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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 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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Common tangent to diameter circles
Stuttgarden   2
N 10 minutes ago by Giant_PT
Source: Spain MO 2025 P2
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
2 replies
Stuttgarden
Mar 31, 2025
Giant_PT
10 minutes ago
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functional equation
hanzo.ei   2
N 22 minutes ago by MathLuis

Find all functions \( f : \mathbb{R} \to \mathbb{R} \) satisfying the equation
\[ (f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}. \]
2 replies
hanzo.ei
5 hours ago
MathLuis
22 minutes ago
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Geometry
youochange   5
N 22 minutes ago by lolsamo
m:}
Let $\triangle ABC$ be a triangle inscribed in a circle, where the tangents to the circle at points $B$ and $C$ intersect at the point $P$. Let $M$ be a point on the arc $AC$ (not containing $B$) such that $M \neq A$ and $M \neq C$. Let the lines $BC$ and $AM$ intersect at point $K$. Let $P'$ be the reflection of $P$ with respect to the line $AM$. The lines $AP'$ and $PM$ intersect at point $Q$, and $PM$ intersects the circumcircle of $\triangle ABC$ again at point $N$.

Prove that the point $Q$ lies on the circumcircle of $\triangle ANK$.
5 replies
youochange
Today at 11:27 AM
lolsamo
22 minutes ago
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Something nice
KhuongTrang   25
N an hour ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
25 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
an hour ago
J
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Two Functional Inequalities
Mathdreams   6
N an hour ago by Assassino9931
Source: 2025 Nepal Mock TST Day 2 Problem 2
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \le x^3$$and $$f(x + y) \le f(x) + f(y) + 3xy(x + y)$$for any real numbers $x$ and $y$.

(Miroslav Marinov, Bulgaria)
6 replies
Mathdreams
Today at 1:34 PM
Assassino9931
an hour ago
J
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Pythagorean new journey
XAN4   2
N an hour ago by mathprodigy2011
Source: Inspired by sarjinius
The number $4$ is written on the blackboard. Every time, Carmela can erase the number $n$ on the black board and replace it with a new number $m$, if and only if $|n^2-m^2|$ is a perfect square. Prove or disprove that all positive integers $n\geq4$ can be written exactly once on the blackboard.
2 replies
XAN4
Today at 3:41 AM
mathprodigy2011
an hour ago
J
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sqrt(2) and sqrt(3) differ in at least 1000 digits
Stuttgarden   2
N an hour ago by straight
Source: Spain MO 2025 P3
We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as \[\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots\]where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$.
2 replies
Stuttgarden
Mar 31, 2025
straight
an hour ago
J
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combinatorics and number theory beautiful problem
Medjl   2
N an hour ago by mathprodigy2011
Source: Netherlands TST for BxMo 2017 problem 4
A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that
$i$ of each $a$ consecutive integers at least one is coloured red;
$ii$ of each $b$ consecutive integers at least one is coloured blue;
$iii$ of each $c$ consecutive integers at least one is coloured green;
$iiii$ of each $d$ consecutive integers at least one is coloured purple.
Determine all good quadruples with $a = 2.$
2 replies
Medjl
Feb 1, 2018
mathprodigy2011
an hour ago
J
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Squence problem
AlephG_64   1
N 2 hours ago by RagvaloD
Source: 2025 Finals Portuguese Math Olympiad P1
Francisco wrote a sequence of numbers starting with $25$. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was $8$, what is the fifth term of the sequence?
1 reply
1 viewing
AlephG_64
Yesterday at 1:19 PM
RagvaloD
2 hours ago
J
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50 points in plane
pohoatza   12
N 2 hours ago by de-Kirschbaum
Source: JBMO 2007, Bulgaria, problem 3
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
12 replies
pohoatza
Jun 28, 2007
de-Kirschbaum
2 hours ago
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beautiful functional equation problem
Medjl   6
N 2 hours ago by Sadigly
Source: Netherlands TST for BxMO 2017 problem 2
Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that :
$i)$$f(p)=1$ for all prime numbers $p$.
$ii)$$f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$
find the smallest $n \geq 2016$ such that $f(n)=n$
6 replies
Medjl
Feb 1, 2018
Sadigly
2 hours ago
J
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Line Combining Fermat Point, Orthocenter, and Centroid
cooljoseph   0
2 hours ago
On triangle $ABC$, draw exterior equilateral triangles on sides $AB$ and $AC$ to obtain $ABC'$ and $ACB'$, respectively. Let $X$ be the intersection of the altitude through $B$ and the median through $C$. Let $Y$ be the intersection of the altitude through $A$ and line $CC'$. Let $Z$ be the intersection of the median through $A$ and the line $BB'$. Prove that $X$, $Y$, and $Z$ lie on a common line.

