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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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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
J
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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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NEPAL TST DAY 2 PROBLEM 2
Tony_stark0094   4
N an hour ago by Mathdreams
Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of $k$ that guarantees that Kritesh's job is possible?
4 replies
Tony_stark0094
Today at 8:37 AM
Mathdreams
an hour ago
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Abelkonkurransen 2025 1a
Lil_flip38   1
N 2 hours ago by MathLuis
Source: abelkonkurransen
Peer and Solveig are playing a game with $n$ coins, all of which show $M$ on one side and $K$ on the opposite side. The coins are laid out in a row on the table. Peer and Solveig alternate taking turns. On his turn, Peer may turn over one or more coins, so long as he does not turn over two adjacent coins. On her turn, Solveig picks precisely two adjacent coins and turns them over. When the game begins, all the coins are showing $M$. Peer plays first, and he wins if all the coins show $K$ simultaneously at any time. Find all $n\geqslant 2$ for which Solveig can keep Peer from winning.
1 reply
Lil_flip38
Mar 20, 2025
MathLuis
2 hours ago
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cos k theta and cos(k + 1) theta are both rational
N.T.TUAN   12
N 2 hours ago by Ilikeminecraft
Source: USA Team Selection Test 2007
Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k + 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta = \pi/6$.
12 replies
N.T.TUAN
Dec 8, 2007
Ilikeminecraft
2 hours ago
J
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Problem 3 IMO 2005 (Day 1)
Valentin Vornicu   119
N 2 hours ago by MTA_2024
Let $x,y,z$ be three positive reals such that $xyz\geq 1$. Prove that
\[ \frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 . \]
Hojoo Lee, Korea
119 replies
Valentin Vornicu
Jul 13, 2005
MTA_2024
2 hours ago
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Rhombus EVAN
62861   71
N 2 hours ago by ihategeo_1969
Source: USA January TST for IMO 2017, Problem 2
Let $ABC$ be a triangle with altitude $\overline{AE}$. The $A$-excircle touches $\overline{BC}$ at $D$, and intersects the circumcircle at two points $F$ and $G$. Prove that one can select points $V$ and $N$ on lines $DG$ and $DF$ such that quadrilateral $EVAN$ is a rhombus.

Danielle Wang and Evan Chen
71 replies
62861
Feb 23, 2017
ihategeo_1969
2 hours ago
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Ihave a minor issue.
CovertQED   1
N 2 hours ago by vanstraelen
The area of triangle ABC is 18，sin2A +sin2B =4sinAsinB.Find the minimum perimeter of triangle ABC.
1 reply
CovertQED
Today at 1:52 PM
vanstraelen
2 hours ago
J
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A and B play a game
EthanWYX2009   3
N 2 hours ago by nitr4m
Source: 2025 TST 23
Let \( n \geq 2 \) be an integer. Two players, Alice and Bob, play the following game on the complete graph \( K_n \): They take turns to perform operations, where each operation consists of coloring one or two edges that have not been colored yet. The game terminates if at any point there exists a triangle whose three edges are all colored.

Prove that there exists a positive number \(\varepsilon\), Alice has a strategy such that, no matter how Bob colors the edges, the game terminates with the number of colored edges not exceeding
\[ \left( \frac{1}{4} - \varepsilon \right) n^2 + n. \]
3 replies
EthanWYX2009
Mar 29, 2025
nitr4m
2 hours ago
J
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geometry parabola problem
smalkaram_3549   8
N 2 hours ago by jb2015007
How would you solve this without using calculus?
8 replies
1 viewing
smalkaram_3549
Yesterday at 9:52 PM
jb2015007
2 hours ago
J
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Problem 3
SlovEcience   1
N 2 hours ago by kokcio
Find all real numbers \( k \) such that the following inequality holds for all \( a, b, c \geq 0 \):

