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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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Leibnitz theorem?
soryn   3
N an hour ago by soryn
If M îs a interior point of the triangle ABC, and Ga,GB,GC are the centoids of triangles MBC, MAC and MAB, respectively, G0 is the centroid of triangle GaGbGc, show that the line MG0 passes through a fixed point.
3 replies
soryn
3 hours ago
soryn
an hour ago
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JEE Related ig?
mikkymini2   14
N 2 hours ago by mikkymini2
Hey everyone,

Just wanted to see if there are any other JEE aspirants on this forum currently prepping for it[mention year if you can]

I am actually entering 10th this year and have decided to try for it...So this year is just going to go in me strengthening my math (IOQM level (heard its enough till Mains part, so will start from there) for the problem solving part, and learn some topics from 11th and 12th as well)

It would be great to connect with others who are going through the same thing - share study strategies, tips, resources, discuss, and maybe even form study groups(not sure how to tho :maybe: ) and motivate each other ig?. :D
So yea, cya later
14 replies
mikkymini2
Apr 10, 2025
mikkymini2
2 hours ago
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Inequalities
sqing   6
N 4 hours ago by sqing
Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$$$$ \frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$$$$ \frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$$Let $ a,b $ be reals such that $ a^2-ab+b^2 =1 $ . Prove that
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
6 replies
sqing
Apr 16, 2025
sqing
4 hours ago
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simplfy this
Miranda2829   6
N 6 hours ago by Miranda2829
4b+13/ -4b+15 = 1/6

anyone can help on the steps to do this ? thank u
6 replies
Miranda2829
Apr 16, 2025
Miranda2829
6 hours ago
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Geometry
AlexCenteno2007   3
N Yesterday at 8:00 PM by vanstraelen
Let ABC be an isosceles triangle with AB = AC and M the midpoint of BC. Consider a point E outside the triangle such that BE = BM and CE perpendicular to AB. The point of intersection of the perpendicular bisector of segment EB with the circumcircle of triangle AMB, which is on the same side as A with respect to BE, is point F. Show that angle FME = 90°
3 replies
AlexCenteno2007
Yesterday at 3:49 AM
vanstraelen
Yesterday at 8:00 PM
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Geometry
AlexCenteno2007   0
Yesterday at 6:28 PM
Let A, B, C, and D be four distinct points on a straight line, in that order. The circles with diameters AC and BD intersect at X and Y. The straight line XY intersects BC at Z. Let P be a point on XY distinct from Z. The straight line CP intersects the circle with diameter AC at C and M, and the straight line BP intersects the circle with diameter BD at B and N. Show that AM, DN, and XY are aligned.
0 replies
AlexCenteno2007
Yesterday at 6:28 PM
0 replies
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Geometry
AlexCenteno2007   2
N Yesterday at 5:47 PM by AlexCenteno2007
Let C be a point on a semicircle of diameter AB and let D be the mid-length of arc AC. Let E be the projection of point D on BC and F the intersection of AE with the semicircle. Prove that BF bisects segment DE.
2 replies
AlexCenteno2007
Yesterday at 3:54 AM
AlexCenteno2007
Yesterday at 5:47 PM
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(IDP),(O),(QEF) have a common point
kyotaro   0
Yesterday at 4:45 PM
Given a non-isosceles triangle ABC inscribed in circle $(O)$. $X, Y, Z$ are the midpoints of $BC, CA, AB$ respectively. $P$ is the midpoint of arc $BAC$, $Q$ is symmetric to $P$ through $O$. Let $I, D, E, F$ be the incenter and tangent of angles $X, Y, Z$ of triangle $XYZ$ respectively. Prove that $(IDP),(O),(QEF)$ have a common point
0 replies
kyotaro
Yesterday at 4:45 PM
0 replies
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Inequalities
sqing   8
N Yesterday at 4:26 PM by sqing
Let $ a,b,c> 0 $ and $ \frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$. Prove that
$$ a+b+c\leq 3 $$
8 replies
sqing
Apr 4, 2025
sqing
Yesterday at 4:26 PM
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Any tips for IOQM(indian olympiad qualifier for math)
Indianmathenthusiast1234   4
N Yesterday at 4:08 PM by Indianmathenthusiast1234
Guys I'm preparing for the IOQM, and i feel like the linear congruences, modular arithmetic and number bases are a bit hard to grasp. Any tips?
4 replies
Indianmathenthusiast1234
Yesterday at 3:49 PM
Indianmathenthusiast1234
Yesterday at 4:08 PM
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idk12345678 Math Contest
idk12345678   22
N Yesterday at 2:21 PM by idk12345678
Welcome to the 1st idk12345678 Math Contest.
You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*


