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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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[*]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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Eazy equation clap
giangtruong13   0
5 hours ago
Find all $x,y,z$ satisfy that: $$\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5x-1$$
0 replies
giangtruong13
5 hours ago
0 replies
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inequality
revol_ufiaw   3
N Today at 2:55 PM by MS_asdfgzxcvb
Prove that that for any real $x \ge 0$ and natural number $n$,
$$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$$
3 replies
revol_ufiaw
Today at 2:05 PM
MS_asdfgzxcvb
Today at 2:55 PM
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How to judge a number is prime or not?
mingzhehu   0
Today at 2:45 PM
A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime
0 replies
mingzhehu
Today at 2:45 PM
0 replies
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What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N Today at 2:40 PM by maxamc
What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.
6 replies
EaZ_Shadow
Dec 28, 2024
maxamc
Today at 2:40 PM
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No more topics!
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right angle, equal ratios, equilateral (2016-17 Savin Competition 7-9 p9)
parmenides51   4
N Jun 20, 2021 by CrazyMathMan
On sides $AB$ and $AC$ of an equilateral triangle $ABC$ are taken points $E$ and $F$, respectively, so that $AE: BE = CF: AF = 2$. Let $P$ be the intersection point of $BF$ and $CE$. Prove that the angle $CPA$ is right.
IMAGE
4 replies
parmenides51
Jun 19, 2021
CrazyMathMan
Jun 20, 2021
J
right angle, equal ratios, equilateral (2016-17 Savin Competition 7-9 p9)
G H J
Reveal topic tags
Equilateral
geometry
right angle
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parmenides51
30629 posts
parmenides51
#1 h  Jun 19, 2021, 6:50 AM • 1 Y
Y by CrazyMathMan
On sides $AB$ and $AC$ of an equilateral triangle $ABC$ are taken points $E$ and $F$, respectively, so that $AE: BE = CF: AF = 2$. Let $P$ be the intersection point of $BF$ and $CE$. Prove that the angle $CPA$ is right.
https://cdn.artofproblemsolving.com/attachments/2/1/e3d2bfc93e8c65aebddedafeb85551f111f8fb.png
This post has been edited 2 times. Last edited by parmenides51, Jun 19, 2021, 7:00 AM
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parmenides51
30629 posts
parmenides51
#2 h  Jun 19, 2021, 6:51 AM
Y by
posted for the image link
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vanstraelen
8949 posts
vanstraelen
#3 h  Jun 19, 2021, 3:37 PM
Y by
Given $A(-a,0),B(0,\sqrt{3}\ a),C(a,0)$.

Conditions $\Rightarrow F(-\frac{a}{3},0),E(-\frac{a}{3},\frac{2\sqrt{3}}{3}\ a)$.

Point $P(-\frac{a}{7},\frac{4\sqrt{3}}{7}\ a)$.

Slope of the line $EC\ :\ m_{EC}=-\frac{\sqrt{3}}{2}$.
Slope of the line $AP\ :\ m_{AP}=\frac{2\sqrt{3}}{3}$, so $EC \bot AP$.
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sunken rock
4379 posts
sunken rock
#4 h  Jun 19, 2021, 5:41 PM
Y by
My solution at https://stanfulger.blogspot.com/2021/06/httpsartofproblemsolvingcomcommunityc4t_19.html

Best regards,
sunken rock
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CrazyMathMan
16 posts
CrazyMathMan
#5 h  Jun 20, 2021, 8:51 AM
Y by
Let M and N b the midpoints of AC and BC, respectively and X be the intersection of PF and MN.
Some observation we can make using simple angle chasing
(1) PNCX is cyclic.
(2) PEAF is cyclic.
(3) MNBA is cyclic.

And AE/EB=AF/FM =>EF//BM.
Now we get angle FPA = angle FEA = angle MBA = angle MNA => XNPA is cyclic.
Threfore, XCNPA is cyclic.
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