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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:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]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
Arrange marbles
FunGuy1   3
N 13 minutes ago by FunGuy1
Source: Own?
Anna has $200$ marbles in $25$ colors such that there are exactly $8$ marbles of each color. She wants to arrange them on $50$ shelves, $4$ marbles on each shelf such that for any $2$ colors there is a shelf that has marbles of those colors.
Can Anna achieve her goal?
3 replies
FunGuy1
3 hours ago
FunGuy1
13 minutes ago
Projective geometry
definite_denny   1
N 15 minutes ago by Funcshun840
Source: IDK
Let ABC be a triangle and let DEF be the tangency point of incircirle with sides BC,CA,AB. Points P,Q are chosen on sides AB,AC such that PQ is parallel to BC and PQ is tangent to the incircle. Let M denote the midpoint of PQ. Let EF intersect BC at T. Prove that TM is tangent to the incircle
1 reply
1 viewing
definite_denny
4 hours ago
Funcshun840
15 minutes ago
Nice problem of concurrency
deraxenrovalo   1
N 33 minutes ago by Funcshun840
Let $(I)$ be an inscribed circle of $\triangle$$ABC$ and touching $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Let $EE'$ and $FF'$ be diameters of $(I)$. Let $X$ and $Y$ be the pole of $DE'$ and $DF'$ with respect to $(I)$, respectively. $BE$ cuts $(I)$ again at $K$. $CF$ cuts $(I)$ again at $L$. The tangent at $K$ of $(I)$ cuts $AX$ at $M$. The tangent at $L$ of $(I)$ cuts $AY$ at $N$. Let $U$ and $V$ be midpoint of $IM$ and $IN$, respectively.

Show that : $UV$, $E'F'$ and perpendicular bisector of $ID$ are concurrent.
1 reply
deraxenrovalo
Today at 4:39 AM
Funcshun840
33 minutes ago
Inspired by old results
sqing   1
N an hour ago by ytChen
Source: Own
Let $  a, b> 0,a + 2b= 1. $ Prove that
$$ \sqrt{a + b^2} +2 \sqrt{b+ a^2} +  |a - b| \geq 2$$Let $  a, b> 0,a + 2b= \frac{3}{4}. $ Prove that
$$ \sqrt{a + (b - \frac{1}{4})^2} +2 \sqrt{b + (a-  \frac{1}{4})^2} + \sqrt{ (a - b)^2+ \frac{1}{4}}  \geq 2$$
1 reply
sqing
May 20, 2025
ytChen
an hour ago
Different Paths Probability
Qebehsenuef   3
N an hour ago by Etkan
Source: OBM
A mouse initially occupies cage A and is trained to change cages by going through a tunnel whenever an alarm sounds. Each time the alarm sounds, the mouse chooses any of the tunnels adjacent to its cage with equal probability and without being affected by previous choices. What is the probability that after the alarm sounds 23 times the mouse occupies cage B?
3 replies
Qebehsenuef
Apr 28, 2025
Etkan
an hour ago
D1036 : Composition of polynomials
Dattier   0
an hour ago
Source: les dattes à Dattier
Find all $A \in \mathbb Q[x]$ with $\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$ and $\deg(A)>1$.
0 replies
Dattier
an hour ago
0 replies
inequality
NTssu   4
N an hour ago by Oksutok
Source: Peking University Mathematics Autumn Camp
For given real number $\theta_1, \theta_2, ......, \theta_l$, prove there exists positive integer $k$ and positive real number $a_1, a_2, ......, a_k$, such that $a_1+a_2+ ......+ a_k=1$, for any $n \leq k$, $m \in \{1,2,......,l\}$, $\left| \sum_{j=1}^n a_j sin(j \theta_m ) \right|< \frac{1}{2018n} $ holds.
4 replies
NTssu
Oct 11, 2019
Oksutok
an hour ago
Nice geometry
gggzul   0
an hour ago
Let $ABC$ be a acute triangle with $\angle BAC=60^{\circ}$. $H, O$ are the orthocenter and excenter. Let $D$ be a point on the same side of $OH$ as $A$, such that $HDO$ is equilateral. Let $P$ be a point on the same side of $BD$ as $A$, such that $BDP$ is equilateral. Let $Q$ be a point on the same side of $CD$ as $A$, such that $CDP$ is equilateral. Let $M$ be the midpoint of $AD$. Prove that $P, M, Q$ are collinear.
0 replies
gggzul
an hour ago
0 replies
Inspired by 2025 KMO
sqing   3
N 2 hours ago by sqing
Source: Own
Let $ a,b,c,d  $ be real numbers satisfying $ a+b+c+d=0 $ and $ a^2+b^2+c^2+d^2= 6 .$ Prove that $$ -\frac{3}{4} \leq abcd\leq\frac{9}{4}$$Let $ a,b,c,d  $ be real numbers satisfying $ a+b+c+d=6 $ and $ a^2+b^2+c^2+d^2= 18 .$ Prove that $$ -\frac{9(2\sqrt{3}+3)}{4} \leq abcd\leq\frac{9(2\sqrt{3}-3)}{4}$$
3 replies
sqing
Yesterday at 2:39 PM
sqing
2 hours ago
Reflections and midpoints in triangle
TUAN2k8   0
2 hours ago
Source: Own
Given an triangle $ABC$ and a line $\ell$ in the plane.Let $A_1,B_1,C_1$ be reflections of $A,B,C$ across the line $\ell$, respectively.Let $D,E,F$ be the midpoints of $B_1C_1,C_1A_1,A_1B_1$, respectively.Let $A_2,B_2,C_2$ be the reflections of $A,B,C$ across $D,E,F$, respectively.Prove that the points $A_2,B_2,C_2$ lie on a line parallel to $\ell$.
0 replies
TUAN2k8
2 hours ago
0 replies
Find the expected END time for the given process
superpi   2
N 2 hours ago by Hello_Kitty
This problem suddenly popped up in my head. But I don't know how to deal with it.

