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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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[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
J
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A point on the midline of BC.
EmersonSoriano   2
N a minute ago by hukilau17
Source: 2017 Peru Southern Cone TST P5
Let $ABC$ be an acute triangle with circumcenter $O$. Draw altitude $BQ$, with $Q$ on side $AC$. The parallel line to $OC$ passing through $Q$ intersects line $BO$ at point $X$. Prove that point $X$ and the midpoints of sides $AB$ and $AC$ are collinear.
2 replies
EmersonSoriano
3 hours ago
hukilau17
a minute ago
J
H
Point that is orthocenter, incenter, and centroid
EmersonSoriano   1
N 8 minutes ago by hukilau17
Source: 2017 Peru Southern Cone TST P9
Let $BXC$ be a triangle and $A_1, A_2, A_3$ points in the same plane such that $X$ is the orthocenter of triangle $A_1BC$, $X$ is the incenter of triangle $A_2BC$, and $X$ is the centroid of triangle $A_3BC$. If line $A_1A_3$ is parallel to $BC$, prove that $A_2$ is the midpoint of segment $A_1A_3$.
1 reply
EmersonSoriano
2 hours ago
hukilau17
8 minutes ago
J
H
Least common multiple from $n+1$ to $n+2016$.
EmersonSoriano   1
N 10 minutes ago by alexheinis
Source: 2017 Peru Southern Cone TST P7
For each positive integer $n$, define
$$P_n = (n+1)(n+2)(n+3)\dots(n+2016)$$and
$$Q_n = \text{lcm}(n+1, n+2, n+3, \dots, n+2016),$$meaning $Q_n$ is the least common multiple of the numbers $n+1, n+2, n+3, \dots, n+2016$. Determine whether or not there exists a constant $C$ such that
$$ \frac{P_n}{Q_n} < C, $$for every positive integer $n$.
1 reply
EmersonSoriano
2 hours ago
alexheinis
10 minutes ago
J
H
Cute NT with bounding
oVlad   2
N 23 minutes ago by Parivash
Source: Russian TST 2015, Day 10 P1 (Groups A & B)
Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.
2 replies
oVlad
Apr 21, 2023
Parivash
23 minutes ago
J
No more topics!
H
J
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concyclic, starting with an inscribed pentagon
parmenides51   2
N May 3, 2019 by AlastorMoody
Source: KJMO 2012 p2
A pentagon $ABCDE$ is inscribed in a circle $O$, and satisfies $\angle A = 90^o, AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ hit $O$ again at $J(\ne B)$, $CE \cap DJ = K$, $BD\cap FK = L$. Prove that $B,L,E,F$ are cyclic.
2 replies
parmenides51
May 3, 2019
AlastorMoody
May 3, 2019
J
concyclic, starting with an inscribed pentagon
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Reveal topic tags
geometry
pentagon
Concyclic
L
G H BBookmark kLocked kLocked NReply
Source: KJMO 2012 p2
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parmenides51
30629 posts
parmenides51
#1 h  May 3, 2019, 12:40 AM • 2 Y
Y by Adventure10, Mango247
A pentagon $ABCDE$ is inscribed in a circle $O$, and satisfies $\angle A = 90^o, AB = CD$. Let $F$ be a point on segment $AE$. Let $BF$ hit $O$ again at $J(\ne B)$, $CE \cap DJ = K$, $BD\cap FK = L$. Prove that $B,L,E,F$ are cyclic.
This post has been edited 1 time. Last edited by parmenides51, Nov 24, 2023, 7:22 PM
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XDDDD_2_the_max
59 posts
XDDDD_2_the_max
#2 h  May 3, 2019, 1:03 AM • 2 Y
Y by Adventure10, Mango247
A complex bash with the real axis perpendicular to $AD$ and $BC$ shows that $KF$ is parallel to $BC$. Angle chase to show that $\angle BLF = \angle BEA$.
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AlastorMoody
2125 posts
AlastorMoody
#3 h  May 3, 2019, 1:45 AM • 2 Y
Y by Adventure10, Mango247
A better way to show $FK||BC$ is:
$$\angle ADC=\angle AEC=\angle BAD=\angle FJK$$and then reim's theorem and apply miquel's theroem to finish off the problem
This post has been edited 1 time. Last edited by AlastorMoody, May 3, 2019, 1:50 AM
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