Easy matrix problem

Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$ .
Determine the size of the set $\{ \det A : A \in M \}$ .
Here $\det A$ denotes the determinant of the matrix $A$ .

3 replies

mathematics2004

Nov 2, 2021

Isolemma

3 hours ago

Number of real roots

girishpimoli 0

5 hours ago

Number of real roots of

$\displaystyle 2\sin(\theta)\cos(3\theta)\sin(5\theta)=-1$

0 replies

girishpimoli

5 hours ago

0 replies

Factorization Ex.28a Q30

Obvious_Wind_1690 1

N 6 hours ago by Lankou

Please help with factorization. Given is the question

$\begin{align*} a(a+1)x^2+(a+b)xy-b(b-1)y^2\\ \end{align*}$
And the given answer is

$\begin{align*} [(a+1)x-(b-1)y][ax+by]\\ \end{align*}$
But I am unable to reach the answer.

1 reply

Obvious_Wind_1690

Today at 4:17 AM

Lankou

6 hours ago

Polynomials

P162008 4

N Today at 4:19 PM by HAL9000sk

If $f(x)$ is a polynomial function such that $f(x) = x\sqrt{1 + (x + 1)\sqrt{1 + (x + 2)\sqrt{1 + (x + 3)\sqrt{1 + \cdots}}}}$ then

A) Degree of $f(x)$ must be greater than $2$

B) $f(-2) = 0$

C) $\sum_{r=1}^{5} \frac{1}{f(r)} = \frac{25}{42}$

D) $\sum_{r=1}^{n} \frac{1}{f(r)} = \frac{n(3n + 5)}{4(n+1)(n+2)}$

4 replies

P162008

Yesterday at 11:18 PM

HAL9000sk

Today at 4:19 PM

Find r_1^2 + r_2^2 + r_3^2

BlackOctopus23 2

N Today at 3:44 PM by BlackOctopus23

Let $r_1$ , $r_2$ , and $r_3$ be the roots of $3x^3 - 8x^2 + 4x - 13$ . Find $r_1^2 + r_2^2 + r_3^2$ Solution

$(r_1 + r_2 + r_3)^2 = (r_1 + r_2 + r_3) (r_1 + r_2 + r_3)$
$= r_1^2 r_1r_2 + r_1r_3 + r_2r_3 + r_2^2 + r_2r_3 + r_3r_1 + r_3r_2 + r_3^2$
$= r_1^2 + r_2^2 + r_3^2 + 2r_1r_2 + 2r_1r_3 + 2r_2r_3$
$r_1^ + r_2^2 + r_3^2$
$= (r_1 + r_2 + r_3)^2 - 2(r-1r_2 + r_1r_3 + r_2r_3)$
$= (\frac{8}{3})^2 - 2(\frac{4}{3})$
$= \frac{64}{9} - \frac{24}{9}$
$= \boxed{\frac{40}{9}}$

2 replies

BlackOctopus23

Today at 1:50 AM

BlackOctopus23

Today at 3:44 PM

hard inequality

revol_ufiaw 10

N Today at 3:43 PM by sqing

Prove that $(a-b)(b-c)(c-d)(d-a)+(a-c)^2 (b-d)^2\ge 0$ for rational $a, b, c, d$ .

10 replies

revol_ufiaw

Today at 1:09 PM

sqing

Today at 3:43 PM

quadratic eq. with integer roots

lakshya2009 2

N Today at 3:31 PM by alexheinis

Find all $a\in \mathbb{Q}$ such that $ax^2+(3a-1)x+1=0$ has integer roots.

2 replies

lakshya2009

Today at 2:45 PM

alexheinis

Today at 3:31 PM

Original Problem-Pigeonhole principle

ondynarilyChezy 1

N Today at 3:22 PM by ondynarilyChezy

Ondy has invented a new mathematical function:
f(x) = x^2 + ax + b
where a and b are real constants known only to him. To challenge his friends Gab, Clyde, TJ, and Rian, he picks a secret subset of 12 distinct integers from the set:
S = {1, 2, 3, ..., 21}
and evaluates f(x) at each selected number. He then tells his friends only the remainders of these outputs modulo 7, i.e., the multiset:
T = { f(x_1) mod 7, f(x_2) mod 7, ..., f(x_{12}) mod 7 }
However, Ondy won’t reveal which x_i gave which output.
Help the friends as they argue whether it’s always guaranteed that among the 12 inputs Ondy picked:
> There exist two numbers x_i ≠ x_j such that:
> f(x_i) ≡ f(x_j) mod 7
> |x_i - x_j| ≤ 6

1 reply

ondynarilyChezy

Yesterday at 4:14 PM

ondynarilyChezy

Today at 3:22 PM

Interesting Geometry

captainmath99 4

N Today at 12:35 PM by captainmath99

Let ABC be a right triangle such that $\angle{C}=90^\circ, CA=6, CB=4$ . A circle O with center C has a radius of 2. Let P be a point on the circle O.

a)What is the minimum value of $(AP+\dfrac{1}{2}BP)$ ?
Answer Check

$\sqrt{37}$

b) What is the minimum value of $(\dfrac{1}{3}AP+BP)$ ?
Answer Check

$\dfrac{3}{4}\sqrt{17}$

4 replies

captainmath99

May 25, 2025

captainmath99

Today at 12:35 PM

[My own problem] logarithms...

jdcuber13 0

Today at 11:09 AM

There exists real numbers $b$ and $n$ , such that $\log_3{(\log_b(21\log_b9) - \log_b189)} = 4$ , and $\log_b(\log_3 n) = 162$ . Find the last two digits of $n$ .

