Found problems: 823
2004 AMC 12/AHSME, 9
The point $ (\minus{}3, 2)$ is rotated $ 90^\circ$ clockwise around the origin to point $ B$. Point $ B$ is then reflected over the line $ y \equal{} x$ to point $ C$. What are the coordinates of $ C$?
$ \textbf{(A)}\ ( \minus{} 3, \minus{} 2)\qquad \textbf{(B)}\ ( \minus{} 2, \minus{} 3)\qquad \textbf{(C)}\ (2, \minus{} 3)\qquad \textbf{(D)}\ (2,3)\qquad \textbf{(E)}\ (3,2)$
2021 IMC, 8
Let $n$ be a positive integer. At most how many distinct unit vectors can be selected in $\mathbb{R}^n$ such that from any three of them, at least two are orthogonal?
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]
2001 District Olympiad, 2
Let $n\in \mathbb{N},\ n\ge 2$. For any matrix $A\in \mathcal{M}_n(\mathbb{C})$, let $m(A)$ be the number of non-zero minors of $A$. Prove that:
a)$m(I_n)=2^n-1$;
b)If $A\in \mathcal{M}_n(\mathbb{C})$ is non-singular, then $m(A)\ge 2^n-1$.
[i]Marius Ghergu[/i]
1988 IMO Shortlist, 5
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?
MIPT student olimpiad spring 2023, 2
Let $A=a_{ij}$ is simetrical real matrix. Prove that :
$\sum_i e^{a_{ii}} \leq tr (e^A)$
1991 Arnold's Trivium, 11
Investigate the convergence of the integral
\[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{dxdy}{1+x^4y^4}\]
1997 AMC 12/AHSME, 8
Mientka Publishing Company prices its bestseller [i]Where's Walter?[/i] as follows: \[C(n) \equal{}
\begin{cases}
12n, &\text{if } 1 \le n \le 24\\
11n, &\text{if } 25 \le n \le 48\\
10n, &\text{if } 49 \le n
\end{cases}\] where $ n$ is the number of books ordered, and $ C(n)$ is the cost in dollars of $ n$ books. Notice that $ 25$ books cost less than $ 24$ books. For how many values of $ n$ is it cheaper to buy more than $ n$ books than to buy exactly $ n$ books?
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 8$
2014 Online Math Open Problems, 15
In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$.
Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_k$, he moves to city $C_{2k}$; otherwise he moves to city $C_{2k+1}$. If the probability that Al is back at city $C_0$ after $10$ moves is $\tfrac{m}{1024}$, find $m$.
[i]Proposed by Ray Li[/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 $a+b+c$.
2019 Miklós Schweitzer, 10
Let $A$ and $B$ be positive self-adjoint operators on a complex Hilbert space $H$. Prove that
\[\limsup_{n \to \infty} \|A^n x\|^{1/n} \le
\limsup_{n \to \infty} \|B^n x\|^{1/n}\]
holds for every $x \in H$ if and only if $A^n \le B^n$ for each positive integer $n$.
2006 VJIMC, Problem 4
Let $A=[a_{ij}]_{n\times n}$ be a matrix with nonnegative entries such that
$$\sum_{i=1}^n\sum_{j=1}^na_{ij}=n.$$
(a) Prove that $|\det A|\le1$.
(b) If $|\det A|=1$ and $\lambda\in\mathbb C$ is an arbitrary eigenvalue of $A$, show that $|\lambda|=1$.
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 $\lim_{n\to \infty}d_n=\infty$.
1987 Traian Lălescu, 1.2
Let be a natural number $ n, $ a complex number $ a, $ and two matrices $ \left( a_{pq}\right)_{1\le q\le n}^{1\le p\le n} ,\left( b_{pq}\right)_{1\le q\le n}^{1\le p\le n}\in\mathcal{M}_n(\mathbb{C} ) $ such that
$$ b_{pq} =a^{p-q}\cdot a_{pq},\quad\forall p,q\in\{ 1,2,\ldots ,n\} . $$
Calculate the determinant of $ B $ (in function of $ a $ and the determinant of $ A $ ).
