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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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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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Perhaps a classic with parameter
mihaig   1
N 14 minutes ago by LLriyue
Find the largest positive constant $r$ such that
$$a^2+b^2+c^2+d^2+2\left(abcd\right)^r\geq6$$for all reals $a\geq1\geq b\geq c\geq d\geq0$ satisfying $a+b+c+d=4.$
1 reply
mihaig
Jan 7, 2025
LLriyue
14 minutes ago
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Interesting inequalities
sqing   2
N 25 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ a^2+b^2+c^2+abc=4$ . Prove that
$$k(a+b+c) -ab-bc\geq 4\sqrt{k(k+1)}-(k+4)$$Where $ k\geq \frac{16}{9}. $
$$ \frac{16}{9}(a+b+c) -ab-bc\geq \frac{28}{9}$$
2 replies
sqing
an hour ago
sqing
25 minutes ago
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Turbo's en route to visit each cell of the board
Lukaluce   17
N an hour ago by Gato_combinatorio
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
17 replies
Lukaluce
Monday at 11:01 AM
Gato_combinatorio
an hour ago
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Connected graph with k edges
orl   26
N an hour ago by Maximilian113
Source: IMO 1991, Day 2, Problem 4, IMO ShortList 1991, Problem 10 (USA 5)
Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.

Note: Graph-Definition. A graph consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x = v_{0},v_{1},v_{2},\cdots ,v_{m} = y\,$ such that each pair $ \,v_{i},v_{i + 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.
26 replies
orl
Nov 11, 2005
Maximilian113
an hour ago
J
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A Projection Theorem
buratinogigle   1
N 2 hours ago by aidan0626
Source: VN Math Olympiad For High School Students P1 - 2025
In triangle $ABC$, prove that
\[ a = b\cos C + c\cos B. \]
1 reply
buratinogigle
3 hours ago
aidan0626
2 hours ago
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A Problem on a Rectangle
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P12 - 2025 - Bonus, MM Problem 2197
Let $ABCD$ be a rectangle and $P$ any point. Let $X, Y, Z, W, S, T$ be the foots of the perpendiculars from $P$ to the lines $AB, BC, CD, DA, AB, BD$, respectively. Let the perpendicular bisectors of $XY$ and $WZ$ intersect at $Q$, and those of $YZ$ and $XW$ intersect at $R$. Prove that the lines $QR$ and $ST$ are parallel.