IMAGE
0 replies
cooljoseph
2 hours ago
0 replies
J
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complete integral values
Medjl   2
N 2 hours ago by Sadigly
Source: Netherlands TST for BxMO 2017 problem 1
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers.
2 replies
Medjl
Feb 1, 2018
Sadigly
2 hours ago
J
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interesting ineq
nikiiiita   5
N 2 hours ago by nikiiiita
Source: Own
Given $a,b,c$ are positive real numbers satisfied $a^3+b^3+c^3=3$. Prove that:
$$\sqrt{2ab+5c^{2}+2a}+\sqrt{2bc+5a^{2}+2b}+\sqrt{2ac+5b^{2}+2c}\le3\sqrt{3\left(a+b+c\right)}$$
5 replies
nikiiiita
Jan 29, 2025
nikiiiita
2 hours ago
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Modular NT
oVlad   3
N Apr 3, 2025 by EVKV
Source: Romania JBMO TST 2024 Day 1 P1
Find all the positive integers $a{}$ and $b{}$ such that $(7^a-5^b)/8$ is a prime number.

Cosmin Manea and Dragoș Petrică
3 replies
oVlad
Jul 31, 2024
EVKV
Apr 3, 2025
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Modular NT
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Reveal topic tags
number theory
modular arithmetic
prime numbers
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Source: Romania JBMO TST 2024 Day 1 P1
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oVlad
1721 posts
oVlad
#1 h  Jul 31, 2024, 12:29 PM • 1 Y
Y by ItsBesi
Find all the positive integers $a{}$ and $b{}$ such that $(7^a-5^b)/8$ is a prime number.

Cosmin Manea and Dragoș Petrică
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sami1618
884 posts
sami1618
#2 h  Jul 31, 2024, 5:30 PM
Y by
Answer: $(a,b)=(2,2)$

Notice that for $7^a\equiv 5^b$ we must have both $a$ and $b$ being even. Let $a=2a'$ and $b=2b'$ then we must have that $$\frac{49^{a'}-25^{b'}}{8}=p$$But as $3$ divides the top we must have that $p=3$ or that $(7^{a'}+5^{b'})(7^{a'}-5^{b'})=24$. But $7^{a'}-5^{b'}$ is at least $2$ and $7^{a'}+5^{b'}$ is at least $12$ so both must be equalities leading to the above solution.
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wizixez
168 posts
wizixez
#3 h  Dec 15, 2024, 3:28 PM
Y by
$49^x-25^y=24$
Since $A(x,y)=49^x-25^y=(7^x-5^y)(7^x+5^y)=24$
checking the cases gives $(a,b)=(2,2)$
Z K Y
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EVKV
43 posts
EVKV
#4 h  Apr 3, 2025, 1:50 AM
Y by
mod 8 gives both a,b as even
So, $(7^{\frac{a}{2}} - 5^{\frac{b}{2}})(7^{\frac{a}{2}} + 5^{\frac{b}{2}}) = 8p$
now checking mod 3 leads to the fact that the prime is 3
So,$(7^{\frac{a}{2}} - 5^{\frac{b}{2}})(7^{\frac{a}{2}} + 5^{\frac{b}{2}}) = 24$
now just casework and u are done

not deserving of a jbmo tst
but follows pattern of random difficulty in jbmo tbh
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