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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.

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0 replies
jlacosta
May 1, 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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Show that 2 triangles are bilogic
kosmonauten3114   0
16 minutes ago
Source: My own
Given a scalene acute triangle $\triangle{ABC}$ with orthic triangle $\triangle{H_AH_BH_C}$, let $P_A$, $P_B$, $P_C$ be points such that $\triangle{AH_BH_C}\cup P_A \sim \triangle{H_ABH_C}\cup P_B \sim \triangle{H_AH_BC}\cup P_C$. Let $\ell_A$ be the trilinear polar of the polar conjugate of $P_A$ wrt $\triangle{ABC}$, and define $\ell_B$ and $\ell_C$ cyclically. Let $\triangle{A'B'C'}$ be the triangle bounded by $\ell_A$, $\ell_B$, $\ell_C$.
Show that $\triangle{ABC}$ and $\triangle{A'B'C'}$ are bilogic.
0 replies
1 viewing
kosmonauten3114
16 minutes ago
0 replies
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Self-evident inequality trick
Lukaluce   22
N 19 minutes ago by sqing
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]When does the equality hold?
22 replies
Lukaluce
May 18, 2025
sqing
19 minutes ago
J
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Cauchy and multiplicative function over a field extension
miiirz30   2
N 20 minutes ago by mohabstudent1
Source: 2025 Euler Olympiad, Round 2
Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$,
$$ f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y), $$where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$.

Proposed by Stijn Cambie, Belgium
2 replies
miiirz30
an hour ago
mohabstudent1
20 minutes ago
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Interesting functions with iterations over integers
miiirz30   0
26 minutes ago
Source: 2025 Euler Olympiad, Round 2
For any subset $S \subseteq \mathbb{Z}^+$, a function $f : S \to S$ is called interesting if the following two conditions hold:

1. There is no element $a \in S$ such that $f(a) = a$.
2. For every $a \in S$, we have $f^{f(a) + 1}(a) = a$ (where $f^{k}$ denotes the $k$-th iteration of $f$).

Prove that:
a) There exist infinitely many interesting functions $f : \mathbb{Z}^+ \to \mathbb{Z}^+$.

b) There exist infinitely many positive integers $n$ for which there is no interesting function
$$ f : \{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}. $$
Proposed by Giorgi Kekenadze, Georgia
0 replies
miiirz30
26 minutes ago
0 replies
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Moving stones on an infinite row
miiirz30   0
40 minutes ago
Source: 2025 Euler Olympiad, Round 2
We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations:

1. Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between.

2. Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right.

3. Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells.

The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player.

IMAGE

Proposed by Luka Tsulaia, Georgia
0 replies
miiirz30
40 minutes ago
0 replies
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Alice, Bob and 6 boxes
Anulick   1
N an hour ago by RPCX
Source: SMMC 2024, B1
Alice has six boxes labelled 1 through 6. She secretly chooses exactly two of the boxes and places a coin inside each. Bob is trying to guess which two boxes contain the coins. Each time Bob guesses, he does so by tapping exactly two of the boxes. Alice then responds by telling him the total number of coins inside the two boxes that he tapped. Bob successfully finds the two coins when Alice responds with the number 2.

What is the smallest positive integer $n$ such that Bob can always find the two coins in at most $n$ guesses?
1 reply
Anulick
Oct 12, 2024
RPCX
an hour ago
J
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functional inequality with equality
miiirz30   0
an hour ago
Source: 2025 Euler Olympiad, Round 2
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that the following two conditions hold:

1. For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$.

2. For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$.

Proposed by Zaza Melikidze, Georgia
0 replies
miiirz30
an hour ago
0 replies
J
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two lines passsing through the midpoint
miiirz30   0
an hour ago
Source: 2025 Euler Olympiad, Round 2
Points $A$, $B$, $C$, and $D$ lie on a line in that order, and points $E$ and $F$ are located outside the line such that $EA=EB$, $FC=FD$ and $EF \parallel AD$. Let the circumcircles of triangles $ABF$ and $CDE$ intersect at points $P$ and $Q$, and the circumcircles of triangles $ACF$ and $BDE$ intersect at points $M$ and $N$. Prove that the lines $PQ$ and $MN$ pass through the midpoint of segment $EF$.

Proposed by Giorgi Arabidze, Georgia
0 replies
miiirz30
an hour ago
0 replies
J
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Intertwined numbers
miiirz30   1
N an hour ago by JARP091
Source: 2025 Euler Olympiad, Round 2
Let a pair of positive integers $(n, m)$ that are relatively prime be called intertwined if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined.

a) Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers.
b) Prove that there does not exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers.
c) Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$.

