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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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Construct the orthocenter by drawing perpendicular bisectors
MarkBcc168   24
N 14 minutes ago by cj13609517288
Source: ELMO 2020 P3
Janabel has a device that, when given two distinct points $U$ and $V$ in the plane, draws the perpendicular bisector of $UV$. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools.

Proposed by Fedir Yudin.
24 replies
MarkBcc168
Jul 28, 2020
cj13609517288
14 minutes ago
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Problem involving Power of centroid
Mahdi_Mashayekhi   1
N 16 minutes ago by sami1618
Given is an triangle $ABC$ with centroid $G$. Let $p$ be the power of $G$ w.r.t circumcircle of $ABC$ and $q$ be the power of $G$ w.r.t incircle of $ABC$. prove that $\frac{a^2+b^2+c^2}{12} \le q-p < \frac{a^2+b^2+c^2}{3}$.
1 reply
Mahdi_Mashayekhi
42 minutes ago
sami1618
16 minutes ago
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Classical-looking inequality
Orestis_Lignos   8
N 33 minutes ago by Baimukh
Source: Greece National Olympiad 2022, Problem 3
The positive real numbers $a,b,c,d$ satisfy the equality
$$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$Find the maximum possible value of $a$.
8 replies
Orestis_Lignos
Feb 26, 2022
Baimukh
33 minutes ago
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BMO 2024 SL C1
GreekIdiot   10
N 33 minutes ago by GreekIdiot
Let $n$, $k$ be positive integers. Julia and Florian play a game on a $2n \times 2n$ board. Julia
has secretly tiled the entire board with invisible dominos. Florian now chooses $k$ cells.
All dominos covering at least one of these cells then turn visible. Determine the minimal
value of $k$ such that Florian has a strategy to always deduce the entire tiling.
10 replies
+1 w
GreekIdiot
Apr 27, 2025
GreekIdiot
33 minutes ago
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BABBAGE'S THEOREM EXTENSION
Mathgloggers   0
3 hours ago
A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$
0 replies
Mathgloggers
3 hours ago
0 replies
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N.S. condition of passing a fixed point for a function
Kunihiko_Chikaya   1
N 4 hours ago by Mathzeus1024
Let $ f(t)$ be a function defined in any real numbers $ t$ with $ f(0)\neq 0.$ Prove that on the $ x-y$ plane, the line $ l_t : tx+f(t) y=1$ passes through the fixed point which isn't on the $ y$ axis in regardless of the value of $ t$ if only if $ f(t)$ is a linear function in $ t$.
1 reply
Kunihiko_Chikaya
Sep 6, 2009
Mathzeus1024
4 hours ago
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Dot product
SomeonecoolLovesMaths   4
N 4 hours ago by quasar_lord
How to prove that dot product is distributive?
4 replies
SomeonecoolLovesMaths
Yesterday at 6:06 PM
quasar_lord
4 hours ago
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About my new website
Samujjal101   21
N 6 hours ago by Samujjal101
Hi everybody!
I'm registering some of the finest minds in math into my website.. it's not completely developed.. but still if you want we would be very grateful to have you!
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Maths-matchmaker is a website for connecting math minds together with a mission to unite together. It uses a matching algorithm to match 1:1 with like minded peers based on their interests or topics in math
21 replies
Samujjal101
Yesterday at 2:28 PM
Samujjal101
6 hours ago
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Inequalities
sqing   6
N Today at 8:58 AM by sqing
Let $a,b,c\geq 0,ab+bc+ca>0$ and $a+b+c=3$. Prove that
$$\frac{8}{3}\leq\frac{(a+b)(b+c)(c+a)}{ab+bc+ca}\leq 3$$$$3\leq\frac{(a+b)(2b+c)(c+a)}{ab+bc+ca}\leq 6$$$$\frac{3}{2}\leq\frac{(a+b)(2b+c)(c+ a)}{ab+bc+2ca}\leq 6$$$$1\leq\frac{(a+b)(2b+c)(c+ a)}{ab+bc+ 3ca}\leq 6$$
6 replies
sqing
Dec 22, 2023
sqing
Today at 8:58 AM
J
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BrUMO 2025 Team Round Problem 3
lpieleanu   2
N Today at 8:57 AM by nehareddyk009
Bruno and Brutus are running on a circular track with a $20$ foot radius. Bruno completes $5$ laps every hour, while Brutus completes $7$ laps every hour. If they start at the same point but run in opposite directions, how far along the track’s circumference (in feet) from the starting point are they when they meet for the sixth time? Note: Do not count the moment they start running as a meeting point.
2 replies
lpieleanu
Sunday at 11:05 PM
nehareddyk009
Today at 8:57 AM
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Inequalities
sqing   6
N Today at 8:42 AM by sqing
Let $x\in(-1,1). $ Prove that
$$ \dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$$$$ \dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$$
6 replies
sqing
Apr 26, 2025
sqing
Today at 8:42 AM
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Inequality, tougher than it looks
tom-nowy   1
N Today at 6:46 AM by thaithuonglaoquan8386
Prove that for $a,b \in \mathbb{R}$
$$ 2(a^2+1)(b^2+1) \geq 3(a+b). $$Is there an elegant way to prove this?
1 reply
tom-nowy
Today at 3:51 AM
thaithuonglaoquan8386
Today at 6:46 AM
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Geometry
AlexCenteno2007   1
N Today at 2:33 AM by AlexCenteno2007
Let ABC be an acute triangle and let D, E and F be the feet of the altitudes from A, B and C respectively. The straight line EF and the circumcircle of ABC intersect at P such that F is between E and P, the straight lines BP and DF intersect at Q. Show that if ED = EP then CQ and DP are parallel.
1 reply
AlexCenteno2007
Yesterday at 3:59 PM
AlexCenteno2007
Today at 2:33 AM
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Product of all even divisors
girishpimoli   4
N Today at 2:29 AM by williamxiao
$(1)$ Product of all even divisors of $9000$

$(2)$ If $4$ dice are rolled, Then number of ways of getting sum at least $13$ is
4 replies
girishpimoli
Yesterday at 2:13 PM
williamxiao
Today at 2:29 AM
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Locus of pole of a line with respect to parabola.
Goutham   3
N Mar 6, 2022 by vanstraelen
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$
3 replies
Goutham
Sep 29, 2010
vanstraelen
Mar 6, 2022
J
Locus of pole of a line with respect to parabola.
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Goutham
3130 posts
Goutham
#1 h  Sep 29, 2010, 5:53 PM • 2 Y
Y by Adventure10, Mango247
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$
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vanstraelen
9001 posts
vanstraelen
#2 h  Oct 3, 2018, 7:27 PM • 2 Y
Y by Adventure10, Mango247
Locus of the poles $M$ is the parabola $P_{2}$.
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RenheMiResembleRice
259 posts
RenheMiResembleRice
#3 h  Jan 22, 2022, 12:50 AM
Y by
vanstraelen wrote:
Locus of the poles $M$ is the parabola $P_{2}$.

Why? ._.
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vanstraelen
9001 posts
vanstraelen
#4 h  Mar 6, 2022, 9:18 PM
Y by
The tangent line $2\lambda x+y-2p\lambda^{2}=0$ in a point $T(2p\lambda,-2p\lambda^{2})$ of the parabola $x^{2}=-2py$.

Two points on this tangent line, $P(0,2p\lambda^{2})$ and $Q(p\lambda,0)$.
The polar lines of these points are $y+2p\lambda^{2}=0$ and $\lambda x-y=0$, cutting in the pole $M$ of the tangent line.

Eliminating the parameter $\lambda$, we find $x^{2}+2py=0$.
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