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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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0 replies
jlacosta
Mar 2, 2025
0 replies
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
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
Hard T^T
Noname23   2
N 2 minutes ago by Noname23
<problem>
2 replies
Noname23
an hour ago
Noname23
2 minutes ago
Yet another configy
Pranav1056   11
N 2 minutes ago by kes0716
Source: India TST 2023 Day 2 P2
In triangle $ABC$, let $D$ be the foot of the perpendicular from $A$ to line $BC$. Point $K$ lies inside triangle $ABC$ such that $\angle KAB = \angle KCA$ and $\angle KAC = \angle KBA$. The line through $K$ perpendicular to like $DK$ meets the circle with diameter $BC$ at points $X,Y$. Prove that $AX \cdot DY = DX \cdot AY$
11 replies
Pranav1056
Jul 9, 2023
kes0716
2 minutes ago
Find the smallest positive integer $a$
ttd.hcmcity   1
N 8 minutes ago by ttd.hcmcity
Find the smallest positive integer $a$ such that there exist integers $b,c,d,e$ so that the polynomial $ax^4+bx^3+cx^2+dx+e$ has 4 distinct roots in the interval $(0;1)$?
1 reply
ttd.hcmcity
Oct 21, 2024
ttd.hcmcity
8 minutes ago
D1010 : How it is possible ?
Dattier   3
N 13 minutes ago by Dattier
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=172840090421781518678763921675392141786000436658021921275090402437796947824966464426797102
59525308036470431210259590181720483369539690621515342820528633073982816814653666658107757108
67856720572225880311472925624694183944650261079955759251769111321319421445397848518597584590
900951222557860592579005088853698315463815905425095325508106272375728975

B=227564340154808184720778276049144229526648735475052708528935496537676518846805227119017278
70644188547893224843051453107076145465733981826429238937805270372241433808862604677609912285
67577953725945090125797351518670892779468968705801340068681556238850340398780828104506916965
606659768601942798676554332768254089685307970609932846902
3 replies
Dattier
Mar 10, 2025
Dattier
13 minutes ago
Number Theory
karasuno   1
N 16 minutes ago by Tkn
Solve the equation $$n!+10^{2014}=m^{4}$$in natural numbers m and n.
1 reply
1 viewing
karasuno
4 hours ago
Tkn
16 minutes ago
Flo0r functi0n
m4thbl3nd3r   5
N 34 minutes ago by pco
Find all positive integers such that $$n=\lfloor \sqrt{n}\rfloor^2+\lfloor \sqrt{n}\rfloor$$
5 replies
m4thbl3nd3r
2 hours ago
pco
34 minutes ago
At least 3 co-prime pairs
eezad3   0
37 minutes ago
You are given a random permutation of the first $n$ integers where $n>10$. Show that there exists at least three positions $i$ such that $gcd(p[i], p[i+1])=1$ (here $p[i]$ refers to the $i$-th position integer in the permutation)
0 replies
eezad3
37 minutes ago
0 replies
Suspicious Quadrilateral Geometry
YaoAOPS   4
N 37 minutes ago by Giabach298
Source: 2025 CTST P8
Let quadrilateral $A_1A_2A_3A_4$ be not cyclic and haves edges not parallel to each other.

Denote $B_i$ as the intersection of the tangent line at $A_i$ with respect to circle $A_{i-1}A_iA_{i+1}$ and the $A_{i+2}$-symmedian with respect to triangle $A_{i+1}A_{i+2}A_{i+3}$ and $C_i$ as the intersection of lines $A_iA_{i+1}$ and $B_iB_{i+1}$, where all indexes taken cyclically.

Prove that $C_1$, $C_2$, $C_3$, and $C_4$ are collinear.
4 replies
1 viewing
YaoAOPS
Mar 10, 2025
Giabach298
37 minutes ago
i need help
MR.1   0
an hour ago
Source: help
can you guys tell me problems about fe in $R+$(i know $R$ well). i want to study so if you guys have some easy or normal problems please send me
0 replies
MR.1
an hour ago
0 replies
Changeable polynomials, can they ever become equal?
mshtand1   4
N 2 hours ago by CHESSR1DER
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 11.5
Initially, two constant polynomials are written on the board: \(0\) and \(1\). At each step, it is allowed to add \(1\) to one of the polynomials and to multiply another one by the polynomial \(45x + 2025\). Can the polynomials become equal at some point?