2010 Laurențiu Panaitopol, Tulcea, 4
Let be an odd integer $ n\ge 3 $ and an $ n\times n $ real matrix $ A $ whose determinant is positive and such that $ A+\text{adj} A=2A^{-1} . $ Prove that $ A^{2010} +\text{adj}^{2010} A =2A^{-2010} . $
[i]Lucian Petrescu[/i]
MIPT student olimpiad autumn 2024, 2
$A,B \in M_{2\times 2}(C)$
Prove that:
$Tr(AAABBABAABBB)=tr(BBBAABABBAAA)$
2011 AMC 12/AHSME, 23
Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ \sqrt{2}-1 \qquad
\textbf{(C)}\ \sqrt{3}-1 \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ 2$
2008 Gheorghe Vranceanu, 2
Consider the $ 4\times 4 $ integer matrices that have the property that each one of them multiplied by its transpose is $
4I. $
[b]a)[/b] Show that the product of the elements of such a matrix is either $ 0, $ either $ 1. $
[b]b)[/b] How many such matrices have the property that the product of its elements is $ 0? $
2005 VJIMC, Problem 2
Let $(a_{i,j})^n_{i,j=1}$ be a real matrix such that $a_{i,i}=0$ for $i=1,2,\ldots,n$. Prove that there exists a set $\mathcal J\subset\{1,2,\ldots,n\}$ of indices such that
$$\sum_{\begin{smallmatrix}i\in\mathcal J\\j\notin\mathcal J\end{smallmatrix}}a_{i,j}+\sum_{\begin{smallmatrix}i\notin\mathcal J\\j\in\mathcal J\end{smallmatrix}}a_{i,j}\ge\frac12\sum_{i,j=1}^na_{i,j}.$$
2007 District Olympiad, 2
Let $A\in \mathcal{M}_n(\mathbb{R}^*)$. If $A\cdot\ ^t A=I_n$, prove that:
a)$|\text{Tr}(A)|\le n$;
b)If $n$ is odd, then $\det(A^2-I_n)=0$.
2022 OMpD, 2
Let $p \geq 3$ be a prime number and let $A$ be a matrix of order $p$ with complex entries. Assume that $\text{Tr}(A) = 0$ and $\det(A - I_p) \neq 0$. Prove that $A^p \neq I_p$.
Note: $\text{Tr}(A)$ is the sum of the main diagonal elements of $A$ and $I_p$ is the identity matrix of order $p$.
1986 IMO Longlists, 46
We wish to construct a matrix with $19$ rows and $86$ columns, with entries $x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86)$, such that:
[i](i)[/i] in each column there are exactly $k$ terms equal to $0$;
[i](ii)[/i] for any distinct $j, k \in \{1, . . . , 86\}$ there is $i \in \{1, . . . , 19\}$ with $x_{ij} + x_{ik} = 3.$
For what values of $k$ is this possible?
1994 IMC, 1
a) Let $A$ be a $n\times n$, $n\geq 2$, symmetric, invertible matrix with real positive elements. Show that $z_n\leq n^2-2n$, where $z_n$ is the number of zero elements in $A^{-1}$.
b) How many zero elements are there in the inverse of the $n\times n$ matrix
$$A=\begin{pmatrix} 1&1&1&1&\ldots&1\\
1&2&2&2&\ldots&2\\
1&2&1&1&\ldots&1\\
1&2&1&2&\ldots&2\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
1&2&1&2&\ldots&\ddots
\end{pmatrix}$$
2016 Romania National Olympiad, 2
Consider a natural number, $ n\ge 2, $ and three $ n\times n $ complex matrices $ A,B,C $ such that $ A $ is invertible, $ B $ is formed by replacing the first line of $ A $ with zeroes, and $ C $ is formed by putting the last $ n-1 $ lines of $ A $ above a line of zeroes. Prove that:
[b]a)[/b] $ \text{rank} \left( A^{-1} B \right) = \text{rank} \left( \left( A^{-1} B\right)^2 \right) =\cdots =\text{rank} \left( \left( A^{-1} B\right)^n \right) $
[b]b)[/b] $ \text{rank} \left( A^{-1} C \right) > \text{rank} \left( \left( A^{-1} C\right)^2 \right) >\cdots >\text{rank} \left( \left( A^{-1} C\right)^n \right) $
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.