2
votes
0answers
40 views

Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a ...
-4
votes
0answers
65 views

Prove that $\sqrt{n}$ is irrational [on hold]

Question: Using fundamental theorem of integers and the fact that every natural number that is not prime, prove that $\sqrt{n}$ is irrational unless $n=m^2$ for some $m\in\mathbb N$. Here is how I ...
1
vote
1answer
25 views

Prove for relatively prime numbers.

Prove that for relatively prime positive integers $a$ and $b$, the equation $ax+by=c$ must have non-negative integer solution if $c>ab-a-b$.
2
votes
1answer
44 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
1
vote
1answer
59 views

If $AB+BA=0$ and $B=AX+XB$, then $B$ is nilpotent.

Suppose $A,B,X \in M_n(\mathbb{R})$ and that $AB+BA=0$ and $B=AX+XA$. Prove that $B$ is a nilpotent matrix.
-2
votes
1answer
73 views

Integral question challenge [duplicate]

I try to find a reasonable solution for this equation but i couldent I try to study lots of material but i couldent solve it. I am a high school student and try to learn. Integral cos(log x)dx
3
votes
2answers
31 views

Find value of $n$ with given conditions

The 4-digit positive number $n$'s digit sum is $20$. The sum of the first two digits is $11$, the sum of the first and the last digit as well. The first digit is the last digit $+3$. What is the ...
6
votes
2answers
279 views

A generalization of IMO 1977 problem 2

Here is the IMO 1977 problem 2: In a finite sequence of real numbers the sum of any seven successive terms is negative, and the sum of any eleven successive terms is positive. Determine the ...
4
votes
0answers
65 views

Smallest value that a certain variable can take in a system of equations.

Consider the solutions $(x,y,z,u)$ of the system of equations: $$\begin{cases} x+y=3(z+u)\\ x+z=4(y+u)\\ x+u=5(y+z)\\ \end{cases}$$ where $x,y,z \text{ and } u$ are positive integers. What ...
1
vote
1answer
134 views

international mathematical competition for college students

I randomly came across with the following problems: Let $A,B \in M_n (\mathbb{C})$ such that $A^2B+B^2A=2ABA.$ Prove that $(AB-BA)^k=0$ for some positive integer $k$. The proof is as follows: Let ...
3
votes
2answers
96 views

Interesting determinant: Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$

Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$, $a_{i,j}=1$ if $i-j\equiv\pm2\pmod n$, and $a_{i,j}=0$ otherwise. Find $\det A$. It seems that the ...
2
votes
2answers
195 views

Solving two simultaneous equations

Suppose that $x$, $y$ and $z$ are three integers (positive,negative or zero) such that we get the following relationships simultaneously $x + y = 1 - z$ and $x^3 + y^3= 1 - z^2$ Find all such ...
0
votes
1answer
85 views

Basis and dimensions for quadratic polynomials

How do I find the basis and dimension for the set of all quadratic polynomials p(x)=ax^2+bx+c that satisfy p(1)=0.
2
votes
2answers
106 views

What is the largest value of $n$ for which $2n + 1$ is a factor of $122 + n^{2}$?

Given that $n$ is a natural number, what is its largest value such that $2n + 1$ is a factor of $122 + n^{2}$?
9
votes
3answers
286 views

The integer $c_n$ in $(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

For non-negative integer $n$, write $$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$$ where $a_n,b_n,c_n$ are integers. For any non-negative integer $m$, prove or disprove ...
3
votes
1answer
263 views

block matrices problem

Let $A,B,C$ and $D$ be n by n matrics such that $AC=CA$. Prove that $\det \begin{pmatrix} A & B\\ C & D \end{pmatrix}=\det(AD-CB)$. The solution is to first assume that $A$ is invertible and ...
3
votes
2answers
60 views

Norms of eigenvalues bigger than 1 implies $|Ax|>x$ for all nonzero $x$?

If all the eigenvalues of $A$ (an n by n real matrix) have norms bigger than 1, is it true that $|Ax|>|x|$ for all nonzero $x\in\mathbb{R}^n$? (This is clearly true if $x$ is an eigenvector ...
4
votes
1answer
149 views

Prove that $f$ is a linear combination of $f_1,f_2,\dots,f_n$.