Answer

$03$ , and inspired by 2017 AIME I Problem 14

personal solution

Simplify the first equation, and we have
$\begin{align*} \log_3{(\log_b(21\log_b9) - \log_b189)} &= 4 \\ \log_b\left(\frac{21\log_b9}{189}\right) = 81 \\ \log_b\left(\frac{\log_b9}{9}\right) = 81 \\ \frac{\log_b9}{9} = \ b^{81} \\ \log_b9 = \ 9b^{81} \\ 9 = \ (b^9)^{b^{81}} \end{align*}$ Now raising to the power of $9$ to both sides and we have $9^9 = (b^{81})^{b^{81}}$ . This means that $b^{81} = 9$ . Using this for the second equation, we now get,
$\begin{align*} \log_b&(\log_3 n) = 162 \\ \log_3n &= b^{162} = (b^{81})^2 \\ \log_3n &= 9^2 = 81 \\ n &= 3^{81} \end{align*}$ Now we just need to find $n = 3^{81}$ mod $100$ . dissecting it into mod $4$ , and mod $25$ , we have $n = 3$ mod $4$ , $n = 3$ mod 25. By CRT, we have $n = 3$ mod $100$ . Therefore, the last two digits of $n$ are $\boxed{03}$ .

0 replies

jdcuber13

Today at 11:09 AM

0 replies

[Sipnayan 2017 SHS] logarithms being defined and not defined

jdcuber13 0

Today at 11:02 AM

For how many positive integers $n$ is $\log(\log(\log n))$ is defined while $\log(\log(\log(\log n)))$ is not?
Note that $\log x = \log_{10}x$ .
Answer

9,999,999,990

personal solution

For a logarithmic function to be defined, the argument must be greater than $0$ . (ie. in $\log_{a}b, \ b > 0$ ).Therefore, the first function must have the argument greater than $0$ .)
For $\log(\log(\log n))$ to be defined,
$\begin{align*} \log(\log n)) &> 0 \\ \log n &> 1 \\ n &> 10 \end{align*}$
Similarly, a logarithmic function that is not defined has its argument less than or equal to 0 (basically the opposite).

$\begin{align*} \log(\log(\log n)) &\leq 0 \\ \log(\log n) &\leq 1 \\ \log n &\leq 10 \\ n &\leq 10^{10} \end{align*}$ Therefore, $10 < n \leq 10^{10}$ to suffice these conditions, so the number of positive integers n to suffice these conditions is $10^{10} - 10 = \boxed{9,999,999,990}$ .

0 replies

jdcuber13

Today at 11:02 AM

0 replies

Easy matrix problem

mihaim 3

N Sep 6, 2015 by loup blanc

Let $A$ and $B$ two matrix of $2\times2$ with real elements such as $A^2+B^2+2AB=O_{2}$ and $det A=detB$ .Compute $\det(A^2-B^2)$ .

3 replies

mihaim

Sep 2, 2015

loup blanc

Sep 6, 2015

Easy matrix problem

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linear algebra

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mihaim

418 posts

mihaim

#1 h Sep 2, 2015, 12:42 PM • 2 Y

Y by Adventure10, Mango247

Let $A$ and $B$ two matrix of $2\times2$ with real elements such as $A^2+B^2+2AB=O_{2}$ and $det A=detB$ .Compute $\det(A^2-B^2)$ .

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wer

1270 posts

wer

#2 h Sep 2, 2015, 1:29 PM • 2 Y

Y by Adventure10, Mango247

We have: $det(A^2+B^2)+det(A^2-B^2)=2(detA^2+detB^2)$ , so $det(A^2-B^2)=0$

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mihaim

418 posts

mihaim

#3 h Sep 2, 2015, 1:34 PM • 2 Y

Y by Adventure10, Mango247

That's my solution too.Well done !

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loup blanc

3601 posts

loup blanc

#4 h Sep 6, 2015, 8:52 AM • 2 Y

Y by Adventure10, Mango247

The hypothesis $A,B$ real and $\det(A)=\det(B)$ are useless.
In fact, if $A,B\in M_2(\mathbb{C})$ satisfy $A^2+B^2+2AB=0_2$ , then $AB=BA$ and $(A+B)^2=0_2$ .

This post has been edited 1 time. Last edited by loup blanc, Sep 6, 2015, 8:52 AM

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