User acat3

7 votes

1 answer

138 views

Prove that $\sum{a^{2}bc}=\binom{n+3}{6}+\binom{n+2}{6}$

asked Sep 8, 2021 at 13:27

5 votes

3 answers

658 views

Elementary proof of tangent half angle formula

asked Mar 26, 2020 at 12:40

5 votes

0 answers

238 views

Number Theory Question from Swedish Maths Olympiad: Find all integers $n\geq 8$ such that $n^{\frac{1}{n-7}}$ is also an integer.

asked Sep 30, 2020 at 21:50

4 votes

3 answers

102 views

Proof that $\sum_{k=1}^{n}{k^{3}\binom{n}{k}^{2}}=2n\binom{n}{2}\binom{2n-3}{n-2}+n^{2}\binom{2n-2}{n-1}$

asked Aug 18, 2021 at 14:13

4 votes

0 answers

268 views

Identity of binomial coefficients: $\binom{n+1}{k+1}=\sum_{i=m}^{n+m-k}{\binom{n-i}{k-m}\binom{i}{m}}$

asked Apr 13, 2020 at 4:24

4 votes

2 answers

180 views

Prove that $\sum_{i=0}^{m}{\binom{n-i}{k}}=\binom{n+1}{k+1}-\binom{n-m}{k+1}$

asked Sep 6, 2021 at 0:29

4 votes

1 answer

103 views

Showing $ \tan\frac{A+B}{2}= \frac{\sin A+\sin B}{\cos A+\cos B}$

asked Jan 9 at 16:06

3 votes

1 answer

192 views

Taking Seats on a Plane: Extension

asked Aug 24, 2021 at 10:53

3 votes

2 answers

114 views

Solution Verification: Show that $\sum_{i=0}^{n}{\binom{n+a}{n-i}\binom{i+a+1}{i}}=2^{n}\binom{n+a}{a}+2^{n-1}\binom{n+a}{a+1}$ for $n\geq 1$

asked Aug 5, 2021 at 11:48

3 votes

1 answer

895 views

Binary Sequences of length $n$ without $k$ consecutive $1$s or $0$s

asked Aug 11, 2021 at 6:27

2 votes

0 answers

32 views

Probability that $3$ non - negative numbers sum up to $1$ and each are less than $m$ or two are greater than $m$

asked Mar 28, 2020 at 10:05

2 votes

3 answers

776 views

Minimum distance between a surface and a point

asked Feb 29, 2020 at 5:58

2 votes

1 answer

560 views

Prove That $\sum\sum\binom{k-m-mi}{i}=\sum\binom{n+1-mj}{j}$

asked Sep 13, 2021 at 1:23

2 votes

2 answers

88 views

Identity for Difference of Sum of Products of Binomial Coefficients

asked Jan 8, 2023 at 15:30

2 votes

2 answers

68 views

Evaluate Sum of Products in The Form of $\sum_{\left\|\mathbf{x}\right\|=C}\prod_{i=0}^{n}\left(2+i\right)^{x_{i}}$

asked Dec 30, 2023 at 0:52

2 votes

1 answer

113 views

Closed Form of A Sequence with Recursive Relation

asked Jan 9 at 14:23

1 vote

1 answer

105 views

Prove that $\left\{{n+1}\atop{k+1}\right\}=\sum_{i=k}^{n}{\left(k+1\right)^{n-i}\left\{{i}\atop{k}\right\}}$

asked Sep 18, 2021 at 5:31

1 vote

2 answers

133 views

Prove that $\left\{{n+k+1}\atop{k}\right\}=\sum_{i=1}^{k}{i\left\{{n+i}\atop{i}\right\}}$

asked Sep 18, 2021 at 6:26

1 vote

1 answer

174 views

Predict Weather Using Markov Chains

asked Oct 30, 2022 at 13:06

1 vote

1 answer

66 views

How to Find Tangent Line Without Tedious Derivative

asked Mar 6, 2020 at 6:46

1 vote

2 answers

94 views

Locus of $\frac{|z-a|}{|z-b|}=c, c<1$ is a circle

asked Apr 2, 2020 at 4:56

1 vote

3 answers

70 views

Prove that this sum of products of binomial coefficient can be simplified like this [duplicate]

