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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!
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0 replies
jlacosta
Mar 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
J
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Circles and Chords
steven_zhang123   1
N 26 minutes ago by steven_zhang123
(1) Let \( A \) , \( B \) and \( C \) be points on circle \( O \) divided into three equal parts. Construct three equal circles \( O_1 \), \( O_2 \), and \( O_3 \) tangent to \( O \) internally at points \( A \), \( B \), and \( C \) respectively. Let \( P \) be any point on arc \( AC \), and draw tangents \( PD \), \( PE \), and \( PF \) to circles \( O_1 \), \( O_2 \), and \( O_3 \) respectively. Prove that \( PE = PD + PF \).

(2) Let \( A_1 \), \( A_2 \), \( \cdots \), \( A_n \) be points on circle \( O \) divided into \( n \) equal parts. Construct \( n \) equal circles \( O_1 \), \( O_2 \), \( \cdots \), \( O_n \) tangent to \( O \) internally at \( A_1 \), \( A_2 \), \( \cdots \), \( A_n \). Let \( P \) be any point on circle \( O \), and draw tangents \( PB_1 \), \( PB_2 \), \( \cdots \), \( PB_n \) to circles \( O_1 \), \( O_2 \), \( \cdots \), \( O_n \). If the sum of \( k \) of \( PB_1 \), \( PB_2 \), \( \cdots \), \( PB_n \) equals the sum of the remaining \( n-k \) (where \( n \geq k \geq 1 \)), find all such \( n \).
1 reply
steven_zhang123
Mar 23, 2025
steven_zhang123
26 minutes ago
J
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Number theory come back
vicentev   1
N 38 minutes ago by Mathzeus1024
Determine the number of pairs of positive integers \( (p, k) \) such that \( p \) is a prime number and \( p^2 + 2k \) is a perfect square less than 2023. A number is called a perfect square if it is the square of an integer.
1 reply
vicentev
Feb 20, 2025
Mathzeus1024
38 minutes ago
J
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A quadrilateral problem
Itsuukaa   0
an hour ago
Given a quadrilateral ABCD where ABD = ∠70°, ∠ADB = 50° and the length of CD = CB. Suppose AB and CD intersect at point P, while AD and BC intersect at point Q. Determine the size of angle ∠APQ - ∠AQP
0 replies
Itsuukaa
an hour ago
0 replies
J
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D1018 : Can you do that ?
Dattier   1
N 2 hours ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Yesterday at 6:01 AM
Dattier
2 hours ago
J
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prove that a chord is tangent to the incircle
ihategeo_1969   3
N 2 hours ago by bin_sherlo
Source: SORY 2019 P6
Let $ABC$ be a triangle with incenter $I$ and intouch triangle $DEF$. Let $P$ be the foot of the perpendicular from $D$ onto $EF$. Assume that $BP$, $CP$ intersect the sides $AC$, $AB$ in $Y,Z$ respectively. Finally, let the rays $IP$, $YZ$ meet the circumcircle of $\triangle ABC$ in $R$, $X$ respectively. Prove that the tangent from $X$ to the incircle and the line $RD$ meet on the circumcircle of $\triangle ABC$.

Proposed by Aditya Khurmi
3 replies
ihategeo_1969
Sunday at 9:01 PM
bin_sherlo
2 hours ago
J
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Inspired by my own results
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 $ and $ ab+bc+2ca=1. $ Prove that$$a^2+b^2+c^2-k(a+2b+c)abc\geq \frac{2\sqrt{k+3}-k-2}{2 }$$Where $ k>0.$
$$a^2+b^2+c^2-(a+2b+c)abc\geq \frac{1}{2}$$$$a^2+b^2+c^2-2(a+2b+c)abc\geq \sqrt{5}-2$$
0 replies
sqing
2 hours ago
0 replies
J
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Hard limits
Snoop76   0
3 hours ago
$a_n$ and $b_n$ satisfies the following recursion formulas: $a_{0}=1, $ $b_{0}=1$, $ a_{n+1}=a_{n}+b_{n}$$ $ and $ $$ b_{n+1}=(2n+3)b_{n}+a_{n}$. Find $ \lim_{n \to \infty} \frac{a_n}{(2n-1)!!}$ $ $ and $ $ $\lim_{n \to \infty} \frac{b_n}{(2n+1)!!}.$
0 replies
Snoop76
3 hours ago
0 replies
J
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Parallel lines in an acute triangle
buratinogigle   1
N 3 hours ago by buratinogigle
Source: VNTST 2025 P2
Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$. Let $D, E, F$ be the reflections of $H$ over $BC, CA, AB$, respectively, and let $A', B', C'$ be the reflections of $A, B, C$ over $BC, CA, AB$, respectively. Let $S$ be the circumcenter of triangle $A'B'C'$, and let $H'$ be the orthocenter of triangle $DEF$. Define $J$ as the center of the circle passing through the three projections of $H$ onto the lines $B'C', C'A', A'B'$. Prove that $HJ$ is parallel to $H'S$.