MM Problem
0 replies
buratinogigle
3 hours ago
0 replies
J
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The difference of the two angles is 180 degrees
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P11 - 2025
In triangle $ABC$, let $D$ be the midpoint of $AB$, and $E$ the midpoint of $CD$. Suppose $\angle ACD = 2\angle DEB$. Prove that
\[ 2\angle AED-\angle DCB =180^\circ. \]
0 replies
1 viewing
buratinogigle
3 hours ago
0 replies
J
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A Segment Bisection Problem
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P9 - 2025
In triangle $ABC$, let the incircle $\omega$ touch sides $BC, CA, AB$ at $D, E, F$, respectively. Let $P$ lie on the line through $D$ perpendicular to $BC$. Let $Q, R$ be the intersections of $PC, PB$ with $EF$, respectively. Let $K, L$ be the projections of $R, Q$ onto line $BC$. Let $M, N$ be the second intersections of $DQ, DR$ with the incircle $\omega$. Let $S$ be the intersection of $KM$ and $LN$. Prove that the line $DS$ bisects segment $QR$.
0 replies
buratinogigle
3 hours ago
0 replies
J
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A Characterization of Rectangles
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P8 - 2025
Prove that if a convex quadrilateral $ABCD$ satisfies the equation
\[ (AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2, \]then $ABCD$ must be a rectangle.
0 replies
buratinogigle
3 hours ago
0 replies
J
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A Generalization of Ptolemy's Theorem
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P7 - 2025
Given a convex quadrilateral $ABCD$, define
\[ \alpha = |\angle ADB - \angle ACB| = |\angle DAC - \angle DBC| \quad\text{and}\quad \beta = |\angle ABD - \angle ACD| = |\angle BAC - \angle BDC|. \]Prove that
\[ AC \cdot BD = AD \cdot BC \cos\alpha + AB \cdot CD \cos\beta. \]
0 replies
buratinogigle
3 hours ago
0 replies
J
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A Cosine-Type Formula for Quadrilaterals
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P6 - 2025
Given a convex quadrilateral $ABCD$, let $\theta$ be the sum of two opposite angles. Prove that
\[ AC^2 \cdot BD^2 = AB^2 \cdot CD^2 + AD^2 \cdot BC^2 - 2AB \cdot CD \cdot AD \cdot BC \cos\theta. \]
0 replies
buratinogigle
3 hours ago
0 replies
J
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On Ptolemy Triangle
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P5 - 2025
Given a convex quadrilateral $ABCD$, construct a point $P$ inside the quadrilateral such that triangles $APB$ and $ADC$ are similar. Prove that triangle $PBD$ has sides $PB, PD, BD$ proportional to $CD\cdot AB$, $BC\cdot AD$, and $BD\cdot AC$, respectively.
0 replies
buratinogigle
3 hours ago
0 replies
J
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An Inequality in a Polygon
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P4 - 2025
Let $\mathcal{A} = A_1A_2\ldots A_n$ be a convex polygon. For $i = 1, \ldots, n$, denote by $\alpha_i$ the internal angle at vertex $A_i$. For any point $P$ inside $\mathcal{A}$, prove that
\[ \cos\frac{\alpha_1}{2}PA_1 + \cos\frac{\alpha_2}{2}PA_2 + \ldots + \cos\frac{\alpha_n}{2}PA_n \ge s, \]where $s$ denotes the semi-perimeter of $\mathcal{A}$.
0 replies
buratinogigle
3 hours ago
0 replies
J
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Cosine Law Extension
buratinogigle   0
3 hours ago
Source: VN Math Olympiad For High School Students P3 - 2025
Given triangle $ABC$, construct parallelogram $ABDC$. Let $P$ be any point in the plane, and denote $\angle BPC = \alpha$, $\angle PAD = \theta$. Prove that
\[ a^2 = b^2 + c^2 - 2PB \cdot PC \cos\alpha - 2PA \cdot AD \cos\theta. \]
0 replies
buratinogigle
3 hours ago
0 replies
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J
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Old problem :(
Drakkur   2
N Apr 3, 2025 by Drakkur
Let a, b, c be positive real numbers. Prove that
$$\dfrac{1}{\sqrt{a^2+bc}}+\dfrac{1}{\sqrt{b^2+ca}}+\dfrac{1}{\sqrt{c^2+ab}}\le \sqrt{2}\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}\right)$$
2 replies
Drakkur
Apr 2, 2025
Drakkur
Apr 3, 2025
J
Old problem :(
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Reveal topic tags
inequalities
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Drakkur
17 posts
Drakkur
#1 h  Apr 2, 2025, 3:38 PM • 1 Y
Y by cubres
Let a, b, c be positive real numbers. Prove that
$$\dfrac{1}{\sqrt{a^2+bc}}+\dfrac{1}{\sqrt{b^2+ca}}+\dfrac{1}{\sqrt{c^2+ab}}\le \sqrt{2}\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}\right)$$
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Quantum-Phantom
259 posts
Quantum-Phantom
#3 h  Apr 2, 2025, 4:21 PM • 1 Y
Y by Drakkur
An interesting but not necessarily useful idea. We apply Ji Chen's result to $p=\tfrac12$ and the two sets of quantities
\begin{align*} x_1,~x_2,~x_3&=\frac1{a^2+bc},~\frac1{b^2+ca},~\frac1{c^2+ab};\\ y_1,~y_2,~y_3&=\frac{(a+2b+c)^2}{2(a+b)^2(b+c)^2},~\frac{(b+2c+a)^2}{2(b+c)^2(c+a)^2},~\frac{(c+2a+b)^2}{2(c+a)^2(a+b)^2}. \end{align*}I used software to check that
\begin{align*}x_1+x_2+x_3&\le y_1+y_2+y_3,\\x_1x_2+x_2x_3+x_3x_1&\le y_1y_2+y_2y_3+y_3y_1,\\x_1x_2x_3&\le y_1y_2y_3\end{align*}are all correct polynomial inequalities.
This post has been edited 1 time. Last edited by Quantum-Phantom, Apr 3, 2025, 7:41 AM
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Drakkur
17 posts
Drakkur
#4 h  Apr 3, 2025, 10:43 AM
Y by
Nice but i found this beautiful solution from @KhuongTrang

We have
$$\left( \displaystyle\sum_{cyc} \dfrac{1}{\sqrt{a^2+bc}}\right)^2\leq \left[ \displaystyle\sum_{cyc} \dfrac{1}{(a+b)(a+c)} \right]\left[ \displaystyle\sum_{cyc} \dfrac{(a+b)(a+c)}{a^2+bc} \right] = \dfrac{2(a+b+c)}{(a+b)(b+c)(c+a)} \left[ \displaystyle\sum_{cyc} \dfrac{a(b+c)}{a^2+bc}+3 \right]$$
Just need to prove that
$$ \dfrac{2(a+b+c)}{(a+b)(b+c)(c+a)} \left[ \displaystyle\sum_{cyc} \dfrac{a(b+c)}{a^2+bc}+3 \right] \leq 2\left( \displaystyle\sum_{cyc} \dfrac{1}{a+b} \right)^2 $$
but this is equivalent to

$$\displaystyle\sum_{cyc} (a-b)(a-c) \left[ \dfrac{1}{a^2+bc} + \dfrac{1}{(b+c)(a+b+c)} \right] \geq 0 $$
which is true.
This post has been edited 1 time. Last edited by Drakkur, Apr 3, 2025, 10:46 AM
Reason: mistake brackets
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