Let $V$ be a vector space and let $f, f_1,f_2,\dots,f_n$ be linear maps from $V$ to $\mathbb{R}$. Suppose that $f(x)=0$ whenever $f_1(x)=f_2(x)=\cdots=f_n(x)=0$. Prove that $f$ is a linear combination ...
1
vote
1answer
124 views

find the value of 1/(2+1/(4+1/(4+1/(…))))

the question is to find the value of this ugly non-stopping fraction $$\frac{1}{2+\frac{1}{4+\frac{1}{4+\frac{1}{\ldots}}}}$$. I have totally no clue; thanks for the help! How am I suppose to solve ...
0
votes
2answers
47 views

Given $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?

Given positive reals $a > b+c$, $e>d+f$, and $i>g+h$, can the quantity $a(ei-hf) + b(-di+fg) - c(dh+eg)$ ever be zero?
0
votes
1answer
264 views

Given a polynomial $P$ find $Q$ such that $Q(x)-Q(x-1)-Q(x-2)=P(x)$ for all $x$

Let $P\in\mathbb{Z}[x]$ be a given polynomial of degree $d$. I want to find the unique polynomial $Q\in\mathbb{Z}[x]$ of degree $d$ such that $Q(x)-Q(x-1)-Q(x-2)=P(x)$. It is possible to construct the ...
6
votes
2answers
106 views

let$ G=\{M_1,M_2,…,M_k\}$ be a finite group if $\sum _{i=0}^k \operatorname{tr} (M_i)=0$ then how prove $\sum _{i=0}^k M_i=0$

Let $G=\{M_1,M_2,...,M_k\}$ be a finite set such that $ M_i\in M_n(\mathbb R)$ and $(G,\;\cdot\:)$ is group with operations of matrix multiplication If $\sum _{i=1}^k \operatorname{tr} (M_i)=0$ ...
4
votes
2answers
613 views

how to prove following matrix is invertible? [duplicate]

how to prove A is invertible or $\ detA\neq 0$ $$A=\begin{pmatrix} \frac11 & \frac12 & \frac13 & \cdots & \frac1n \\ \frac12 & \frac13 & \frac14 & \cdots & ...
-3
votes
2answers
139 views

Lucky Lattice Points

How many lattice points lie on the sphere given by following equation ? $$x^2+y^2+z^2=2013$$ Hint: A lattice point has integer coordinates.
1
vote
2answers
201 views

Finding the distance between the $x$-intercepts of two lines

A line with slope $4$ intersects a line with slope $7$ at the point $(10,28)$. What is the distance between the $x$-intercepts of these two lines? This question was asked in a Math Competition in ...
3
votes
2answers
159 views

Finding the number of different ordered quadruples $(a,b,c,d)$ of complex numbers

Find the number of different ordered quadruples $(a,b,c,d)$ of complex numbers such that: $$a^2=1$$ $$b^3=1$$ $$c^4=1$$ $$d^6=1$$ $$a+b+c+d=0$$
2
votes
1answer
62 views

If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.

Let $H$ be an $n\times n$ matrix with entries $\pm1$. Its rows are mutually orthogonal. If $H$ has an $a\times b$ submatrix of all $1$s, please prove that $ab\le n$.
4
votes
2answers
206 views

Prove that function is bijective

Let $n \in \mathbb{N} \setminus \{ 0 \} $ and $A \in M_n(\mathbb{R})$ with $m \in \mathbb{N} \setminus \{ 0 \}$ as $A^m= \alpha \times I_n$, with $ \alpha \in \mathbb{R} \setminus \{ -1,1 \}$. ...
5
votes
3answers
214 views

Show that $\operatorname{rank}(A^2+A+I_3)=1$

If $A \in M_3(\mathbb{R}), A \ne I_3 $ and $A^3=I_3$ Show that $\operatorname{rank}(A^2+A+I_3)=1$. What I have reached so far is that $\operatorname{rank}(A-I_3)+\operatorname{rank}(A^2+A+I_3)\le ...
0
votes
2answers
245 views

Solving $\frac{1}{a}+\frac{1}{b}=\frac{1}{200}\;$

How many ordered pairs of integers $(a,b)$ are there such that $$\frac{1}{a}+\frac{1}{b}=\frac{1}{200}\;?$$
4
votes
2answers
71 views

let $A\in M_n\mathbb R$ .how prove these statements with following condition?

