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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 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
Inequalities
sqing   12
N 13 minutes ago by sqing
Let $ a, b, c $ be real numbers such that $ a + b + c = 0 $ and $ abc = -16 $. Prove that$$ \frac{a^2 + b^2}{c} + \frac{b^2 + c^2}{a} + \frac{c^2 + a^2}{b} -a^2\geq 2$$$$ \frac{a^2 + b^2}{c} + \frac{b^2 + c^2}{a} + \frac{c^2 + a^2}{b}-a^2-bc\geq -2$$Equality holds when $a=-4,b=c=2.$
12 replies
1 viewing
sqing
Jul 9, 2025
sqing
13 minutes ago
2-variable inequalities
sqing   2
N 37 minutes ago by sqing
Source: Own
Let $ a,b>0,a(a+b)^2=5 $. Prove that
$$   a^2( (a^2+ab+b^2)^2+8b) \geq 24$$$$ a^2( (a^2+ab+b^2)^2+9b)\geq  \frac{99}{4} $$
2 replies
sqing
Today at 3:33 AM
sqing
37 minutes ago
what accursed word contains "ZAX"???
Scilyse   18
N 39 minutes ago by AndreiVila
Source: 2024 IMOSL G5
Let $ABC$ be a triangle with incentre $I$, and let $\Omega$ be the circumcircle of triangle $BIC$. Let $K$ be a point in the interior of segment $BC$ such that $\angle BAK < \angle KAC$. The angle bisector of $\angle BKA$ intersects $\Omega$ at points $W$ and $X$ such that $A$ and $W$ lie on the same side of $BC$, and the angle bisector of $\angle CKA$ intersects $\Omega$ at points $Y$ and $Z$ such that $A$ and $Y$ lie on the same side of $BC$.

Prove that $\angle WAY = \angle ZAX$.

Proposed by David Brodsky, Uzbekistan
18 replies
Scilyse
Jul 16, 2025
AndreiVila
39 minutes ago
Inequalities
sqing   17
N 40 minutes ago by sqing
Let $ a,b,c\geq 0  . $ Prove that
$$ \sqrt{ a^3+b^3+c^3+\frac{1}{4}} +  \frac{9}{5}abc+\frac{1}{2} \geq a+b+c$$O706
17 replies
sqing
Jul 19, 2025
sqing
40 minutes ago
APMO 2015 P1
aditya21   64
N an hour ago by lpieleanu
Source: APMO 2015
Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively.
Prove that $AB = V W$

Proposed by Warut Suksompong, Thailand
64 replies
aditya21
Mar 30, 2015
lpieleanu
an hour ago
[PMO23 Qualifying I.12] Biased Coins
kae_3   1
N an hour ago by Siopao_Enjoyer
Alice tosses two biased coins, each of which has a probability $p$ of obtaining a head, simultaneously and repeatedly until she gets two heads. Suppose that this happens on the $r$th toss for some integer $r\geq1$. Given that there is $36\%$ chance that $r$ is even, what is the value of $p$?

$\text{(a) }\dfrac{\sqrt{7}}{4}\qquad\text{(b) }\dfrac{2}{3}\qquad\text{(c) }\dfrac{\sqrt{2}}{2}\qquad\text{(d) }\dfrac{3}{4}$

Answer Confirmation
1 reply
kae_3
Feb 9, 2025
Siopao_Enjoyer
an hour ago
2024 PMO Part 1 #3
orangefronted   10
N an hour ago by Siopao_Enjoyer
The Department of Education orders public schools to have class sizes between 15 and 65, inclusive. Suppose that all public schools comply with this order, and that each possible class size is attained by some class in some public school. What is the largest integer N with the property that there is certainly some class in some public school in the country that has at least N students with the same birth month?
10 replies
orangefronted
Jan 16, 2024
Siopao_Enjoyer
an hour ago
I miss Turbo
sarjinius   36
N an hour ago by cielblue
Source: 2025 IMO P6
Consider a $2025\times2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.

Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.

Proposed by Zhao Yu Ma and David Lin Kewei, Singapore
36 replies
sarjinius
Jul 16, 2025
cielblue
an hour ago
Bonza functions
KevinYang2.71   63
N an hour ago by cielblue
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.

Proposed by Lorenzo Sarria, Colombia
63 replies
KevinYang2.71
Jul 15, 2025
cielblue
an hour ago
IMO 2025 P2
sarjinius   88
N an hour ago by cielblue
Source: 2025 IMO P2
Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C, M, N, D$ lie on $MN$ in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E\neq A$ and meets $\Gamma$ again at $F\neq A$. Let $H$ be the orthocentre of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.

