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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.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

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
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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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Polynomials and powers
rmtf1111   26
N a minute ago by ihategeo_1969
Source: RMM 2018 Day 1 Problem 2
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying
$$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$
26 replies
rmtf1111
Feb 24, 2018
ihategeo_1969
a minute ago
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Inspired by my own results
sqing   2
N 3 minutes ago by invisibleman
Source: Own
Let $ a , b $ be reals such that $ a+b+ab=1. $ Show that$$ 1-\frac{1 }{\sqrt2}\le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le 1+\frac{1 }{\sqrt2} $$Let $ a , b\geq 0 $ and $ a+b+ab=1. $ Show that$$ \frac{3}{2}\le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le 1+\frac{1 }{\sqrt2} $$
2 replies
sqing
Yesterday at 8:32 AM
invisibleman
3 minutes ago
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A problem
kwin   3
N 8 minutes ago by kwin
Let x, y, z > 0 and sqrt(x) + sqrt(y) + sqrt(z) >= 1. Prove that:
x^2+y^2+z^2 + 7(xy+yz+zx) >= 2.sqrt(2).sqrt((a+b)(b+c)(c+a)
Sorry, This is the first time I post so I don't know how to text as mathtype.
3 replies
kwin
Feb 10, 2025
kwin
8 minutes ago
J
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number theory
MuradSafarli   8
N 26 minutes ago by Sadigly
Find all natural numbers \( k \) such that

\[ 4k^3 + 4k + 1 \]
is a perfect square.
8 replies
MuradSafarli
Mar 14, 2025
Sadigly
26 minutes ago
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Friendsss
Titibuuu   3
N 29 minutes ago by Titibuuu
Source: Cono Sur
Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$. Two subsets $A$ and $B$ of $S$ are said to be friends if the following conditions are true:

