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Inequality em981
oldbeginner   21
N 26 minutes ago by sqing
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
21 replies
+1 w
oldbeginner
Sep 22, 2016
sqing
26 minutes ago
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Find the minimum
sqing   29
N 28 minutes ago by sqing
Source: Zhangyanzong
Let $a,b$ be positive real numbers such that $a^2b^2+\frac{4a}{a+b}=4.$ Find the minimum value of $a+2b.$
29 replies
1 viewing
sqing
Sep 4, 2018
sqing
28 minutes ago
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Interesting inequality
sqing   3
N 31 minutes ago by sqing
Source: Own
Let $ (a+b)^2+(a-b)^2=1. $ Prove that
$$0\geq (a+b-1)(a-b+1)\geq -\frac{3}{2}-\sqrt 2$$$$ -\frac{9}{2}+2\sqrt 2\geq (a+b-2)(a-b+2)\geq -\frac{9}{2}-2\sqrt 2$$
3 replies
sqing
an hour ago
sqing
31 minutes ago
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Simple triangle geometry [a fixed point]
darij grinberg   50
N 32 minutes ago by ezpotd
Source: German TST 2004, IMO ShortList 2003, geometry problem 2
Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.
50 replies
darij grinberg
May 18, 2004
ezpotd
32 minutes ago
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Inspired by RMO 2006
sqing   6
N 32 minutes ago by sqing
Source: Own
Let $ a,b >0 . $ Prove that
$$ \frac {a^{2}+1}{b+k}+\frac { b^{2}+1}{ka+1}+\frac {2}{a+kb} \geq \frac {6}{k+1} $$Where $k\geq 0.03 $
$$ \frac {a^{2}+1}{b+1}+\frac { b^{2}+1}{a+1}+\frac {2}{a+b} \geq 3 $$
6 replies
sqing
Saturday at 3:24 PM
sqing
32 minutes ago
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IMO 2009, Problem 2
orl   143
N 37 minutes ago by ezpotd
Source: IMO 2009, Problem 2
Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP = OQ.$

Proposed by Sergei Berlov, Russia
143 replies
orl
Jul 15, 2009
ezpotd
37 minutes ago
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Inequalities from SXTX
sqing   24
N an hour ago by sqing
T702. Let $ a,b,c>0 $ and $ a+2b+3c=\sqrt{13}. $ Prove that $$ \sqrt{a^2+1} +2\sqrt{b^2+1} +3\sqrt{c^2+1} \geq 7$$S
T703. Let $ a,b $ be real numbers such that $ a+b\neq 0. $. Find the minimum of $ a^2+b^2+(\frac{1-ab}{a+b} )^2.$
T704. Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that $$ \frac{a^2+7}{(c+a)(a+b)} + \frac{b^2+7}{(a+b)(b+c)} +\frac{c^2+7}{(b+c)(c+a)} \geq 6$$S
24 replies
sqing
Feb 18, 2025
sqing
an hour ago
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A sharp one with 3 var (2)
mihaig   0
an hour ago
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a+b+c+\sqrt{abc}\geq4.$$
0 replies
mihaig
an hour ago
0 replies
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f(1)f(2)...f(n) has at most n prime factors
MarkBcc168   39
N an hour ago by cursed_tangent1434
Source: 2020 Cyberspace Mathematical Competition P2
Let $f(x) = 3x^2 + 1$. Prove that for any given positive integer $n$, the product
$$f(1)\cdot f(2)\cdot\dots\cdot f(n)$$has at most $n$ distinct prime divisors.

Proposed by Géza Kós
39 replies
MarkBcc168
Jul 15, 2020
cursed_tangent1434
an hour ago
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a tst 2013 test
Math2030   9
N 2 hours ago by Shan3t
Given the sequence $(a_n): a_1=1, a_2=11$ and $a_{n+2}=a_{n+1}+5a_{n}, n \geq 1$
. Prove that $a_n $not is a perfect square for all $n > 3$.
9 replies
Math2030
Saturday at 5:26 AM
Shan3t
2 hours ago
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Inspired by 2025 Beijing
sqing   11
N 2 hours ago by sqing
Source: Own
Let $ a,b,c,d >0 $ and $ (a^2+b^2+c^2)(b^2+c^2+d^2)=36. $ Prove that
$$ab^2c^2d \leq 8$$$$a^2bcd^2 \leq 16$$$$ ab^3c^3d \leq \frac{2187}{128}$$$$ a^3bcd^3 \leq \frac{2187}{32}$$
11 replies
sqing
Saturday at 4:56 PM
sqing
2 hours ago
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A functional equation
super1978   1
N 2 hours ago by CheerfulZebra68
Source: Somewhere
Find all functions $f: \mathbb R \to \mathbb R$ such that:$$ f(f(x)(y+f(y)))=xf(y)+f(xy) $$for all $x,y \in \mathbb R$
1 reply
super1978
3 hours ago
CheerfulZebra68
2 hours ago
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Sequence
lgx57   8
N Apr 30, 2025 by Vivaandax
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
8 replies
lgx57
Apr 27, 2025
Vivaandax
Apr 30, 2025
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Sequence
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lgx57
48 posts
lgx57
#1 h  Apr 27, 2025, 12:56 PM
Y by
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.
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lgx57
48 posts
lgx57
#2 h  Apr 27, 2025, 12:57 PM
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I can only find that $a_n \sim \sqrt{2n}$.
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aops-g5-gethsemanea2
3468 posts
aops-g5-gethsemanea2
#3 h  Apr 27, 2025, 2:05 PM
Y by
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?
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lgx57
48 posts
lgx57
#4 h  Apr 27, 2025, 2:19 PM
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aops-g5-gethsemanea2 wrote:
lgx57 wrote:
$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$. Find the general term of $\{a_n\}$.

do you mean closed form or explicit formula of $a_n$?

Just find a function $f$ ,s.t. $a_n=f(n)$
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steve4916
67 posts
steve4916
#5 h  Apr 27, 2025, 2:24 PM
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now prove me if im wrong but there is no simple closed form for this
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lgx57
48 posts
lgx57
#6 h  Apr 27, 2025, 2:48 PM
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steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?
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johnnie.walker
2 posts
johnnie.walker
#7 h  Apr 29, 2025, 1:42 PM
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jasperE3
11384 posts
jasperE3
#8 h  Apr 29, 2025, 3:38 PM
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lgx57 wrote:
steve4916 wrote:
now prove me if im wrong but there is no simple closed form for this

Why?

why would there be a closed form
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Vivaandax
89 posts
Vivaandax
#9 h  Apr 30, 2025, 5:08 AM
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You can bound the value of a_n quite well (consider IMO Shortlist 1975 Problem 14), but there is not an explicit formula to calculate a_n.
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