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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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IMO 2014 Problem 4
ipaper   168
N 17 minutes ago by Bonime
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.

Proposed by Giorgi Arabidze, Georgia.
168 replies
ipaper
Jul 9, 2014
Bonime
17 minutes ago
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function Z to Z..
Jackson0423   3
N 29 minutes ago by jasperE3
Let \( f : \mathbb{Z} \to \mathbb{Z} \) be a function satisfying
\[ f(f(x)) = x^2 - 6x + 6 \quad \text{for all} \quad x \in \mathbb{Z}. \]Given that
\[ f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5, \]find the value of
\[ f(0) + f(1) + f(2) + \cdots + f(6). \]
3 replies
Jackson0423
Yesterday at 2:49 PM
jasperE3
29 minutes ago
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Functional equation
tenplusten   9
N an hour ago by Jakjjdm
Source: Bulgarian NMO 2015 P4
Find all functions $f:\mathbb{R^+}\to\mathbb {R^+} $ such that for all $x,y\in R^+$ the followings hold:
$i) $ $f (x+y)\ge f (x)+y $
$ii) $ $f (f (x))\le x $
9 replies
tenplusten
Apr 29, 2018
Jakjjdm
an hour ago
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Prove DK and BC are perpendicular.
yunxiu   61
N an hour ago by Avron
Source: 2012 European Girls’ Mathematical Olympiad P1
Let $ABC$ be a triangle with circumcentre $O$. The points $D,E,F$ lie in the interiors of the sides $BC,CA,AB$ respectively, such that $DE$ is perpendicular to $CO$ and $DF$ is perpendicular to $BO$. (By interior we mean, for example, that the point $D$ lies on the line $BC$ and $D$ is between $B$ and $C$ on that line.)
Let $K$ be the circumcentre of triangle $AFE$. Prove that the lines $DK$ and $BC$ are perpendicular.

Netherlands (Merlijn Staps)
61 replies
yunxiu
Apr 13, 2012
Avron
an hour ago
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Angle oriented geometry
Problems_eater   0
3 hours ago
Let $A, B, C,D$ be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?

1.If lines $AB$ and $CD$ are distinct and parallel, then
the oriented angle $ABC$ is equal to the oriented angle DCB.

2.If $B$ lies on the segment $AC$, then
the oriented angle $DBA$ plus the oriented angle $DBC $equals $180°$.

3.If the oriented angle$ ABC$ plus the oriented angle $BCD$ equals 0°, then
lines $AB $and $CD$ are parallel.

4.If the oriented angle $ABC$ plus the oriented angle $BCD$ equals $180°$, then
lines $AB$ and $CD$are parallel.
0 replies
Problems_eater
3 hours ago
0 replies
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how many quadrilaterals ?
Ecrin_eren   6
N Today at 5:31 PM by mathprodigy2011
"All the diagonals of an 11-gon are drawn. How many quadrilaterals can be formed using these diagonals as sides? (The vertices of the quadrilaterals are selected from the vertices of the 11-gon.)"
6 replies
Ecrin_eren
Apr 13, 2025
mathprodigy2011
Today at 5:31 PM
J
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Plane geometry problem with inequalities
ReticulatedPython   3
N Today at 2:48 PM by vanstraelen
Let $A$ and $B$ be points on a plane such that $AB=1.$ Let $P$ be a point on that plane such that $$\frac{AP^2+BP^2}{(AP)(BP)}=3.$$Prove that $$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$
Source: Own
3 replies
ReticulatedPython
Apr 10, 2025
vanstraelen
Today at 2:48 PM
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Inequalities
sqing   1
N Today at 1:55 PM by sqing
Let $ a,b $ be reals such that $ a^2-ab+b^2 =3$ . Prove that
$$ \frac{13}{ 10 }> \frac{1}{ a^2+1 }+ \frac{1}{ b^2+1 } \geq \frac{1}{ 2 }$$$$ \frac{6}{ 5 }>\frac{1}{ a^4+1 }+ \frac{1}{ b^4+1 } \geq \frac{1}{ 5 }$$$$ \frac{1}{ a^6+1 }+ \frac{1}{ b^6+1 } \geq \frac{1}{ 14 }$$
1 reply
sqing
Today at 8:59 AM
sqing
Today at 1:55 PM
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idk12345678 Math Contest
idk12345678   21
N Today at 1:25 PM by idk12345678
Welcome to the 1st idk12345678 Math Contest.
You have 4 hours. You do not have to prove your answers.
Post \signup username to sign up. Post your answers in a hide tag and I will tell you your score.*


