Found problems: 823
2025 District Olympiad, P4
Find all triplets of matrices $A,B,C\in\mathcal{M}_2(\mathbb{R})$ which satisfy \begin{align*}
A=BC-CB \\
B=CA-AC \\
C=AB-BA
\end{align*}
[i]Proposed by David Anghel[/i]
2005 IMC, 6
6. If $ p,q$ are rationals, $r=p+\sqrt{7}q$, then prove there exists a matrix
$\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in M_{2}(Z)- ( \pm I_{2})$
for which $\frac{ar+b}{cr+d}=r$ and $det(A)=1$
2007 Nicolae Păun, 2
For a given natural number, $ n\ge 2, $ consider two matrices $ A,B\in\mathcal{M}_n(\mathbb{C}) $ that commute and such that $ A $ is invertible and that the function $ M:\mathbb{C}\longrightarrow\mathbb{C} ,M(x)=\det (A+xB) $ is bounded above or below.
Prove that $ B^n=0. $
[i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]
1959 Putnam, B4
Given the following matrix
$$\begin{pmatrix}
11& 17 & 25& 19& 16\\
24 &10 &13 & 15&3\\
12 &5 &14& 2&18\\
23 &4 &1 &8 &22 \\
6&20&7 &21&9
\end{pmatrix},$$
choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible.
2018 Korea USCM, 2
Suppose a $n\times n$ real matrix $A$ satisfies $\text{tr}(A)=2018$, $\text{rank}(A)=1$. Prove that $A^2=2018 A$.
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 $a.$
1995 Putnam, 3
To each number with $n^2$ digits, we associate the $n\times n$ determinant of the matrix obtained by writing the digits of the number in order along the rows. For example : $8617\mapsto \det \left(\begin{matrix}{\;8}& 6\;\\ \;1 &{ 7\;}\end{matrix}\right)=50$.
Find, as a function of $n$, the sum of all the determinants associated with $n^2$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n = 2$, there are $9000$ determinants.)
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 $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$.
1976 IMO Longlists, 25
We consider the following system
with $q=2p$:
\[\begin{matrix} a_{11}x_{1}+\ldots+a_{1q}x_{q}=0,\\ a_{21}x_{1}+\ldots+a_{2q}x_{q}=0,\\ \ldots ,\\ a_{p1}x_{1}+\ldots+a_{pq}x_{q}=0,\\ \end{matrix}\]
in which every coefficient is an element from the set $\{-1,0,1\}$$.$ Prove that there exists a solution $x_{1}, \ldots,x_{q}$ for the system with the properties:
[b]a.)[/b] all $x_{j}, j=1,\ldots,q$ are integers$;$
[b]b.)[/b] there exists at least one j for which $x_{j} \neq 0;$
[b]c.)[/b] $|x_{j}| \leq q$ for any $j=1, \ldots ,q.$
1990 Brazil National Olympiad, 5
Let
$f(x)=\frac{ax+b}{cx+d}$
$F_n(x)=f(f(f...f(x)...))$ (with $n\ f's$)
Suppose that $f(0) \not =0$, $f(f(0)) \not = 0$, and for some $n$ we have $F_n(0)=0$,
show that $F_n(x)=x$ (for any valid x).
2009 All-Russian Olympiad, 6
Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)\equal{}a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)\equal{}b$, $ f(b)\equal{}a$.
2002 India IMO Training Camp, 11
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.
1988 IMO, 2
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?
2009 District Olympiad, 1
Let $A,B,C\in \mathcal{M}_3(\mathbb{R})$ such that $\det A=\det B=\det C$ and $\det(A+iB)=\det(C+iA)$. Prove that $\det (A+B)=\det (C+A)$.
2006 District Olympiad, 2
Let $G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}$ and $H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}$. Prove that $G$ and $H$ together with the operation of matrix multiplication are two non-isomorphical groups.
