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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 combinatorial geometry problem
xyz123456   4
N a minute ago by xyz123456
$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$
4 replies
xyz123456
Mar 3, 2025
xyz123456
a minute ago
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a combinatoric problem
xyz123456   0
4 minutes ago
Source: Noga Alon
$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$ $prove\ that:\exists \ c_1{,}c_2>0{,}c_1n^{\frac{6}{5}}<f\left(n\right)<c_2n^{\frac{6}{5}}\ln n$
0 replies
xyz123456
4 minutes ago
0 replies
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Turbo's en route to visit each cell of the board
Lukaluce   18
N 10 minutes ago by kamatadu
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
18 replies
2 viewing
Lukaluce
Apr 14, 2025
kamatadu
10 minutes ago
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Shortlist 2017/G4
fastlikearabbit   120
N 13 minutes ago by Frd_19_Hsnzde
Source: Shortlist 2017, Romanian TST 2018
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
120 replies
fastlikearabbit
Jul 10, 2018
Frd_19_Hsnzde
13 minutes ago
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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
J
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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
J
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J
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no right triangle with vertices lattice points from 3 different colors
parmenides51   0
Sep 20, 2022
Source: Rioplatense 2008 L1 P6
Is it possible to color the points in the plane that have integer coordinates with three colors (all three colors must be used) such that there is no right triangle with all three vertices of different colors?
0 replies
parmenides51
Sep 20, 2022
0 replies
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no right triangle with vertices lattice points from 3 different colors
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Reveal topic tags
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Source: Rioplatense 2008 L1 P6
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parmenides51
30629 posts
parmenides51
#1 h  Sep 20, 2022, 5:31 PM
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Is it possible to color the points in the plane that have integer coordinates with three colors (all three colors must be used) such that there is no right triangle with all three vertices of different colors?
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