Found problems: 823
2025 VJIMC, 4
Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.
2002 District Olympiad, 3
a)Find a matrix $A\in \mathcal{M}_3(\mathbb{C})$ such that $A^2\neq O_3$ and $A^3=O_3$.
b)Let $n,p\in\{2,3\}$. Prove that if there is bijective function $f:\mathcal{M}_n(\mathbb{C})\rightarrow \mathcal{M}_p(\mathbb{C})$ such that $f(XY)=f(X)f(Y),\ \forall X,Y\in \mathcal{M}_n(\mathbb{C})$, then $n=p$.
[i]Ion Savu[/i]
2003 China Team Selection Test, 2
In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.
2012 District Olympiad, 2
Let $ A,B\in\mathcal{M} \left( \mathbb{R} \right) $ that satisfy $ AB=O_3. $ Prove that:
[b]a)[/b] The function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ defined as $ f(x)=\det \left( A^2+B^2+xBA \right) $ is a polynomial one, of degree at most $ 2. $
[b]b)[/b] $ \det\left( A^2+B^2 \right)\ge 0. $
1976 Czech and Slovak Olympiad III A, 4
Determine all solutions of the linear system of equations
\begin{align*}
&x_1& &-x_2& &-x_3& &-\cdots& &-x_n& &= 2a, \\
-&x_1& &+3x_2& &-x_3& &-\cdots& &-x_n& &= 4a, \\
-&x_1& &-x_2& &+7x_3& &-\cdots& &-x_n& &= 8a, \\
&&&&&&&&&&&\vdots \\
-&x_1& &-x_2& &-x_3& &-\cdots& &+\left(2^n-1\right)x_n& &= 2^na,
\end{align*}
with unknowns $x_1,\ldots,x_n$ and a real parameter $a.$
2000 Irish Math Olympiad, 5
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
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
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.
1996 IMC, 3
The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if
$A^{2}=I$, where $I$ is the identity operator. Let $\dim V=n$.
i) Prove that for every involution $A$ on $V$, there exists a basis of $V$ consisting of eigenvectors
of $A$.
ii) Find the maximal number of distinct pairwise commuting involutions on $V$.
2007 Nicolae Coculescu, 4
Let $ n\in{N^*}$,$ n\ge{3}$ and $ a_1,a_2,...,a_n\in{R^*}$, so that $ |a_i|\neq{|a_j|}$, for every $ i,j\in{\{1,2,...,n\}}, i\neq{j}$.
Find $ p\in{S_n}$ with the property:
$ a_ia_j < \equal{} a_{p(i)}a_{p(j)}$, for every $ i,j\in{\{1,2,....n\}}$,$ i\neq{j}$ (Teodor Radu)
2014 Putnam, 2
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 $\det(A).$
2019 Ramnicean Hope, 3
Let be two $ 2\times 2 $ real matrices $ A,B, $ such that $ AB=\begin{pmatrix} 1&1\\1&2 \end{pmatrix} . $
Calculate $ \left((BA)^{-1} +BA\right)^{2019 } . $
[i]Dan Nedeianu[/i]
2011 Miklós Schweitzer, 5
Let n, k be positive integers. Let $f_a(x) := ||x - a||^{2n}$ , where the vectors $x = (x_1, ..., x_k) , a\in R^k$ , and ||·|| is the Euclidean norm. Let the vector space $Q_{n, k}$ be generated by the functions $f_a$ ($a\in R^k$). What is the largest integer N for which $Q_{n, k}$ contains all polynomials of $x_1, ..., x_k$ whose total degree is at most N?
2005 District Olympiad, 1
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
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 ${{A}^{k}}_{1}+{{A}^{k}}_{2}+.....+{{A}^{k}}_{m}\ne {{O}_{n}}$
Gheorghe Țițeica 2024, P4
Let $n\geq 2$. Find all matrices $A\in\mathcal{M}_n(\mathbb{C})$ such that $$\text{rank}(A^2)+\text{rank}(B^2)\geq 2\text{rank}(AB),$$ for all $B\in\mathcal{M}_n(\mathbb{C})$.
[i]Cristi Săvescu[/i]
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 1
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
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
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
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
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
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
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.