Found problems: 823
1967 IMO Shortlist, 5
Solve the system of equations:
$
\begin{matrix}
x^2 + x - 1 = y \\
y^2 + y - 1 = z \\
z^2 + z - 1 = x.
\end{matrix}
$
2007 Princeton University Math Competition, 7
Given two sequences $x_n$ and $y_n$ defined by $x_0 = y_0 = 7$,
\[x_n = 4x_{n-1}+3y_{n-1}, \text{ and}\]\[y_n = 3y_{n-1}+2x_{n-1},\]
find $\lim_{n \to \infty} \frac{x_n}{y_n}$.
2002 AMC 10, 20
Let $ a$, $ b$, and $ c$ be real numbers such that $ a \minus{} 7b \plus{} 8c \equal{} 4$ and $ 8a \plus{} 4b \minus{} c \equal{} 7$. Then $ a^2 \minus{} b^2 \plus{} c^2$ is
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$
2013 Putnam, 5
Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$
[Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]
1987 Putnam, B5
Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \dots, r_{2n}$, there are complex numbers $w_1, w_2, \dots, w_n$ such that
\[
\mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array}
\right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n
\end{array} \right).
\]
(Note: if $C$ is a matrix of complex numbers, $\mathrm{re}(C)$ is the matrix whose entries are the real parts of the entries of $C$.)
2021 Romania National Olympiad, 2
Let $n \ge 2$ and $ a_1, a_2, \ldots , a_n $, nonzero real numbers not necessarily distinct. We define matrix $A = (a_{ij})_{1 \le i,j \le n} \in M_n( \mathbb{R} )$ , $a_{i,j} = max \{ a_i, a_j \}$, $\forall i,j \in \{ 1,2 , \ldots , n \} $. Show that $\mathbf{rank}(A) $= $\mathbf{card} $ $\{ a_k | k = 1,2, \ldots n \} $
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 $\det (A+B)=\det (C+A)$.
2008 IMC, 5
Let $ n$ be a positive integer, and consider the matrix $ A \equal{} (a_{ij})_{1\leq i,j\leq n}$ where $ a_{ij} \equal{} 1$ if $ i\plus{}j$ is prime and $ a_{ij} \equal{} 0$ otherwise.
Prove that $ |\det A| \equal{} k^2$ for some integer $ k$.
1994 IMO Shortlist, 2
In a certain city, age is reckoned in terms of real numbers rather than integers. Every two citizens $x$ and $x'$ either know each other or do not know each other. Moreover, if they do not, then there exists a chain of citizens $x = x_0, x_1, \ldots, x_n = x'$ for some integer $n \geq 2$ such that $ x_{i-1}$ and $x_i$ know each other. In a census, all male citizens declare their ages, and there is at least one male citizen. Each female citizen provides only the information that her age is the average of the ages of all the citizens she knows. Prove that this is enough to determine uniquely the ages of all the female citizens.
2005 SNSB Admission, 1
[b]a)[/b] Let be three vectorial spaces $ E,F,G, $ where $ F $ has finite dimension, and $ E $ is a subspace of $ F. $ Prove that if the function $ T:F\longrightarrow G $ is linear, then
$$ \dim TF -\dim TE\le \dim F-\dim E. $$
[b]b)[/b] Let $ A,B,C $ be matrices of real numbers. Prove that
$$ \text{rang} (AB) +\text{rang} (BC) \le \text{rang} (ABC) +\text{rang} (B) . $$
2021 Brazil Undergrad MO, Problem 1
Consider the matrices like
$$M=
\left(
\begin{array}{ccc}
a & b & c \\
c & a & b \\
b & c & a
\end{array}
\right)$$
such that $det(M) = 1$.
Show that
a) There are infinitely many matrices like above with $a,b,c \in \mathbb{Q}$
b) There are finitely many matrices like above with $a,b,c \in \mathbb{Z}$
2010 Contests, 4
Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.
