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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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[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 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
J
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Interesting inequality
sqing   1
N 5 minutes ago by Mathzeus1024
Let $ a,b> 0 $ and $ a+b+ab=1. $ Prove that
$$\frac{1}{1+a^2} + \frac{1}{1+b^2} +a+b\leq \frac{5}{\sqrt{2}}-1 $$
1 reply
sqing
Today at 3:35 AM
Mathzeus1024
5 minutes ago
J
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AM-GM FE ineq
navi_09220114   3
N 6 minutes ago by navi_09220114
Source: Own. Malaysian IMO TST 2025 P3
Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$.

Proposed by Ivan Chan Kai Chin
3 replies
navi_09220114
Mar 22, 2025
navi_09220114
6 minutes ago
J
H
Ez inequality
m4thbl3nd3r   3
N 16 minutes ago by lbh_qys
Let $a,b,c>0$. Prove that $$\sum \frac{ab^2}{a^2+2b^2+c^2}\le \frac{a+b+c}{4}$$
3 replies
m4thbl3nd3r
Yesterday at 3:57 PM
lbh_qys
16 minutes ago
J
H
Integer polynomial commutes with sum of digits
cjquines0   40
N an hour ago by ihategeo_1969
Source: 2016 IMO Shortlist N1
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
Proposed by Warut Suksompong, Thailand
40 replies
cjquines0
Jul 19, 2017
ihategeo_1969
an hour ago
J
No more topics!
H
J
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Two sets of diagonals in 7 gon are equal
Goutham   1
N Jun 19, 2011 by Batominovski
Source: ToT 2003-JO-2
In $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1$ and $A_7A_2$ are congruent to each other and diagonals $A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2$ and $A_7A_3$ are also congruent to each other. Is the polygon necessarily regular?
1 reply
Goutham
Jun 19, 2011
Batominovski
Jun 19, 2011
J
Two sets of diagonals in 7 gon are equal
G H J
Reveal topic tags
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Source: ToT 2003-JO-2
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Goutham
3130 posts
Goutham
#1 h  Jun 19, 2011, 5:13 AM • 2 Y
Y by Adventure10, Mango247
In $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1$ and $A_7A_2$ are congruent to each other and diagonals $A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2$ and $A_7A_3$ are also congruent to each other. Is the polygon necessarily regular?
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Batominovski
1608 posts
Batominovski
#2 h  Jun 19, 2011, 11:01 AM • 2 Y
Y by Adventure10, Mango247
Yes. First note that $\triangle A_{i-2}A_iA_{i+2}$'s are all equivalent. You can now work out that $\angle A_{i-2} A_i A_{i+2} = \frac{3\pi}{7}$ for every $i$. Now, you can show that the heptagon is cyclic, i.e., circumscribable. The rest is easy.
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