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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.

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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
Geometry
Lukariman   9
N 13 minutes ago by lbh_qys
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
9 replies
Lukariman
Tuesday at 12:43 PM
lbh_qys
13 minutes ago
I need the technique
DievilOnlyM   15
N an hour ago by sqing
Let a,b,c be real numbers such that: $ab+7bc+ca=188$.
FInd the minimum value of: $5a^2+11b^2+5c^2$
15 replies
DievilOnlyM
May 23, 2019
sqing
an hour ago
Linear colorings mod 2^n
vincentwant   1
N an hour ago by vincentwant
Let $n$ be a positive integer. The ordered pairs $(x,y)$ where $x,y$ are integers in $[0,2^n)$ are each labeled with a positive integer less than or equal to $2^n$ such that every label is used exactly $2^n$ times and there exist integers $a_1,a_2,\dots,a_{2^n}$ and $b_1,b_2,\dots,b_{2^n}$ such that the following property holds: For any two lattice points $(x_1,y_1)$ and $(x_2,y_2)$ that are both labeled $t$, there exists an integer $k$ such that $x_2-x_1-ka_t$ and $y_2-y_1-kb_t$ are both divisible by $2^n$. How many such labelings exist?
1 reply
vincentwant
Apr 30, 2025
vincentwant
an hour ago
sqrt(n) or n+p (Generalized 2017 IMO/1)
vincentwant   1
N an hour ago by vincentwant
Let $p$ be an odd prime. Define $f(n)$ over the positive integers as follows:
$$f(n)=\begin{cases}
\sqrt{n}&\text{ if n is a perfect square} \\
n+p&\text{ otherwise}
\end{cases}$$
Let $p$ be chosen such that there exists an ordered pair of positive integers $(n,k)$ where $n>1,p\nmid n$ such that $f^k(n)=n$. Prove that there exists at least three integers $i$ such that $1\leq i\leq k$ and $f^i(n)$ is a perfect square.
1 reply
vincentwant
Apr 30, 2025
vincentwant
an hour ago
Cyclic Quads and Parallel Lines
gracemoon124   16
N 4 hours ago by ohiorizzler1434
Source: 2015 British Mathematical Olympiad?
Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$. Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$. Prove that the lines $PQ$ and $AB$ are parallel.
16 replies
gracemoon124
Aug 16, 2023
ohiorizzler1434
4 hours ago
Radical Center on the Euler Line (USEMO 2020/3)
franzliszt   37
N 4 hours ago by Ilikeminecraft
Source: USEMO 2020/3
Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $\Gamma$ denote the circumcircle of triangle $ABC$, and $N$ the midpoint of $OH$. The tangents to $\Gamma$ at $B$ and $C$, and the line through $H$ perpendicular to line $AN$, determine a triangle whose circumcircle we denote by $\omega_A$. Define $\omega_B$ and $\omega_C$ similarly.
Prove that the common chords of $\omega_A$,$\omega_B$ and $\omega_C$ are concurrent on line $OH$.

Proposed by Anant Mudgal
37 replies
franzliszt
Oct 24, 2020
Ilikeminecraft
4 hours ago
Line passes through fixed point, as point varies
Jalil_Huseynov   60
N 6 hours ago by Rayvhs
Source: APMO 2022 P2
Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.
60 replies
Jalil_Huseynov
May 17, 2022
Rayvhs
6 hours ago
Tangent to two circles
Mamadi   2
N Yesterday at 9:33 PM by A22-
Source: Own
Two circles \( w_1 \) and \( w_2 \) intersect each other at \( M \) and \( N \). The common tangent to two circles nearer to \( M \) touch \( w_1 \) and \( w_2 \) at \( A \) and \( B \) respectively. Let \( C \) and \( D \) be the reflection of \( A \) and \( B \) respectively with respect to \( M \). The circumcircle of the triangle \( DCM \) intersect circles \( w_1 \) and \( w_2 \) respectively at points \( E \) and \( F \) (both distinct from \( M \)). Show that the line \( EF \) is the second tangent to \( w_1 \) and \( w_2 \).
2 replies
Mamadi
May 2, 2025
A22-
Yesterday at 9:33 PM
perpendicularity involving ex and incenter
Erken   20
N Yesterday at 7:48 PM by Baimukh
Source: Kazakhstan NO 2008 problem 2
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
20 replies
Erken
Dec 24, 2008
Baimukh
Yesterday at 7:48 PM
Isosceles Triangle Geo
oVlad   4
N Yesterday at 7:43 PM by Double07
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$
4 replies
oVlad
Apr 12, 2025
Double07
Yesterday at 7:43 PM
Geometry
Lukariman   1
N Yesterday at 7:34 PM by Primeniyazidayi
Given acute triangle ABC ,AB=b,AC=c . M is a variable point on side AB. The circle circumscribing triangle BCM intersects AC at N.

a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.

