Question

minimum value of $x^{\frac{x}{x+y+z+t}}\cdot y^{\frac{y}{x+y+z+t}}\cdot z^{\frac{z}{x+y+z+t}}\cdot t^{\frac{t}{x+y+z+t}}$

score 87 · Answer 1 · 2026-04-15 17:41:12

minimum value of $x^{\frac{x}{x+y+z+t}}\cdot y^{\frac{y}{x+y+z+t}}\cdot z^{\frac{z}{x+y+z+t}}\cdot t^{\frac{t}{x+y+z+t}}$

score 132 · Answer 2 · 2026-04-15 13:20:38

132

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If $a,b,c$ are positive reals satisfying $a+b+c \geq abc$, prove that: $a^2+b^2+c^2 \geq \sqrt 3abc$

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score 74 · Answer 3 · 2026-04-17 11:56:13

74

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Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $

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score 550 · Answer 4 · 2026-04-10 21:24:04

550

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score 57 · Answer 5 · 2026-04-16 22:44:22

57

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score 80 · Answer 6 · 2026-04-16 10:48:48

80

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3.9k

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56

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321

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34

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score 41 · Answer 12 · 2026-04-10 18:36:12

41

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score 128 · Answer 13 · 2026-03-25 06:13:06

128

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Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\leq 1$.

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score 62 · Answer 14 · 2026-04-15 08:09:45

62

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score 261 · Answer 15 · 2026-04-15 17:55:56

261

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List Question

minimum value of $x^{\frac{x}{x+y+z+t}}\cdot y^{\frac{y}{x+y+z+t}}\cdot z^{\frac{z}{x+y+z+t}}\cdot t^{\frac{t}{x+y+z+t}}$

If $a,b,c$ are positive reals satisfying $a+b+c \geq abc$, prove that: $a^2+b^2+c^2 \geq \sqrt 3abc$

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $

Difference between the equation inequalities and absolute value inequalities

Prove that $\sqrt{c-d} \ge \sqrt{c} - \sqrt{d}$ for $c \geq d \geq 0$

Prove $ \sum_{cyc}\frac{x}{\sqrt{x^2+8yz}} \ge 1, \forall x,y,z\gt 0 $

Important Olympiad-inequalities

Prove inequality: $|x-y|^r \le 2^r (x^r+y^r)$

Limit inferior of product of two non negative bounded sequences

What am I doing wrong here? involving inequality

What is the geometric interpretation of reverse triangle inequality?

maximal point of a combination of continuous sub-additive monotone function

Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\leq 1$.

I want to prove that two norms are equivalent but I am struggling with an upper bound

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