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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
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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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Inequality
nguyentlauv   3
N an hour ago by NguyenVanHoa29
Source: Own
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=3$ and $k\ge 0$, prove that
$$\frac{\sqrt{a+1}}{b+c+k}+\frac{\sqrt{b+1}}{c+a+k}+\frac{\sqrt{c+1}}{a+b+k} \geq \frac{3\sqrt{2}}{k+2}.$$
3 replies
1 viewing
nguyentlauv
May 6, 2025
NguyenVanHoa29
an hour ago
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schur weighted
Ducksohappi   1
N 2 hours ago by truongngochieu
Schur-weighted:
let a,b,c be positive. Prove that:
$a^3+b^3+c^3+3abc\ge \sum ab\sqrt{a^2+b^2}$
1 reply
Ducksohappi
3 hours ago
truongngochieu
2 hours ago
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forced vertices in graphs
Davdav1232   1
N 2 hours ago by CBMaster
Source: Israel TST 7 2025 p2
Let \( G \) be a graph colored using \( k \) colors. We say that a vertex is forced if it has neighbors in all the other \( k - 1 \) colors.

Prove that for any \( 2024 \)-regular graph \( G \), there exists a coloring using \( 2025 \) colors such that at least \( 1013 \) of the colors have a forced vertex of that color.

Note: The graph coloring must be valid, this means no \( 2 \) vertices of the same color may be adjacent.
1 reply
Davdav1232
May 8, 2025
CBMaster
2 hours ago
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Cyclic inequality with rational functions
MathMystic33   1
N 2 hours ago by Nguyenhuyen_AG
Source: 2025 Macedonian Team Selection Test P3
Let \(x_1,x_2,x_3,x_4\) be positive real numbers. Prove the inequality
\[ \frac{x_1 + 3x_2}{x_2 + x_3} \;+\; \frac{x_2 + 3x_3}{x_3 + x_4} \;+\; \frac{x_3 + 3x_4}{x_4 + x_1} \;+\; \frac{x_4 + 3x_1}{x_1 + x_2} \;\ge\;8. \]
1 reply
MathMystic33
Yesterday at 6:00 PM
Nguyenhuyen_AG
2 hours ago
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I got stuck in this combinatorics
artjustinhere237   2
N 3 hours ago by artjustinhere237
Let $S = \{1, 2, 3, \ldots, k\}$, where $k \geq 4$ is a positive integer.
Prove that there exists a subset of $S$ with exactly $k - 2$ elements such that the sum of its elements is a prime number.
2 replies
artjustinhere237
Yesterday at 4:56 PM
artjustinhere237
3 hours ago
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d1-d2 divides n for all divisors d1, d2
a_507_bc   5
N 3 hours ago by Assassino9931
Source: Romania 3rd JBMO TST 2023 P1
Determine all natural numbers $n \geq 2$ with at most four natural divisors, which have the property that for any two distinct proper divisors $d_1$ and $d_2$ of $n$, the positive integer $d_1-d_2$ divides $n$.
5 replies
a_507_bc
May 20, 2023
Assassino9931
3 hours ago
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Something weird with this one FE in integers (probably challenging, maybe not)
Gaunter_O_Dim_of_math   2
N 3 hours ago by aaravdodhia
Source: Pang-Cheng-Wu, FE, problem number 52.
During FE problems' solving I found a very specific one:

Find all such f that f: Z -> Z and for all integers a, b, c
f(a^3 + b^3 + c^3) = f(a)^3 + f(b)^3 + f(c)^3.

