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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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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Menelau's theorem
noneofyou34   1
N 12 minutes ago by ethan2011
Please can someone help me prove that orthocenter of a triangle exists by using Menelau's Theorem!
1 reply
noneofyou34
an hour ago
ethan2011
12 minutes ago
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Hard to approach it !
BogG   129
N 38 minutes ago by OronSH
Source: Swiss Imo Selection 2006
Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.
129 replies
BogG
May 25, 2006
OronSH
38 minutes ago
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Simple triangle geometry [a fixed point]
darij grinberg   48
N 42 minutes ago by OronSH
Source: German TST 2004, IMO ShortList 2003, geometry problem 2
Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.
48 replies
darij grinberg
May 18, 2004
OronSH
42 minutes ago
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IMO ShortList 1998, geometry problem 4
orl   14
N 44 minutes ago by OronSH
Source: IMO ShortList 1998, geometry problem 4
Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that
\[ \angle MAB = \angle NAC\quad \mbox{and}\quad \angle MBA = \angle NBC. \]
Prove that
\[ \frac {AM \cdot AN}{AB \cdot AC} + \frac {BM \cdot BN}{BA \cdot BC} + \frac {CM \cdot CN}{CA \cdot CB} = 1. \]
14 replies
orl
Oct 22, 2004
OronSH
44 minutes ago
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Functional equation
Nima Ahmadi Pour   98
N an hour ago by ezpotd
Source: ISl 2005, A2, Iran prepration exam
We denote by $\mathbb{R}^+$ the set of all positive real numbers.

Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property:
\[f(x)f(y)=2f(x+yf(x))\]
for all positive real numbers $x$ and $y$.

Proposed by Nikolai Nikolov, Bulgaria
98 replies
Nima Ahmadi Pour
Apr 24, 2006
ezpotd
an hour ago
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Geometry
noneofyou34   0
an hour ago
Please can someone help me prove that orthocenter of a triangle exists by using Menelau's Theorem!
0 replies
noneofyou34
an hour ago
0 replies
J
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Hard combi
EeEApO   0
an hour ago
In a quiz competition, there are a total of $100 $questions, each with $4$ answer choices. A participant who answers all questions correctly will receive a gift. To ensure that at least one member of my family answers all questions correctly, how many family members need to take the quiz?

Now, suppose my spouse and I move into a new home. Every year, we have twins. Starting at the age of $16$, each of our twin children also begins to have twins every year. If this pattern continues, how many years will it take for my family to grow large enough to have the required number of members to guarantee winning the quiz gift?
0 replies
EeEApO
an hour ago
0 replies
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Inequality with mathematical means
StefanSebez   12
N an hour ago by Sh309had
Source: Serbia JBMO TST 2022 P1
Prove that for all positive real numbers $a$, $b$ the following inequality holds:
\begin{align*} \sqrt{\frac{a^2+b^2}{2}}+\frac{2ab}{a+b}\ge \frac{a+b}{2}+ \sqrt{ab} \end{align*}When does equality hold?
12 replies
StefanSebez
Jun 1, 2022
Sh309had
an hour ago
J
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Really fun geometry problem
Sadigly   4
N 2 hours ago by Double07
Source: Azerbaijan Senior MO 2025 P6
In the acute triangle $ABC$ with $AB<AC$, the foot of altitudes from $A,B,C$ to the sides $BC,CA,AB$ are $D,E,F$, respectively. $H$ is the orthocenter. $M$ is the midpoint of segment $BC$. Lines $MH$ and $EF$ intersect at $K$. Let the tangents drawn to circumcircle $(ABC)$ from $B$ and $C$ intersect at $T$. Prove that $T;D;K$ are colinear
4 replies
Sadigly
3 hours ago
Double07
2 hours ago
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Orthocenter
jayme   8
N 2 hours ago by cj13609517288
Dear Mathlinkers,

1. ABC an acuatangle triangle
2. H the orthcenter of ABC
3. DEF the orthic triangle of ABC
4. A* the midpoint of AH
5. X the point of intersection of AH and EF.

