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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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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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GMO P6 2024
Z4ADies   4
N 9 minutes ago by ihategeo_1969
Source: Geometry Mains Olympiad (GMO) 2024 P6
Given a triangle $\triangle ABC$ with circumcircle $\Omega$, excircle $\Gamma$ that is tangent to segment $BC$ at $T$. $A$-mixtillinear incircle $\omega$ of $ABC$ is tangent to $AB, AC $ at $D,E$. Suppose $N$ is the midpoint of the arc $BAC$, and $G \in AT \cap \Omega$. Show that if the circles $(NDE), \omega$ have the same radius, then tangents to $\Omega$ at $N, G$ intersect on $DE$.

Author:Mykhailo Sydorenko (Ukraine)
4 replies
Z4ADies
Oct 20, 2024
ihategeo_1969
9 minutes ago
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Perfect Square Function
Miku3D   15
N 43 minutes ago by bin_sherlo
Source: 2021 APMO P5
Determine all Functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(f(a)-b)+bf(2a)$ is a perfect square for all integers $a$ and $b$.
15 replies
Miku3D
Jun 9, 2021
bin_sherlo
43 minutes ago
J
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Symmetric FE
Phorphyrion   7
N 2 hours ago by megarnie
Source: 2023 Israel TST Test 7 P1
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds:
\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]
7 replies
Phorphyrion
May 9, 2023
megarnie
2 hours ago
J
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Unusual Hexagon Geo
oVlad   2
N 2 hours ago by Double07
Source: Romania Junior TST 2025 Day 1 P4
Let $ABCDEF$ be a convex hexagon, such that the triangles $ABC$ and $DEF$ are equilateral and the diagonals $AD, BE$ and $CF$ are concurrent. Prove that $AC\parallel DF$ or $BE=AD+CF.$
2 replies
oVlad
Apr 12, 2025
Double07
2 hours ago
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Centroid Distance Identity in Triangle
zeta1   5
N Today at 4:03 PM by DottedCaculator
Let M be any point inside triangle ABC, and let G be the centroid of triangle ABC. Prove that:

\[ |MA|^2 + |MB|^2 + |MC|^2 = |GA|^2 + |GB|^2 + |GC|^2 + 3|MG|^2 \]
5 replies
zeta1
Today at 12:28 PM
DottedCaculator
Today at 4:03 PM
J
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A Segment Bisection Problem
buratinogigle   3
N Today at 1:54 PM by AGCN
Source: VN Math Olympiad For High School Students P9 - 2025
In triangle $ABC$, let the incircle $\omega$ touch sides $BC, CA, AB$ at $D, E, F$, respectively. Let $P$ lie on the line through $D$ perpendicular to $BC$. Let $Q, R$ be the intersections of $PC, PB$ with $EF$, respectively. Let $K, L$ be the projections of $R, Q$ onto line $BC$. Let $M, N$ be the second intersections of $DQ, DR$ with the incircle $\omega$. Let $S$ be the intersection of $KM$ and $LN$. Prove that the line $DS$ bisects segment $QR$.
3 replies
buratinogigle
Today at 1:36 AM
AGCN
Today at 1:54 PM
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AMM Geo Problem
FireBreathers   0
Today at 1:27 PM
Source: Proposed by Tran Quang Hung
Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$. Let $EBC$ and $FAD$ be similar isosceles triangles with $EB = EC$ and $FA = FD$ erected externally to $ABCD$. Let $P$ be the point such that $EP$ is perpendicular to $DB$ and $FP$ is perpendicular to $AC$. Prove $P A = P B$
0 replies
FireBreathers
Today at 1:27 PM
0 replies
J
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Middle geo from UKR TST
mshtand1   5
N Today at 12:59 PM by jrpartty
Source: Ukraine IMO 2023 TST P2
$BE$ and $CF$ are the altitudes of the acute scalene $\triangle ABC$, $O$ is its circumcenter and $M$ is the midpoint of the side $BC$. If point, which is symmetric to $M$ with respect to $O$, lies on the line $EF$, find all possible values of the ratio $\dfrac{AM}{AO}$.
Proposed by Fedir Yudin
5 replies
mshtand1
May 5, 2023
jrpartty
Today at 12:59 PM
J
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Geometry
MathsII-enjoy   1
N Today at 12:41 PM by aidenkim119
Given triangle $ABC$ inscribed in $(O)$, $S$ is the midpoint of arc $BAC$ of $(O)$. The perpendicular bisector $BO$ intersects $BS$ at $I$. $(I;IB)$ intersects $AB$ at $U$ different from $B$. $H$ is the orthocenter of triangle $ABC$. Prove that $UH$ = $US$
1 reply
MathsII-enjoy
Yesterday at 3:03 PM
aidenkim119
Today at 12:41 PM
J
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Perpendicular bisector meets the circumcircle of another triangle
steppewolf   3
N Today at 12:17 PM by Omerking
Source: 2023 Junior Macedonian Mathematical Olympiad P4
We are given an acute $\triangle ABC$ with circumcenter $O$ such that $BC<AB$. The bisector of $\angle ACB$ meets the circumcircle of $\triangle ABC$ at a second point $D$. The perpendicular bisector of $AC$ meets the circumcircle of $\triangle BOD$ for the second time at $E$. The line $DE$ meets the circumcircle of $\triangle ABC$ for the second time at $F$. Prove that the lines $CF$, $OE$ and $AB$ are concurrent.