Proposed by Stijn Cambie, Belgium
1 reply
miiirz30
an hour ago
JARP091
an hour ago
J
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Find the value
sqing   11
N an hour ago by mathematical-forest
Source: 2024 China Fujian High School Mathematics Competition
Let $f(x)=a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0,$ $a_i\in\{-1,1\} ,i=0,1,2,\cdots,6 $ and $f(2)=-53 .$ Find the value of $f(1).$
11 replies
sqing
Jun 22, 2024
mathematical-forest
an hour ago
J
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three circles
barasawala   8
N 2 hours ago by FrancoGiosefAG
Source: Mexico 2003
$A, B, C$ are collinear with $B$ betweeen $A$ and $C$. $K_{1}$ is the circle with diameter $AB$, and $K_{2}$ is the circle with diameter $BC$. Another circle touches $AC$ at $B$ and meets $K_{1}$ again at $P$ and $K_{2}$ again at $Q$. The line $PQ$ meets $K_{1}$ again at $R$ and $K_{2}$ again at $S$. Show that the lines $AR$ and $CS$ meet on the perpendicular to $AC$ at $B$.
8 replies
barasawala
Mar 2, 2007
FrancoGiosefAG
2 hours ago
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d(2025^{a_i}-1) divides a_{n+1}
navi_09220114   3
N 2 hours ago by quacksaysduck
Source: TASIMO 2025 Day 2 Problem 5
Let $a_n$ be a strictly increasing sequence of positive integers such that for all positive integers $n\ge 1$
\[d(2025^{a_n}-1)|a_{n+1}.\]Show that for any positive real number $c$ there is a positive integers $N_c$ such that $a_n>n^c$ for all $n\geq N_c$.

Note. Here $d(m)$ denotes the number of positive divisors of the positive integer $m$.
3 replies
navi_09220114
May 19, 2025
quacksaysduck
2 hours ago
J
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Interesting inequalities
sqing   6
N 2 hours ago by sqing
Source: Own
Let $ a,b,c,d\geq 0 , a+b+c+d \leq 4.$ Prove that
$$a(bc+bd+cd) \leq \frac{256}{81}$$$$ ab(a+2c+2d ) \leq \frac{256}{27}$$$$ ab(a+3c+3d ) \leq \frac{32}{3}$$$$ ab(c+d ) \leq \frac{64}{27}$$
6 replies
sqing
Yesterday at 1:25 PM
sqing
2 hours ago
J
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Trapezium with two right-angles: prove < AKB = 90° and more
Leonardo   6
N 2 hours ago by FrancoGiosefAG
Source: Mexico 2002
Let $ABCD$ be a quadrilateral with $\measuredangle DAB=\measuredangle ABC=90^{\circ}$. Denote by $M$ the midpoint of the side $AB$, and assume that $\measuredangle CMD=90^{\circ}$. Let $K$ be the foot of the perpendicular from the point $M$ to the line $CD$. The line $AK$ meets $BD$ at $P$, and the line $BK$ meets $AC$ at $Q$. Show that $\angle{AKB}=90^{\circ}$ and $\frac{KP}{PA}+\frac{KQ}{QB}=1$.

[Moderator edit: The proposed solution can be found at http://erdos.fciencias.unam.mx/mexproblem3.pdf .]
6 replies
Leonardo
May 8, 2004
FrancoGiosefAG
2 hours ago
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J
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Number Theory
TUAN2k8   4
N Apr 19, 2025 by vsamc
Prove that there are infinitely many of positive integers m such that $m+1 | 3^m+1$
4 replies
TUAN2k8
Apr 18, 2025
vsamc
Apr 19, 2025
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Number Theory
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TUAN2k8
30 posts
TUAN2k8
#1 h  Apr 18, 2025, 3:25 PM
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Prove that there are infinitely many of positive integers m such that $m+1 | 3^m+1$
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Roger.Moore
5 posts
Roger.Moore
#2 h  Apr 18, 2025, 6:06 PM
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TUAN2k8
30 posts
TUAN2k8
#3 h  Apr 18, 2025, 11:08 PM
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Any solution?
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epl1
157 posts
epl1
#4 h  Apr 19, 2025, 2:44 AM
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Woah, self made?
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vsamc
3789 posts
vsamc
#5 h  Apr 19, 2025, 2:51 AM • 2 Y
Y by kiyoras_2001, centslordm
$m = 3^{3^k}$ works for all $k\geq 0$, since $$3^{3^k} + 1 \mid (3^{3^k})^{3^{3^k-k}} + 1^{3^{3^k-k}} = 3^{3^{3^k}} + 1.$$
This post has been edited 2 times. Last edited by vsamc, Apr 19, 2025, 2:51 AM
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