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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)!

Be sure to mark your calendars for the following upcoming events:
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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
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
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
Preparing for Putnam level entrance examinations
Cats_on_a_computer   4
N 10 minutes ago by Cats_on_a_computer
Non American high schooler in the equivalent of grade 12 here. Where I live, two the best undergraduates program in the country accepts students based on a common entrance exam. The first half of the exam is “screening”, with 4 options being presented per question, each of which one has to assign a True or False. This first half is about the difficulty of an average AIME, or JEE Adv paper, and it is a requirement for any candidate to achieve at least 24/40 on this half for the examiners to even consider grading the second part. The second part consists of long form questions, and I have, no joke, seen them literally rip off, verbatim, Putnam A6s. Some of the problems are generally standard textbook problems in certain undergrad courses but obviously that doesn’t translate it to being doable for high school students. I’ve effectively got to prepare for a slightly nerfed Putnam, if you will, and so I’ve been looking for resources (not just problems) for Putnam level questions. Does anyone have any suggestions?
4 replies
Cats_on_a_computer
Yesterday at 8:32 AM
Cats_on_a_computer
10 minutes ago
Hard geometry
Lukariman   0
33 minutes ago
Given triangle ABC, a line d intersects the sides AB, AC and the line BC at D, E, F respectively.

(a) Prove that the circles circumscribing triangles ADE, BDF and CEF pass through a point P and P belongs to the circumcircle of triangle ABC.

(b) Prove that the centers of the circles circumscribing triangles ADE, BDF, CEF and ABC are all on the circle.

(c) Let $O_a$,$ O_b$, $O_c$ be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle $O_a$$O_b$$O_c$ belongs to d.

(d) Prove that the orthocenters of triangles ADE, ABC, BDF, CEF are collinear.
0 replies
Lukariman
33 minutes ago
0 replies
CGMO5: Carlos Shine's Fact 5
v_Enhance   61
N 43 minutes ago by Adywastaken
Source: 2012 China Girl's Mathematical Olympiad
As shown in the figure below, the in-circle of $ABC$ is tangent to sides $AB$ and $AC$ at $D$ and $E$ respectively, and $O$ is the circumcenter of $BCI$. Prove that $\angle ODB = \angle OEC$.
IMAGE
61 replies
v_Enhance
Aug 13, 2012
Adywastaken
43 minutes ago
Grid combi with T-tetrominos
Davdav1232   1
N an hour ago by NO_SQUARES
Source: Israel TST 8 2025 p1
Let \( f(N) \) denote the maximum number of \( T \)-tetrominoes that can be placed on an \( N \times N \) board such that each \( T \)-tetromino covers at least one cell that is not covered by any other \( T \)-tetromino.

Find the smallest real number \( c \) such that
\[
f(N) \leq cN^2
\]for all positive integers \( N \).
1 reply
Davdav1232
Thursday at 8:29 PM
NO_SQUARES
an hour ago
pqr/uvw convert
Nguyenhuyen_AG   10
N an hour ago by Nguyenhuyen_AG
Source: https://github.com/nguyenhuyenag/pqr_convert
Hi everyone,
As we know, the pqr/uvw method is a powerful and useful tool for proving inequalities. However, transforming an expression $f(a,b,c)$ into $f(p,q,r)$ or $f(u,v,w)$ can sometimes be quite complex. That's why I’ve written a program to assist with this process.
I hope you’ll find it helpful!

Download: pqr_convert

Screenshot:
IMAGE
IMAGE
10 replies
1 viewing
Nguyenhuyen_AG
Apr 19, 2025
Nguyenhuyen_AG
an hour ago
interesting functional
Pomegranat   2
N an hour ago by Pomegranat
Source: I don't know sorry
Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that for all positive real numbers \( x \) and \( y \), the following equation holds:
\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]
2 replies
Pomegranat
3 hours ago
Pomegranat
an hour ago
Function equation algebra
TUAN2k8   1
N an hour ago by TUAN2k8
Source: Balkan MO 2025
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x \in \mathbb{R}$ and $y \in \mathbb{R}$,
\begin{align}
f(x+yf(x))+y=xy+f(x+y).
\end{align}
1 reply
TUAN2k8
2 hours ago
TUAN2k8
an hour ago
Functional equation with a twist (it's number theory)
Davdav1232   1
N an hour ago by NO_SQUARES
Source: Israel TST 8 2025 p2
Prove that for all primes \( p \) such that \( p \equiv 3 \pmod{4} \) or \( p \equiv 5 \pmod{8} \), there exist integers
\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]such that
\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]
1 reply
+1 w
Davdav1232
Thursday at 8:32 PM
NO_SQUARES
an hour ago
Coolabra
Titibuuu   3
N 2 hours ago by sqing
Let \( a, b, c \) be distinct real numbers such that
\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]Find the maximum possible value of \( a \).
3 replies
Titibuuu
Today at 2:21 AM
sqing
2 hours ago
Is the result of this is the same as cauchy?
ItzsleepyXD   0
2 hours ago
Source: curiosity
Prove or disprove that for all continuous or monotonic function $f : \mathbb{R}^2 \to \mathbb{R}$ .The solution to $$f(a,x)+f(b,y)=f(a+b,x+y) \text{ for all }a,b,x,y \in \mathbb{R}$$is only $f(x,y)=cx+dy$ for some $c,d \in \mathbb{R}$
0 replies
ItzsleepyXD
2 hours ago
0 replies
a fractions problem
kjhgyuio   1
N 2 hours ago by Ash_the_Bash07
.........
1 reply
kjhgyuio
3 hours ago
Ash_the_Bash07
2 hours ago
Find the marginal profit..
ArmiAldi   1
N 3 hours ago by Juno_34
Source: can someone help me
The total profit selling x units of books is P(x) = (6x - 7)(9x - 8) .
Find the marginal average profit function?
1 reply
ArmiAldi
Mar 2, 2008
Juno_34
3 hours ago
Marginal Profit
NC4723   1
N 3 hours ago by Juno_34
Please help me solve this
1 reply
NC4723
Dec 11, 2015
Juno_34
3 hours ago
Romania NMO 2023 Grade 11 P1
DanDumitrescu   15
N Today at 5:46 AM by anudeep
Source: Romania National Olympiad 2023
Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation

\[
    \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
    \]
15 replies
DanDumitrescu
Apr 14, 2023
anudeep
Today at 5:46 AM
Finite solution for x
Rohit-2006   1
N Apr 21, 2025 by Filipjack
$P(t)$ be a non constant polynomial with real coefficients. Prove that the system of simultaneous equations —
$$\int_{0}^{x} P(t)sin t dt =0$$$$\int_{0}^{x}P(t) cos t dt=0$$has finitely many solutions $x$.
1 reply
Rohit-2006
Apr 21, 2025
Filipjack
Apr 21, 2025
Finite solution for x
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Rohit-2006
242 posts
#1
Y by
$P(t)$ be a non constant polynomial with real coefficients. Prove that the system of simultaneous equations —
$$\int_{0}^{x} P(t)sin t dt =0$$$$\int_{0}^{x}P(t) cos t dt=0$$has finitely many solutions $x$.
This post has been edited 1 time. Last edited by Rohit-2006, Apr 21, 2025, 4:20 AM
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Filipjack
873 posts
#2
Y by
One can show by induction on the degree of $P$ that \begin{align*} \int P(t) \sin t \mathrm{d}t &=F'(t)\sin t - F(t)\cos t + \mathcal{C}, \\ \int P(t) \cos t \mathrm{d}t &= F'(t) \cos t + F(t)\sin(t) + \mathcal{C}, \end{align*}where $F(t)=P(t)-P''(t)+P''''(t)- \ldots.$

Thus, our system is equivalent to $$\begin{cases}F'(x)\sin x - F(x)\cos x  + F(0) = 0  \\ F'(x) \cos x + F(x)\sin x - F'(0) = 0  \end{cases}.$$
Multiplying the first equation by $-\cos x$ and the second one by $\sin x,$ and then adding them, we get $$F(x)-F(0) \cos x - F'(0) \sin x=0.$$
It would therefore be enough to show that an equation of the form $F(x)+a \sin x + b \cos x=0$ with $F$ a nonconstant polynomial has finitely many solutions. If $a=b=0$ this is clear, so let's assume $a^2+b^2 \neq 0.$ Then the equation can be rewritten as $\frac{F(x)}{\sqrt{a^2+b^2}}+ \frac{a}{\sqrt{a^2+b^2}} \sin(x+ \varphi) = 0,$ where $\varphi$ satisfies $\cos \varphi = \frac{a}{\sqrt{a^2+b^2}}$ and $\sin \varphi = \frac{b}{\sqrt{a^2+b^2}}.$

So now it is enough to show that an equation of the form $G(x) + a \sin (x+b)=0$ has finitely many solutions, where $G$ is a nonconstant polinomial. We will prove this by induction on the degree of $G.$

Let $h(x) = G(x) + a \sin (x+b).$ If $\deg G = 1,$ then $h'(x)=c+a \cos(x+b),$ where $c$ is some constant. The zero set of $h'$ is either the empty set or a discrete set. If it is the empty set, then $h$ is strictly monotonic, so it has at most one zero. If it is discrete, then so is the zero set of $h$ (as a corollary of Rolle's theorem, between any two consecutive zeroes of $h'$ there is at most one zero of $h$). On the other hand, $\lim_{x \to \infty} |h(x)|= \infty,$ so there is some $C$ such that $|h(x)| \ge 1$ for any $x \ge C.$ Thus, the zero set of $h$ is contained in the interval $(-C,C).$ Since the zero set of $h$ is bounded and discrete, it is finite.

Now assume the claim is true for $\deg G = k.$ Then $h'(x)=0$ has finitely many solutions by the induction hypothesis (notice that the equation $G'(x) + a \cos (x+b)$ is still of the form $A(x) + \alpha \sin ( x+ \beta)=0$ because $\cos(x+b)=\sin(x+b + \pi/2)$). With the same corollary of Rolle's theorem we conclude that $h$ has finitely many zeroes, which concludes the proof.
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