 2

Tags: algebra , polynomial , linear algebra

Show that any real coefficient polynomial $f(x,y)$ is a linear combination of polynomials of the form $(x+ay)^k$. ($a$ is a real number and $k$ is a non-negative integer.)

2022 Putnam, B2

Tags: linear algebra , combinatorial geometry

Let $\times$ represent the cross product in $\mathbb{R}^3.$ For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $$S=\{v \times w: v, w \in S\}?$$

2010 Romania National Olympiad, 1

Tags: linear algebra , matrix

Let $a,b\in \mathbb{R}$ such that $b>a^2$. Find all the matrices $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A^2-2aA+bI_2)=0$.

2004 Romania Team Selection Test, 7

Tags: linear algebra , matrix , binomial coefficient , polynomial , induction , algebra , number theory

Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have \[ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . \] Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.

2004 Germany Team Selection Test, 2

Tags: linear algebra , matrix , combinatorics

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.

2021 Miklós Schweitzer, 1

Tags: linear algebra , number theory

Let $n, m \in \mathbb{N}$; $a_1,\ldots, a_m \in \mathbb{Z}^n$. Show that nonnegative integer linear combinations of these vectors give exactly the whole $\mathbb{Z}^n$ lattice, if $m \ge n$ and the following two statements are satisfied: [list] [*] The vectors do not fall into the half-space of $\mathbb{R}^n$ containing the origin (i.e. they do not fall on the same side of an $n-1$ dimensional subspace), [*] the largest common divisor (not pairwise, but together) of $n \times n$ minor determinants of the matrix $(a_1,\ldots, a_m)$ (which is of size $m \times n$ and the $i$-th column is $a_i$ as a column vector) is $1$. [/list]

2005 Miklós Schweitzer, 6

Tags: linear algebra , matrix , inequalities

$SU_2(\mathbb{C})=\left\{\begin{pmatrix} z & w \\ -\bar{w} & \bar{z} \end{pmatrix} : z,w\in\mathbb{C} , z\bar{z}+w\bar{w}=1\right\}$ A and B are 2 elements of the above matrix group and have eigenvalues $e^{i\theta_1}$ , $e^{-i\theta_1}$ and $e^{i\theta_2}$ , $e^{-i\theta_2}$respectively, where $0\leq\theta_i\leq\pi$ . Prove that if AB has eigenvalue $e^{i\theta_3}$ , then $\theta_3$ satisfies the inequality $|\theta_1-\theta_2|\leq\theta_3\leq \min\{\theta_1+\theta_2 , 2\pi-(\theta_1+\theta_2)\}$

2005 China Northern MO, 4

Tags: combinatorics , linear algebra , induction , matrix

Let $A$ be the set of $n$-digit integers whose digits are all from $\{ 1, 2, 3, 4, 5 \}$. $B$ is subset of $A$ such that it contains digit $5$, and there is no digit $3$ in front of digit $5$ (i.e. for $n = 2$, $35$ is not allowed, but $53$ is allowed). How many elements does set $B$ have?

2011 Putnam, A4

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

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?

1971 IMO Shortlist, 11

Tags: matrix , linear algebra , inequality , combinatorial inequality

The matrix \[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\] satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]

2004 Romania National Olympiad, 3

Tags: algebra , function , integration , linear algebra , calculus , inequalities , polynomial

Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \] [i]Ion Rasa[/i]

1997 Iran MO (3rd Round), 3

Tags: linear algebra , algebra

Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$. We consider pairwise distances between numbers in $S$. If every distance that appears, except the distance $1$, occurs at least twice, prove that all the $x_i$ are rational.

