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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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
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Polynomials
CuriousBabu   6
N 33 minutes ago by Pengu14
\[ \frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 \]
Find the number of real solutions.
6 replies
CuriousBabu
Apr 14, 2025
Pengu14
33 minutes ago
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Inequalities
lgx57   5
N an hour ago by jjmmxx
Let $0 < a,b,c < 1$. Prove that

$$a(1-b)+b(1-c)+c(1-a)<1$$
5 replies
lgx57
Mar 19, 2025
jjmmxx
an hour ago
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ez problem....
Cobedangiu   0
an hour ago
Let $x,y \in Z$ and $xy \cancel \vdots7$
Find $n \in Z^+$.
$x^2+y^2+xy=7^n$
0 replies
Cobedangiu
an hour ago
0 replies
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A Loggy Problem from Pythagoras
Mathzeus1024   0
2 hours ago
Prove or disprove: $\exists x \in \mathbb{R}^{+}$ such that $\ln(x), \ln(2x), \ln(3x)$ are the lengths of a right triangle.
0 replies
Mathzeus1024
2 hours ago
0 replies
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hard number theory
eric201291   2
N 3 hours ago by eric201291
Prove:There are no integers x, y, that y^2+9998587980=x^3.
2 replies
eric201291
Wednesday at 2:17 PM
eric201291
3 hours ago
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Inequalities
sqing   9
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
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
9 replies
sqing
Apr 16, 2025
sqing
4 hours ago
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Geometry
AlexCenteno2007   4
N 4 hours ago by sunken rock
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°
4 replies
AlexCenteno2007
Yesterday at 3:49 AM
sunken rock
4 hours ago
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JEE Related ig?
mikkymini2   16
N 5 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
16 replies
mikkymini2
Apr 10, 2025
mikkymini2
5 hours ago
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Leibnitz theorem?
soryn   2
N Today at 4:53 AM 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.
2 replies
soryn
Today at 3:11 AM
soryn
Today at 4:53 AM
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simplfy this
Miranda2829   6
N Today at 12:26 AM 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
Today at 12:26 AM
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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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line from which 3 points are equidistant (Greece Junior 2000)
parmenides51   1
N Jul 12, 2019 by Bashy99
Given are three non collinear points in plane. Find a line from which all three points are equidistant. How many such lines of the plane exist?
1 reply
parmenides51
Jul 12, 2019
Bashy99
Jul 12, 2019
J
line from which 3 points are equidistant (Greece Junior 2000)
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Reveal topic tags
geometry
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parmenides51
30629 posts
parmenides51
#1 h  Jul 12, 2019, 8:09 AM • 1 Y
Y by Adventure10
Given are three non collinear points in plane. Find a line from which all three points are equidistant. How many such lines of the plane exist?
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Bashy99
698 posts
Bashy99
#2 h  Jul 12, 2019, 8:19 AM • 1 Y
Y by Adventure10
Solution.
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