Assume $A\in M_n(\mathbb{R})$, $A\neq 0$ such that: \begin{align*} A=(a_{ij}),\ 1\le i,j\le n,\\ a_{ik}a_{jk}=a_{kk}a_{ij},\ \forall\, i,j \end{align*} How to prove that: ...
2
votes
1answer
79 views

Assume $A_1,A_2,…,A_n\in M_{m×m}(F)$ that satisfy the following conditions, how to prove that $A_1A_2…A_n=0$?

Assume $A_1,A_2,...,A_n\in M_{m×m}(F)$ (where $F$ is a field) such that $A_jA_i=A_iA_j$ $A_i^2=0, \;\;\forall 1\leq i \leq n.$ If $m\lt2^n$ then how to prove that $A_1A_2...A_n=0.$ Thanks in ...
10
votes
2answers
239 views

Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?

Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advance
26
votes
3answers
563 views

Square matrices satisfying certain relations must have dimension divisible by $3$

I saw this tucked away in a MathOverflow comment and am asking this question to preserve (and advertise?) it. It's a nice problem! Problem: Suppose $A$ and $B$ are real $n\times n$ matrices with ...
0
votes
2answers
155 views

Help understanding train problem

A train $150$ $m$ long passes a km stone in $15$ seconds and another train of the same length traveling in opposite direction in $8$ seconds. The speed of the ...
2
votes
2answers
167 views

let A,B be complex matrics and $2A(B-A)=A+B$ how prove $AB=BA$

let $A,B\in M_n(\mathbb C)$ $\mathbb C$ is complex field such that $$2A(B-A)=A+B$$ how prove $AB=BA$ thanks in advance
6
votes
2answers
109 views

let $A,B\in M_{n}(C)$ such that c is complex field and $AB^2-B^2A=B$ how prove $B^n=0$

Let $A,B\in M_{n}(C)$ such that $C$ is complex field and $AB^2-B^2A=B$. How prove $B^n=0$. thanks in advance
2
votes
1answer
93 views

how prove $A_1$+$(-1)^nA_n$ is scalar matrix with following condition

let $A_i\in M_n (\mathbb{R})$ ,$i=1,2,...,n$ $$A_1\cdot A_2 \cdot...\cdot A_n=I\hspace{5pt}\&\hspace{5pt}\det A_1=...=\det A_n=1$$ Assume that $A_1-A_k$ for $k=1,2,..,n-1$ are none zero and ...
2
votes
2answers
509 views

how prove GL(n,R) is not connected subset and open subset of$M_n (\mathbb{R})$with this distance

let n>1 be natural and fix number, $S:=${A : $M_n (\mathbb{R})$ be all real matrix,define this meter for all $A=[a_{ij}]$ $B=[b_{ij}]$ d(A,B):=max{|$a_{ij}-b_{ij}$|:i,j=1,2,2...,n} and GL(n,R) is ...
1
vote
2answers
154 views

How to prove that this linear operator is nilpotent?

Let $A\in M_n(\mathbb{C})$ be an arbitrary matrix , $\mathbb{C}$ is complex fields, and $L$ a mapping that is defined by $L:M_n(\mathbb{C})\to M_n(\mathbb{C})$, $L(X):=AX+XA$. How can we show that ...
5
votes
3answers
407 views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
2
votes
1answer
250 views

Applying a Function to Square Matrices

Moderator Note: This question is from a contest which ended 1 Dec 2012. Consider a polynomial $f$ with complex coefficients. Call such $f$ broken if we can find a square matrix $M$ such that $M ...
0
votes
1answer
297 views

Dimension of Vector space

This problem is taken from International Mathematics Competition for University Students 2009 (IMC 2009), Day 2, Problem 5. Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a ...