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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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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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each triangle P_iP_jP_k has area <= 1, triangle exist with area <=4, interior
parmenides51   1
N Jul 17, 2019 by Pluto1708
Source: Australian 1991
Let $P_1, P_2, ..., P_n$ be $n$ different points in a given plane such that each triangle $P_iP_jP_k$ ($i \ne j \ne k \ne i$) has an area not greater than $1$. Prove that there exists a triangle $\Delta $ in this plane such that
(a) $\Delta $ has an area not greater than $4$; and
(b) each of the points $P_1, P_2, ..., P_n$ lies in the interior or on the boundary of $\Delta $.
1 reply
parmenides51
Jul 17, 2019
Pluto1708
Jul 17, 2019
J
each triangle P_iP_jP_k has area <= 1, triangle exist with area <=4, interior
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Reveal topic tags
geometry
combinatorial geometry
area of a triangle
areas
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Source: Australian 1991
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parmenides51
30630 posts
parmenides51
#1 h  Jul 17, 2019, 11:41 AM • 1 Y
Y by Adventure10
Let $P_1, P_2, ..., P_n$ be $n$ different points in a given plane such that each triangle $P_iP_jP_k$ ($i \ne j \ne k \ne i$) has an area not greater than $1$. Prove that there exists a triangle $\Delta $ in this plane such that
(a) $\Delta $ has an area not greater than $4$; and
(b) each of the points $P_1, P_2, ..., P_n$ lies in the interior or on the boundary of $\Delta $.
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Pluto1708
1107 posts
Pluto1708
#2 h  Jul 17, 2019, 11:51 AM • 1 Y
Y by Adventure10
Isn't this a classic extremely combinatorics problem?
Anyways for a solution take three poi ta A, B, c such that ABC has maximum area. Consider the Antimedial triangle of ABC ,XYZ then XYZ area is less than equal to 4 and an easy argument oroves it validity
This post has been edited 1 time. Last edited by Pluto1708, Jul 17, 2019, 11:56 AM
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