User acat3 - Mathematics Stack Exchange

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

12 votes

Accepted

Numbers from 1 to 10 put in a circle

answered Mar 15, 2020 at 12:18

11 votes

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

answered Mar 4, 2020 at 14:10

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

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

10 votes

Accepted

how to use matrix to prove this identity?

answered Mar 24, 2020 at 5:54

10 votes

Accepted

Olympiad Chessboard Problem

answered Aug 28, 2021 at 10:00

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

8 votes

Combinatorics counting problem

answered Mar 28, 2020 at 13:05

8 votes

Accepted

Geometry problem without trigonometry

answered Aug 4, 2021 at 1:50

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

How do I prove this identity for complex numbers?

answered Mar 22, 2020 at 13:52

7 votes

Accepted

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

answered Nov 30, 2023 at 7:24

7 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

Proof of the Laplace Expansion?

answered Aug 16, 2021 at 8:19

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

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

answered Nov 28, 2023 at 15:36

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

5 votes

Help with solving for $a$ in $\ln(x-a) = \frac12 \ln(x-b) + \frac12 \ln(x+b)$

answered Aug 1, 2020 at 5:49

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

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

Accepted

Approximation of $\arcsin(\sqrt{1-x^2})$ by $2\arcsin(\sqrt{\frac{1-x}2})$

answered Dec 12, 2020 at 1:02

5 votes

Accepted

How to derive simple formula for $\frac{1^3}{1^4 + 4} - \frac{3^3}{3^4 + 4} + \frac{5^3}{5^4 + 4} - ··· +\frac{(-1)^n(2n+1)^3}{(2n+1)^4 + 4}$?

answered Aug 3, 2021 at 3:37

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

Binomial Identity and Counting

answered Nov 14, 2020 at 13:14

5 votes

Accepted

Find the minimum value of $x^2+y^2$, where $x,y$ are nonnegative integers and $x+y=k$.

answered Mar 9, 2020 at 10:57

5 votes

Accepted

Is it true that $\int_a^b \min\{f,g\}(x)\,dx \leq \min\left\{\int_a^b f(x)\,dx, \int_a^b g(x)\,dx \right\}$ for continuous $f,g$ on $[a,b]?$

answered Feb 18, 2020 at 11:04

5 votes

$|x+y|^p \leq |x|^p + |y|^p$ for $0 < p < 1$.

answered Feb 22, 2020 at 10:27

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526 Answers

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

Numbers from 1 to 10 put in a circle

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

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

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

how to use matrix to prove this identity?

Olympiad Chessboard 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??

Combinatorics counting problem

Geometry problem without trigonometry

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

How do I prove this identity for complex numbers?

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

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

Proof of the Laplace Expansion?

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

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

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

Help with solving for $a$ in $\ln(x-a) = \frac12 \ln(x-b) + \frac12 \ln(x+b)$

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

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

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

Approximation of $\arcsin(\sqrt{1-x^2})$ by $2\arcsin(\sqrt{\frac{1-x}2})$

How to derive simple formula for $\frac{1^3}{1^4 + 4} - \frac{3^3}{3^4 + 4} + \frac{5^3}{5^4 + 4} - ··· +\frac{(-1)^n(2n+1)^3}{(2n+1)^4 + 4}$?

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 $a + b + c = 0$

Binomial Identity and Counting

Find the minimum value of $x^2+y^2$, where $x,y$ are nonnegative integers and $x+y=k$.

Is it true that $\int_a^b \min\{f,g\}(x)\,dx \leq \min\left\{\int_a^b f(x)\,dx, \int_a^b g(x)\,dx \right\}$ for continuous $f,g$ on $[a,b]?$

$|x+y|^p \leq |x|^p + |y|^p$ for $0 < p < 1$.