2005 Romania Team Selection Test, 3
Let $\mathbb{N}=\{1,2,\ldots\}$. Find all functions $f: \mathbb{N}\to\mathbb{N}$ such that for all $m,n\in \mathbb{N}$ the number $f^2(m)+f(n)$ is a divisor of $(m^2+n)^2$.
2012 Pre-Preparation Course Examination, 3
Suppose that $T,U:V\longrightarrow V$ are two linear transformations on the vector space $V$ such that $T+U$ is an invertible transformation. Prove that
$TU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n$.
1978 Romania Team Selection Test, 7
[b]a)[/b] Prove that for any natural number $ n\ge 1, $ there is a set $ \mathcal{M} $ of $ n $ points from the Cartesian plane such that the barycenter of every subset of $ \mathcal{M} $ has integral coordinates (both coordinates are integer numbers).
[b]b)[/b] Show that if a set $ \mathcal{N} $ formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of $ \mathcal{N} , $ the barycenter of which has non-integral coordinates.
1996 Moscow Mathematical Olympiad, 6
Eight students solved $8$ problems.
a) It turned out that each problem was solved by $5$ students. Prove that there are two students such that each problem is solved by at least one of them.
b) If it turned out that each problem was solved by $4$ students, it can happen that there is no pair of students such that each problem is solved by at least one of them. (Give an example.)
Proposed by S. Tokarev
2003 Romania National Olympiad, 4
Let be a $ 3\times 3 $ real matrix $ A. $ Prove the following statements.
[b]a)[/b] $ f(A)\neq O_3, $ for any polynomials $ f\in\mathbb{R} [X] $ whose roots are not real.
[b]b)[/b] $ \exists n\in\mathbb{N}\quad \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0 $
[i]Laurențiu Panaitopol[/i]
2010 IMC, 4
Let $A$ be a symmetric $m\times m$ matrix over the two-element field all of whose diagonal entries are zero. Prove that for every positive integer $n$ each column of the matrix $A^n$ has a zero entry.
2001 District Olympiad, 1
Let $A\in \mathcal{M}_2(\mathbb{R})$ such that $\det(A)=d\neq 0$ and $\det(A+dA^*)=0$. Prove that $\det(A-dA^*)=4$.
[i]Daniel Jinga[/i]
2010 Mediterranean Mathematics Olympiad, 4
Let $p$ be a positive integer, $p>1.$ Find the number of $m\times n$ matrices with entries in the set $\left\{ 1,2,\dots,p\right\} $ and such that the sum of elements on each row and each column is not divisible by $p.$
1994 Putnam, 4
For $n\ge 1$ let $d_n$ be the $\gcd$ of the entries of $A^n-\mathcal{I}_2$ where
\[ A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}\quad \text{ and }\quad \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix}\]
Show that $\lim_{n\to \infty}d_n=\infty$.
1979 IMO Longlists, 7
$M = (a_{i,j} ), \ i, j = 1, 2, 3, 4$, is a square matrix of order four. Given that:
[list]
[*] [b](i)[/b] for each $i = 1, 2, 3,4$ and for each $k = 5, 6, 7$,
\[a_{i,k} = a_{i,k-4};\]\[P_i = a_{1,}i + a_{2,i+1} + a_{3,i+2} + a_{4,i+3};\]\[S_i = a_{4,i }+ a_{3,i+1} + a_{2,i+2} + a_{1,i+3};\]\[L_i = a_{i,1} + a_{i,2} + a_{i,3} + a_{i,4};\]\[C_i = a_{1,i} + a_{2,i} + a_{3,i} + a_{4,i},\]
[*][b](ii)[/b] for each $i, j = 1, 2, 3, 4$, $P_i = P_j , S_i = S_j , L_i = L_j , C_i = C_j$, and
[*][b](iii)[/b] $a_{1,1} = 0, a_{1,2} = 7, a_{2,1} = 11, a_{2,3} = 2$, and $a_{3,3} = 15$.[/list]
find the matrix M.