Proposed by Trần Quang Hùng, Vietnam
88 replies
sarjinius
Jul 15, 2025
cielblue
an hour ago
Sunny lines
sarjinius   52
N an hour ago by cielblue
Source: 2025 IMO P1
A line in the plane is called $sunny$ if it is not parallel to any of the $x$axis, the $y$axis, or the line $x+y=0$.

Let $n\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:
[list]
[*] for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and
[*] exactly $k$ of the $n$ lines are sunny.
[/list]
Proposed by Linus Tang, USA
52 replies
sarjinius
Jul 15, 2025
cielblue
an hour ago
easy and not easy; 4 vars symetric
perfect_square   1
N 2 hours ago by mihaig
1. Let $ a,b,c,d \ge 0 $ which satisfy: $a^2+b^2+c^2+d^2=4$
Find maximum value of $ P=a+b+c+d-abcd$;
2. Let $ a,b,c,d \ge 0 $ which satisfy: $a^8+b^8+c^8+d^8=4$
Find maximum value of $ P=a+b+c+d-abcd$;
1 reply
1 viewing
perfect_square
4 hours ago
mihaig
2 hours ago
Divisibility by 3 of sum
Razorrizelim   1
N 2 hours ago by Mathzeus1024
Source: 2025 turkey national olympiad first round
For how many of the integers \( n = 195, 196, 197, 198, 199, 200 \), is the number \( 1^n + 2^{n-1} + 3^{n-2} + \cdots + n^1 \) divisible by 3?\[ \textbf{(A)}\ 2 \quad \textbf{(B)}\ 3 \quad \textbf{(C)}\ 4 \quad \textbf{(D)}\ 5 \quad \textbf{(E)}\ 6 \]
1 reply
Razorrizelim
Jul 18, 2025
Mathzeus1024
2 hours ago
Yes or No?
EthanWYX2009   1
N 2 hours ago by EthanWYX2009
Source: 2025 May 谜之竞赛-4
A prime number \( p \) is called good if there exist \( p \) not-all-equal positive integers \( a_1, a_2, \cdots, a_p \) such that for every integer \( 1 \leq k \leq p \), $\sum_{i=1}^p \frac{a_i}{a_{i + k}}$ is an integer.

Question: Are there infinitely many good primes?

Proposed by Zhenyu Dong and Chunji Wang
1 reply
EthanWYX2009
3 hours ago
EthanWYX2009
2 hours ago
trigonometric functions
VivaanKam   16
N May 16, 2025 by Shan3t
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
16 replies
VivaanKam
Apr 29, 2025
Shan3t
May 16, 2025
trigonometric functions
G H J
G H BBookmark kLocked kLocked NReply
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VivaanKam
198 posts
#1 • 2 Y
Y by PikaPika999, linjiah
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
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Lijin
235 posts
#2 • 2 Y
Y by PikaPika999, linjiah
Are you talking about graphing them or just the basic ratios?
Z K Y
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Yiyj1
1295 posts
#4 • 2 Y
Y by PikaPika999, linjiah
Basic ratios: draw a right triangle. I'm terrible at asy so i can't draw one D:. Anyways, label one of the angles $\theta$. Then, label the hypotenuse with $H$, the leg adjacent to $\theta$ as $A$ (for adjacent) and the other leg $O$ (for opposite). Then, just remember this: SOH CAH TOA:\[\sin\theta=\dfrac{O}{H}, \cos\theta=\dfrac{A}{H}, \tan\theta=\dfrac{O}{A}.\]Then there are like $\sec, \csc, \cot$, which are the reciprocals of $\cos, \sin, \tan$. IMPORTANT: $\sec$ is the reciprocal of $\cos$ and $\csc$ is the reciprocal of $\sin$, not the other way around.
Z K Y
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aok
353 posts
#5 • 1 Y
Y by linjiah
To solve things such as $\sin 150$ or $\cos 270$ you can draw a circle of radius 1 around the origin, rotate a line from the positive x line counterclockwise by $\theta^\circ$ to form a new line. To solve for cos,sin, and tan $\theta,$ where the $(x,y)$ hits circle after rotation and $x = \cos\theta$,$y = \sin\theta$, and $\frac{y}{x} = \tan\theta.$
This post has been edited 4 times. Last edited by aok, Apr 29, 2025, 10:48 PM
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VivaanKam
198 posts
#6 • 1 Y
Y by linjiah
Yiyj1 wrote:
Basic ratios: draw a right triangle. I'm terrible at asy so i can't draw one D:. Anyways, label one of the angles $\theta$. Then, label the hypotenuse with $H$, the leg adjacent to $\theta$ as $A$ (for adjacent) and the other leg $O$ (for opposite). Then, just remember this: SOH CAH TOA:\[\sin\theta=\dfrac{O}{H}, \cos\theta=\dfrac{A}{H}, \tan\theta=\dfrac{O}{A}.\]Then there are like $\sec, \csc, \cot$, which are the reciprocals of $\cos, \sin, \tan$. IMPORTANT: $\sec$ is the reciprocal of $\cos$ and $\csc$ is the reciprocal of $\sin$, not the other way around.

