User aarbee

11 votes

6 answers

5k views

Why every prime (>3) is represented as $6k\pm1$

number-theory

asked Dec 23, 2013 at 3:42

10 votes

2 answers

772 views

Intuition for connectedness of positive definite matrices

matrices positive-definite

asked Dec 25, 2016 at 6:15

8 votes

4 answers

35k views

Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd [closed]

factoring

asked Jan 17, 2014 at 9:32

7 votes

2 answers

5k views

Average of 3 consecutive odd numbers

algebra-precalculus

asked Nov 14, 2013 at 5:50

7 votes

3 answers

11k views

$(x^{2022}+1)(1+x^2+x^4+...+x^{2020})=2022\cdot x^{2021}$

sequences-and-series

asked Aug 10, 2019 at 19:17

6 votes

2 answers

88 views

Solve for $x$: $\log_3(x-2)\ge\log_5(4-x)$

inequality logarithms

asked Jun 5, 2021 at 21:04

5 votes

2 answers

79 views

Let two subsets $P,Q$ be selected from the set $A=\{1,2,3,4,5\}$. Find the probability... [duplicate]

probability combinatorics combinations

asked Feb 2 at 11:35

5 votes

2 answers

119 views

Finding the number of complex numbers satisfying $|z|=\text{max}\{|z-1|,|z+1|\}$

complex-numbers

asked Feb 17 at 12:15

5 votes

4 answers

183 views

Find the range of $\frac{\sqrt{(x-1)(x+3)}}{x+2}$

calculus functions

asked May 10 at 17:34

5 votes

6 answers

26k views

Count of 3-digit numbers with at least one digit as 9

combinatorics permutations number-systems

asked Oct 9, 2013 at 7:45

4 votes

1 answer

2k views

Cyclic subgroup of a quotient group

abstract-algebra group-theory cyclic-groups

asked Jul 25, 2013 at 6:55

4 votes

4 answers

80k views

Common chord of two circles

geometry

asked Feb 25, 2014 at 10:18

4 votes

1 answer

585 views

Sum of two sets

general-topology

asked Dec 27, 2016 at 7:11

4 votes

1 answer

7k views

Finding medians from vectors

vector-spaces vectors triangles vector-analysis inner-products

asked Oct 23, 2017 at 11:13

4 votes

1 answer

75 views

Determining a sequence's $n$-th term from its first and second differences, when latter is in arithmetic or geometric progression

sequences-and-series

asked Jul 26, 2019 at 10:41

4 votes

1 answer

70 views

$a+b+c=-3\sqrt{ac}$

sequences-and-series

asked Aug 1, 2019 at 16:42

4 votes

2 answers

116 views

Domain of $7^{\log_7(x^2-4x+5)}$

inequality complex-numbers logarithms quadratics

asked Aug 6, 2019 at 17:15

4 votes

0 answers

80 views

Let $f(x)=\begin{cases}2x+a;&x\ge-1\\bx^2+3;&x\lt-1\end{cases}$ and $g(x)=\begin{cases}x+4;&0\le x\le4\\-3x-2;&-2\lt x\lt0\end{cases}$ then $g(f(x))$

functions inequality conic-sections piecewise-continuity

asked Feb 23 at 5:57

4 votes

2 answers

91 views

Find the value of $\lim_{ n \to \infty} \left(\frac{1^2+1}{1-n^3}+\frac{2^2+2}{2-n^3}+\frac{3^2+3}{3-n^3}+...+\frac{n^2+n}{n-n^3}\right)$

limits definite-integrals

asked Jul 8, 2021 at 8:23

4 votes

2 answers

120 views

Evaluate $\lim_{x\to(e^{-1})^+}\frac{e^{\frac{\ln(1+\ln x)}x}}{x-e^{-1}}$

calculus limits logarithms contest-math

asked Aug 4, 2021 at 19:06

3 votes

3 answers

101 views

Evaluate $\int_1^{\sqrt3}\left( x^{2x^2+1}+\ln x^{\left(2x^{2x^2+1}\right)}\right)dx$

calculus integration definite-integrals

asked Sep 2, 2021 at 4:56

3 votes

4 answers

96 views

If $\alpha\ne1,\alpha^6=1$ and $\sum_{r=1}^6 {^6}C_r\alpha^{r-1}=x,$ then find the value of $|x|$.

