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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:
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0 replies
1 viewing
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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Number of sign change in cos ka
Rohit-2006   2
N 2 minutes ago by Rohit-2006
Let $0\leq\alpha\leq\pi$. Denote by $V_n(\alpha)$ the number of changes of signs in the
sequence
$$1, cos \alpha, cos 2\alpha, . . . , cos n\alpha.$$Then prove that
$$\lim_{n\rightarrow\infty}\frac{V_n(\alpha)}{n}=\frac{\alpha}{\pi}$$.
2 replies
Rohit-2006
31 minutes ago
Rohit-2006
2 minutes ago
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Collinearity with orthocenter
liberator   178
N 4 minutes ago by endless_abyss
Source: IMO 2013 Problem 4
Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes from $B$ and $C$, respectively. Denote by $\omega_1$ is the circumcircle of $BWN$, and let $X$ be the point on $\omega_1$ such that $WX$ is a diameter of $\omega_1$. Analogously, denote by $\omega_2$ the circumcircle of triangle $CWM$, and let $Y$ be the point such that $WY$ is a diameter of $\omega_2$. Prove that $X,Y$ and $H$ are collinear.

Proposed by Warut Suksompong and Potcharapol Suteparuk, Thailand
178 replies
liberator
Jan 4, 2016
endless_abyss
4 minutes ago
J
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Obsolete NT
GreekIdiot   1
N 11 minutes ago by amapstob
Source: older isl
Find all $n \in \mathbb{N}$ greater than $1$, such that, if $gcd(a,b)=1$, then $a \equiv b \: mod \: n \iff ab \equiv 1 \: mod \: n$
1 reply
GreekIdiot
2 hours ago
amapstob
11 minutes ago
J
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2x+1 is a perfect square but the following x+1 integers are not.
Sumgato   6
N 16 minutes ago by MuradSafarli
Source: Spain Mathematical Olympiad 2018 P1
Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares.
6 replies
Sumgato
Mar 17, 2018
MuradSafarli
16 minutes ago
J
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Nice problem
hanzo.ei   0
21 minutes ago
Given two positive integers \( m, n \) satisfying \( m > n \) and their sum is an even number, consider the quadratic polynomial:

\[ P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1). \]
Prove that all roots of \( P(x) \) are positive integers but are not perfect squares.
0 replies
hanzo.ei
21 minutes ago
0 replies
J
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Checkerboard
Ecrin_eren   4
N 23 minutes ago by Ecrin_eren
On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)
4 replies
Ecrin_eren
Mar 21, 2025
Ecrin_eren
23 minutes ago
J
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a^2+b^2+c^2=2(ab+bc+ca)
sqing   13
N 25 minutes ago by sqing
Source: Own
Let $a,b,c>0 $ and $a^2+b^2+c^2=2(ab+bc+ca).$ Prove that
$$a^3+b^3+c^3\geq \frac{33}{2}abc$$$$a^2(b+c)+c^2(a+b)\geq \frac{5}{2}abc$$$$a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2)\geq \frac{21}{2}abc$$$$\left( \frac{a}{b} + \frac{b}{c} \right) \left( \frac{b}{a} + \frac{a}{c} \right) \geq\frac{25}{16}$$$$\left( \frac{a}{b} + \frac{b}{c} \right) \left( \frac{b}{a} + \frac{a}{c} \right)\left( \frac{c}{a} + \frac{c}{b} \right)\geq \frac{25}{2}$$
13 replies
sqing
Jul 17, 2022
sqing
25 minutes ago
J
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1/sqrt(5) ???
navi_09220114   3
N 31 minutes ago by math_comb01
Source: Own. Malaysian IMO TST 2025 P12
Two circles $\omega_1$ and $\omega_2$ are externally tangent at a point $A$. Let $\ell$ be a line tangent to $\omega_1$ at $B\neq A$ and $\omega_2$ at $C\neq A$. Let $BX$ and $CY$ be diameters in $\omega_1$ and $\omega_2$ respectively. Suppose points $P$ and $Q$ lies on $\omega_2$ such that $XP$ and $XQ$ are tangent to $\omega_2$, and points $R$ and $S$ lies on $\omega_1$ such that $YR$ and $YS$ are tangent to $\omega_1$.

a) Prove that the points $P$, $Q$, $R$, $S$ lie on a circle $\Gamma$.

b) Prove that the four segments $XP$, $XQ$, $YR$, $YS$ determine a quadrilateral with an incircle $\gamma$, and its radius is $\displaystyle\frac{1}{\sqrt{5}}$ times the radius of $\Gamma$.

