Found problems: 823
2003 Gheorghe Vranceanu, 1
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. $
2009 APMO, 2
Let $ a_1$, $ a_2$, $ a_3$, $ a_4$, $ a_5$ be real numbers satisfying the following equations:
$ \frac{a_1}{k^2\plus{}1}\plus{}\frac{a_2}{k^2\plus{}2}\plus{}\frac{a_3}{k^2\plus{}3}\plus{}\frac{a_4}{k^2\plus{}4}\plus{}\frac{a_5}{k^2\plus{}5} \equal{} \frac{1}{k^2}$ for $ k \equal{} 1, 2, 3, 4, 5$
Find the value of $ \frac{a_1}{37}\plus{}\frac{a_2}{38}\plus{}\frac{a_3}{39}\plus{}\frac{a_4}{40}\plus{}\frac{a_5}{41}$ (Express the value in a single fraction.)
2004 Putnam, A3
Define a sequence $\{u_n\}_{n=0}^{\infty}$ by $u_0=u_1=u_2=1,$ and thereafter by the condition that
$\det\begin{vmatrix} u_n & u_{n+1} \\ u_{n+2} & u_{n+3} \end{vmatrix}=n!$
for all $n\ge 0.$ Show that $u_n$ is an integer for all $n.$ (By convention, $0!=1$.)
2012 Miklós Schweitzer, 2
Call a subset $A$ of the cyclic group $(\mathbb{Z}_n,+)$ [i]rich[/i] if for all $x,y \in \mathbb{Z}_n$ there exists $r \in \mathbb{Z}_n$ such that $x-r,x+r,y-r,y+r$ are all in $A$. For what $\alpha$ is there a constant $C_\alpha>0$ such that for each odd positive integer $n$, every rich subset $A \subset \mathbb{Z}_n$ has at least $C_\alpha n^\alpha$ elements?
2019 Korea USCM, 2
Matrices $A$, $B$ are given as follows.
\[A=\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 12\end{pmatrix}\]
Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$.
2001 All-Russian Olympiad, 2
In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one. Prove that the sum of all these vectors is zero. (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)
2020 Brazil Undergrad MO, Problem 3
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 $5$
1975 Czech and Slovak Olympiad III A, 4
Determine all real values of parameter $p$ such that the equation \[|x-2|+|y-3|+y=p\] is an equation of a ray in the plane $xy.$
2020 Simon Marais Mathematics Competition, B1
Let $\mathcal{M}$ be the set of $5\times 5$ real matrices of rank $3$. Given a matrix in $\mathcal{M}$, the set of columns of $A$ has $2^5-1=31$ nonempty subsets. Let $k_A$ be the number of these subsets that are linearly independent.
Determine the maximum and minimum values of $k_A$, as $A$ varies over $\mathcal{M}$.
[i]The rank of a matrix is the dimension of the span of its columns.[/i]
2008 AIME Problems, 9
A particle is located on the coordinate plane at $ (5,0)$. Define a [i]move[/i] for the particle as a counterclockwise rotation of $ \pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$-direction. Given that the particle's position after $ 150$ moves is $ (p,q)$, find the greatest integer less than or equal to $ |p|\plus{}|q|$.
1988 IMO Longlists, 7
Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?
1986 Miklós Schweitzer, 2
Show that if $k\leq \frac n2$ and $\mathcal F$ is a family $k\times k$ submatrices of an $n\times n$ matrix such that any two intersect then
$$|\mathcal F|\leq \binom{n-1}{k-1}^2$$
[Gy. Katona]
1995 Putnam, 6
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 $a = b = c$.
2013 IMC, 3
Suppose that $\displaystyle{{v_1},{v_2},...,{v_d}}$ are unit vectors in $\displaystyle{{{\Bbb R}^d}}$. Prove that there exists a unitary vector $\displaystyle{u}$ such that $\displaystyle{\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}}$ for $\displaystyle{i = 1,2,...,d}$.
[b]Note.[/b] Here $\displaystyle{ \cdot }$ denotes the usual scalar product on $\displaystyle{{{\Bbb R}^d}}$.
