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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 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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Trigonometry
pythagorazz   2
N 20 minutes ago by vanstraelen
Let \( A \), \( B \), and \( C \) be the angles of an acute triangle such that
\[ \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C = \frac{34}{25}, \]and
\[ \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A = \frac{34}{25}. \]Find
\[ \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B. \]
2 replies
pythagorazz
Today at 9:03 AM
vanstraelen
20 minutes ago
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Equation in naturals
Ahiles   50
N 30 minutes ago by ray66
Source: BMO 2009 Problem 1
Solve the equation
\[ 3^x - 5^y = z^2.\]
in positive integers.

Greece
50 replies
1 viewing
Ahiles
Apr 30, 2009
ray66
30 minutes ago
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Vietnam Mock Test
imnotgoodatmathsorry   4
N 33 minutes ago by no_room_for_error
Second Entrance Mock test for grade 10 specialized in Mathematics at High School for Gifted Students, HNUE, Vietnam
13/4/2025

Problem 1:
1) Let $a,b$ be positive reals. Prove that: $\frac{a}{a+1} + \frac{b}{b+2} < \frac{\sqrt{a} + \sqrt{b}}{2}$
2) In a small garden there are $3$ rabbits and $3$ carrots. Each rabbit will choose randomly a carrot to eat. Find the probability of a carrot was chose by less than $2$ rabbit.
Problem 2:
1) Solve the equation system: $(x+y)(x^2+y^2)=567$ and $\sqrt{xy}(x+y)^2=243$
2) Let $a,b,c$ be positive rational numbers such that: $a+b+c=2\sqrt{abc}$
Problem 3:
Let triangle $ABC$ ($\angle A$, $\angle B$, $\angle C < 90$) with excircle $(O)$ and incircle $(I)$. Incircle $(I)$ touches $BC,CA,AB$ at $D,E,F$. The excircle with the diameter of $AI$ cuts excircle $(O)$ at $K$ ($K \neq A$). $KD$ cuts the excircle with the diameter of $AI$ at $P$ ($P \neq K$) and $AK$ cuts $BC$ at $Q$. Prove that:
1) $\Delta KEC$ ~ $\Delta KFB$ and $KD$ is the bisector of $\angle BKC$
2) $AP \bot BC$
3) $IQ$ is the tangent line of the excircle of $\Delta IBC$
Problem 4,5: (will type tomorrow)
4 replies
imnotgoodatmathsorry
Yesterday at 4:13 PM
no_room_for_error
33 minutes ago
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Turbo's en route to visit each cell of the board
Lukaluce   12
N 38 minutes ago by cj13609517288
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
12 replies
Lukaluce
Today at 11:01 AM
cj13609517288
38 minutes ago
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I found this question really easy, but it is a P4...
Sadigly   3
N 39 minutes ago by grupyorum
Take a sequence $(a_n)_{n=1}^\infty$ such that

$a_1=3$

$a_n=a_1a_2a_3...a_{n-1}-1$

a) Prove that there exists infitely many primes that divides at least 1 term of the sequence.
b) Prove that there exists infitely many primes that doesn't divide any term of the sequence.
3 replies
Sadigly
Yesterday at 7:17 PM
grupyorum
39 minutes ago
J
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EGMO magic square
Lukaluce   6
N 41 minutes ago by Euclid9876
Source: EGMO 2025 P6
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?

Proposed by Paulius Aleknavičius, Lithuania
6 replies
Lukaluce
Today at 11:03 AM
Euclid9876
41 minutes ago
J
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Parallelograms and concyclicity
Lukaluce   17
N 42 minutes ago by cj13609517288
Source: EGMO 2025 P4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
17 replies
Lukaluce
Today at 10:59 AM
cj13609517288
42 minutes ago
J
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US Puzzle Championship Scorecards
djmathman   1
N 42 minutes ago by MathRook7817
Some of the discussion in the Contests & Programs SMT thread reminded me of Scorecards puzzles from the US Puzzle Championship. They behave similarly to "24 game" puzzles, but the allowable operations are slightly expanded.

[quote="Scorecards rules"]
Operations are limited to addition ("+"), subtraction ("-"), multiplication ("x"), division ("/"), and exponentiation ("^").
Decimal points may be used; ...; use minus sign ("-") to indicate negative values. Use parentheses if needed to disambiguate operator precedence.
[/quote]
As an example, the puzzle $11 \leftarrow 5,8,9$ would have answer $8 + 9^{.5} = 11$.

These have appeared on the USPC three years: 2018, 2022, and 2023. Try your hand at these! Some of them are much more devious than they first appear.