Authored by Petar Filipovski
3 replies
steppewolf
Jun 10, 2023
Omerking
Today at 12:17 PM
J
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Quad formed by orthocenters has same area (all 7's!)
v_Enhance   35
N Today at 12:05 PM by Wictro
Source: USA January TST for the 55th IMO 2014
Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.
35 replies
v_Enhance
Apr 28, 2014
Wictro
Today at 12:05 PM
J
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Parallelograms and concyclicity
Lukaluce   26
N Today at 8:13 AM by AshAuktober
Source: EGMO 2025 P4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
26 replies
Lukaluce
Apr 14, 2025
AshAuktober
Today at 8:13 AM
J
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A Characterization of Rectangles
buratinogigle   1
N Today at 7:39 AM by lbh_qys
Source: VN Math Olympiad For High School Students P8 - 2025
Prove that if a convex quadrilateral $ABCD$ satisfies the equation
\[ (AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2, \]then $ABCD$ must be a rectangle.
1 reply
buratinogigle
Today at 1:35 AM
lbh_qys
Today at 7:39 AM
J
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Constant Angle Sum
i3435   6
N Today at 6:35 AM by bin_sherlo
Source: AMASCIWLOFRIAA1PD (mock oly geo contest) P3
Let $ABC$ be a triangle with circumcircle $\Omega$, $A$-angle bisector $l_A$, and $A$-median $m_A$. Suppose that $l_A$ meets $\overline{BC}$ at $D$ and meets $\Omega$ again at $M$. A line $l$ parallel to $\overline{BC}$ meets $l_A$, $m_A$ at $G$, $N$ respectively, so that $G$ is between $A$ and $D$. The circle with diameter $\overline{AG}$ meets $\Omega$ again at $J$.

As $l$ varies, show that $\angle AMN + \angle DJG$ is constant.

MP8148
6 replies
i3435
May 11, 2021
bin_sherlo
Today at 6:35 AM
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J
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2 var inquality
sqing   4
N Apr 8, 2025 by sqing
Source: Own
Let $ a,b>0 . $ Prove that
$$\dfrac{1}{a^2+b^2}+\dfrac{1}{4ab} \ge \dfrac{\dfrac{3}{2} +\sqrt 2}{(a+b)^2} $$$$\dfrac{1}{a^3+b^3}+\dfrac{1}{4ab(a+b)}\ge \dfrac{\dfrac{7}{4} +\sqrt 3}{(a+b)^3} $$
4 replies
sqing
Apr 7, 2025
sqing
Apr 8, 2025
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2 var inquality
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inequalities proposed
algebra
inequalities
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Source: Own
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sqing
41636 posts
sqing
#1 h  Apr 7, 2025, 9:28 AM
Y by
Let $ a,b>0 . $ Prove that
$$\dfrac{1}{a^2+b^2}+\dfrac{1}{4ab} \ge \dfrac{\dfrac{3}{2} +\sqrt 2}{(a+b)^2} $$$$\dfrac{1}{a^3+b^3}+\dfrac{1}{4ab(a+b)}\ge \dfrac{\dfrac{7}{4} +\sqrt 3}{(a+b)^3} $$
This post has been edited 2 times. Last edited by sqing, Apr 7, 2025, 12:17 PM
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lbh_qys
526 posts
lbh_qys
#2 h  Apr 7, 2025, 9:34 AM • 1 Y
Y by teomihai
sqing wrote:
Let $ a,b>0 . $ Prove that
$$\dfrac{1}{a^2+b^2}+\dfrac{1}{4ab} \ge \dfrac{\dfrac{3}{2} +\sqrt 2}{(a+b)^2} $$

By C-S,
\[ \frac{1}{a^2+b^2}+\frac{1}{4ab} = \frac{1}{a^2 + b^2} + \frac{\frac 12}{2ab}\ge\frac{\left(1+\sqrt{\frac 12}\right)^2}{a^2+b^2+2ab}=\frac{\frac 32+\sqrt{2}}{(a+b)^2}. \]
Z K Y
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sqing
41636 posts
sqing
#3 h  Apr 7, 2025, 11:44 AM
Y by
Nice.Thank lbh_qys.
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pieMax2713
4177 posts
pieMax2713
#4 h  Apr 8, 2025, 4:05 AM
Y by
sqing wrote:
Let $ a,b>0 . $ Prove that
$$\dfrac{1}{a^3+b^3}+\dfrac{1}{4ab(a+b)}\ge \dfrac{\dfrac{7}{4} +\sqrt 3}{(a+b)^3} $$
similarly, by C-S
\[\frac{1}{a^3+b^3}+\frac{1}{4ab(a+b)}=\frac{1}{a^3+b^3}+\frac{\frac 34}{3ab(a+b)}\ge\frac{\left(1+\sqrt{\frac 34}\right)^2}{a^3+b^3+3ab(a+b)}=\frac{\frac 74+\sqrt3}{(a+b)^3}\]
Z K Y
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sqing
41636 posts
sqing
#5 h  Apr 8, 2025, 8:55 AM
Y by
Nice.Thanks.
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