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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.

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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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Show a inequalitie from another
NiltonCesar   0
28 minutes ago
Mathematics Olympiad Problem from the State of São Paulo – Brazil

It is known that
$e^x > 1 + x$ for all $x \in \mathbb{R}*$. Using this inequality, prove that:

(a) For all $x > 0$,
$e^x > 1 + x + \frac{x^2}{4}$

(b) There exists a real constant $x_0$ such that, for all $x > x_0$,
$e^x > x^{2003}$.
0 replies
NiltonCesar
28 minutes ago
0 replies
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Cute NT Problem
M11100111001Y1R   6
N an hour ago by X.Allaberdiyev
Source: Iran TST 2025 Test 4 Problem 1
A number \( n \) is called lucky if it has at least two distinct prime divisors and can be written in the form:
\[ n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k} \]where \( p_1, \dots, p_k \) are distinct prime numbers that divide \( n \). (Note: it is possible that \( n \) has other prime divisors not among \( p_1, \dots, p_k \).) Prove that for every prime number \( p \), there exists a lucky number \( n \) such that \( p \mid n \).
6 replies
M11100111001Y1R
May 27, 2025
X.Allaberdiyev
an hour ago
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China MO 2021 P6
NTssu   23
N 2 hours ago by bin_sherlo
Source: CMO 2021 P6
Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$
23 replies
NTssu
Nov 25, 2020
bin_sherlo
2 hours ago
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Cyclic Sum
P162008   1
N 2 hours ago by aaravdodhia
If $\sum_{cyc} \alpha = 0$ and $\sum_{cyc} \frac{\alpha^4}{2\alpha^2 + \beta\gamma} = 1$ then find the greatest possible value of $\sum_{cyc}\alpha^4.$
1 reply
P162008
May 26, 2025
aaravdodhia
2 hours ago
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Prove that the circumcentres of the triangles are collinear
orl   19
N 2 hours ago by Ilikeminecraft
Source: IMO Shortlist 1997, Q9
Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i = 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i+1}$ and $ A_iA_{i+2}$ (the addition of indices being mod 3). Let $ B_i, i = 1, 2, 3,$ be the second point of intersection of $ C_{i+1}$ and $ C_{i+2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.
19 replies
orl
Aug 10, 2008
Ilikeminecraft
2 hours ago
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c^a + a = 2^b
Havu   9
N 2 hours ago by Havu
Find $a, b, c\in\mathbb{Z}^+$ such that $a,b,c$ coprime, $a + b = 2c$ and $c^a + a = 2^b$.
9 replies
Havu
May 10, 2025
Havu
2 hours ago
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Original Problem
wonderboy807   1
N 2 hours ago by jasperE3
f(0)=f(1)=1. \frac{f(n)f(n-m+1)}{f(n-m)} + \frac{f(n+1)f(n-m)}{f(m-n)} = \frac{f(n+2)f(n-m)f(m-n)}{f(n-m+1)f(m-n+1)}. Find f(10).

Answer: Click to reveal hidden text

Solution: Click to reveal hidden text
1 reply
wonderboy807
5 hours ago
jasperE3
2 hours ago
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An algorithm for discovering prime numbers?
Lukaluce   4
N 2 hours ago by alexanderhamilton124
Source: 2025 Junior Macedonian Mathematical Olympiad P3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
4 replies
Lukaluce
May 18, 2025
alexanderhamilton124
2 hours ago
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Orthocentroidal circle, orthotransversal, concurrent lines
kosmonauten3114   0
2 hours ago
Source: My own
Let $\triangle{ABC}$ be a scalene oblique triangle, and $P$ a point on the orthocentroidal circle of $\triangle{ABC}$ ($P \notin \text{X(4)}$).
Prove that the orthotransversal of $P$, trilinear polar of the polar conjugate ($\text{X(48)}$-isoconjugate) of $P$, Droz-Farny axis of $P$ are concurrent.