asked Feb 23, 2020 at 15:22

1 vote

1 answer

118 views

Pairing 8 numbers, counting probability with or without permutation

asked Mar 23, 2020 at 1:21

0 votes

1 answer

134 views

Sum of three power of seventh roots of unity form a regular 7 - gon inside a circle

asked Mar 25, 2020 at 16:51

0 votes

0 answers

192 views

Another proof of Jensen’s Inequality (finite form)

asked Apr 8, 2020 at 4:20

0 votes

2 answers

41 views

Proving $\sum_{i=0}^{p}{\sum_{j=0}^{q+i}{\sum_{k=0}^{r+j}{\binom{p}{i}\binom{q+i}{j}\binom{r+j}{k}}}}=4^{p}3^{q}2^{r}$

asked Mar 17, 2020 at 5:17

0 votes

1 answer

62 views

Transforming circle in z - plane to w - plane

asked Mar 21, 2020 at 15:09

0 votes

3 answers

116 views

Triangle's circumcircle inequality: Is it suitable for high schoolers?

asked Mar 5, 2020 at 0:39

0 votes

1 answer

100 views

On the proof that $\binom{2n}{2}=2\binom{n}{2}+n^{2}$

asked Jul 4, 2023 at 18:25

Stack Exchange Network

29 Questions

Prove that $\sum{a^{2}bc}=\binom{n+3}{6}+\binom{n+2}{6}$

Elementary proof of tangent half angle formula

Number Theory Question from Swedish Maths Olympiad: Find all integers $n\geq 8$ such that $n^{\frac{1}{n-7}}$ is also an integer.

Proof that $\sum_{k=1}^{n}{k^{3}\binom{n}{k}^{2}}=2n\binom{n}{2}\binom{2n-3}{n-2}+n^{2}\binom{2n-2}{n-1}$

Identity of binomial coefficients: $\binom{n+1}{k+1}=\sum_{i=m}^{n+m-k}{\binom{n-i}{k-m}\binom{i}{m}}$

Prove that $\sum_{i=0}^{m}{\binom{n-i}{k}}=\binom{n+1}{k+1}-\binom{n-m}{k+1}$

Showing $ \tan\frac{A+B}{2}= \frac{\sin A+\sin B}{\cos A+\cos B}$

Taking Seats on a Plane: Extension

Solution Verification: Show that $\sum_{i=0}^{n}{\binom{n+a}{n-i}\binom{i+a+1}{i}}=2^{n}\binom{n+a}{a}+2^{n-1}\binom{n+a}{a+1}$ for $n\geq 1$

Binary Sequences of length $n$ without $k$ consecutive $1$s or $0$s

Probability that $3$ non - negative numbers sum up to $1$ and each are less than $m$ or two are greater than $m$

Minimum distance between a surface and a point

Prove That $\sum\sum\binom{k-m-mi}{i}=\sum\binom{n+1-mj}{j}$

Identity for Difference of Sum of Products of Binomial Coefficients

Evaluate Sum of Products in The Form of $\sum_{\left\|\mathbf{x}\right\|=C}\prod_{i=0}^{n}\left(2+i\right)^{x_{i}}$

Closed Form of A Sequence with Recursive Relation

Prove that $\left\{{n+1}\atop{k+1}\right\}=\sum_{i=k}^{n}{\left(k+1\right)^{n-i}\left\{{i}\atop{k}\right\}}$

Prove that $\left\{{n+k+1}\atop{k}\right\}=\sum_{i=1}^{k}{i\left\{{n+i}\atop{i}\right\}}$

Predict Weather Using Markov Chains

How to Find Tangent Line Without Tedious Derivative

Locus of $\frac{|z-a|}{|z-b|}=c, c<1$ is a circle

Prove that this sum of products of binomial coefficient can be simplified like this [duplicate]

Pairing 8 numbers, counting probability with or without permutation

Sum of three power of seventh roots of unity form a regular 7 - gon inside a circle

Another proof of Jensen’s Inequality (finite form)

Proving $\sum_{i=0}^{p}{\sum_{j=0}^{q+i}{\sum_{k=0}^{r+j}{\binom{p}{i}\binom{q+i}{j}\binom{r+j}{k}}}}=4^{p}3^{q}2^{r}$

Transforming circle in z - plane to w - plane

Triangle's circumcircle inequality: Is it suitable for high schoolers?

On the proof that $\binom{2n}{2}=2\binom{n}{2}+n^{2}$