[list=1]
[*] 2018
[list=a]
[*] $30\leftarrow 4, 5, 8$
[*] $25\leftarrow 2,6,9$
[*] $23\leftarrow 4,5,9$
[/list]
[*] 2022
[list=a]
[*] $13\leftarrow 3,6,7$
[*] $24\leftarrow 3,8,8$
[*] $25\leftarrow 3,3,4$
[/list]
[*] 2023
[list=a]
[*] $26\leftarrow 5,5,6$
[*] $15\leftarrow 2,6,8$
[*] $11\leftarrow 2,8,9$
[/list]
[/list]
1 reply
+1 w
djmathman
2 hours ago
MathRook7817
42 minutes ago
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Find the angle
pythagorazz   4
N an hour ago by pythagorazz
Let $X$ be a point inside equilateral triangle $ABC$ such that $AX=\sqrt{2},BX=3$, and $CX=\sqrt{5}$. Find the measure of $\angle{AXB}$ in degrees.
4 replies
pythagorazz
Today at 9:07 AM
pythagorazz
an hour ago
J
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sequence infinitely similar to central sequence
InterLoop   17
N an hour ago by juckter
Source: EGMO 2025/2
An infinite increasing sequence $a_1 < a_2 < a_3 < \dots$ of positive integers is called central if for every positive integer $n$, the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1$, $b_2$, $b_3$, $\dots$ of positive integers such that for every central sequence $a_1$, $a_2$, $a_3$, $\dots$, there are infinitely many positive integers $n$ with $a_n = b_n$.
17 replies
InterLoop
Yesterday at 12:38 PM
juckter
an hour ago
J
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Weighted Activity Selection Algorithm
Maximilian113   2
N an hour ago by Maximilian113
An interesting problem:

There are $n$ events $E_1, E_2, \cdots, E_n$ that are each continuous and last on a certain time interval. Each event has a weight $w_i.$ However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:

For each $E_i$ consider $x_i,$ the sum of $w_j$ over all $j$ such that $E_j$ and $E_i$ are not compatible.
1. At each step, delete the event that has the maximal $x_i.$ If there are multiple such events, delete the event with the minimal weight.
2. Update all $x_i$
3. Repeat until all $x_i$ are $0.$
2 replies
Maximilian113
Yesterday at 12:30 AM
Maximilian113
an hour ago
J
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pairwise coprime sum gcd
InterLoop   31
N an hour ago by juckter
Source: EGMO 2025/1
For a positive integer $N$, let $c_1 < c_2 < \dots < c_m$ be all the positive integers smaller than $N$ that are coprime to $N$. Find all $N \ge 3$ such that
$$\gcd(N, c_i + c_{i+1}) \neq 1$$for all $1 \le i \le m - 1$.
31 replies
InterLoop
Yesterday at 12:34 PM
juckter
an hour ago
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EGMO Genre Predictions
ohiorizzler1434   20
N 2 hours ago by khina
Everybody, with EGMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
20 replies
ohiorizzler1434
Mar 28, 2025
khina
2 hours ago
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Sum of squared areas of polyhedron's faces...
Miquel-point   1
N 2 hours ago by Miquel-point
Source: KoMaL B. 5453
The faces of a convex polyhedron are quadrilaterals $ABCD$, $ABFE$, $CDHG$, $ADHE$ and $EFGH$ according to the diagram. The edges from points $A$ and $G$, respectively are pairwise perpendicular. Prove that \[[ABCD]^2+[ABFE]^2+[ADHE]^2=[BCGF]^2+[CDHG]^2+[EFGH]^2,\]where $[XYZW]$ denotes the area of quadrilateral $XYZW$.

Proposed by Géza Kós, Budapest
1 reply
Miquel-point
2 hours ago
Miquel-point
2 hours ago
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2014 preRMO p3, ABCD with perp. diagonals, AB=20, BC=70, CD=90, DA=?
parmenides51   4
N Jan 16, 2021 by RMOAspirantFaraz
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals.
If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?
4 replies
parmenides51
Aug 9, 2019
RMOAspirantFaraz
Jan 16, 2021
J
2014 preRMO p3, ABCD with perp. diagonals, AB=20, BC=70, CD=90, DA=?
G H J
Reveal topic tags
geometry
perpendicular
diagonals
L
G H BBookmark kLocked kLocked NReply
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parmenides51
30630 posts
parmenides51
#1 h  Aug 9, 2019, 12:06 AM • 1 Y
Y by Adventure10
Let $ABCD$ be a convex quadrilateral with perpendicular diagonals.
If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?
Z K Y
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a87321dc0
917 posts
a87321dc0
#2 h  Aug 9, 2019, 4:44 AM • 2 Y
Y by Adventure10, Mango247
Solution
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tensor.exe
8 posts
tensor.exe
#3 h  Jan 16, 2021, 1:14 PM
Y by
what is this theorem called
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tensor.exe
8 posts
tensor.exe
#4 h  Jan 16, 2021, 1:16 PM
Y by
tensor.exe wrote:
what is this theorem called

leave it i got it
its kinda like pythogoras theorem but for orthogonal quads
Z K Y
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RMOAspirantFaraz
543 posts
RMOAspirantFaraz
#5 h  Jan 16, 2021, 1:38 PM • 1 Y
Y by Ambareesh
Straightforward
This post has been edited 2 times. Last edited by RMOAspirantFaraz, Jan 16, 2021, 1:39 PM
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