Everything what I've got is that f is odd, f(n) = n or -n or 0
for all n from 0 to 11 (just bash it), but it is very simple and do not give the main idea.
I actually have spent not so much time on this problem, but definitely have no clue. As far as I see, number theory here or classical FE solving or advanced methods, which I know, do not work at all.
Is here a normal solution (I mean, without bashing and something with a huge number of ugly and weird inequalities)?
Or this is kind of rubbish, which was put just for bash?
2 replies
Gaunter_O_Dim_of_math
Yesterday at 8:10 PM
aaravdodhia
3 hours ago
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Bulgaria 8
orl   9
N 3 hours ago by Assassino9931
Source: IMO LongList 1959-1966 Problem 34
Find all pairs of positive integers $\left( x;\;y\right) $ satisfying the equation $2^{x}=3^{y}+5.$
9 replies
orl
Sep 2, 2004
Assassino9931
3 hours ago
J
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P (x^2) = P (x) P (x + 2) for any complex x
parmenides51   8
N 3 hours ago by Wildabandon
Source: 2008 Brazil IMO TST 4.2
Find all polynomials $P (x)$ with complex coefficients such that $$P (x^2) = P (x) · P (x + 2)$$for any complex number $x.$
8 replies
parmenides51
Jul 24, 2021
Wildabandon
3 hours ago
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Brazilian Locus
kraDracsO   16
N 3 hours ago by Giant_PT
Source: IberoAmerican, Day 2, P4
Let $B$ and $C$ be two fixed points in the plane. For each point $A$ of the plane, outside of the line $BC$, let $G$ be the barycenter of the triangle $ABC$. Determine the locus of points $A$ such that $\angle BAC + \angle BGC = 180^{\circ}$.

Note: The locus is the set of all points of the plane that satisfies the property.
16 replies
kraDracsO
Sep 9, 2023
Giant_PT
3 hours ago
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Plz help
Bet667   3
N Yesterday at 6:50 PM by K1mchi_
f:R-->R for any integer x,y
f(yf(x)+f(xy))=(x+f(x))f(y)
find all function f
(im not good at english)
3 replies
Bet667
Jan 28, 2024
K1mchi_
Yesterday at 6:50 PM
J
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2019 SMT Team Round - Stanford Math Tournament
parmenides51   17
N Yesterday at 6:40 PM by Rombo
p1. Given $x + y = 7$, find the value of x that minimizes $4x^2 + 12xy + 9y^2$.


p2. There are real numbers $b$ and $c$ such that the only $x$-intercept of $8y = x^2 + bx + c$ equals its $y$-intercept. Compute $b + c$.



p3. Consider the set of $5$ digit numbers $ABCDE$ (with $A \ne 0$) such that $A+B = C$, $B+C = D$, and $C + D = E$. What’s the size of this set?


p4. Let $D$ be the midpoint of $BC$ in $\vartriangle ABC$. A line perpendicular to D intersects $AB$ at $E$. If the area of $\vartriangle ABC$ is four times that of the area of $\vartriangle BDE$, what is $\angle ACB$ in degrees?


p5. Define the sequence $c_0, c_1, ...$ with $c_0 = 2$ and $c_k = 8c_{k-1} + 5$ for $k > 0$. Find $\lim_{k \to \infty} \frac{c_k}{8^k}$.


p6. Find the maximum possible value of $|\sqrt{n^2 + 4n + 5} - \sqrt{n^2 + 2n + 5}|$.


p7. Let $f(x) = \sin^8 (x) + \cos^8(x) + \frac38 \sin^4 (2x)$. Let $f^{(n)}$ (x) be the $n$th derivative of $f$. What is the largest integer $a$ such that $2^a$ divides $f^{(2020)}(15^o)$?


p8. Let $R^n$ be the set of vectors $(x_1, x_2, ..., x_n)$ where $x_1, x_2,..., x_n$ are all real numbers. Let $||(x_1, . . . , x_n)||$ denote $\sqrt{x^2_1 +... + x^2_n}$. Let $S$ be the set in $R^9$ given by $$S = \{(x, y, z) : x, y, z \in R^3 , 1 = ||x|| = ||y - x|| = ||z -y||\}.$$If a point $(x, y, z)$ is uniformly at random from $S$, what is $E[||z||^2]$?


p9. Let $f(x)$ be the unique integer between $0$ and $x - 1$, inclusive, that is equivalent modulo $x$ to $\left( \sum^2_{i=0} {{x-1} \choose i} ((x - 1 - i)! + i!) \right)$. Let $S$ be the set of primes between $3$ and $30$, inclusive. Find $\sum_{x\in S}^{f(x)}$.