The definition of the Droz-Farny axis of $P$ with respect to $\triangle{ABC}$ is as follows:
For a point $P \neq \text{X(4)}$, there exists a pair of orthogonal lines $\ell_1$, $\ell_2$ through $P$ such that the midpoints of the 3 segments cut off by $\ell_1$, $\ell_2$ from the sidelines of $\triangle{ABC}$ are collinear. The line through these 3 midpoints is the Droz-Farny axis of $P$ wrt $\triangle{ABC}$.
0 replies
kosmonauten3114
2 hours ago
0 replies
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inequality
Hoapham235   0
2 hours ago
Let $ 0 \leq a, b, c \leq 1$. Find the maximum of \[P=\dfrac{a}{\sqrt{2bc+1}}+\dfrac{b}{\sqrt{2ca+1}}+\dfrac{c}{\sqrt{2ab+1}}.\]
0 replies
Hoapham235
2 hours ago
0 replies
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Find x^2 + y^2
Darealzolt   3
N 2 hours ago by jasperE3
Let \(x,y\) be positive real numbers that fulfill
\[ \frac{x^2}{y^2}+\frac{4x^2-3xy-4y^2}{2xy-5y^2}=2 \]Hence find the value of \(x^2+y^2\)
3 replies
Darealzolt
5 hours ago
jasperE3
2 hours ago
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3^n + 61 is a square
VideoCake   28
N 3 hours ago by Jupiterballs
Source: 2025 German MO, Round 4, Grade 11/12, P6
Determine all positive integers \(n\) such that \(3^n + 61\) is the square of an integer.
28 replies
VideoCake
May 26, 2025
Jupiterballs
3 hours ago
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Centroid, altitudes and medians, and concyclic points
BR1F1SZ   5
N 3 hours ago by AshAuktober
Source: Austria National MO Part 1 Problem 2
Let $\triangle{ABC}$ be an acute triangle with $BC > AC$. Let $S$ be the centroid of triangle $ABC$ and let $F$ be the foot of the perpendicular from $C$ to side $AB$. The median $CS$ intersects the circumcircle $\gamma$ of triangle $\triangle{ABC}$ at a second point $P$. Let $M$ be the point where $CS$ intersects $AB$. The line $SF$ intersects the circle $\gamma$ at a point $Q$, such that $F$ lies between $S$ and $Q$. Prove that the points $M,P,Q$ and $F$ lie on a circle.

(Karl Czakler)
5 replies
BR1F1SZ
May 5, 2025
AshAuktober
3 hours ago
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An easy number theory problem
TUAN2k8   0
4 hours ago
Source: Own
Find all positive integers $n$ such that there exist positive integers $a$ and $b$ with $a \neq b$ satifying the condition that,
$1) \frac{a^n}{b} + \frac{b^n}{a}$ is an integer.
$2) \frac{a^n}{b} + \frac{b^n}{a} | a^{10}+b^{10}$.
0 replies
TUAN2k8
4 hours ago
0 replies
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Max and min of a/5+b/4+c/3+d/2 under a/4+b/3+c/2+d=1/2 and a^2/7+b^2/5+c^2/3+...
tom-nowy   1
N Apr 27, 2025 by steve4916
Let $a, b, c, d$ be real numbers satisfying the following two conditions:
\begin{align*} &\frac{a}{4}+\frac{b}{3}+\frac{c}{2}+d=\frac{1}{2}, \\ &\frac{a^2}{7}+\frac{b^2}{5}+\frac{c^2}{3}+d^2 + \frac{ab}{3}+\frac{2ac}{5}+\frac{ad}{2} +\frac{bc}{2}+\frac{2bd}{3}+\frac{cd}{2}=\frac{1}{3} . \end{align*}Find the possible maximum and minimum values of
$$\frac{a}{5}+\frac{b}{4}+\frac{c}{3}+\frac{d}{2}.$$
1 reply
tom-nowy
Apr 27, 2025
steve4916
Apr 27, 2025
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Max and min of a/5+b/4+c/3+d/2 under a/4+b/3+c/2+d=1/2 and a^2/7+b^2/5+c^2/3+...
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Reveal topic tags
maximum and minimum
inequalities proposed
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tom-nowy
134 posts
tom-nowy
#1 h  Apr 27, 2025, 11:48 AM
Y by
Let $a, b, c, d$ be real numbers satisfying the following two conditions:
\begin{align*} &\frac{a}{4}+\frac{b}{3}+\frac{c}{2}+d=\frac{1}{2}, \\ &\frac{a^2}{7}+\frac{b^2}{5}+\frac{c^2}{3}+d^2 + \frac{ab}{3}+\frac{2ac}{5}+\frac{ad}{2} +\frac{bc}{2}+\frac{2bd}{3}+\frac{cd}{2}=\frac{1}{3} . \end{align*}Find the possible maximum and minimum values of
$$\frac{a}{5}+\frac{b}{4}+\frac{c}{3}+\frac{d}{2}.$$
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steve4916
82 posts
steve4916
#2 h  Apr 27, 2025, 2:27 PM • 1 Y
Y by tom-nowy
Use cauchy?
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