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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.
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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
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Is this NT?
navi_09220114   0
13 minutes ago
Source: Malaysian IMO TST 2025 P11
Let $n$, $d$ be positive integers such that $d>\frac{n}{2}$. Suppose $a_1, a_2,\cdots,a_{d+2}$ is a sequence of integers satisfying $a_{d+1}=a_1$, $a_{d+2}=a_2$, and for all indices $1\le i_1<i_2<\cdots <i_s\le d$, $$a_{i_1}+a_{i_2}+\cdots+a_{i_s}\not\equiv 0\pmod n$$Prove that there exists $1\le i\le d$ such that $$a_{i+1}\equiv a_i \pmod n \quad \text{or} \quad a_{i+1}\equiv a_i+a_{i+2} \pmod n$$
Proposed by Yeoh Zi Song
0 replies
navi_09220114
13 minutes ago
0 replies
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(a,b)-cuts for circles
navi_09220114   0
14 minutes ago
Source: Malaysian IMO TST 2025 P10
Let $m$ and $n$ be positive integers. Find all pairs of non-negative integers $a$ and $b$ that always satisfy the following condition:

Given any configuration of $m$ white dots and $n$ black dots on a circle, there always exist a line cutting the circle into two arcs, one of which consists of exactly $a$ white dots and $b$ black dots.

Proposed by Tan Min Heng
0 replies
navi_09220114
14 minutes ago
0 replies
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sum divides n-th moment
navi_09220114   0
15 minutes ago
Source: Own. Malaysian IMO TST 2025 P9
Given four distinct positive integers $a<b<c<d$ such that $\gcd(a,b,c,d)=1$, find the maximum possible number of integers $1\le n\le 2025$ such that $$a+b+c+d\mid a^n+b^n+c^n+d^n$$
Proposed by Ivan Chan Kai Chin
0 replies
navi_09220114
15 minutes ago
0 replies
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Equilateral triangle fun
navi_09220114   0
17 minutes ago
Source: Own. Malaysian IMO TST 2025 P8
Let $ABC$ be an equilateral triangle, and $P$ is a point on its incircle. Let $\omega_a$ be the circle tangent to $AB$ passing through $P$ and $A$. Similarly, let $\omega_b$ be the circle tangent to $BC$ passing through $P$ and $B$, and $\omega_c$ be the circle tangent to $CA$ passing through $P$ and $C$.

Prove that the circles $\omega_a$, $\omega_b$, $\omega_c$ has a common tangent line.

Proposed by Ivan Chan Kai Chin
0 replies
navi_09220114
17 minutes ago
0 replies
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Equality case being all distinct reals?
navi_09220114   0
18 minutes ago
Source: Own. Malaysian IMO TST 2025 P7
Given a real polynomial $P(x)=a_{2024}x^{2024}+\cdots+a_1x+a_0$ with degree $2024$, such that for all positive reals $b_1, b_2,\cdots, b_{2025}$ with product $1$, then; $$P(b_1)+P(b_2)+\cdots +P(b_{2025})\ge 0$$Suppose there exist positive reals $c_1, c_2, \cdots, c_{2025}$ with product $1$, such that; $$P(c_1)+P(c_2)+ \cdots +P(c_{2025})=0$$Is it possible that the values $c_1, c_2, \cdots, c_{2025}$ are all distinct?

Proposed by Ivan Chan Kai Chin
0 replies
navi_09220114
18 minutes ago
0 replies
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a+b+c=3 inequality
jokehim   0
18 minutes ago
Source: my problem
Problem. Given non-negative real numbers $a,b,c$ satisfying $a+b+c=3.$ Prove that $$\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}\le \frac{9}{ab+bc+ca+6}.$$Proposed by Phan Ngoc Chau
0 replies
jokehim
18 minutes ago
0 replies
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Round up to the nearest power of two
navi_09220114   0
19 minutes ago
Source: Own. Malaysian IMO TST 2025 P6
A sequence $2^{a_1}, 2^{a_2}, \cdots,2^{a_m}$ is called \textit{good}, if $a_i$ are non-negative integers, and $a_{i+1}-a_{i}$ is either $0$ or $1$ for all $1\le i\le m-1$.

Fix a positive integer $n$, and Ivan has a whiteboard with some ones written on it. In each step, he may erase any good sequence $2^{a_1}, 2^{a_2}, \cdots,2^{a_m}$ that appears on the whiteboard, and then he writes the number $2^k$ such that $$2^{k-1}<2^{a_1}+2^{a_2}+\cdots+2^{a_m}\le 2^{k}$$Suppose Ivan starts with the least possible number of ones to obtain $2^n$ after some steps, determine the minimum number of steps he will need in order to do so.

Proposed by Ivan Chan Kai Chin
0 replies
navi_09220114
19 minutes ago
0 replies
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Whiteboard magic again
navi_09220114   0
21 minutes ago
Source: Malaysian IMO TST 2025 P5
Fix positive integers $n$ and $k$, and $2n$ positive (not neccesarily distinct) real numbers $a_1,\cdots, a_n$, $b_1, \cdots, b_n$. An equation is written on a whiteboard: $$t=*\times*\times\cdots\times*$$where $t$ is a fixed positive real number, with exactly $k$ asterisks.

Ebi fills each asterisk with a number from $a_1, a_2,\cdots, a_n$, while Rubi fills each asterisk with a number from $b_1, b_2,\cdots, b_n$, so that the equation on the whiteboard is correct. Suppose for every positive real number $t$, the number of ways for Ebi and Rubi to do so are equal.

Prove that the sequences $a_1,\cdots, a_n$ and $b_1, \cdots, b_n$ are permutations of each other.

(Note: $t=a_1a_2a_3$ and $t=a_2a_3a_1$ are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example $t=a_1a_1a_2$.)