[list]
[*] They do not share any elements.
[*] They both have the same number of elements.
[*] The product of all elements from $A$ equals the product of all elements from $B$.
[/list]
Prove that there are two subsets of $S$ that are friends such that each one of them contains at least $738$ elements.[/quote]
3 replies
Titibuuu
Yesterday at 8:58 AM
Titibuuu
29 minutes ago
J
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Another Looooong Geo for Opening to Day 3
AlperenINAN   0
29 minutes ago
Source: Turkey TST 2025 P7
Let $\omega$ be a circle on the plane. Let $\omega_1$ and $\omega_2$ be circles which are internally tangent to $\omega$ at points $A$ and $B$ respectively. Let the centers of $\omega_1$ and $\omega_2$ be $O_1$ and $O_2$ respectively and let the intersection points of $\omega_1$ and $\omega_2$ be $X$ and $Y$. Assume that $X$ lies on the line $AB$. Let the common external tangent of $\omega_1$ and $\omega_2$ that is closer to point $Y$ be tangent to the circles $\omega_1$ and $\omega_2$ at $K$ and $L$ respectively. Let the second intersection point of the line $AK$ and $\omega$ be $P$ and let the second intersection point of the circumcircle of $PKL$ and $\omega$ be $S$. Let the circumcenter of $AKL$ be $Q$ and let the intersection points of $SQ$ and $O_1O_2$ be $R$. Prove that
$$\frac{\overline{O_1R}}{\overline{RO_2}}=\frac{\overline{AX}}{\overline{XB}}$$
0 replies
AlperenINAN
29 minutes ago
0 replies
J
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Looooong Geo Finale for Day 2
AlperenINAN   0
41 minutes ago
Source: Turkey TST 2025 P6
Let $ABC$ be a scalene triangle with incenter $I$ and incircle $\omega$. Let the tangency points of $\omega$ to $BC,AC\text{ and } AB$ be $D,E,F$ respectively. Let the line $EF$ intersect the circumcircle of $ABC$ at the points $G, H$. Assume that $E$ lies between the points $F$ and $G$. Let $\Gamma$ be a circle that passes through $G$ and $H$ and that is tangent to $\omega$ at the point $M$ which lies on different semi-planes with $D$ with respect to the line $EF$. Let $\Gamma$ intersect $BC$ at points $K$ and $L$ and let the second intersection point of the circumcircle of $ABC$ and the circumcircle of $AKL$ be $N$. Prove that the intersection point of $NM$ and $AI$ lies on the circumcircle of $ABC$ if and only if the intersection point of $HB$ and $GC$ lies on $\Gamma$.
0 replies
AlperenINAN
41 minutes ago
0 replies
J
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USAMO 1985 #4
Mrdavid445   6
N an hour ago by anudeep
There are $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\left\lfloor\frac{n}{2}\right\rfloor-1$ of them, each of whom either knows both or else knows neither of the two. Assume that knowing is a symmetric relation, and that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.
6 replies
Mrdavid445
Jul 26, 2011
anudeep
an hour ago
J
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Polygon formed by the edges of an infinite chessboard
AlperenINAN   0
an hour ago
Source: Turkey TST 2025 P5
Let $P$ be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers $a_1,a_2,a_3$ represent the number of unit squares that have exactly $1,2\text{ or } 3$ edges on the boundary of $P$ respectively. Find the largest real number $k$ such that the inequality $a_1+a_2>ka_3$ holds for each polygon constructed with these conditions.
0 replies
AlperenINAN
an hour ago
0 replies
J
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Nice FE as the First Day Finale
swynca   0
an hour ago
Source: 2025 Turkey TST P3
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{R}-\{0\}$,
$$ f(x) \neq 0 \text{ and } \frac{f(x)}{f(y)} + \frac{f(y)}{f(x)} - f \left( \frac{x}{y}-\frac{y}{x} \right) =2 $$
0 replies
swynca
an hour ago
0 replies
J
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Arithmetic Sequence about Number of Primes
swynca   0
an hour ago
Source: 2025 Turkey TST P2
For all positive integers $n$, the function $\gamma: \mathbb{Z}^+ \to \mathbb{Z}_{\geq 0}$ is defined as, $\gamma(1) = 0$ and for all $n > 1$, if the prime factorization of $n$ is $n = p_1^{\alpha_1} p_2^{\alpha_2} \dots p_k^{\alpha_k},$ then $\gamma(n) = \alpha_1 + \alpha_2 + \dots + \alpha_k$. We have an arithmetic sequence $X = \{x_i\}_{i=1}^{\infty}$. If for a positive integer $a > 1$, the sequence $\{ \gamma(a^{x_i} -1) \}$ is also an arithmetic sequence, show that the sequence $X$ has to be constant.
0 replies
+1 w
swynca
an hour ago
0 replies
J
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unnecessary wrapped FE on Q
iStud   1
N an hour ago by jasperE3
Source: Monthly Contest KTOM March 2025 P3 Essay
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that
\[f(f(f(\frac{x+y}{2}))+x+y)=f(x)+f(y)+f(\frac{x+y}{2})\]for all rational numbers $x,y$.

