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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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>=512 different isosceles triangles whose vertices have the same color
parmenides51   2
N 6 minutes ago by cooljoseph
Source: Mathematics Regional Olympiad of Mexico West 2016 P6
The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.
2 replies
+1 w
parmenides51
Sep 7, 2022
cooljoseph
6 minutes ago
J
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Fourth power ineq
Project_Donkey_into_M4   1
N 2 hours ago by sqing
Source: 2018 Mock RMO tdp and kayak P1
Let $a,b,c,d \in \mathbb{R}^+$ such that $a+b+c+d \leq 1$. Prove that\[\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq 255\cdot abcd.\]
1 reply
Project_Donkey_into_M4
Yesterday at 6:20 PM
sqing
2 hours ago
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Is this FE solvable?
ItzsleepyXD   0
2 hours ago
Source: Original
Let $c_1,c_2 \in \mathbb{R^+}$. Find all $f : \mathbb{R^+} \rightarrow \mathbb{R^+}$ such that for all $x,y \in \mathbb{R^+}$ $$f(x+c_1f(y))=f(x)+c_2f(y)$$
0 replies
ItzsleepyXD
2 hours ago
0 replies
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Dear Sqing: So Many Inequalities...
hashtagmath   36
N 2 hours ago by sqing
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)
36 replies
hashtagmath
Oct 30, 2024
sqing
2 hours ago
J
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In a school of $800$ students, $224$ students play cricket, $240$ students play
Vulch   1
N 4 hours ago by RollingPanda4616
Hello everyone,
In a school of $800$ students, $224$ students play cricket, $240$ students play hockey and $336$ students play basketball. $64$ students play both basketball and hockey, $80$ students play both cricket and basketball, $40$ students play both cricket and hockey, and $24$ students play all three: basketball, hockey, and cricket. Find the number of students who do not play any game.

Edit:
In the above problem,I just want to know that why the number of students who don't play any game shouldn't be 0, because,if we add 224,240 and 336 it comes out to be 800.I have solution,but I just want to know how to explain it without theoretically.Thank you!
1 reply
Vulch
5 hours ago
RollingPanda4616
4 hours ago
J
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100th post
MathJedi108   1
N 5 hours ago by mdk2013
Well I guess this is my 100th post, it would be really funny if it isn't can yall share your favorite experience on AoPS here?
1 reply
MathJedi108
6 hours ago
mdk2013
5 hours ago
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Find all triples
pedronis   2
N 6 hours ago by Kempu33334
Find all triples of positive integers $(n, r, s)$ such that $n^2 + n + 1$ divides $n^r + n^s + 1$.
2 replies
pedronis
Apr 19, 2025
Kempu33334
6 hours ago
J
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Median geometry
Sedro   4
N Yesterday at 10:01 PM by Sedro
In triangle $ABC$, points $D$, $E$, and $F$ are the midpoints of sides $BC$, $CA$, and $AB$, respectively. Prove that the area of the triangle with side lengths $AD$, $BE$, and $CF$ has area $\tfrac{3}{4}[ABC]$.
4 replies
Sedro
Yesterday at 6:03 PM
Sedro
Yesterday at 10:01 PM
J
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Combinatorial proof
MathBot101101   5
N Yesterday at 9:56 PM by Kempu33334
Is there a way to prove
\frac{1}{(1+1)!}+\frac{2}{(2+1)!}+...+\frac{n}{(n+1)!}
without induction and using only combinatorial arguments?

Induction proof wasn't quite as pleasing for me.
5 replies
MathBot101101
Yesterday at 7:37 AM
Kempu33334
Yesterday at 9:56 PM
J
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geometry
carvaan   1
N Yesterday at 6:38 PM by Lankou
The difference between two angles of a triangle is 24°. All angles are numerically double digits. Find the number of possible values of the third angle.
1 reply
carvaan
Yesterday at 5:46 PM
Lankou
Yesterday at 6:38 PM
J
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weird permutation problem
Sedro   1
N Yesterday at 6:07 PM by Sedro
Let $\sigma$ be a permutation of $1,2,3,4,5,6,7$ such that there are exactly $7$ ordered pairs of integers $(a,b)$ satisfying $1\le a < b \le 7$ and $\sigma(a) < \sigma(b)$. How many possible $\sigma$ exist?
1 reply
Sedro
Yesterday at 2:09 AM
Sedro
Yesterday at 6:07 PM
J
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Recursion
Sid-darth-vater   6
N Yesterday at 5:59 PM by vanstraelen
Help, I can't characterize ts and I dunno what to do
6 replies
Sid-darth-vater
Yesterday at 3:02 AM
vanstraelen
Yesterday at 5:59 PM
J
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geometry
carvaan   0
Yesterday at 5:48 PM
OABC is a trapezium with OC // AB and ∠AOB = 37°. Furthermore, A, B, C all lie on the circumference of a circle centred at O. The perpendicular bisector of OC meets AC at D. If ∠ABD = x°, find last 2 digit of 100x.
0 replies
carvaan
Yesterday at 5:48 PM
0 replies
J
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Graph of polynomials
Ecrin_eren   1
N Yesterday at 5:36 PM by vanstraelen
The graph of the quadratic polynomial with real coefficients y = px^2 + qx + r, called G1, intersects the graph of the polynomial y = x^2, called G2, at points A and B. The lines tangent to G2 at points A and B intersect at point C. It is known that point C lies on G1. What is the value of p?
1 reply
Ecrin_eren
Yesterday at 3:00 PM
vanstraelen
Yesterday at 5:36 PM
J
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J
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nice fe with non-linear function being the answer
jjkim0336   3
N Yesterday at 8:17 AM by Lufin
Source: own
f:R+ -> R+

f(xf(y)+y) = y f(y^2 +x)
3 replies
jjkim0336
Apr 8, 2025
Lufin
Yesterday at 8:17 AM
J
nice fe with non-linear function being the answer
G H J
Reveal topic tags
algebra
functional equation
L
G H BBookmark kLocked kLocked NReply
Source: own
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jjkim0336
20 posts
jjkim0336
#1 h  Apr 8, 2025, 2:25 PM
Y by
f:R+ -> R+

f(xf(y)+y) = y f(y^2 +x)
Z K Y
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aidenkim119
32 posts
aidenkim119
#2 h  Apr 8, 2025, 3:10 PM
Y by
f(x) = 1/x
Z K Y
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jjkim0336
20 posts
jjkim0336
#3 h  Apr 18, 2025, 7:45 AM
Y by
bump…. looking for solutions….
Z K Y
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Lufin
10 posts
Lufin
#4 h  Yesterday at 8:17 AM
Y by
Here is my solution
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