\[ ab + bc + ca \leq \frac{(a + b + c)^2}{3} + k \cdot \max \{ (a - b)^2, (b - c)^2, (c - a)^2 \} \leq a^2 + b^2 + c^2 \]
1 reply
SlovEcience
Apr 9, 2025
kokcio
2 hours ago
J
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Inequality with a,b,c
GeoMorocco   8
N 2 hours ago by GeoMorocco
Source: Morocco Training 2025
Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{a\sqrt{3+bc}}{b+c}+\frac{b\sqrt{3+ca}}{c+a}+\frac{c\sqrt{3+ab}}{a+b}\ge a+b+c $$
8 replies
GeoMorocco
Apr 10, 2025
GeoMorocco
2 hours ago
J
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prove that any quadrilateral satisfying this inequality is a trapezoid
mqoi_KOLA   1
N 2 hours ago by vgtcross
Prove that any quadrilateral satisfying this inequality is a Trapezoid/trapzium $$ |r - p| < q + s < r + p $$where $p,r$ are lengths of parallel sides and $q,s$ are other two sides.
1 reply
mqoi_KOLA
Today at 3:48 AM
vgtcross
2 hours ago
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Prove that there exists a convex 1990-gon
orl   13
N 3 hours ago by akliu
Source: IMO 1990, Day 2, Problem 6, IMO ShortList 1990, Problem 16 (NET 1)
Prove that there exists a convex 1990-gon with the following two properties :

a.) All angles are equal.
b.) The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
13 replies
orl
Nov 11, 2005
akliu
3 hours ago
J
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Indonesia Regional MO 2019 Part A
parmenides51   13
N Today at 3:09 PM by SomeonecoolLovesMaths
Indonesia Regional MO
Year 2019 Part A

Time: 90 minutes Rules


p1. In the bag there are $7$ red balls and $8$ white balls. Audi took two balls at once from inside the bag. The chance of taking two balls of the same color is ...


p2. Given a regular hexagon with a side length of $1$ unit. The area of the hexagon is ...


p3. It is known that $r, s$ and $1$ are the roots of the cubic equation $x^3 - 2x + c = 0$. The value of $(r-s)^2$ is ...


p4. The number of pairs of natural numbers $(m, n)$ so that $GCD(n,m) = 2$ and $LCM(m,n) = 1000$ is ...


p5. A data with four real numbers $2n-4$, $2n-6$, $n^2-8$, $3n^2-6$ has an average of $0$ and a median of $9/2$. The largest number of such data is ...


p6. Suppose $a, b, c, d$ are integers greater than $2019$ which are four consecutive quarters of an arithmetic row with $a <b <c <d$. If $a$ and $d$ are squares of two consecutive natural numbers, then the smallest value of $c-b$ is ...


p7. Given a triangle $ABC$, with $AB = 6$, $AC = 8$ and $BC = 10$. The points $D$ and $E$ lies on the line segment $BC$. with $BD = 2$ and $CE = 4$. The measure of the angle $\angle DAE$ is ...


p8. Sequqnce of real numbers $a_1,a_2,a_3,...$ meet $\frac{na_1+(n-1)a_2+...+2a_{n-1}+a_n}{n^2}=1$ for each natural number $n$. The value of $a_1a_2a_3...a_{2019}$ is ....


p9. The number of ways to select four numbers from $\{1,2,3, ..., 15\}$ provided that the difference of any two numbers at least $3$ is ...


p10. Pairs of natural numbers $(m , n)$ which satisfies $$m^2n+mn^2 +m^2+2mn = 2018m + 2019n + 2019$$are as many as ...


p11. Given a triangle $ABC$ with $\angle ABC =135^o$ and $BC> AB$. Point $D$ lies on the side $BC$ so that $AB=CD$. Suppose $F$ is a point on the side extension $AB$ so that $DF$ is perpendicular to $AB$. The point $E$ lies on the ray $DF$ such that $DE> DF$ and $\angle ACE = 45^o$. The large angle $\angle AEC$ is ...


p12. The set of $S$ consists of $n$ integers with the following properties: For every three different members of $S$ there are two of them whose sum is a member of $S$. The largest value of $n$ is ....


p13. The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....


p14. The polynomial P satisfies the equation $P (x^2) = x^{2019} (x+ 1) P (x)$ with $P (1/2)= -1$ is ....


p15. Look at a chessboard measuring $19 \times 19$ square units. Two plots are said to be neighbors if they both have one side in common. Initially, there are a total of $k$ coins on the chessboard where each coin is only loaded exactly on one square and each square can contain coins or blanks. At each turn. You must select exactly one plot that holds the minimum number of coins in the number of neighbors of the plot and then you must give exactly one coin to each neighbor of the selected plot. The game ends if you are no longer able to select squares with the intended conditions. The smallest number of $k$ so that the game never ends for any initial square selection is ....
13 replies
parmenides51
Nov 11, 2021
SomeonecoolLovesMaths
Today at 3:09 PM
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Diophantine Equation in (x^2+4) set
Johann Peter Dirichlet   0
Today at 3:01 PM
Let $S=\{n^2+4 | n \in \mathbb{Z}\}$.