Hint
1 reply
iStud
4 hours ago
jasperE3
an hour ago
J
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Minimal Grouping in a Complete Graph
swynca   0
an hour ago
Source: 2025 Turkey TST P1
In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.
0 replies
swynca
an hour ago
0 replies
J
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Like Father Like Son... (or Like Grandson?)
AlperenINAN   0
an hour ago
Source: Turkey TST 2025 P4
Let $a,b,c$ be given pairwise coprime positive integers where $a>bc$. Let $m<n$ be positive integers. We call $m$ to be a grandson of $n$ if and only if, for all possible piles of stones whose total mass adds up to $n$ and consist of stones with masses $a,b,c$, it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of $m$. Find the greatest possible number that doesn't have any grandsons.
0 replies
AlperenINAN
an hour ago
0 replies
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J
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2 var inquality
sqing   6
N Yesterday at 1:40 PM by ionbursuc
Source: Own
Let $ a , b\geq 0 $ and $ \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{3}{2}. $ Show that$$ a+b+ab\geq1$$Let $ a , b\geq 0 $ and $ \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{5}{6}. $ Show that$$ a+b+ab\geq2$$
6 replies
sqing
Yesterday at 4:06 AM
ionbursuc
Yesterday at 1:40 PM
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2 var inquality
G H J
Reveal topic tags
inequalities proposed
inequalities
L
G H BBookmark kLocked kLocked NReply
Source: Own
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sqing
41055 posts
sqing
#1 h  Yesterday at 4:06 AM
Y by
Let $ a , b\geq 0 $ and $ \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{3}{2}. $ Show that$$ a+b+ab\geq1$$Let $ a , b\geq 0 $ and $ \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{5}{6}. $ Show that$$ a+b+ab\geq2$$
This post has been edited 2 times. Last edited by sqing, Yesterday at 12:32 PM
Z K Y
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sqing
41055 posts
sqing
#2 h  Yesterday at 4:13 AM
Y by
Let $ a , b\geq 0 $ and $ \frac{1}{a^2+1}+\frac{1}{b^2+1}\le \frac{15+\sqrt 5 }{10}. $ Show that$$ a+b+a^2 +b^2 \geq 1$$Let $ a , b\geq 0 $ and $ \frac{1}{a^2+1}+\frac{1}{b^2+1}\ge \frac{9 }{5} . $ Show that$$ a+b+a^2+ab+b^2 \leq1$$
This post has been edited 1 time. Last edited by sqing, Yesterday at 12:32 PM
Z K Y
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ionbursuc
944 posts
ionbursuc
#4 h  Yesterday at 12:25 PM
Y by
sqing wrote:
Let $ a,b >0 . $ Prove that
$$ \frac{a+b}{2b}+\frac{2b}{a+b}+1\geq \sqrt{3\left(\frac{a+b}{2b}+\frac{2b}{a+b}+1\right)} $$

$\frac{a+b}{2b}+\frac{2b}{a+b}+1\ge \sqrt{3\left( \frac{a+b}{2b}+\frac{2b}{a+b}+1 \right)}\Leftrightarrow \frac{a+b}{2b}+\frac{2b}{a+b}\ge 2$-it's true
Z K Y
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ionbursuc
944 posts
ionbursuc
#5 h  Yesterday at 12:26 PM • 1 Y
Y by MihaiT
sqing wrote:
Let $ a,b >0 . $ Prove that
$$ \frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}+1\geq \sqrt{3\left(\frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}+1\right)} $$Where $ k>0. $

$\frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}+1\ge \sqrt{3\left( \frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}+1 \right)}\Leftrightarrow \frac{a+kb}{(k+1)b}+\frac{(k+1)b}{a+kb}\ge 2$
Z K Y
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sqing
41055 posts
sqing
#6 h  Yesterday at 12:30 PM • 1 Y
Y by MihaiT
Good.Thanks.
Let $ a,b,c>0 . $ Prove that$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq\sqrt{3\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right)}$$
This post has been edited 1 time. Last edited by sqing, Yesterday at 1:14 PM
Z K Y
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ionbursuc
944 posts
ionbursuc
#7 h  Yesterday at 1:29 PM
Y by
Let $a,b>0.$Prove that $\left( \frac{a+b}{2a}+\frac{2a}{a+b}+2 \right)\left( \frac{a+b}{2b}+\frac{2b}{a+b}+2 \right)\ge 12+2\left( \sqrt[4]{\frac{a}{b}}+\sqrt[4]{\frac{b}{a}} \right)$
Z K Y
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ionbursuc
944 posts
ionbursuc
#8 h  Yesterday at 1:40 PM
Y by
Let $a,b>0.$Prove that $\left( \frac{a+b}{2a}+\frac{2a}{a+b}+1 \right)\left( \frac{a+b}{2b}+\frac{2b}{a+b}+1 \right)\ge 5+2\left( \sqrt[4]{\frac{a}{b}}+\sqrt[4]{\frac{b}{a}} \right)$
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