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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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
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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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Nice one
imnotgoodatmathsorry   4
N 11 minutes ago by lbh_qys
Source: Own
With $x,y,z >0$.Prove that: $\frac{xy}{4y+4z+x} + \frac{yz}{4z+4x+y} +\frac{zx}{4x+4y+z} \le \frac{x+y+z}{9}$
4 replies
imnotgoodatmathsorry
May 2, 2025
lbh_qys
11 minutes ago
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Continued fraction
tapir1729   11
N an hour ago by Mathandski
Source: TSTST 2024, problem 2
Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and
\[ 1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 + (p-1)x}}}=\frac{P(x)}{Q(x)}. \]Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are not divisible by $p$.

Andrew Gu
11 replies
tapir1729
Jun 24, 2024
Mathandski
an hour ago
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Cycle in a graph with a minimal number of chords
GeorgeRP   1
N an hour ago by Photaesthesia
Source: Bulgaria IMO TST 2025 P3
In King Arthur's court every knight is friends with at least $d>2$ other knights where friendship is mutual. Prove that King Arthur can place his knights around a round table in such a way that every knight is friends with the $2$ people adjacent to him and between them there are at least $\frac{d^2}{10}$ friendships of knights that are not adjacent to each other.
1 reply
GeorgeRP
Yesterday at 7:51 AM
Photaesthesia
an hour ago
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Japan MO Finals 2021 P4
maple116   2
N an hour ago by Gauler
Source: Japan MO Finals 2021 P4
Let $a_1,a_2,\dots,a_{2021}$ be $2021$ integers which satisfy
\[ a_{n+5}+a_n>a_{n+2}+a_{n+3}\]for all integers $n=1,2,\dots,2016$. Find the minimum possible value of the difference between the maximum value and the minimum value among $a_1,a_2,\dots,a_{2021}$.
2 replies
maple116
Feb 14, 2021
Gauler
an hour ago
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Tangents to a cyclic quadrilateral
v_Enhance   23
N an hour ago by Ilikeminecraft
Source: ELMO Shortlist 2013: Problem G9, by Allen Liu
Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$.

Proposed by Allen Liu
23 replies
v_Enhance
Jul 23, 2013
Ilikeminecraft
an hour ago
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Strange angle condition and concyclic points
lminsl   128
N an hour ago by Giant_PT
Source: IMO 2019 Problem 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
128 replies
lminsl
Jul 16, 2019
Giant_PT
an hour ago
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Two lines meet at circle
mathpk   51
N 2 hours ago by Ilikeminecraft
Source: APMO 2008 problem 3
Let $ \Gamma$ be the circumcircle of a triangle $ ABC$. A circle passing through points $ A$ and $ C$ meets the sides $ BC$ and $ BA$ at $ D$ and $ E$, respectively. The lines $ AD$ and $ CE$ meet $ \Gamma$ again at $ G$ and $ H$, respectively. The tangent lines of $ \Gamma$ at $ A$ and $ C$ meet the line $ DE$ at $ L$ and $ M$, respectively. Prove that the lines $ LH$ and $ MG$ meet at $ \Gamma$.
51 replies
mathpk
Mar 22, 2008
Ilikeminecraft
2 hours ago
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Hard geometry
Lukariman   5
N 2 hours ago by Lukariman
Given circle (O) and chord AB with different diameters. The tangents of circle (O) at A and B intersect at point P. On the small arc AB, take point C so that triangle CAB is not isosceles. The lines CA and BP intersect at D, BC and AP intersect at E. Prove that the centers of the circles circumscribing triangles ACE, BCD and OPC are collinear.
5 replies
Lukariman
Yesterday at 4:28 AM
Lukariman
2 hours ago
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P,Q,B are collinear
MNJ2357   29
N 2 hours ago by cj13609517288
Source: 2020 Korea National Olympiad P2
$H$ is the orthocenter of an acute triangle $ABC$, and let $M$ be the midpoint of $BC$. Suppose $(AH)$ meets $AB$ and $AC$ at $D,E$ respectively. $AH$ meets $DE$ at $P$, and the line through $H$ perpendicular to $AH$ meets $DM$ at $Q$. Prove that $P,Q,B$ are collinear.
29 replies
MNJ2357
Nov 21, 2020
cj13609517288
2 hours ago
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Factorising and prime numbers...
Sadigly   5
N 3 hours ago by ektorasmiliotis
Source: Azerbaijan Senior MO 2025 P4
Prove that for any $p>2$ prime number, there exists only one positive number $n$ that makes the equation $n^2-np$ a perfect square of a positive integer
5 replies
Sadigly
May 8, 2025
ektorasmiliotis
3 hours ago
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IMO 2011 Problem 3
Amir Hossein   85
N 3 hours ago by NerdyNashville
Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies
\[f(x + y) \leq yf(x) + f(f(x))\]
for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$.

Proposed by Igor Voronovich, Belarus
85 replies
Amir Hossein
Jul 18, 2011
NerdyNashville
3 hours ago
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Angle Relationships in Triangles
steven_zhang123   0
4 hours ago
In $\triangle ABC$, $AB > AC$. The internal angle bisector of $\angle BAC$ and the external angle bisector of $\angle BAC$ intersect the ray $BC$ at points $D$ and $E$, respectively. Given that $CE - CD = 2AC$, prove that $\angle ACB = 2\angle ABC$.
0 replies
steven_zhang123
4 hours ago
0 replies
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acute triangle and its circumcenter and orthocenter
N.T.TUAN   6
N 4 hours ago by MathLuis
Source: USA TST 2005, Problem 2
Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that
\[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]
6 replies
N.T.TUAN
May 14, 2007
MathLuis
4 hours ago
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Imtersecting two regular pentagons
Miquel-point   2
N 5 hours ago by ohiorizzler1434
Source: KoMaL B. 5093
The intersection of two congruent regular pentagons is a decagon with sides of $a_1,a_2,\ldots ,a_{10}$ in this order. Prove that
\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]
2 replies
Miquel-point
Yesterday at 6:27 PM
ohiorizzler1434
5 hours ago
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Easy Quadric equation
VicKmath7   2
N Apr 17, 2025 by RagvaloD
Source: Archimedes Junior 2015
Find all values of the real parameter $a$, so that the equation $x^2+(a-2)x-(a-1)(2a-3)=0$ has two real roots, so that the one is the square of the other.
2 replies
VicKmath7
Mar 16, 2020
RagvaloD
Apr 17, 2025
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Easy Quadric equation
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Reveal topic tags
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Source: Archimedes Junior 2015
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VicKmath7
1390 posts
VicKmath7
#1 h  Mar 16, 2020, 5:54 PM
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Find all values of the real parameter $a$, so that the equation $x^2+(a-2)x-(a-1)(2a-3)=0$ has two real roots, so that the one is the square of the other.
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Rainbow1971
35 posts
Rainbow1971
#2 h  Apr 17, 2025, 1:27 PM
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This post has been edited 1 time. Last edited by Rainbow1971, Apr 17, 2025, 8:07 PM
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RagvaloD
4916 posts
RagvaloD
#3 h  Apr 17, 2025, 6:02 PM
Y by
Roots are $a-1,3-2a$

So enough to prove $a-1=(3-2a)^2$ or $3-2a=(a-1)^2$ and so $a=2,\frac{5}{4}, \sqrt{2}, -\sqrt{2}$
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