p10. In the Cartesian plane, consider a box with vertices $(0, 0)$,$\left( \frac{22}{7}, 0\right)$,$(0, 24)$,$\left( \frac{22}{7}, 4\right)$. We pick an integer $a$ between $1$ and $24$, inclusive, uniformly at random. We shoot a puck from $(0, 0)$ in the direction of $\left( \frac{22}{7}, a\right)$ and the puck bounces perfectly around the box (angle in equals angle out, no friction) until it hits one of the four vertices of the box. What is the expected number of times it will hit an edge or vertex of the box, including both when it starts at $(0, 0)$ and when it ends at some vertex of the box?


p11. Sarah is buying school supplies and she has $\$2019$. She can only buy full packs of each of the following items. A pack of pens is $\$4$, a pack of pencils is $\$3$, and any type of notebook or stapler is $\$1$. Sarah buys at least $1$ pack of pencils. She will either buy $1$ stapler or no stapler. She will buy at most $3$ college-ruled notebooks and at most $2$ graph paper notebooks. How many ways can she buy school supplies?


p12. Let $O$ be the center of the circumcircle of right triangle $ABC$ with $\angle ACB = 90^o$. Let $M$ be the midpoint of minor arc $AC$ and let $N$ be a point on line $BC$ such that $MN \perp BC$. Let $P$ be the intersection of line $AN$ and the Circle $O$ and let $Q$ be the intersection of line $BP$ and $MN$. If $QN = 2$ and $BN = 8$, compute the radius of the Circle $O$.


p13. Reduce the following expression to a simplified rational $$\frac{1}{1 - \cos \frac{\pi}{9}}+\frac{1}{1 - \cos \frac{5 \pi}{9}}+\frac{1}{1 - \cos \frac{7 \pi}{9}}$$

p14. Compute the following integral $\int_0^{\infty} \log (1 + e^{-t})dt$.


p15. Define $f(n)$ to be the maximum possible least-common-multiple of any sequence of positive integers which sum to $n$. Find the sum of all possible odd $f(n)$


PS. You should use hide for answers. Collected here.
17 replies
1 viewing
parmenides51
Feb 6, 2022
Rombo
Yesterday at 6:40 PM
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A problem of collinearity.
Raul_S_Baz   0
Yesterday at 4:19 PM
Î am the author.
IMAGE
P.S: How can I verify that it is an original problem? Thanks!
0 replies
Raul_S_Baz
Yesterday at 4:19 PM
0 replies
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Inequalities
nhathhuyyp5c   2
N Yesterday at 4:11 PM by alexheinis
Let $x,y$ be positive reals such that $3x-2xy\leq 1$. Find $\min$ \[ M = \frac{1 - x^2}{x^2} + 2y^2 + 3x + \frac{24}{y} + 2025. \]

2 replies
nhathhuyyp5c
Yesterday at 3:31 PM
alexheinis
Yesterday at 4:11 PM
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k functional equation
Tony_stark0094   4
N Mar 31, 2025 by jasperE3
solve for $f:R \rightarrow R$ such that
$$f(x+f(y))=y+f(x+1)$$
4 replies
Tony_stark0094
Mar 31, 2025
jasperE3
Mar 31, 2025
J
functional equation
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Reveal topic tags
functional equation
algebra
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Tony_stark0094
69 posts
Tony_stark0094
#1 h  Mar 31, 2025, 1:29 PM
Y by
solve for $f:R \rightarrow R$ such that
$$f(x+f(y))=y+f(x+1)$$
Z Y
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Mathzeus1024
884 posts
Mathzeus1024
#2 h  Mar 31, 2025, 2:40 PM
Y by
It works for $\textcolor{red}{f(x) = 1 \pm x}$.
Z Y
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maromex
189 posts
maromex
#3 h  Mar 31, 2025, 2:51 PM
Y by
This is Albania TST 2008. Little spoiler
Z Y
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Tony_stark0094
69 posts
Tony_stark0094
#4 h  Mar 31, 2025, 4:34 PM
Y by
maromex wrote:
This is Albania TST 2008. Little spoiler

can you share the link to the original post?
Z Y
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jasperE3
11333 posts
jasperE3
#5 h  Mar 31, 2025, 5:08 PM • 1 Y
Y by maromex
https://artofproblemsolving.com/community/c6h1354361p7406603
for example
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=
a