2010 Contests, 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 $p.$
2004 Unirea, 2
Let be two matrices $ A,N\in\mathcal{M}_2(\mathbb{R}) $ that commute and such that $ N $ is nilpotent. Show that:
[b]a)[/b] $ \det (A+N)=\det (A) $
[b]b)[/b] if $ A $ is general linear, then the matrix $ A+N $ is invertible and $ (A+N)^{-1}=(A-N)A^{-2} . $
2012 Romania National Olympiad, 2
[color=darkred]Let $n$ and $k$ be two natural numbers such that $n\ge 2$ and $1\le k\le n-1$ . Prove that if the matrix $A\in\mathcal{M}_n(\mathbb{C})$ has exactly $k$ minors of order $n-1$ equal to $0$ , then $\det (A)\ne 0$ .[/color]
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]
2014 Romania National Olympiad, 4
Let $ A\in\mathcal{M}_4\left(\mathbb{R}\right) $ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $ A^2+I $ is singular if and only if there exists a nonzero matrix in $ \mathcal{M}_4\left( \mathbb{R} \right) $ that anti-commutes with it.
2005 Miklós Schweitzer, 11
Let $E: R^n \backslash \{0\} \to R^+$ be a infinitely differentiable, quadratic positive homogeneous (that is, for any λ>0 and $p \in R^n \backslash \{0\}$ , $E (\lambda p) = \lambda^2 E (p)$). Prove that if the second derivative of $E''(p): R^n \times R^n \to R$ is a non-degenerate bilinear form at any point $p \in R^n \backslash \{0\}$, then $E''(p)$ ($p \in R^n \backslash \{0\}$) is positive definite.
2007 Purple Comet Problems, 15
The alphabet in its natural order $\text{ABCDEFGHIJKLMNOPQRSTUVWXYZ}$ is $T_0$. We apply a permutation to $T_0$ to get $T_1$ which is $\text{JQOWIPANTZRCVMYEGSHUFDKBLX}$. If we apply the same permutation to $T_1$, we get $T_2$ which is $\text{ZGYKTEJMUXSODVLIAHNFPWRQCB}$. We continually apply this permutation to each $T_m$ to get $T_{m+1}$. Find the smallest positive integer $n$ so that $T_n=T_0$.
2009 Hong Kong TST, 2
Find the total number of solutions to the following system of equations:
$ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\
b(a + d)\equiv b \pmod{37} \\
c(a + d)\equiv c \pmod{37} \\
bc + d^2\equiv d \pmod{37} \\
ad - bc\equiv 1 \pmod{37} \end{array}$
2005 iTest, 36
Find the determinant of this matrix: $\begin{bmatrix}
2 & 2 & 2 & 2 & 2 & 2 \\
4 & 2 & 2 & 2 & 2 & 2 \\
4 & 4 & 2 & 2 & 2 & 2 \\
4 & 4 & 4 & 2 & 2 & 2 \\
4 & 4 & 4 & 4 & 2 & 2 \\
4 & 4 & 4& 4 & 4 & 2
\end{bmatrix} $
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 $p.$
1998 IMC, 1
$V$ is a real vector space and $ f, f_{i}: V \rightarrow \mathbb{R} $ are linear for $i = 1, 2, ... , k.$ Also $f $ is zero at all points for which all of $ f_{i }$ are zero. Show that $ f $ is a linear combination of the $f_{i}$.
1950 Miklós Schweitzer, 4
Put
$ M\equal{}\begin{pmatrix}p&q&r\\
r&p&q\\q&r&p\end{pmatrix}$
where $ p,q,r>0$ and $ p\plus{}q\plus{}r\equal{}1$. Prove that
$ \lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\
\frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}$
2004 IMC, 6
For $ n\geq 0$ define the matrices $ A_n$ and $ B_n$ as follows: $ A_0 \equal{} B_0 \equal{} (1)$, and for every $ n>0$ let
\[ A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\
A_{n \minus{} 1} & B_{n \minus{} 1} \\
\end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\
A_{n \minus{} 1} & 0 \\
\end{array} \right).
\]
Denote by $ S(M)$ the sum of all the elements of a matrix $ M$. Prove that $ S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1})$, for all $ n,k\geq 2$.