1 reply
1 viewing
buratinogigle
3 hours ago
buratinogigle
3 hours ago
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2 degree polynomial
PrimeSol   2
N 3 hours ago by pco
Let $P_{1}(x)= x^2 +b_{1}x +c_{1}, ... , P_{n}(x)=x^2+ b_{n}x+c_{n}$, $P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$ $\forall i,j ,1 \leq i<j \leq n : P_{i}(x) \ne P_{j}(x)$.
$\forall i,j, 1\leq i<j \leq n : Q_{i,j}(x)= P_{i}(x) + P_{j}(x)$ polynomial with only one root.
$max(n)=?$
2 replies
PrimeSol
Yesterday at 6:13 AM
pco
3 hours ago
J
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Inspired by my own results
sqing   0
3 hours ago
Source: Own
Let $ a,b,c\geq 0 $ and $ kab+bc+kca=1. $ Prove that$$a^2+b^2+c^2-(a+kb+kc)abc\geq \frac{2}{2k+1}$$Where $ k\in N^+.$
Let $ a,b,c\geq 0 $ and $ ab+bc+ca=1. $ Prove that$$a^2+b^2+c^2-(a+b+c)abc\geq \frac{2}{3}$$
0 replies
sqing
3 hours ago
0 replies
J
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D1015 : A strange EF for polynomials
Dattier   5
N 4 hours ago by alexheinis
Source: les dattes à Dattier
Find all $P \in \mathbb R[x,y]$ with $P \not\in \mathbb R[x] \cup \mathbb R[y]$ and $\forall g,f$ homeomorphismes of $\mathbb R$, $P(f,g)$ is an homoemorphisme too.
5 replies
Dattier
Mar 16, 2025
alexheinis
4 hours ago
J
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Equation on rational numbers
chrono223   4
N 4 hours ago by Davdav1232
Source: KMO 2022 P8
$p$ is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers $(x,y)$ that satisfy the following equation.
$$p^2 x^4-6px^2+1=y^2$$
4 replies
chrono223
Oct 29, 2022
Davdav1232
4 hours ago
J
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Double factorial inequality
Snoop76   3
N 4 hours ago by Snoop76
Source: own
Show that: $$2n \cdot \sum_{k=0}^n (2k-1)!!{n\choose k}>\sum_{k=0}^n (2k+1)!!{n\choose k}$$Note: consider $(-1)!!=1$ and $n>1$
3 replies
Snoop76
Feb 7, 2025
Snoop76
4 hours ago
J
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An interesting inequality
JK1603JK   1
N 4 hours ago by JK1603JK
Source: unknown
Let a,b,c>=0 and a^2+b^2+c^2+abc=4 then prove \frac{1}{a+b+2}+\frac{1}{b+c+2}+\frac{1}{c+a+2} \le \frac{6-(a+b+c)}{4}
When does equality occur?
1 reply
JK1603JK
Mar 21, 2025
JK1603JK
4 hours ago
J
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J
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Divisibility
RenheMiResembleRice   2
N Mar 23, 2025 by navier3072
Source: Byer
Prove that for all n ∈ ℕ, 133|($11^{\left(n+2\right)}+12^{\left(2n+1\right)}$).
2 replies
RenheMiResembleRice
Mar 23, 2025
navier3072
Mar 23, 2025
J
Divisibility
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Reveal topic tags
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RenheMiResembleRice
241 posts
RenheMiResembleRice
#1 h  Mar 23, 2025, 3:07 AM
Y by
Prove that for all n ∈ ℕ, 133|($11^{\left(n+2\right)}+12^{\left(2n+1\right)}$).
This post has been edited 1 time. Last edited by RenheMiResembleRice, Mar 23, 2025, 3:07 AM
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CM1910
42 posts
CM1910
#2 h  Mar 23, 2025, 9:08 AM
Y by
Information
$$133 | 11^{n + 2} + 12^{2n + 1} \iff 7 | 11^{n + 2} + 12^{2n + 1} \text{ } \wedge \text{ } 19 | 11^{n + 2} + 12^{2n + 1}$$To prove both of them, we look at values of $n \mod 3$.
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navier3072
105 posts
navier3072
#3 h  Mar 23, 2025, 9:50 AM
Y by
$133 \mid 133 \cdot 11^n =121 \cdot 11^n + 12 \cdot 11^n =11^{n+2} +12 \cdot 144^n = 11^{n+2}+12^{2n+1}$
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