User River Li - Mathematics Stack Exchange

42 votes

6 answers

2k views

About the inequality $x^{x^{x^{x^{x^x}}}} \ge \frac12 x^2 + \frac12$

asked Jun 7, 2021 at 5:56

19 votes

1 answer

682 views

The limit and asymptotic analysis of $a_n^2 - n$ from $a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n}$

asked Aug 24, 2020 at 6:21

18 votes

5 answers

641 views

Prove that $a^{4/a} + b^{4/b} + c^{4/c} \ge 3$

asked Jul 28, 2022 at 23:46

15 votes

1 answer

256 views

Prove/Disprove $|1 + z_1| + |1 + z_2| + |1 + r z_1 z_2| \ge 1 + \min(r, 1/r)$

asked Jul 21, 2022 at 1:38

14 votes

1 answer

262 views

Prove that $16(a\sin a + \cos a - 1)^2 \le 2a^4 + a^3 \sin 2a, \ \forall a\ge 0$

asked Jan 29, 2021 at 6:31

12 votes

4 answers

562 views

Upper bounds for $\frac{x_1}{1+x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + x_2^2 + \cdots + x_n^2}$

asked Nov 26, 2020 at 3:14

9 votes

2 answers

203 views

Prove $\tan x + \tan y + \tan z \le \frac{17}{6}$ for reals $\sum\limits_{\mathrm{cyc}} \sin x = 2, \sum\limits_{\mathrm{cyc}} \cos x = \frac{11}{5}$

asked Jun 18, 2023 at 15:30

9 votes

0 answers

228 views

If continuity condition is necessary for Miklós Schweitzer 2015 Problem 8

asked May 4, 2020 at 10:58

8 votes

3 answers

284 views

Prove or disprove $\sum\limits_{1\le i < j \le n} \frac{x_ix_j}{1-x_i-x_j} \le \frac18$ for $\sum\limits_{i=1}^n x_i = \frac12$($x_i\ge 0, \forall i$)

asked Jun 2, 2022 at 14:09

7 votes

3 answers

361 views

Prove $f(-\frac12) \le \frac{3}{16}$ if all roots of $f(x) = x^4 - x^3 + a x + b$ are real

asked Aug 3, 2022 at 15:29

7 votes

5 answers

802 views

Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$

asked Jun 7, 2020 at 14:17

7 votes

1 answer

193 views

Prove that $a^{\lambda b} + b^{\lambda a} + a^{\lambda b^2} + b^{\lambda a^2} \le 2$ for positive reals $a+b=1$

asked Oct 1, 2020 at 3:45

6 votes

3 answers

399 views

Prove: $(\forall m, n\in\Bbb N_{>0})(\exists x\in\Bbb R)$ s. t. $2\sin n x \cos m x \ge 1$

asked Jun 11, 2020 at 13:18

5 votes

2 answers

346 views

Prove $\sqrt{a + ab} + \sqrt{b} + \sqrt{c} \ge 3$ for $c = \min(a, b, c)$ and $ab + bc + ca = 2$

asked Nov 8, 2020 at 2:35

5 votes

1 answer

139 views

Best $C$ such that $\sum\limits_{1\leq i<j\leq n}|z_i-z_j|\leq C\sum\limits^n_{i=1}|z_i|$ subject to $\sum\limits_{i=1}^n z_i = 0$

asked Aug 20, 2022 at 14:24

4 votes

1 answer

113 views

Whether $\lim_{n\to \infty} \frac{2}{\mathsf{e}}(\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{k}(1-\frac{2k}{n})^{n-1})^{-1/n}$ exists

asked Jun 25, 2020 at 15:52

3 votes

1 answer

140 views

Prove or disprove $\det(6(A^3 + B^3 + C^3)+(A+B+C)^3) \ge 5^2\det(A^2 + B^2 + C^2)\cdot\det(A + B+ C)$

asked Jun 27, 2023 at 2:41

3 votes

0 answers

96 views

Prove that $\sin(x + y + z) \ge \frac45$ for reals with $\sum\limits_{\mathrm{cyc}} \sin x = 2$ and $\sum\limits_{\mathrm{cyc}} \cos x = \frac{11}{5}$

asked Aug 25, 2023 at 1:24

3 votes

3 answers

299 views

Prove $2 < 1/a + 1/b < \mathrm{e}$ provided $b\ln a - a\ln b = a - b$

asked Jun 10, 2021 at 1:28

2 votes

1 answer

243 views

Rational rank one decomposition of symmetric positive semidefinite integer matrices

asked Feb 23, 2021 at 17:03

2 votes

1 answer

64 views

Prove or disprove that $(1+\mathrm{e}s)\int_0^\infty \frac{\mathrm{e}^{-sx}}{2x\sqrt{2\pi}} \mathrm{e}^{-\frac{(\ln x + 1)^2}{8}}\mathrm{d} x \ge 1$

asked Jul 8, 2022 at 15:11

2 votes

0 answers

65 views

A cyclic homogeneous inequality in three variables involving 4th root radicals

asked Jun 3, 2020 at 6:16

2 votes

0 answers

111 views

Prove $\frac{4}{a^2 + 2b^2 + 3c^2 + 10} \le \frac{5a + 3b + c + 7d}{16(a+b+c+d)}$ for positive reals $abcd=1$

asked Oct 9, 2020 at 3:31

2 votes

1 answer

81 views

More terms in asymptotic expansion of $\sum_{n=-\infty}^\infty \arctan \frac{D}{2n+1} \log\left(\frac{D}{|2n+1|}\right) \frac{1}{n+3/4}$