The contest is attached to the post

Clarifications

*I mightve done them wrong feel free to ask about an answer
22 replies
idk12345678
Apr 10, 2025
idk12345678
Yesterday at 2:21 PM
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Inequalities
sqing   13
N Yesterday at 2:20 PM by sqing
Let $ a,b,c $ be real numbers so that $ a+2b+3c=2 $ and $ 2ab+6bc+3ca =1. $ Show that
$$-\frac{1}{6} \leq ab-bc+ ca\leq \frac{1}{2}$$$$\frac{5-\sqrt{61}}{9} \leq a-b+c\leq \frac{5+\sqrt{61}}{9} $$
13 replies
sqing
Apr 9, 2025
sqing
Yesterday at 2:20 PM
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Solve an equation
lgx57   4
N Yesterday at 1:32 PM by lgx57
Find all positive integers $n$ and $x$ such that:
$$2^{2n+1}-7=x^2$$
4 replies
lgx57
Mar 12, 2025
lgx57
Yesterday at 1:32 PM
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Identity Proof
jjsunpu   3
N Yesterday at 11:33 AM by jjsunpu
Hi this is my identity I name it Excalibur

for example

for k^2

it's 2n - 1

for k^3

it's 3n^2 -3n + 1
3 replies
jjsunpu
Wednesday at 10:35 AM
jjsunpu
Yesterday at 11:33 AM
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min sum AP+PM in a square ABCD (2001 Denmark MO - Mohr Contest p3)
parmenides51   2
N Aug 17, 2020 by Destiny-wolf
In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$.
IMAGE
2 replies
parmenides51
Aug 16, 2020
Destiny-wolf
Aug 17, 2020
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min sum AP+PM in a square ABCD (2001 Denmark MO - Mohr Contest p3)
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parmenides51
30629 posts
parmenides51
#1 h  Aug 16, 2020, 9:59 AM
Y by
In the square $ABCD$ of side length $2$ the point $M$ is the midpoint of $BC$ and $P$ a point on $DC$. Determine the smallest value of $AP+PM$.
https://1.bp.blogspot.com/-WD8WXIE6DK4/XzcC9GYsa6I/AAAAAAAAMXg/vl2OrbAdChEYrRpemYmj6DiOrdOSqj_IgCLcBGAsYHQ/s178/2001%2BMohr%2Bp3.png
This post has been edited 1 time. Last edited by parmenides51, Aug 16, 2020, 10:01 AM
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sunken rock
4383 posts
sunken rock
#2 h  Aug 16, 2020, 10:16 AM
Y by
Let $A'$ reflection of $A$ in $CD$; the sum is min. when $M-P-A'$ are collinear!

Best regards,
sunken rock
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Destiny-wolf
5 posts
Destiny-wolf
#3 h  Aug 17, 2020, 4:08 PM
Y by
The answer sunken rock gave is correct. You could even solve this using Fermat's Principle of Least Time.
Wikipedia Page Link: https://en.wikipedia.org/wiki/Fermat%27s_principle
Light always take the shortest path in a single medium so if you imagine the square ABCD being a single medium and the side DC as a plane mirror then to reach Point M from Point A light would take the shortest distance.
So if you shoot light from A and you are sure that it hits point M then light follows this principle of reflection that the angle of reflection is equal to angle of reflection so angle DPA is equal to angle CPM. Then AP+PM would be the shortest.

Regards,
DESTINY-WOLF
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