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Interesting inequalities
sqing   1
N 17 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , a(b+c)=k.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{4\sqrt{k}-6}{ k-2}$$Where $5\leq k\in N^+.$
Let $ a,b,c\geq 0 , a(b+c)=9.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{6}{7}$$
1 reply
+1 w
sqing
an hour ago
sqing
17 minutes ago
J
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Inequality Involving Complex Numbers with Modulus Less Than 1
tom-nowy   0
24 minutes ago
Let $x,y,z$ be complex numbers such that $|x|<1, |y|<1,$ and $|z|<1$.
Prove that $$ |x+y+z|^2 +3>|xy+yz+zx|^2+3|xyz|^2 .$$
0 replies
tom-nowy
24 minutes ago
0 replies
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Inequality
nguyentlauv   2
N 24 minutes ago by nguyentlauv
Source: Own
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=3$ and $k\ge 0$, prove that
$$\frac{\sqrt{a+1}}{b+c+k}+\frac{\sqrt{b+1}}{c+a+k}+\frac{\sqrt{c+1}}{a+b+k} \geq \frac{3\sqrt{2}}{k+2}.$$
2 replies
nguyentlauv
May 6, 2025
nguyentlauv
24 minutes ago
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japan 2021 mo
parkjungmin   0
25 minutes ago

The square box question

Is there anyone who can release it
0 replies
parkjungmin
25 minutes ago
0 replies
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easy sequence
Seungjun_Lee   17
N 30 minutes ago by GreekIdiot
Source: KMO 2023 P1
A sequence of positive reals $\{ a_n \}$ is defined below. $$a_0 = 1, a_1 = 3, a_{n+2} = \frac{a_{n+1}^2+2}{a_n}$$Show that for all nonnegative integer $n$, $a_n$ is a positive integer.
17 replies
Seungjun_Lee
Nov 4, 2023
GreekIdiot
30 minutes ago
J
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Japan MO Finals 2023
parkjungmin   0
37 minutes ago
It's hard. Help me
0 replies
parkjungmin
37 minutes ago
0 replies
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I Brazilian TST 2007 - Problem 4
e.lopes   77
N an hour ago by alexanderhamilton124
Source: 2007 Brazil TST, Russia TST, and AIMO; also SL 2006 N5
Find all integer solutions of the equation \[\frac {x^{7} - 1}{x - 1} = y^{5} - 1.\]
77 replies
e.lopes
Mar 11, 2007
alexanderhamilton124
an hour ago
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Japan MO Finals 2024
parkjungmin   0
an hour ago
Source: Please tell me the question
Please tell me the question
0 replies
parkjungmin
an hour ago
0 replies
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k^2/p for k =1 to (p-1)/2
truongphatt2668   0
an hour ago
Let $p$ be a prime such that: $p = 4k+1$. Simplify:
$$\sum_{k=1}^{\frac{p-1}{2}}\begin{Bmatrix}\dfrac{k^2}{p}\end{Bmatrix}$$
0 replies
truongphatt2668
an hour ago
0 replies
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Interesting inequalities
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 0 , ab+bc+kca =k+2.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{k^2+k+2+2k\sqrt{k+2}}{(k+1)^2}$$Where $ k\in N^+.$
Let $ a,b,c\geq 0 , ab+bc+ca =3.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq 1+\frac{\sqrt 3}{2}$$Let $ a,b,c\geq 0 , ab+bc+2ca =4.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{16}{9}$$
1 reply
sqing
an hour ago
sqing
an hour ago
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