Prove : X is the orthocenter of A*BC.

Sincerely
Jean-Louis
8 replies
jayme
Mar 25, 2015
cj13609517288
2 hours ago
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Constructing graphs satisfying conditions on degrees
jlammy   19
N 2 hours ago by de-Kirschbaum
Source: EGMO 2017 P4
Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:

(i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$.

(ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.
19 replies
jlammy
Apr 9, 2017
de-Kirschbaum
2 hours ago
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Hard Inequality
William_Mai   14
N 2 hours ago by hi2024IMOp14
Given $a, b, c \in \mathbb{R}$ such that $a^2 + b^2 + c^2 = 1$.
Find the minimum value of $P = ab + 2bc + 3ca$.

Source: Pham Le Van
14 replies
William_Mai
May 3, 2025
hi2024IMOp14
2 hours ago
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exist solutions?
teomihai   4
N 2 hours ago by teomihai
Find how many perfect squares of five different digits there are, with elements from the set ${0,1,4,6,9}$.
4 replies
teomihai
2 hours ago
teomihai
2 hours ago
J
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Geometry
MTA_2024   7
N 2 hours ago by hi2024IMOp14
Let $ABC$ be a triangle such that $AB=3$,$BC=5$ and $AC=6$.Let $D$ be a point on side $AC$ and $E$ one on side $BC$ so that the line $DE$ is tangent to the incircle of $\triangle ABC$ .
Evaluate the perimeter of triangle $\triangle CDE$.
7 replies
MTA_2024
Yesterday at 6:52 PM
hi2024IMOp14
2 hours ago
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BMT 2018 Algebra Round Problem 7
IsabeltheCat   5
N Apr 22, 2025 by P162008
Let $$h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$$Find $$\sum_{n=0}^\infty \frac{h_n}{n!}.$$
5 replies
IsabeltheCat
Dec 3, 2018
P162008
Apr 22, 2025
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BMT 2018 Algebra Round Problem 7
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Reveal topic tags
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IsabeltheCat
4242 posts
IsabeltheCat
#1 h  Dec 3, 2018, 3:36 AM • 2 Y
Y by Adventure10, Mango247
Let $$h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$$Find $$\sum_{n=0}^\infty \frac{h_n}{n!}.$$
This post has been edited 1 time. Last edited by IsabeltheCat, Dec 3, 2018, 11:11 PM
Reason: wrong dummy variable was used previously
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rzlng
99 posts
rzlng
#2 h  Dec 3, 2018, 4:51 AM • 3 Y
Y by Srofller, Adventure10, Mango247
sketch
@below, thanks. fixed
This post has been edited 1 time. Last edited by rzlng, Dec 3, 2018, 5:11 AM
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TomCalc
1635 posts
TomCalc
#3 h  Dec 3, 2018, 5:05 AM • 1 Y
Y by Adventure10
IsabeltheCat wrote:
Find $$\sum_{k=0}^\infty \frac{h_n}{n!}.$$
The second sum has a wrong dummy variable and @above the answer is correct.
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NikoIsLife
9657 posts
NikoIsLife
#4 h  Dec 3, 2018, 10:59 AM • 3 Y
Y by Adventure10, Mango247, soryn
Solution
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IsabeltheCat
4242 posts
IsabeltheCat
#5 h  Dec 3, 2018, 11:11 PM • 2 Y
Y by Adventure10, Mango247
TomCalc wrote:
IsabeltheCat wrote:
Find $$\sum_{k=0}^\infty \frac{h_n}{n!}.$$
The second sum has a wrong dummy variable and @above the answer is correct.

Thanks for pointing that out. I fixed the error in the original post.
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P162008
186 posts
P162008
#6 h  Apr 22, 2025, 6:56 AM
Y by
Storage
This post has been edited 3 times. Last edited by P162008, Apr 28, 2025, 9:40 PM
Reason: Typo
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