So like this?

[asy]

draw((0,0)--(3,0)--(0,2)--cycle);
label("$\theta$", (2.7,0.1),W);
label("$A$", (1.5,0), S);
label("$O$", (0,1.205), W);
label("$H$", (1.2,1.1), NE);
[/asy]
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VivaanKam
198 posts
#7 • 1 Y
Y by linjiah
That’s cool! So if you have the lengths of a triangle you can find its angles?
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VivaanKam
198 posts
#8 • 1 Y
Y by linjiah
aok wrote:
To solve things such as $\sin 150$ or $\cos 270$ you can draw a circle of radius 1 around the origin, rotate a line from the positive x line counterclockwise by $\theta^\circ$ to form a new line. To solve for cos,sin, and tan $\theta,$ where the $(x,y)$ hits circle after rotation and $x = \cos\theta$,$y = \sin\theta$, and $\frac{y}{x} = \tan\theta.$

are they like polar quardinits ?
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VivaanKam
198 posts
#9 • 1 Y
Y by linjiah
but the wouldn't $\cos x$ have 2 values because on a circle there are two quordinates with the same $x$ position?
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lpieleanu
3120 posts
#10 • 1 Y
Y by linjiah
Yes, you can find the side lengths of a triangle given its angles. (If it is right, you can just use the standard ratio definitions of $\sin, \cos, \tan$ and use inverse trigonometric functions, and if it is not right, then you can use the Law of Cosines to find each angle.)

The point in rectangular coordinates $(\cos(\theta), \sin(\theta))$ corresponds to the point in polar coordinates $(1, \theta),$ i.e. $(\cos(\theta), \sin(\theta))$ is the point on the unit circle at an angle of $\theta$ radians counterclockwise of the positive $x$-axis.

Yes, the equation $\cos(x)=a$ has two solutions in $[0, 2\pi)$ for all $-1<a<1.$

Also, reminder that you can combine all of your questions into the same post. :)
This post has been edited 1 time. Last edited by lpieleanu, Apr 30, 2025, 6:39 PM
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aok
353 posts
#11 • 1 Y
Y by linjiah
that is correct, cos x = a has 2 solutions (generally)
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aok
353 posts
#12 • 1 Y
Y by linjiah
for x btw
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aok
353 posts
#13 • 1 Y
Y by linjiah
VivaanKam wrote:
That’s cool! So if you have the lengths of a triangle you can find its angles?

Correct, use the opposite of those functions.
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aok
353 posts
#14 • 1 Y
Y by linjiah
*use the cos theorem to find cos(x) then use the cos^-1
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BlackOctopus23
350 posts
#15 • 1 Y
Y by linjiah
The Unit Circle is also vital in trigonometry and in understanding the functions. This video helped me understand it a lot! Click to reveal hidden text. The unit circle is basically a circle of radius one. Remember that $cos$ is the $x$ and $sin$ is the $y$ if we are viewing it in the perspective of a graph.
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aok
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#16
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Using unit circle as stated.
aok wrote:
To solve things such as $\sin 150$ or $\cos 270$ you can draw a circle of radius 1 around the origin, rotate a line from the positive x line counterclockwise by $\theta^\circ$ to form a new line. To solve for cos,sin, and tan $\theta,$ where the $(x,y)$ hits circle after rotation and $x = \cos\theta$,$y = \sin\theta$, and $\frac{y}{x} = \tan\theta.$
This post has been edited 1 time. Last edited by aok, May 16, 2025, 12:42 AM
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Shan3t
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#17
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might be a bit advanced but Ceva's Theorem, and Extended LoS
This post has been edited 1 time. Last edited by Shan3t, May 16, 2025, 1:03 AM
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Shan3t
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#18
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Shan3t wrote:
might be a bit advanced but Ceva's Theorem, and Extended LoS

also SAS(for area, side angle side), and Ceva's branches off to Menelaus's
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