[i]Proposed by Tomasz Tkocz, University of Warwick.[/i]
1957 Putnam, B1
Consider the determinant of the matrix $(a_{ij})_{ij}$ with $1\leq i,j \leq 100$ and $a_{ij}=ij.$ Prove that if the absolute value of each of the $100!$ terms in the expansion of this determinant is divided by $101,$ then the remainder is always $1.$
2024 Mexican University Math Olympiad, 2
Let \( A \) and \( B \) be two square matrices with complex entries such that \( A + B = AB \), \( A = A^* \), and \( A \) has all distinct eigenvalues. Prove that there exists a polynomial \( P \) with complex coefficients such that \( P(A) = B \).
MathLinks Contest 7th, 6.3
Let $ \Omega$ be the circumcircle of triangle $ ABC$. Let $ D$ be the point at which the incircle of $ ABC$ touches its side $ BC$. Let $ M$ be the point on $ \Omega$ such that the line $ AM$ is parallel to $ BC$. Also, let $ P$ be the point at which the circle tangent to the segments $ AB$ and $ AC$ and to the circle $ \Omega$ touches $ \Omega$. Prove that the points $ P$, $ D$, $ M$ are collinear.
2014 SEEMOUS, Problem 1
Let $n$ be a nonzero natural number and $f:\mathbb R\to\mathbb R\setminus\{0\}$ be a function such that $f(2014)=1-f(2013)$. Let $x_1,x_2,x_3,\ldots,x_n$ be real numbers not equal to each other. If
$$\begin{vmatrix}1+f(x_1)&f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&1+f(x_2)&f(x_3)&\cdots&f(x_n)\\f(x_1)&f(x_2)&1+f(x_3)&\cdots&f(x_n)\\\vdots&\vdots&\vdots&\ddots&\vdots\\f(x_1)&f(x_2)&f(x_3)&\cdots&1+f(x_n)\end{vmatrix}=0,$$prove that $f$ is not continuous.
1947 Putnam, A6
A $3\times 3$ matrix has determinant $0$ and the cofactor of any element is equal to the square of that element. Show that every element in the matrix is $0.$
2024 Romania National Olympiad, 3
Let $A,B \in \mathcal{M}_n(\mathbb{R}).$ We consider the function $f:\mathcal{M}_n(\mathbb{C}) \to \mathcal{M}_n(\mathbb{C}),$ defined by $f(Z)=AZ+B\overline{Z},$ $Z \in \mathcal{M}_n(\mathbb{C}),$ where $\overline{Z}$ is the matrix whose entries are the conjugates of the corresponding entries of $Z.$ Prove that the following statements are equivalent:
$(1)$ the function $f$ is injective;
$(2)$ the function $f$ is surjective;
$(3)$ the matrices $A+B$ and $A-B$ are invertible.
2019 LIMIT Category C, Problem 4
Which of the following are true?
$\textbf{(A)}~\exists A\in M_3(\mathbb R)\text{ such that }A^2=-I_3$
$\textbf{(B)}~\exists A,B\in M_3(\mathbb R)\text{ such that }AB-BA=I_3$
$\textbf{(C)}~\forall A\in M_4,\det\left(I_4+A^2\right)\ge0$
$\textbf{(D)}~\text{None of the above}$
2019 SEEMOUS, 2
Let $A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})$. Prove that there exist $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}$ such that:
$$\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2) $$
2012 European Mathematical Cup, 4
Olja writes down $n$ positive integers $a_1, a_2, \ldots, a_n$ smaller than $p_n$ where $p_n$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $xy$. If there are two equal numbers Oleg wins. Can Oleg guarantee a win?
[i]Proposed by Matko Ljulj.[/i]
2001 IMC, 4
Let $A=(a_{k,l})_{k,l=1,...,n}$ be a complex $n \times n$ matrix such that for each $m \in \{1,2,...,n\}$ and $1 \leq j_{1} <...<j_{m}$ the determinant of the matrix $(a_{j_{k},j_{l}})_{k,l=1,...,n}$ is zero. Prove that $A^{n}=0$ and that there exists a permutation $\sigma \in S_{n}$ such that the matrix $(a_{\sigma(k),\sigma(l)})_{k,l=1,...,n}$ has all of its nonzero elements above the diagonal.
2006 Petru Moroșan-Trident, 3
Let be a $ 2\times 2 $ real matrix such that $ \det \left( A^6+64I \right) =0. $
Show that $ \det A=4. $
[i]Viorel Botea[/i]