Found problems: 823
2012 Centers of Excellency of Suceava, 1
Let be a natural number $ n $ and a $ n\times n $ nilpotent real matrix $ A. $
Prove that $ 0=\det\left( A+\text{adj} A \right) . $
[i]Neculai Moraru[/i]
2004 District Olympiad, 2
a) Let $x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R}$ and $a_{ij}=\sin(x_i-y_j),\ i,j=\overline{1,3}$ and $A=(a_{ij})\in \mathcal{M}_3$ Prove that $\det A=0$.
b) Let $z_1,z_2,\ldots,z_{2n}\in \mathbb{C}^*,\ n\ge 3$ such that $|z_1|=|z_2|=\ldots=|z_{n+3}|$ and $\arg z_1\ge \arg z_2\ge \ldots\ge \arg(z_{n+3})$. If $b_{ij}=|z_i-z_{j+n}|,\ i,j=\overline{1,n}$ and $B=(b_{ij})\in \mathcal{M}_n$, prove that $\det B=0$.
2012 Uzbekistan National Olympiad, 4
Given $a,b$ and $c$ positive real numbers with $ab+bc+ca=1$. Then prove that
$\frac{a^3}{1+9b^2ac}+\frac{b^3}{1+9c^2ab}+\frac{c^3}{1+9a^2bc} \geq \frac{(a+b+c)^3}{18}$
2008 All-Russian Olympiad, 8
We are given $ 3^{2k}$ apparently identical coins,one of which is fake,being lighter than the others. We also dispose of three apparently identical balances without weights, one of which is broken (and yields outcomes unrelated to the actual situations). How can we find the fake coin in $ 3k\plus{}1$ weighings?
2017 Korea USCM, 4
For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix
$$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$
is positive definite.
2016 District Olympiad, 2
Let A,B,C,D four matrices of order n with complex entries, n>=2 and let k real number such that AC+kBD=I and AD=BC. Prove that CA+kDB=I and DA=CB.
2012 Mediterranean Mathematics Olympiad, 3
Consider a binary matrix $M$(all entries are $0$ or $1$) on $r$ rows and $c$ columns, where every row and every column contain at least one entry equal to $1$. Prove that there exists an entry $M(i,j) = 1$, such that the corresponding row-sum $R(i)$ and column-sum $C(j)$ satisfy $r R(i)\ge c C(j)$.
(Proposed by Gerhard Woeginger, Austria)
1995 Putnam, 5
Let $x_1,x_2,\cdots, x_n$ be real valued differentiable functions of a variable $t$ which satisfy
\begin{align*}
& \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\
& \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\
& \;\qquad \vdots \\
& \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\
\end{align*}
For some constants $a_{ij}>0$. Suppose that $\lim_{t \to \infty}x_i(t)=0$ for all $1\le i \le n$. Are the functions $x_i$ necessarily linearly dependent?
2009 Putnam, A1
Let $ f$ be a real-valued function on the plane such that for every square $ ABCD$ in the plane, $ f(A)\plus{}f(B)\plus{}f(C)\plus{}f(D)\equal{}0.$ Does it follow that $ f(P)\equal{}0$ for all points $ P$ in the plane?
2002 Iran Team Selection Test, 10
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
2004 Romania National Olympiad, 3
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]
2014 AIME Problems, 14
Let $m$ be the largest real solution to the equation \[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\] There are positive integers $a,b,c$ such that $m = a + \sqrt{b+\sqrt{c}}$. Find $a+b+c$.
2005 Germany Team Selection Test, 3
For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.
2011 District Olympiad, 2
Consider the matrices $A\in \mathcal{M}_{m,n}(\mathbb{C})$ and $B\in \mathcal{M}_{n,m}(\mathbb{C})$ with $n\le m$. It is given that $\text{rank}(AB)=n$ and $(AB)^2=AB$.
a)Prove that $(BA)^3=(BA)^2$.
b)Find $BA$.
1990 Putnam, B3
Let $S$ be a set of $ 2 \times 2 $ integer matrices whose entries $a_{ij}(1)$ are all squares of integers and, $(2)$ satisfy $a_{ij} \le 200$. Show that $S$ has more than $ 50387 (=15^4-15^2-15+2) $ elements, then it has two elements that commute.
2006 Grigore Moisil Urziceni, 2
Let be two matrices $ A,B\in\mathcal{M}_2\left( \mathbb{C} \right) $ satisfying $ AB-BA=A. $ Show that:
[b]a)[/b] $ \text{tr} (A) =\det (A) =0 $
[b]b)[/b] $ AB^nA=0, $ for any natural number $ n $
2010 VJIMC, Problem 3
Let $A$ and $B$ be two $n\times n$ matrices with integer entries such that all of the matrices
$$A,\enspace A+B,\enspace A+2B,\enspace A+3B,\enspace\ldots,\enspace A+(2n)B$$are invertible and their inverses have integer entries, too. Show that $A+(2n+1)B$ is also invertible and that its inverse has integer entries.
2002 Putnam, 4
In Determinant Tic-Tac-Toe, Player $1$ enters a $1$ in an empty $3 \times 3$ matrix. Player $0$ counters with a $0$ in a vacant position and play continues in turn intil the $ 3 \times 3 $ matrix is completed with five $1$’s and four $0$’s. Player $0$ wins if the determinant is $0$ and player $1$ wins otherwise. Assuming both players pursue optimal strategies, who will win and how?
1999 IberoAmerican, 3
Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud.
For each positive integer $k$, find all the positive integers $n$ such that every possible ($n,k$)-cloud has two mutually exterior tangent circumferences of the same color.
2007 Nicolae Păun, 1
Prove that $ \exists X,Y,Z\in \mathcal{M}_n(\mathbb{C})$ such that
a)$ X^2\plus{}Y^2\equal{}A$
b) $ X^3\plus{}Y^3\plus{}Z^3\equal{}A$ , where $ A\in \mathcal{M}_n(\mathbb{C})$
2024 IMC, 3
For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0,1\}$, such that $A^2$ is the matrix of all ones?
2021 IMO Shortlist, A6
Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.
2003 District Olympiad, 3
a)Prove that any matrix $A\in \mathcal{M}_4(\mathbb{C})$ can be written as a sum of four matrices $B_1,B_2,B_3,B_4\in \mathcal{M}_4(\mathbb{C})$ with the rank equal to $1$.
b)$I_4$ can't be written as a sum of less than four matrices with the rank equal to $1$.
[i]Manuela Prajea & Ion Savu[/i]
2016 Vietnam Team Selection Test, 6
Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$. For each polynomial $P$, denote \[ V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). \] Prove that there is a monic polynomial $Q$, $\deg Q=8$ satisfying:
i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$.
ii) $Q$ has $8$ real roots (including multiplicity).
2008 Romania National Olympiad, 3
Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\equal{}1$ has a unique solution in $ A$, $ x\equal{}1$. Prove that
a) $ 0$ is the only nilpotent element of $ A$;
b) there exists an integer $ k\geq 2$, such that the equation $ x^k\equal{}x$ has $ n$ solutions in $ A$.