The contest is attached to the post

Clarifications

*I mightve done them wrong feel free to ask about an answer
21 replies
idk12345678
Apr 10, 2025
idk12345678
Today at 1:25 PM
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purple comet math competition question
AVY2024   4
N Today at 1:02 PM by K1mchi_
Given that (1 + tan 1)(1 + tan 2). . .(1 + tan 45) = 2n, find n
4 replies
AVY2024
Today at 11:00 AM
K1mchi_
Today at 1:02 PM
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Inequalities
sqing   25
N Today at 12:06 PM by sqing
Let $ a,b,c,d>0 $ and $(a+c)(b+d)=ac+\frac{3}{2}bd.$ Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq \frac{20-\sqrt{10}}{3}$$Let $ a,b,c,d>0 $ and $(a+c)(b+d)=ac+\frac{4}{3}bd.$ Prove that
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq \frac{21-\sqrt{6}}{3}$$
25 replies
sqing
Dec 3, 2024
sqing
Today at 12:06 PM
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Polynomials
CuriousBabu   3
N Today at 11:40 AM by osszhangbanvan
\[ \frac{(x+y+z)^5 - x^5 - y^5 - z^5}{(x+y)(y+z)(z+x)} = 0 \]
Find the number of real solutions.
3 replies
CuriousBabu
Yesterday at 4:09 PM
osszhangbanvan
Today at 11:40 AM
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Combination
aria123   0
Today at 10:59 AM
Prove that three squares of side length $4$ cannot completely cover a square of side length $5$, if the three smaller squares do not overlap in their interiors (i.e., they may touch at edges or corners, but no part of one lies over another).
0 replies
aria123
Today at 10:59 AM
0 replies
J
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Geo Mock #6
Bluesoul   3
N Today at 3:16 AM by dudade
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
3 replies
Bluesoul
Apr 1, 2025
dudade
Today at 3:16 AM
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One nice inequlity 1
prof.   1
N Apr 2, 2025 by sqing
If $a,b,c$ are positiv real number, such that $a^2+b^2+c^2=3abc$ prove inequality $$\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2}\ge\frac{9}{a+b+c}.$$
1 reply
prof.
Apr 2, 2025
sqing
Apr 2, 2025
J
One nice inequlity 1
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prof.
442 posts
prof.
#1 h  Apr 2, 2025, 12:15 PM • 1 Y
Y by Bet667
If $a,b,c$ are positiv real number, such that $a^2+b^2+c^2=3abc$ prove inequality $$\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2}\ge\frac{9}{a+b+c}.$$
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sqing
41616 posts
sqing
#2 h  Apr 2, 2025, 12:41 PM
Y by
prof. wrote:
If $a,b,c$ are positiv real number, such that $a^2+b^2+c^2=3abc$ prove inequality $$\frac{a}{b^2c^2}+\frac{b}{c^2a^2}+\frac{c}{a^2b^2}\ge\frac{9}{a+b+c}.$$
Moldova 2003
https://artofproblemsolving.com/community/c6h486457p2725666
https://artofproblemsolving.com/community/c4h1734956p11260260
This post has been edited 1 time. Last edited by sqing, Apr 2, 2025, 12:43 PM
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