There are N bulbs. All the bulbs' available time follows same exponential distribution with parameter lambda(Or any arbitrary distribution with mean $\mu$). We do following operations
1. First, turn on the all $N$ bulbs
2. For each $k >= 2$ bulbs goes out, append ONE NEW BULB and turn on (This step starts and finishes immediately when kth bulb goes out)
3. Repeat 2 until all the bulbs goes out

What is the expected terminate time for the above process for given $N, k, \lambda$?

Or, is there any more conditions to complete the problem?

2 replies
superpi
Yesterday at 4:33 PM
Hello_Kitty
2 hours ago
a exhaustive question
shrayagarwal   19
N 2 hours ago by SomeonecoolLovesMaths
Source: number theory
If $ a$ and $ b$ are natural numbers such that $ a+13b$ is divisible by $ 11$ and $ a+11b$ is divisible by $ 13$, then find the least possible value of $ a+b$.
19 replies
shrayagarwal
Dec 4, 2006
SomeonecoolLovesMaths
2 hours ago
36x⁴ + 12x² - 36x + 13 > 0
fxandi   4
N 5 hours ago by wh0nix
Prove that for any real $x \geq 0$ holds inequality $36x^4 + 12x^2 - 36x + 13 > 0.$
4 replies
fxandi
May 5, 2025
wh0nix
5 hours ago
2024 Miklós-Schweitzer problem 3
Martin.s   3
N Today at 1:30 AM by naenaendr
Do there exist continuous functions $f, g: \mathbb{R} \to \mathbb{R}$, both nowhere differentiable, such that $f \circ g$ is differentiable?
3 replies
Martin.s
Dec 5, 2024
naenaendr
Today at 1:30 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
245 posts
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$P(t)$ 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
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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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