Proposed by Ivan Chan Kai Chin
3 replies
navi_09220114
Yesterday at 1:10 PM
math_comb01
31 minutes ago
J
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Very hard math problem
slimshady360   0
39 minutes ago

In a chess tournament with n ≥ 5 players, each player played all other players. One gets a point for a
win, half a point for a draw, and zero points for a loss. At the end of the tournament, each player had
a different number of points. Prove that the second and third ranked players had together more points
than the winner of the tournament.
0 replies
slimshady360
39 minutes ago
0 replies
J
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Interesting inequality
sqing   2
N 44 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0,(ab+c^2)(ac+b^2)\neq 0 $ and $ a+b+c=3 . $ Prove that
$$ \frac{1}{ab+c^2}+\frac{1}{ac+b^2} \geq\frac{3}{4} $$$$ \frac{1}{ab+2c^2}+\frac{1}{ac+2b^2} \geq\frac{4}{9} $$
2 replies
+1 w
sqing
an hour ago
sqing
44 minutes ago
J
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The three lines AA', BB' and CC' meet on the line IO
WakeUp   43
N an hour ago by cursed_tangent1434
Source: Romanian Master Of Mathematics 2012
Let $ABC$ be a triangle and let $I$ and $O$ denote its incentre and circumcentre respectively. Let $\omega_A$ be the circle through $B$ and $C$ which is tangent to the incircle of the triangle $ABC$; the circles $\omega_B$ and $\omega_C$ are defined similarly. The circles $\omega_B$ and $\omega_C$ meet at a point $A'$ distinct from $A$; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA',BB'$ and $CC'$ are concurrent at a point on the line $IO$.

(Russia) Fedor Ivlev
43 replies
WakeUp
Mar 3, 2012
cursed_tangent1434
an hour ago
J
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Simson line intersecting radius
XAN4   0
an hour ago
Source: own
Point $P$ lies on $\triangle{ABC}$. The Simson line of $P$ intersects Line $OP$ at $Q$. $A'$ and $B'$ are the reflections of $A$ and $B$ over $Q$. Prove that the circles $\odot(A'BC)$ and $\odot(AB'C)$ intersects at a point on line $OP$.
0 replies
+1 w
XAN4
an hour ago
0 replies
J
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D1015 : A strange EF for polynomials
Dattier   4
N an hour ago by Dattier
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.
4 replies
Dattier
Mar 16, 2025
Dattier
an hour ago
J
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Elegant inequality
SunnyEvan   4
N an hour ago by SunnyEvan
Source: proposed by Zhenping An
Let $a$, $b$, $c$, $d$ be non-negative real numbers such that
\[2a+2b+2c+2d+ab+bc+cd+da+3=abcd.\]prove that : \[\sqrt[4]{abc}+\sqrt[4]{bcd}+\sqrt[4]{cda}+\sqrt[4]{dab}\le\sqrt[4]{27(1+a)(1+b)(1+c)(1+d)}.\]
4 replies
SunnyEvan
Yesterday at 11:32 AM
SunnyEvan
an hour ago
J
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J
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Inequality
Marry123   4
N Apr 8, 2017 by envision2017
If $n\geq2$ show this for $a_1,a_2,...,a_n\in(0,1)$:
$(1-a_1)(1-a_2)....(1-a_n)>1-a_1-a_2-....-a_n$
4 replies
Marry123
Apr 4, 2017
envision2017
Apr 8, 2017
J
Inequality
G H J
Reveal topic tags
inequalities
L
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Marry123
136 posts
Marry123
#1 h  Apr 4, 2017, 1:02 AM • 1 Y
Y by Adventure10
If $n\geq2$ show this for $a_1,a_2,...,a_n\in(0,1)$:
$(1-a_1)(1-a_2)....(1-a_n)>1-a_1-a_2-....-a_n$
This post has been edited 3 times. Last edited by Marry123, Apr 4, 2017, 1:07 AM
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m.yekta
57 posts
m.yekta
#2 h  Apr 4, 2017, 1:23 AM • 1 Y
Y by Adventure10
See this:
http://mathworld.wolfram.com/WeierstrassProductInequality.html
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Marry123
136 posts
Marry123
#3 h  Apr 4, 2017, 1:31 AM • 2 Y
Y by Adventure10, Mango247
Great!
Another solution?
Z K Y
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rezareza14
259 posts
rezareza14
#4 h  Apr 7, 2017, 3:23 PM • 2 Y
Y by Adventure10, Mango247
Marry123 wrote:
Great!
Another solution?

If $a_1+a_2+...+a_n\ge 1$ we are done!(because $RHS\le 0$ but $LHS>0$)
If $a_1+a_2+...+a_n<1$ Try Induction!
Good Luck :)
This post has been edited 1 time. Last edited by rezareza14, Apr 7, 2017, 3:23 PM
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envision2017
101 posts
envision2017
#5 h  Apr 8, 2017, 6:10 AM • 1 Y
Y by Adventure10
this is Bernoull inequality
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