Proposed by Oleksii Masalitin
4 replies
mshtand1
Yesterday at 12:47 AM
CHESSR1DER
2 hours ago
Guessing polynomial by its maximum values on segments
NO_SQUARES   3
N 2 hours ago by pco
Source: Kvant 2025 no. 1 M2828 and The XIX Southern Mathematical Tournament
Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps?
M. Didin
3 replies
NO_SQUARES
Yesterday at 3:21 PM
pco
2 hours ago
easy number theory
MuradSafarli   1
N 2 hours ago by Tuvshuu
\[
v_p(n!) \leq \frac{n}{p - 1}
\]
1 reply
MuradSafarli
2 hours ago
Tuvshuu
2 hours ago
Inspired by Kazakhstan 2017
sqing   1
N 2 hours ago by sqing
Source: Own
Let $a,b,c\ge \frac{1}{2}$ and $a+b+c=2. $ Prove that
$$\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge 1$$Let $a,b,c\ge \frac{1}{3}$ and $a+b+c=1. $ Prove that
$$\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge 9$$
1 reply
1 viewing
sqing
3 hours ago
sqing
2 hours ago
Collinear geometry problem with incircle
ilovemath0402   1
N 2 hours ago by deraxenrovalo
Given acute $\triangle ABC$ not isosceles, the incircle $(I)$. $D,E,F$ is the intersection of $(I)$ with $BC,CA,AB$. $P$ is the projection of $D$ onto $EF$. $DP$ cut $(I)$ at the second point $K$. $L$ is the projection of $A$ onto $IK$. $(LEC), (LFB)$ cut $(I)$ at the second point $M,N$ respectively. Prove $M,N,P$ are collinear
1 reply
ilovemath0402
Jul 22, 2023
deraxenrovalo
2 hours ago
Problem 4 - Perfect Squares
Ln142   5
N Jun 6, 2020 by Al3jandro0000
Source: 2020 Austrian National Competition for Advanced Students, Part 1 problem 4
Determine all positive integers $N$ such that
$$2^N-2N$$is a perfect square.

(Walther Janous)
5 replies
Ln142
Jun 6, 2020
Al3jandro0000
Jun 6, 2020
Problem 4 - Perfect Squares
G H J
G H BBookmark kLocked kLocked NReply
Source: 2020 Austrian National Competition for Advanced Students, Part 1 problem 4
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Ln142
36 posts
#1
Y by
Determine all positive integers $N$ such that
$$2^N-2N$$is a perfect square.

(Walther Janous)
Z K Y
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yefangzhou
257 posts
#2 • 1 Y
Y by pavel kozlov
if $N=1(mod 2)$ and $N\ge 3$
then $2^N-2N=2(mod 4)$ is not a perfect square.
if $N=0(mod 2)$ and $N\ge 8$,assume that $N=2k$,
then $(2^k-1)^2<2^N-2N=2^{2k}-4k<(2^k)^2$ is not a perfect square.
Z K Y
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Ln142
36 posts
#3
Y by
yefangzhou wrote:
if $N=1(mod 2)$ and $N\ge 3$
then $2^N-2N=2(mod 4)$ is not a perfect square.
if $N=0(mod 2)$ and $N\ge 8$,assume that $N=2k$,
then $(2^k-1)^2<2^N-2N=2^{2k}-4k<(2^k)^2$ is not a perfect square.
How would you prove $\left({2^k-1}\right)^2 < 2^{2k}-4k$ ? I used an induction proof at the competition.
Z K Y
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yefangzhou
257 posts
#4
Y by
derivative could help.
Z K Y
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Ln142
36 posts
#5
Y by
I proved it in this manner:
$\left({2^k-1}\right)^2 < 2^{2k}-4k$ is equivalent to $2^{k+1} > 4k+1$

Base:
$k=3$
$16 > 13$

Induction hypothesis:
$2^{k+1} > 4k+1$ for all positive integers $k \leq 3$

Inductive step:
We have to show:
$2^{k+2} > 4k+5$

From the induction hypothesis we know:
$2^{k+2} >8k+2$

If we can show:
$8k+2 > 4k +5$ then $2^{k+2} > 4k+5$ must also be true.

This is equal to $k> 3/4$ which is obviously true for $ k \geq 3$
Thus, the equation $2^{k+1} > 4k+1$ must be true for $k \geq 3$

Checking the remaining even cases yields the solution 0 for $N=2$, checking the case $N=1$ yields the solution 1. (the case $N=1$ has to be cheked because $2^N-2N$ is not an even number for $N=1$)

Thus $N=1$ and $N=2$ are the only solutions.
Z K Y
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Al3jandro0000
804 posts
#6
Y by
@above maybe this is more common: Let $P(k): 2^{k+1}>4k+1: k\ge 3$. Easily $P (3) $ works so assume $P(k) $ works for some $k\ge 3$.
$P^{-1}(k+1):4(k+1)+1=(4k+1)+4<2^{k+1}+4 <2^{k+1}+2^{k+1}=2^{k+2} $

So $P (k+1) $ is true and the induction on $P$ holds
Z K Y
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