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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
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[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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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PJ // AC iff BC^2 = AC· QC
parmenides51   1
N 6 minutes ago by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1998 OMM P5
The tangents at points $B$ and $C$ on a given circle meet at point $A$. Let $Q$ be a point on segment $AC$ and let $BQ$ meet the circle again at $P$. The line through $Q $ parallel to $AB$ intersects $BC$ at $J$. Prove that $PJ$ is parallel to $AC$ if and only if $BC^2 = AC\cdot QC$.
1 reply
parmenides51
Jul 28, 2018
FrancoGiosefAG
6 minutes ago
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Self-evident inequality trick
Lukaluce   10
N 37 minutes ago by ytChen
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]When does the equality hold?
10 replies
+1 w
Lukaluce
Yesterday at 3:34 PM
ytChen
37 minutes ago
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Power Of Factorials
Kassuno   181
N 41 minutes ago by SomeonecoolLovesMaths
Source: IMO 2019 Problem 4
Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \]Proposed by Gabriel Chicas Reyes, El Salvador
181 replies
Kassuno
Jul 17, 2019
SomeonecoolLovesMaths
41 minutes ago
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Gergonne point Harmonic quadrilateral
niwobin   4
N an hour ago by on_gale
Triangle ABC has incircle touching the sides at D, E, F as shown.
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)
4 replies
niwobin
May 17, 2025
on_gale
an hour ago
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NCG Returns!
blacksheep2003   64
N an hour ago by SomeonecoolLovesMaths
Source: USEMO 2020 Problem 1
Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\]for positive integers $x$, $y$, $z$?
64 replies
blacksheep2003
Oct 24, 2020
SomeonecoolLovesMaths
an hour ago
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Binomial stuff
Arne   2
N 2 hours ago by Speedysolver1
Source: Belgian IMO preparation
Let $p$ be prime, let $n$ be a positive integer, show that \[ \gcd\left({p - 1 \choose n - 1}, {p + 1 \choose n}, {p \choose n + 1}\right) = \gcd\left({p \choose n - 1}, {p - 1 \choose n}, {p + 1 \choose n + 1}\right). \]
2 replies
Arne
Apr 4, 2006
Speedysolver1
2 hours ago
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Geometry hard problem
noneofyou34   1
N 2 hours ago by Lil_flip38
Let ABC be a triangle with incircle Γ. The tangency points of Γ with sides BC, CA, AB are A1, B1, C1 respectively. Line B1C1 intersects line BC at point A2. Similarly, points B2 and C2 are constructed. Prove that the perpendicular lines from A2, B2, C2 to lines AA1, BB1, CC1 respectively are concurret.
1 reply
noneofyou34
Today at 3:13 PM
Lil_flip38
2 hours ago
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segment of projections is half as sidelength, right triangle inscribed in right
parmenides51   3
N 2 hours ago by NumberzAndStuff
Source: 2020 Austrian Federal Competition For Advanced Students, Part 1, p2
Let $ABC$ be a right triangle with a right angle in $C$ and a circumcenter $U$. On the sides $AC$ and $BC$, the points $D$ and $E$ lie in such a way that $\angle EUD = 90 ^o$. Let $F$ and $G$ be the projection of $D$ and $E$ on $AB$, respectively. Prove that $FG$ is half as long as $AB$.

(Walther Janous)
3 replies
parmenides51
Nov 22, 2020
NumberzAndStuff
2 hours ago
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Austrian Regional MO 2025 P2
BR1F1SZ   3
N 2 hours ago by NumberzAndStuff
Source: Austrian Regional MO
Let $\triangle{ABC}$ be an isosceles triangle with $AC = BC$ and circumcircle $\omega$. The line through $B$ perpendicular to $BC$ is denoted by $\ell$. Furthermore, let $M$ be any point on $\ell$. The circle $\gamma$ with center $M$ and radius $BM$ intersects $AB$ once more at point $P$ and the circumcircle $\omega$ once more at point $Q$. Prove that the points $P,Q$ and $C$ lie on a straight line.

(Karl Czakler)
3 replies
BR1F1SZ
Apr 18, 2025
NumberzAndStuff
2 hours ago
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Computing functions
BBNoDollar   5
N 2 hours ago by ICE_CNME_4
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[ f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0. \](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
5 replies
BBNoDollar
Yesterday at 5:25 PM
ICE_CNME_4
2 hours ago
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Reflections of lines through reflections of excenters
cjquines0   39
N 3 hours ago by awesomeming327.
Source: 2016 IMO Shortlist G7
Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$.
[list=a]
[*] Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$.
[*] Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$.
[/list]
39 replies
cjquines0
Jul 19, 2017
awesomeming327.
3 hours ago
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2xy is perfect square and x^2 + y^2 is prime
parmenides51   4
N 3 hours ago by LeYohan
Source: Dutch NMO 2020 p4
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.
4 replies
parmenides51
Nov 23, 2020
LeYohan
3 hours ago
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Really classical inequatily from canada
shobber   79
N 3 hours ago by sharknavy75
Source: Canada 2002
Prove that for all positive real numbers $a$, $b$, and $c$,
\[ \frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c \]
and determine when equality occurs.
79 replies
shobber
Mar 5, 2006
sharknavy75
3 hours ago
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Functional equation
Pmshw   18
N 3 hours ago by jasperE3
Source: Iran 2nd round 2022 P2
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have:
$$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$
18 replies
Pmshw
May 8, 2022
jasperE3
3 hours ago
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2019 Polynomial problem
srnjbr   2
N Mar 30, 2025 by srnjbr
suppose t is a member of the interval (1,2). show that there exists a polynomial p with coefficients +-1 such that |p(t)-2019|<=1
2 replies
srnjbr
Mar 25, 2025
srnjbr
Mar 30, 2025
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2019 Polynomial problem
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Reveal topic tags
polynomial
algebra
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srnjbr
61 posts
srnjbr
#1 h  Mar 25, 2025, 6:15 PM
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suppose t is a member of the interval (1,2). show that there exists a polynomial p with coefficients +-1 such that |p(t)-2019|<=1
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pco
23515 posts
pco
#2 h  Mar 27, 2025, 11:19 AM
Y by
srnjbr wrote:
suppose t is a member of the interval (1,2). show that there exists a polynomial p with coefficients +-1 such that |p(t)-2019|<=1
Preliminary claim : $\forall A\in[0,t^n)$ ($n$ positive integer), $\exists Q(x)$ polynomial with coefficients $0,1$ and degree $<n$ such that $A-Q(t)\in[0,t)$
Proof

Let then $n$ such that $\sum_{i=0}^nt^i\ge 2018>\sum_{i=0}^{n-1}t^i$
Let $A=\frac{\sum_{i=0}^nt^i-2018}2$ : $A\in[0,\frac{t^n}2)\subset [0,t^{n-1})$
And so, using claim, $\exists Q(x)$ polynomial with coefficients $0,1$ and degree $<n-1$ such that $A-Q(t)\in[0,t)$
So $2A-2Q(t)\in[0,2t)$
Which is $\sum_{i=0}^nt^i-2018-2Q(t)\in[0,2t)$
$\implies$ $\sum_{i=0}^nt^i-2Q(t)\in[2018,2018+2t)\subset[2018,2020]$
$\implies$ $\sum_{i=0}^nt^i-2Q(t)-2019\in[-1,1]$
And $P(x)=\sum_{i=0}^nx^i-2Q(x)$ is a polynomial whose all coefficients are $\pm 1$

Q.E.D
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srnjbr
61 posts
srnjbr
#3 h  Mar 30, 2025, 12:58 PM
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THANK YOU
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