2013 Bogdan Stan, 2

Tags: linear algebra , matrix

For a $ n\times n $ real matrix $ M, $ prove that [b]a)[/b] $ M=0 $ if $ \text{tr} \left(M^tM\right) =0. $ [b]b)[/b] $ ^tM=M $ if $M^tM=M^2. $ [b]c)[/b] $ ^tM=-M $ if $ M^tM=-M^2. $ [b]d)[/b] Give example of a $ 2\times 2 $ real matrix $ A $ satisfying the following: $ \text{(i)} ^tA\cdot A^2=A^3 $ and $ ^tA\neq A $ $ \text{(ii)} ^tA\cdot A^2=-A^3 $ and $ ^tA\neq -A $ [i]Vasile Pop[/i]

2021 Romania National Olympiad, 4

Tags: linear algebra

Let $n \ge 2$ and matrices $A,B \in M_n(\mathbb{R})$. There exist $x \in \mathbb{R} \backslash \{0,\frac{1}{2}, 1 \}$, such that $ xAB + (1-x)BA = I_n$. Show that $(AB-BA)^n = O_n$.

2011 Today's Calculation Of Integral, 696

Tags: algebra , inequalities , linear algebra , polynomial , calculus , integration , function

Let $P(x),\ Q(x)$ be polynomials such that : \[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\] Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$.

1977 IMO, 2

Tags: algebra , linear 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.

1969 Miklós Schweitzer, 12

Tags: vector , linear algebra , matrix , probability and stats

Let $ A$ and $ B$ be nonsingular matrices of order $ p$, and let $ \xi$ and $ \eta$ be independent random vectors of dimension $ p$. Show that if $ \xi,\eta$ and $ \xi A\plus{} \eta B$ have the same distribution, if their first and second moments exist, and if their covariance matrix is the identity matrix, then these random vectors are normally distributed. [i]B. Gyires[/i]

2013 VTRMC, Problem 6

Tags: linear algebra , matrix

Let \begin{align*}X&=\begin{pmatrix}7&8&9\\8&-9&-7\\-7&-7&9\end{pmatrix}\\Y&=\begin{pmatrix}9&8&-9\\8&-7&7\\7&9&8\end{pmatrix}.\end{align*}Let $A=Y^{-1}X$ and let $B$ be the inverse of $X^{-1}+A^{-1}$. Find a matrix $M$ such that $M^2=XY-BY$ (you may assume that $A$ and $X^{-1}+A^{-1}$ are invertible).

2002 IMC, 11

Tags: linear algebra , matrix , complex number

Let $A$ be a complex $n \times n$ Matrix for $n >1$. Let $A^{H}$ be the conjugate transpose of $A$. Prove that $A\cdot A^{H} =I_{n}$ if and only if $A=S\cdot (S^{H})^{-1}$ for some complex Matrix $S$.

1996 Miklós Schweitzer, 3

Tags: linear algebra

Let $1\leq a_1 < a_2 <... < a_{2n} \leq 4n-2$ be integers, such that their sum is even. Prove that for all sufficiently large n, there exist $\varepsilon_1 , ..., \varepsilon_{2n} = \pm1$ such that $$\sum\varepsilon_i = \sum\varepsilon_i a_i = 0$$

2022 SEEMOUS, 1

Tags: linear algebra , matrices , sylvester inequality , eigenvalue

Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$. Prove that: a) $(AB)^2 = AB.$ b) $(AB - BA)^3 = O_n.$

2001 India Regional Mathematical Olympiad, 4

Tags: matrix , linear algebra

Consider an $n \times n$ array of numbers $a_{ij}$ (standard notation). Suppose each row consists of the $n$ numbers $1,2,\ldots n$ in some order and $a_{ij} = a_{ji}$ for $i , j = 1,2, \ldots n$. If $n$ is odd, prove that the numbers $a_{11}, a_{22} , \ldots a_{nn}$ are $1,2,3, \ldots n$ in some order.

2004 Alexandru Myller, 3

Tags: linear algebra

Let $A$ and $B$ be $2\times 2$ matrices with integer entries, such that $AB=BA$ and $\det B=1$. Prove tht if $\det(A^3+B^3)=1$, then $A^2=O$.