asked Sep 24, 2020 at 4:21

1 vote

2 answers

2k views

KKT conditions strict inequality constraints

asked Aug 15, 2018 at 13:38

1 vote

1 answer

70 views

Prove the identity $\sum_{i=0}^{k-1} p^{i-1}(1-p)^{n-1-i}\binom{n}{i} [p(n-i) + (k-i)(i-np)] = 0$

asked Sep 9, 2020 at 2:11

1 vote

0 answers

129 views

Lower bounds for $|\sin 1| + |\sin 2| + \cdots + |\sin (3n)|$

asked Nov 24, 2020 at 16:43

1 vote

2 answers

214 views

Sum of Squares for $a^2+b^2+c^2+d^2+abcd+1\ge ab+bc+cd+da + ac+bd$

asked Jun 9, 2020 at 3:17

1 vote

4 answers

184 views

Can any cyclic polynomial in $a, b, c$ be expressed in terms of $a^2b+b^2c+c^2a$, $a+b+c$, $ab+bc+ca$ and $abc$?

asked Jun 10, 2020 at 4:03

1 vote

0 answers

134 views

About $\sum_{m=1}^k m |\sin m| = \int_0^k x |\sin x| \mathrm{d} x + O(k)$

asked Jul 2, 2020 at 16:20

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36 Questions

About the inequality $x^{x^{x^{x^{x^x}}}} \ge \frac12 x^2 + \frac12$

The limit and asymptotic analysis of $a_n^2 - n$ from $a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n}$

Prove that $a^{4/a} + b^{4/b} + c^{4/c} \ge 3$

Prove/Disprove $|1 + z_1| + |1 + z_2| + |1 + r z_1 z_2| \ge 1 + \min(r, 1/r)$

Prove that $16(a\sin a + \cos a - 1)^2 \le 2a^4 + a^3 \sin 2a, \ \forall a\ge 0$

Upper bounds for $\frac{x_1}{1+x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + x_2^2 + \cdots + x_n^2}$

Prove $\tan x + \tan y + \tan z \le \frac{17}{6}$ for reals $\sum\limits_{\mathrm{cyc}} \sin x = 2, \sum\limits_{\mathrm{cyc}} \cos x = \frac{11}{5}$

If continuity condition is necessary for Miklós Schweitzer 2015 Problem 8

Prove or disprove $\sum\limits_{1\le i < j \le n} \frac{x_ix_j}{1-x_i-x_j} \le \frac18$ for $\sum\limits_{i=1}^n x_i = \frac12$($x_i\ge 0, \forall i$)

Prove $f(-\frac12) \le \frac{3}{16}$ if all roots of $f(x) = x^4 - x^3 + a x + b$ are real

Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$

Prove that $a^{\lambda b} + b^{\lambda a} + a^{\lambda b^2} + b^{\lambda a^2} \le 2$ for positive reals $a+b=1$

Prove: $(\forall m, n\in\Bbb N_{>0})(\exists x\in\Bbb R)$ s. t. $2\sin n x \cos m x \ge 1$

Prove $\sqrt{a + ab} + \sqrt{b} + \sqrt{c} \ge 3$ for $c = \min(a, b, c)$ and $ab + bc + ca = 2$

Best $C$ such that $\sum\limits_{1\leq i<j\leq n}|z_i-z_j|\leq C\sum\limits^n_{i=1}|z_i|$ subject to $\sum\limits_{i=1}^n z_i = 0$

Whether $\lim_{n\to \infty} \frac{2}{\mathsf{e}}(\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{k}(1-\frac{2k}{n})^{n-1})^{-1/n}$ exists

Prove or disprove $\det(6(A^3 + B^3 + C^3)+(A+B+C)^3) \ge 5^2\det(A^2 + B^2 + C^2)\cdot\det(A + B+ C)$

Prove that $\sin(x + y + z) \ge \frac45$ for reals with $\sum\limits_{\mathrm{cyc}} \sin x = 2$ and $\sum\limits_{\mathrm{cyc}} \cos x = \frac{11}{5}$

Prove $2 < 1/a + 1/b < \mathrm{e}$ provided $b\ln a - a\ln b = a - b$

Rational rank one decomposition of symmetric positive semidefinite integer matrices

Prove or disprove that $(1+\mathrm{e}s)\int_0^\infty \frac{\mathrm{e}^{-sx}}{2x\sqrt{2\pi}} \mathrm{e}^{-\frac{(\ln x + 1)^2}{8}}\mathrm{d} x \ge 1$

A cyclic homogeneous inequality in three variables involving 4th root radicals

Prove $\frac{4}{a^2 + 2b^2 + 3c^2 + 10} \le \frac{5a + 3b + c + 7d}{16(a+b+c+d)}$ for positive reals $abcd=1$

More terms in asymptotic expansion of $\sum_{n=-\infty}^\infty \arctan \frac{D}{2n+1} \log\left(\frac{D}{|2n+1|}\right) \frac{1}{n+3/4}$

KKT conditions strict inequality constraints

Prove the identity $\sum_{i=0}^{k-1} p^{i-1}(1-p)^{n-1-i}\binom{n}{i} [p(n-i) + (k-i)(i-np)] = 0$

Lower bounds for $|\sin 1| + |\sin 2| + \cdots + |\sin (3n)|$

Sum of Squares for $a^2+b^2+c^2+d^2+abcd+1\ge ab+bc+cd+da + ac+bd$

Can any cyclic polynomial in $a, b, c$ be expressed in terms of $a^2b+b^2c+c^2a$, $a+b+c$, $ab+bc+ca$ and $abc$?

About $\sum_{m=1}^k m |\sin m| = \int_0^k x |\sin x| \mathrm{d} x + O(k)$