User acat3

13 votes

Given that $x_0$ is a real root of $x^3+px + q = 0$, how can I show that $p^2 \geq 4x_0q$?

answered Aug 1, 2020 at 5:14

11 votes

Is the volume of a cube the greatest among rectangular-faced shapes of the same perimeter?

answered Jul 29, 2020 at 4:09

11 votes

How find tangent line of the given curve at this point?

answered Mar 4, 2020 at 14:10

11 votes

Accepted

Numbers from 1 to 10 put in a circle

answered Mar 15, 2020 at 12:18

10 votes

Accepted

how to use matrix to prove this identity?

answered Mar 24, 2020 at 5:54

10 votes

Show by counting two ways that $\sum_{i=1}^{n}i(n-i)=\sum_{i=1}^{n}{i\choose 2}={n+1 \choose 3}$?

answered Jul 29, 2020 at 9:09

9 votes

Accepted

Olympiad Chessboard Problem

answered Aug 28, 2021 at 10:00

8 votes

Accepted

Geometry problem without trigonometry

answered Aug 4, 2021 at 1:50

8 votes

Combinatorics counting problem

answered Mar 28, 2020 at 13:05

8 votes

Chance of generating two zeros in a row, compared to chance of generating a zero and a one in a row when generating random numbers between 0 and 5??

answered Dec 25, 2023 at 1:57

7 votes

Accepted

How do I prove this identity for complex numbers?

answered Mar 22, 2020 at 13:52

7 votes

Show not possible to find positive whole numbers $m,n$ such that $m^2 − n^2 = 6$.

answered Apr 5, 2020 at 5:59

7 votes

Accepted

counting sequences of elements of the set {1,2,3,4} with given property

answered Nov 30, 2023 at 7:24

6 votes

Accepted

Let $a,b,c \in \mathbb{R}^{+}$ and $f(x)=\sqrt{a^2+x^2}+\sqrt{(b-x)^2+c^2}$. Then $\min f={?}$

answered Aug 16, 2020 at 7:44

6 votes

Accepted

A bag with 3 black balls and 33 white balls, find expected number of pulls

answered Sep 15, 2021 at 9:33

6 votes

Accepted

If a point is selected inside a rectangle what's the probability that the point is closer to center than vertex?

answered Apr 1, 2022 at 7:19

6 votes

doubt regarding a step proof of Cauchy-Schwarz inequality. Is it valid?

answered Nov 28, 2023 at 15:36

5 votes

Accepted

Solving $f\left( n \right) = \sum\limits_{i > j \ge 0} {{}^{n + 1}{C_i}{}^n{C_j}} $

answered Jan 6, 2023 at 7:45

5 votes

The number of subsets of a set $S$ that fulfil certain criteria.

answered Mar 4, 2023 at 13:32

5 votes

Accepted

Range of a function, algebraically

answered Mar 5, 2023 at 14:39

5 votes

Counting the number of ways of organizing an odd number of objects into 3 odd groups, elegant method?

answered Mar 30, 2023 at 9:17

5 votes

Accepted

Proving an identity relating to binomial coefficients

answered Sep 21, 2021 at 13:55

5 votes

How many ordered quadruples $(a,b,c,d)$ satisfy $a+b+c+d=18$ under various conditions?

answered Aug 9, 2020 at 0:13

5 votes

Accepted

Combinatoric proof of $\binom{j-1}{m-1}=\sum_{k=m}^j(-1)^{k-m}\binom{j}{k}$?

answered Aug 26, 2020 at 7:08

5 votes

Show that $|a| + |b| + |c| \leq |a - |b - c|| + |b - |c - a|| + |c - |a - b||$ where $a, b, c \in \mathbb{R}$ and $a + b + c = 0$

answered Sep 3, 2020 at 8:01

5 votes

Accepted

The identity $\binom{n+k}{k} = \sum_i \binom{n}{i}\binom{k}{i}$

answered Apr 14, 2020 at 12:57

5 votes

Accepted

Solve the summation $\sum_{n_1+n_2+n_3=n}n_1\binom{n}{n_1,n_2,n_3}$

answered Aug 12, 2021 at 9:39

5 votes

Proof of the Laplace Expansion?

answered Aug 16, 2021 at 8:19

5 votes

Binomial Identity and Counting

answered Nov 14, 2020 at 13:14

5 votes

Accepted

Which one is the larger value of $\min(x,y)$ and $\max(0,x+y-1)$, for all $x,y\in [0,1]$?

answered Nov 28, 2020 at 3:30

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525 Answers

Given that $x_0$ is a real root of $x^3+px + q = 0$, how can I show that $p^2 \geq 4x_0q$?

Is the volume of a cube the greatest among rectangular-faced shapes of the same perimeter?

How find tangent line of the given curve at this point?

Numbers from 1 to 10 put in a circle

how to use matrix to prove this identity?

Show by counting two ways that $\sum_{i=1}^{n}i(n-i)=\sum_{i=1}^{n}{i\choose 2}={n+1 \choose 3}$?

Olympiad Chessboard Problem

Geometry problem without trigonometry

Combinatorics counting problem

Chance of generating two zeros in a row, compared to chance of generating a zero and a one in a row when generating random numbers between 0 and 5??

How do I prove this identity for complex numbers?

Show not possible to find positive whole numbers $m,n$ such that $m^2 − n^2 = 6$.

counting sequences of elements of the set {1,2,3,4} with given property

Let $a,b,c \in \mathbb{R}^{+}$ and $f(x)=\sqrt{a^2+x^2}+\sqrt{(b-x)^2+c^2}$. Then $\min f={?}$

A bag with 3 black balls and 33 white balls, find expected number of pulls

If a point is selected inside a rectangle what's the probability that the point is closer to center than vertex?

doubt regarding a step proof of Cauchy-Schwarz inequality. Is it valid?

Solving $f\left( n \right) = \sum\limits_{i > j \ge 0} {{}^{n + 1}{C_i}{}^n{C_j}} $

The number of subsets of a set $S$ that fulfil certain criteria.

Range of a function, algebraically

Counting the number of ways of organizing an odd number of objects into 3 odd groups, elegant method?

Proving an identity relating to binomial coefficients

How many ordered quadruples $(a,b,c,d)$ satisfy $a+b+c+d=18$ under various conditions?

Combinatoric proof of $\binom{j-1}{m-1}=\sum_{k=m}^j(-1)^{k-m}\binom{j}{k}$?

Show that $|a| + |b| + |c| \leq |a - |b - c|| + |b - |c - a|| + |c - |a - b||$ where $a, b, c \in \mathbb{R}$ and $a + b + c = 0$

The identity $\binom{n+k}{k} = \sum_i \binom{n}{i}\binom{k}{i}$

Solve the summation $\sum_{n_1+n_2+n_3=n}n_1\binom{n}{n_1,n_2,n_3}$

Proof of the Laplace Expansion?

Binomial Identity and Counting

Which one is the larger value of $\min(x,y)$ and $\max(0,x+y-1)$, for all $x,y\in [0,1]$?