Proposed by Wong Jer Ren
0 replies
navi_09220114
21 minutes ago
0 replies
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Isogonal from antipodes
navi_09220114   0
22 minutes ago
Source: Own. Malaysian IMO TST 2025 P4
Let $ABC$ be a triangle, with incenter $I$ and $A$-excenter $J$. The lines $BI$, $CI$, $BJ$ and $CJ$ intersect the circumcircle of $ABC$ at $P$, $Q$, $R$ and $S$ respectively. Let $IM$, $JN$ be diameters in the circumcircles of triangles $IPQ$ and $JRS$ respectively.

Prove that $\angle BAM+\angle CAN=180^{\circ}$.

Proposed by Ivan Chan Kai Chin
0 replies
navi_09220114
22 minutes ago
0 replies
J
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AM-GM FE ineq
navi_09220114   0
24 minutes ago
Source: Own. Malaysian IMO TST 2025 P3
Let $\mathbb R$ be the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ where there exist a real constant $c\ge 0$ such that $$x^3+y^2f(y)+zf(z^2)\ge cf(xyz)$$holds for all reals $x$, $y$, $z$ that satisfy $x+y+z\ge 0$.

Proposed by Ivan Chan Kai Chin
0 replies
navi_09220114
24 minutes ago
0 replies
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a^{2000}+b^{2000}=a^{1998}+b^{1998} (Greece Junior 1999 p1)
parmenides51   2
N an hour ago by ali123456
Show that if $a,b$ are positive real numbers such that $a^{2000}+b^{2000}=a^{1998}+b^{1998}$ then $a^2+b^2 \le 2$.
2 replies
parmenides51
Mar 17, 2020
ali123456
an hour ago
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Number Theory
Foxellar   0
2 hours ago
Evaluate the following expression:

\[ 19^{17^{15^{13^{11^{9^{7^{5^3}}}}}}} \mod 100 \]
0 replies
Foxellar
2 hours ago
0 replies
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High School Math
Foxellar   1
N 2 hours ago by Sivin
\[ \textbf{Problem:} \]
Let \( r_1, r_2, \dots, r_{20} \) be the roots of the polynomial

\[ P(x) = x^{20} - 7x^3 + 1. \]
Find the value of

\[ S = \frac{1}{r_1^2 + 1} + \frac{1}{r_2^2 + 1} + \dots + \frac{1}{r_{20}^2 + 1} \]
in the form \( \frac{m}{n} \), where \( m \) and \( n \) are coprime positive integers. Determine the value of \( m + 2n \).
1 reply
Foxellar
3 hours ago
Sivin
2 hours ago
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An inequality
jokehim   3
N 2 hours ago by Indpsolver
Let $a,b,c \in \mathbb{R}: a+b+c=3$ then prove $$\color{black}{\frac{a^2}{a^{2}-2a+3}+\frac{b^2}{b^{2}-2b+3}+\frac{c^2}{c^{2}-2c+3}\ge \frac{3}{2}.}$$
3 replies
jokehim
Yesterday at 3:05 PM
Indpsolver
2 hours ago
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Trig Identity
mithu542   5
N Yesterday at 6:40 AM by Euler-epii10
Prove the following identity:
$$\dfrac{1+\tan \theta}{1-\tan \theta}=\tan \left(\theta+\dfrac{\pi}{4}\right),$$where $\theta$ is in radians.
5 replies
mithu542
Thursday at 4:09 PM
Euler-epii10
Yesterday at 6:40 AM
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Trig Identity
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Reveal topic tags
trigonometry
tangent
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mithu542
1557 posts
mithu542
#1 h  Thursday at 4:09 PM
Y by
Prove the following identity:
$$\dfrac{1+\tan \theta}{1-\tan \theta}=\tan \left(\theta+\dfrac{\pi}{4}\right),$$where $\theta$ is in radians.
This post has been edited 3 times. Last edited by mithu542, Thursday at 8:49 PM
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Soupboy0
164 posts
Soupboy0
#2 h  Thursday at 4:17 PM
Y by
using tan(x+y) the identity becomes true
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mathprodigy2011
231 posts
mathprodigy2011
#4 h  Thursday at 6:54 PM • 1 Y
Y by aidan0626
Soupboy0 wrote:
using tan(x+y) the identity becomes true

you cant really use that because tan pi/2 is undefined, you have to do sin(x+pi/2)/cos(x+pi/2)
This post has been edited 1 time. Last edited by mathprodigy2011, Thursday at 6:55 PM
Reason: .
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mithu542
1557 posts
mithu542
#6 h  Thursday at 9:13 PM
Y by
Oops I typed it wrong I meant $\pi/4$ (sorry) so @Soupboy0 would be the right idea.
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franklin2013
174 posts
franklin2013
#7 h  Thursday at 10:19 PM
Y by
Using the $\tan(x+y)$ identity as @Soupboy0 said, we get:
$$\tan\left(\theta+\frac{\pi}{4}\right)=\frac{\tan(\theta)+\tan\left(\frac{\pi}{4}\right)}{1-\tan(\theta)\tan\left(\frac{\pi}{4}\right)}=\frac{1+\tan(\theta)}{1-\tan(\theta)},$$as desired.
This post has been edited 1 time. Last edited by franklin2013, Thursday at 10:20 PM
Reason: minor formatting edit
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Euler-epii10
1 post
Euler-epii10
#8 h  Yesterday at 6:40 AM
Y by
Alternative solution:
\[\frac{1+\tan x}{1-\tan x}=\frac{\cos x+\sin x}{\cos x-\sin x}=\frac{\sin \left( x+\frac{\pi }{4} \right)}{\cos \left( x+\frac{\pi }{4} \right)}=\tan \left( x+\frac{\pi }{4} \right)\]
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