Find all $p,q,r \in S$ so that $pq-r=4$.
0 replies
Johann Peter Dirichlet
Today at 3:01 PM
0 replies
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J
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largest roof, gable or four-pitched? (Oral Moscow 2008 VII p5 )
parmenides51   9
N Sep 1, 2020 by bobthefam
Ivan Ivanovich built a blockhouse, square at the base, and is going to cover it with a roof. He chooses between two roofs of the same height: a gable and a four-pitched (see pictures). Which of these roofs will need more sheet metal?
IMAGE
original wording in Russian
9 replies
parmenides51
Sep 1, 2020
bobthefam
Sep 1, 2020
J
largest roof, gable or four-pitched? (Oral Moscow 2008 VII p5 )
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Reveal topic tags
geometry
3D geometry
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parmenides51
30630 posts
parmenides51
#1 h  Sep 1, 2020, 11:52 AM • 1 Y
Y by Mango247
Ivan Ivanovich built a blockhouse, square at the base, and is going to cover it with a roof. He chooses between two roofs of the same height: a gable and a four-pitched (see pictures). Which of these roofs will need more sheet metal?
https://1.bp.blogspot.com/-FVJW7fkS5e0/XziBQdiNfJI/AAAAAAAAMaE/e7DcZURu-cIt0W7NmQkUUbg6ugh2h9itQCLcBGAsYHQ/s0/oral%2B2009%2B8.5.png
original wording in Russian
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HumanCalculator9
6231 posts
HumanCalculator9
#2 h  Sep 1, 2020, 11:55 AM • 2 Y
Y by Mango247, Mango247
What is the height?
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SharpApricot123
1270 posts
SharpApricot123
#3 h  Sep 1, 2020, 1:42 PM
Y by
follow up question: Is the roof the shaded part, or is it all the sides excluding the base?
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Supermathking
270 posts
Supermathking
#4 h  Sep 1, 2020, 1:47 PM
Y by
HumanCalculator9 wrote:
What is the height?

We don't know what the height is, and I don't think it matters.
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HumanCalculator9
6231 posts
HumanCalculator9
#5 h  Sep 1, 2020, 1:54 PM
Y by
SharpApricot123 wrote:
follow up question: Is the roof the shaded part, or is it all the sides excluding the base?

the roof is the shaded parts I believe
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natmath
8219 posts
natmath
#6 h  Sep 1, 2020, 2:38 PM
Y by
Maybe they have to cover the same height?

So if the sidelength of the square is $s$ and the height $h$, we can find the length of the hypotenuse of the right triangle with legs of $\frac{s}{2}$ and $h$. call this length $l$. We have the area of the gable as $2sl$ and the area of the four-pitch as $4(sl/2)$, so they are both equal.
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bobthefam
917 posts
bobthefam
#7 h  Sep 1, 2020, 3:02 PM
Y by
HumanCalculator9 wrote:
SharpApricot123 wrote:
follow up question: Is the roof the shaded part, or is it all the sides excluding the base?

the roof is the shaded parts I believe

I believe that it is everything except the bottom square.
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bobthefam
917 posts
bobthefam
#8 h  Sep 1, 2020, 3:09 PM
Y by
My Sol

Overall, the wording needs to be changed to make it more clear. We have to make a lot of assumptions.
This post has been edited 4 times. Last edited by bobthefam, Sep 1, 2020, 3:12 PM
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parmenides51
30630 posts
parmenides51
#9 h  Sep 1, 2020, 3:28 PM
Y by
Everytime I have doubts about the translation given by Google translate, I add the original wording as above. Obviously base and height are equal at both figures. So either adding the base or not does not matter. My only guess is that in the left figure (gable), it cannot have a hole because it might flood in a possible rain.
This post has been edited 2 times. Last edited by parmenides51, Sep 1, 2020, 3:30 PM
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bobthefam
917 posts
bobthefam
#10 h  Sep 1, 2020, 4:18 PM • 3 Y
Y by Mango247, Mango247, Mango247
parmenides51 wrote:
Everytime I have doubts about the translation given by Google translate, I add the original wording as above. Obviously base and height are equal at both figures. So either adding the base or not does not matter. My only guess is that in the left figure (gable), it cannot have a hole because it might flood in a possible rain.

Sol based on @above
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