complex-numbers summation contest-math

asked Jul 11, 2021 at 16:14

3 votes

1 answer

117 views

If $\cos\alpha+\cos\beta+\cos\gamma=\sin\alpha+\sin\beta+\sin\gamma=0$ and $\cos3\alpha=\dfrac34$ then find $\cos^8\alpha+\cos^8\beta+\cos^8\gamma$

trigonometry inequality complex-numbers

asked Jun 14, 2021 at 18:18

3 votes

3 answers

125 views

The quadratic equation whose roots are $\sec^2\theta$ and $\csc^2\theta$ can be

trigonometry roots quadratics

asked Jun 14, 2021 at 21:37

3 votes

1 answer

78 views

If $f:R\to R$ satisfies $f(x+2xy)=f(x)+2f(xy)\forall x,y\in R$ and $f(10)=11$ then prove the following

functions

asked Jul 5, 2021 at 19:44

3 votes

1 answer

55 views

Solve $\int\frac{a^2-x^2}{\sqrt{(a^2-x^2)^2-e^2}}dx$

integration

asked Jul 6, 2021 at 10:23

3 votes

4 answers

542 views

Find the equations of circles passing through $(1, -1)$ touching the lines $4x+3y+5=0$ and $3x-4y-10=0$

geometry circles transformation

asked May 31, 2021 at 17:32

3 votes

3 answers

127 views

If $x^3-\frac1{x^3}=108+76\sqrt2$, find $x-\frac1x$

algebra-precalculus

asked Jun 5, 2021 at 19:33

3 votes

4 answers

209 views

A six digit number is formed randomly using digits $\{1,2,3\}$ with repetitions. Choose the correct option(s):

probability combinatorics solution-verification proof-explanation

asked Jun 8, 2021 at 13:08

3 votes

1 answer

488 views

Finding $|A|^2+|B|^2$ if $A=\text{adj} (B)-B^T$ and $B=\text{adj} (A)-A^T$

matrices determinant

asked May 9, 2021 at 12:18

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286 Questions

Why every prime (>3) is represented as $6k\pm1$

Intuition for connectedness of positive definite matrices

Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd [closed]

Average of 3 consecutive odd numbers

$(x^{2022}+1)(1+x^2+x^4+...+x^{2020})=2022\cdot x^{2021}$

Solve for $x$: $\log_3(x-2)\ge\log_5(4-x)$

Let two subsets $P,Q$ be selected from the set $A=\{1,2,3,4,5\}$. Find the probability... [duplicate]

Finding the number of complex numbers satisfying $|z|=\text{max}\{|z-1|,|z+1|\}$

Find the range of $\frac{\sqrt{(x-1)(x+3)}}{x+2}$

Count of 3-digit numbers with at least one digit as 9

Cyclic subgroup of a quotient group

Common chord of two circles

Sum of two sets

Finding medians from vectors

Determining a sequence's $n$-th term from its first and second differences, when latter is in arithmetic or geometric progression

$a+b+c=-3\sqrt{ac}$

Domain of $7^{\log_7(x^2-4x+5)}$

Let $f(x)=\begin{cases}2x+a;&x\ge-1\\bx^2+3;&x\lt-1\end{cases}$ and $g(x)=\begin{cases}x+4;&0\le x\le4\\-3x-2;&-2\lt x\lt0\end{cases}$ then $g(f(x))$

Find the value of $\lim_{ n \to \infty} \left(\frac{1^2+1}{1-n^3}+\frac{2^2+2}{2-n^3}+\frac{3^2+3}{3-n^3}+...+\frac{n^2+n}{n-n^3}\right)$

Evaluate $\lim_{x\to(e^{-1})^+}\frac{e^{\frac{\ln(1+\ln x)}x}}{x-e^{-1}}$

Evaluate $\int_1^{\sqrt3}\left( x^{2x^2+1}+\ln x^{\left(2x^{2x^2+1}\right)}\right)dx$

If $\alpha\ne1,\alpha^6=1$ and $\sum_{r=1}^6 {^6}C_r\alpha^{r-1}=x,$ then find the value of $|x|$.

If $\cos\alpha+\cos\beta+\cos\gamma=\sin\alpha+\sin\beta+\sin\gamma=0$ and $\cos3\alpha=\dfrac34$ then find $\cos^8\alpha+\cos^8\beta+\cos^8\gamma$

The quadratic equation whose roots are $\sec^2\theta$ and $\csc^2\theta$ can be

If $f:R\to R$ satisfies $f(x+2xy)=f(x)+2f(xy)\forall x,y\in R$ and $f(10)=11$ then prove the following

Solve $\int\frac{a^2-x^2}{\sqrt{(a^2-x^2)^2-e^2}}dx$

Find the equations of circles passing through $(1, -1)$ touching the lines $4x+3y+5=0$ and $3x-4y-10=0$

If $x^3-\frac1{x^3}=108+76\sqrt2$, find $x-\frac1x$

A six digit number is formed randomly using digits $\{1,2,3\}$ with repetitions. Choose the correct option(s):

Finding $|A|^2+|B|^2$ if $A=\text{adj} (B)-B^T$ and $B=\text{adj} (A)-A^T$