b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.
1 reply
Lukariman
Yesterday at 4:02 PM
Primeniyazidayi
Yesterday at 7:34 PM
Incentre-excentre geometry
oVlad   2
N Yesterday at 7:17 PM by Double07
Source: Romania Junior TST 2025 Day 2 P2
Consider a scalene triangle $ABC$ with incentre $I$ and excentres $I_a,I_b,$ and $I_c$, opposite the vertices $A,B,$ and $C$ respectively. The incircle touches $BC,CA,$ and $AB$ at $E,F,$ and $G$ respectively. Prove that the circles $IEI_a,IFI_b,$ and $IGI_c$ have a common point other than $I$.
2 replies
oVlad
Tuesday at 12:54 PM
Double07
Yesterday at 7:17 PM
Great similarity
steven_zhang123   4
N Yesterday at 6:55 PM by khina
Source: a friend
As shown in the figure, there are two points $D$ and $E$ outside triangle $ABC$ such that $\angle DAB = \angle CAE$ and $\angle ABD + \angle ACE = 180^{\circ}$. Connect $BE$ and $DC$, which intersect at point $O$. Let $AO$ intersect $BC$ at point $F$. Prove that $\angle ACE = \angle AFC$.
4 replies
steven_zhang123
Yesterday at 2:13 PM
khina
Yesterday at 6:55 PM
Concurrent lines
MathChallenger101   4
N Yesterday at 3:34 PM by oVlad
Let $A B C D$ be an inscribed quadrilateral. Circles of diameters $A B$ and $C D$ intersect at points $X_1$ and $Y_1$, and circles of diameters $B C$ and $A D$ intersect at points $X_2$ and $Y_2$. The circles of diameters $A C$ and $B D$ intersect in two points $X_3$ and $Y_3$. Prove that the lines $X_1 Y_1, X_2 Y_2$ and $X_3 Y_3$ are concurrent.
4 replies
MathChallenger101
Feb 8, 2025
oVlad
Yesterday at 3:34 PM
Overlapping game
Kei0923   3
N Apr 30, 2025 by CrazyInMath
Source: 2023 Japan MO Finals 1
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
3 replies
Kei0923
Feb 11, 2023
CrazyInMath
Apr 30, 2025
Overlapping game
G H J
G H BBookmark kLocked kLocked NReply
Source: 2023 Japan MO Finals 1
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Kei0923
95 posts
#1 • 2 Y
Y by GeoKing, itslumi
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
This post has been edited 1 time. Last edited by Kei0923, Feb 11, 2023, 11:22 AM
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Tintarn
9042 posts
#2 • 2 Y
Y by SPHS1234, GuvercinciHoca
Answer
Solution
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hectorleo123
344 posts
#4 • 1 Y
Y by GeoKing
Kei0923 wrote:
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.
$\color{blue}\boxed{\textbf{Answer: 24}}$
$\color{blue}\boxed{\textbf{Proof:}}$
$\color{blue}\rule{24cm}{0.3pt}$
$\text{Let us consider the following coloring:}$
[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
fill((1,1)--(2,1)--(2,2)--(1,2)--cycle, black);
fill((1,3)--(2,3)--(2,4)--(1,4)--cycle, black);
fill((3,3)--(4,3)--(4,4)--(3,4)--cycle, black);
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle, black);
[/asy]
$$\text{Let's note that each tetromino covers exactly one black square, as there can be at most 2 tiles per square and there are 4 black squares}$$$$\Rightarrow \text{Number of tiles}\le 2\times 4=8$$$\text{Let us consider the following coloring:}$
[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, blue);
fill((0,2)--(0,3)--(1,3)--(1,2)--cycle, blue);
fill((0,4)--(0,5)--(1,5)--(1,4)--cycle, blue);
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle, blue);
fill((2,2)--(3,2)--(3,3)--(2,3)--cycle, blue);
fill((2,4)--(3,4)--(3,5)--(2,5)--cycle, blue);
fill((4,0)--(5,0)--(5,1)--(4,1)--cycle, blue);
fill((4,2)--(5,2)--(5,3)--(4,3)--cycle, blue);
fill((4,4)--(4,5)--(5,5)--(5,4)--cycle, blue);
[/asy]
$$\text{Let's keep in mind that each tetromino covers exactly one blue square, since there are 9 blue squares and there are at most 8 tiles,}$$$$\text{then at least 1 square remains uncovered}$$$$\text{Number of squares covered by at least one tile}\le 24$$$\color{blue}\rule{24cm}{0.3pt}$
$\color{blue}\boxed{\textbf{Example:}}$
$\color{blue}\rule{24cm}{0.3pt}$
$\text{We fill the board with two layers:}$
[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((1,0)--(1,1)--(0,1));
draw((0,3)--(1,3)--(1,2)--(2,2)--(2,0));
draw((2,2)--(2,4)--(1,4)--(1,5));
draw((5,0)--(4,0)--(4,1)--(3,1)--(3,5));
draw((4,5)--(4,4)--(5,4));
draw((3,3)--(4,3)--(4,2)--(5,2));
[/asy]
[asy]
draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0));
draw((0,1)--(1,1)--(1,2)--(3,2)--(3,1)--(2,1)--(2,0));
draw((3,2)--(5,2));
draw((5,1)--(4,1)--(4,0));
draw((1,5)--(1,4)--(0,4));
draw((0,3)--(2,3)--(2,4)--(3,4)--(3,5));
draw((2,3)--(4,3)--(4,4)--(5,4));
[/asy]
$$\text{Note that there are 24 squares that meet the order}$$$\color{blue}\rule{24cm}{0.3pt}$
$\color{green}\boxed{\textbf{Conclusion:}}$
$\color{green}\rule{24cm}{0.3pt}$
$$\boxed{\textbf{There are at most 24 squares covered by at least one tile}}_\blacksquare$$$\color{green}\rule{24cm}{0.3pt}$
Z K Y
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CrazyInMath
457 posts
#6
Y by
The answer is $24$. Consider that two S-tetromino can form a $2\times 5$ rectangle that has two corners missing
Using four of those can cover everything but the center square.
Consider color the board like this

ABABA
CDCDC
ABABA
CDCDC
ABABA

then each piece would cover one of each A, B, C, D.
As there are only four Ds, we can use at most eight S-tetrominoes
As there are nine A's, there would at least be one uncovered A, so one cannot cover all squares.
Z K Y
N Quick Reply
G
H
=
a