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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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Calculate the distance of chess king!!
egxa   1
N 14 minutes ago by ItzsleepyXD
Source: All Russian 2025 9.4
A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
1 reply
egxa
42 minutes ago
ItzsleepyXD
14 minutes ago
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Nice problem about the Lemoine point of triangle JaBC and OI line
Ktoan07   0
16 minutes ago
Source: Own
Let \(\triangle ABC\) be an acute-angled, non-isosceles triangle with circumcenter \(O\) and incenter \(I\), such that

\[ \prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0, \]
where \(a = BC\), \(b = CA\), and \(c = AB\).

Let \(J_a\), \(J_b\), and \(J_c\) be the excenters opposite to vertices \(A\), \(B\), and \(C\), respectively, and let \(L_a\), \(L_b\), and \(L_c\) be the Lemoine points of triangles \(J_aBC\), \(J_bCA\), and \(J_cAB\), respectively.

Prove that the circles \((L_aBC)\), \((L_bCA)\), and \((L_cAB)\) all pass through a common point \(P\). Moreover, the isogonal conjugate of \(P\) with respect to \(\triangle ABC\) lies on the line \(OI\).

Note (Hint)
0 replies
Ktoan07
16 minutes ago
0 replies
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Rectangular line segments in russia
egxa   0
40 minutes ago
Source: All Russian 2025 9.1
Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?
0 replies
egxa
40 minutes ago
0 replies
J
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external tangents of circumcircles
egxa   0
41 minutes ago
Source: All Russian 2025 9.2
The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). The points of tangency of the circumcircles of triangles \(ABE\) and \(CDE\) with their common external tangents lie on a circle \(\omega\). The points of tangency of the circumcircles of triangles \(ADE\) and \(BCE\) with their common external tangents lie on a circle \(\gamma\). Prove that the centers of circles \(\omega\) and \(\gamma\) coincide.
0 replies
egxa
41 minutes ago
0 replies
J
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Writing letters in grid
egxa   0
an hour ago
Source: All russian 2025 10.1
Petya and Vasya are playing a game on an initially empty \(100 \times 100\) grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?
0 replies
egxa
an hour ago
0 replies
J
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Existence of a circle tangent to four lines
egxa   0
an hour ago
Source: All Russian 2025 10.2
Inside triangle \(ABC\), point \(P\) is marked. Point \(Q\) is on segment \(AB\), and point \(R\) is on segment \(AC\) such that the circumcircles of triangles \(BPQ\) and \(CPR\) are tangent to line \(AP\). Lines are drawn through points \(B\) and \(C\) passing through the center of the circumcircle of triangle \(BPC\), and through points \(Q\) and \(R\) passing through the center of the circumcircle of triangle \(PQR\). Prove that there exists a circle tangent to all four drawn lines.
0 replies
egxa
an hour ago
0 replies
J
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Existence of perfect squares
egxa   0
an hour ago
Source: All Russian 2025 10.3
Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number
\[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \]is a perfect square.
0 replies
egxa
an hour ago
0 replies
J
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Weighted graph problem
egxa   0
an hour ago
Source: All Russian 2025 10.4
In the plane, $106$ points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than $1$. For every group of $6$ marked points, he calculated the sum of the numbers on all $15$ connecting segments. It turned out that the absolute value of each such sum is at least \(C\), and there are both positive and negative such sums. What is the maximum possible value of \(C\)?
0 replies
+1 w
egxa
an hour ago
0 replies
J
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Board problem with complex numbers
egxa   0
an hour ago
Source: All Russian 2025 11.1
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[ a^2 + b^2 + 1 = 2ab \]Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[ c^2 + d^2 + 2025 = 2cd \]
0 replies
egxa
an hour ago
0 replies
J
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two 3D problems in one day
egxa   0
an hour ago
Source: All Russian 2025 11.2
A right prism \(ABCA_1B_1C_1\) is given. It is known that triangles \(A_1BC\), \(AB_1C\), \(ABC_1\), and \(ABC\) are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle \(ABC\), lie on the same sphere.
0 replies
egxa
an hour ago
0 replies
J
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Important pairs of polynomials
egxa   0
an hour ago
Source: All Russian 2025 11.3
A pair of polynomials \(F(x, y)\) and \(G(x, y)\) with integer coefficients is called $\emph{important}$ if from the divisibility of both differences \(F(a, b) - F(c, d)\) and \(G(a, b) - G(c, d)\) by $100$, it follows that both \(a - c\) and \(b - d\) are divisible by 100. Does there exist such an important pair of polynomials \(P(x, y)\), \(Q(x, y)\), such that the pair \(P(x, y) - xy\) and \(Q(x, y) + xy\) is also important?
0 replies
egxa
an hour ago
0 replies
J
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3D russian combo
egxa   0
an hour ago
Source: All Russian 2025 11.4
A natural number \(N\) is given. A cube with side length \(2N + 1\) is made up of \((2N + 1)^3\) unit cubes, each of which is either black or white. It turns out that among any $8$ cubes that share a common vertex and form a \(2 \times 2 \times 2\) cube, there are at most $4$ black cubes. What is the maximum number of black cubes that could have been used?
0 replies
egxa
an hour ago
0 replies
J
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cubefree divisibility
DottedCaculator   57
N an hour ago by tastyl
Source: 2021 ISL N1
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$
57 replies
DottedCaculator
Jul 12, 2022
tastyl
an hour ago
J
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ARMO 2025
Oksutok   0
an hour ago
All-Russian MO 2025 Day 1
0 replies
Oksutok
an hour ago
0 replies
J
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J
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Kinda lookimg Like AM-GM
Atillaa   1
N Apr 13, 2025 by Natrium
Show that for all positive real numbers \( a, b, c \), the following inequality always holds:
\[ \frac{ab}{b+1} + \frac{bc}{c+1} + \frac{ca}{a+1} \geq \frac{3abc}{1 + abc} \]
1 reply
Atillaa
Apr 13, 2025
Natrium
Apr 13, 2025
J
Kinda lookimg Like AM-GM
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Reveal topic tags
inequalities
AM-GM
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G H BBookmark kLocked kLocked NReply
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Atillaa
2 posts
Atillaa
#1 h  Apr 13, 2025, 7:58 AM
Y by
Show that for all positive real numbers \( a, b, c \), the following inequality always holds:
\[ \frac{ab}{b+1} + \frac{bc}{c+1} + \frac{ca}{a+1} \geq \frac{3abc}{1 + abc} \]
This post has been edited 2 times. Last edited by Atillaa, Apr 13, 2025, 8:00 AM
Z K Y
The post below has been deleted. Click to close.
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Natrium
49 posts
Natrium
#2 h  Apr 13, 2025, 8:42 AM
Y by
Multiplying through by the denominators and simplifying, we see that the inequality is equivalent with:

$$\sum{a^3b^2c}+\sum{a^3b^2c}+\sum{a^2b}+\sum{ab}-2\sum{a^2bc}-2\sum{a^2b^2c}\ge 0$$$$\iff\sum{ac(c+1)(ab-1)^2}\ge 0$$
So we are done :D
This post has been edited 1 time. Last edited by Natrium, Apr 13, 2025, 8:45 AM
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