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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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0 replies
jlacosta
Apr 2, 2025
0 replies
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A property of divisors
rightways   11
N 7 minutes ago by clarkculus
Source: Kazakhstan NMO 2016, P1
Prove that one can arrange all positive divisors of any given positive integer around a circle so that for any two neighboring numbers one is divisible by another.
11 replies
1 viewing
rightways
Mar 17, 2016
clarkculus
7 minutes ago
J
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Inspired by giangtruong13
sqing   0
26 minutes ago
Source: Own
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ 61\leq a+b+2c+d\leq \frac{265}{3}$$$$- \frac{2121}{2}\leq ab+bc-2cd+da\leq \frac{14045}{12}$$$$\frac{519506-7471\sqrt{7471}}{27}\leq ab+bc-2cd+3da\leq 33620$$
0 replies
sqing
26 minutes ago
0 replies
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ineq.trig.
wer   14
N 33 minutes ago by mpcnotnpc
If a, b, c are the sides of a triangle, show that: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{r}{R}\le2$
14 replies
wer
Jul 5, 2014
mpcnotnpc
33 minutes ago
J
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Synthetic Geometry Olympiad
kooooo   1
N 35 minutes ago by kaede_Arcadia
Source: yyaa(me) and kaede_Arcadia
We are posting the problems of the Synthetic Geometry Olympiad, which was recently concluded and hosted by kaede_Arcadia and myself.

Problem 1
Let \( \triangle ABC \) be a triangle with its 9-point center \( N \) and excentral triangle \( \triangle I_A I_B I_C \). Denote the tangency points of the \( A \)-excircle with sides \( BC \), \( CA \), and \( AB \) as \( D_A, D_B, D_C \), respectively. Similarly, define \( E_A, E_B, E_C \) and \( F_A, F_B, F_C \) for the \( B \)- and \( C \)-excircles.
Let \( E_CE_A \cap F_AF_B = X \), \( F_AF_B \cap D_BD_C = Y \), and \( D_BD_C \cap E_CE_A = Z \). Let \( T \) be the radical center of the circles \( \odot(D_AYZ) \), \( \odot(E_BZX) \), and \( \odot(F_CXY) \).
Prove that the lines \( I_AX \), \( I_BY \), \( I_CZ, NT \) are concurrent.

Problem 2
Let \( \triangle ABC \) be a triangle with circumcenter \( O \), incenter \( I \) and incentral triangle \( \triangle DEF \). Let the line \( AI \) intersect \( \odot(AEF) \) again at \( X \). Similarly, define \( Y \) and \( Z \).
Let \( N_1 \) and \( N_2 \) be the 9-point centers of \( \triangle DEF \) and \( \triangle XYZ \), respectively.
Prove that the points \( O, I \), \( N_1, N_2 \) are collinear.

Problem 3
Let \( \triangle ABC \) be a triangle, and let \( (P, Q) \) be an isogonal conjugate pair. Suppose the line through \( P \) and perpendicular to \( AP \) intersects \( \odot(PBC) \) again at \( P_A \). Similarly, define \( P_B, P_C \). Suppose the line through \( Q \) and perpendicular to \( AQ \) intersects \( \odot(QBC) \) again at \( Q_A \). Similarly, define \( Q_B, Q_C \).
Let \( H_P \) and \( H_Q \) be the orthocenters of \( \triangle P_AP_BP_C \) and \( \triangle Q_AQ_BQ_C \), respectively. Define \( T = BP_B \cap CP_C \) and \( U = BQ_B \cap CQ_C \). Let \( T' \) and \( U' \) be the isogonal conjugates of \( T \) and \( U \) with respect to \( \triangle P_AP_BP_C \) and \( \triangle Q_AQ_BQ_C \), respectively.
Prove that the lines \( P_AQ_A, P_BQ_B, P_CQ_C, H_PH_Q, TU, T'U' \) are concurrent.
1 reply
kooooo
Feb 11, 2025
kaede_Arcadia
35 minutes ago
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No more topics!
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inequalities hard
Cobedangiu   5
N Apr 2, 2025 by Primeniyazidayi
problem
5 replies
Cobedangiu
Mar 31, 2025
Primeniyazidayi
Apr 2, 2025
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inequalities hard
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Reveal topic tags
inequalities
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Cobedangiu
42 posts
Cobedangiu
#1 h  Mar 31, 2025, 11:45 AM
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problem
Attachments:
This post has been edited 1 time. Last edited by Cobedangiu, Mar 31, 2025, 2:50 PM
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Cobedangiu
42 posts
Cobedangiu
#4 h  Apr 2, 2025, 8:44 AM
Y by
....................
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IceyCold
196 posts
IceyCold
#5 h  Apr 2, 2025, 2:29 PM
Y by
Cobedangiu wrote:
problem

...

Equality occurs at $z=2x=2y=2.$Cauchy your way there.
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sqing
41522 posts
sqing
#6 h  Apr 2, 2025, 2:55 PM
Y by
Let $ a,b,c>0 $ and $ abc=2. $ Prove that $$ \dfrac{a}{2a^2+ b^2+5} + \dfrac{2b}{6b^2+ c^2+6} + \dfrac{4c}{3c^2+ 4a^2+16} \leq \dfrac{1}{2} $$Let $ a,b,c>0 $ and $ a+b+c=4. $ Prove that $$ \dfrac{a}{2a^2+ b^2+5} + \dfrac{2b}{6b^2+ c^2+6} + \dfrac{4c}{3c^2+ 4a^2+16} \leq \dfrac{1}{2} $$
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Cobedangiu
42 posts
Cobedangiu
#7 h  Apr 2, 2025, 4:47 PM
Y by
sqing wrote:
Let $ a,b,c>0 $ and $ abc=2. $ Prove that $$ \dfrac{a}{2a^2+ b^2+5} + \dfrac{2b}{6b^2+ c^2+6} + \dfrac{4c}{3c^2+ 4a^2+16} \leq \dfrac{1}{2} $$Let $ a,b,c>0 $ and $ a+b+c=4. $ Prove that $$ \dfrac{a}{2a^2+ b^2+5} + \dfrac{2b}{6b^2+ c^2+6} + \dfrac{4c}{3c^2+ 4a^2+16} \leq \dfrac{1}{2} $$

I need solution....
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Primeniyazidayi
50 posts
Primeniyazidayi
#8 h  Apr 2, 2025, 8:31 PM
Y by
I think setting x=a,y=2b and z=c might work
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