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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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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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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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2018 Sipnayan Junior Highscool Semifinals A Average.1
wonderboy807   0
18 minutes ago
Let f(1) = 2016 , f(2) = 2018 ,

f(n) = [f(n-1)]^2 + [f(n-2)]^2 \quad \text{for all } n \geq 3.

What is the units digit of f(2018) ?

Answer: Click to reveal hidden text

Solution: Click to reveal hidden text
0 replies
wonderboy807
18 minutes ago
0 replies
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PMO 2020 Qualifying Stage I.6
wonderboy807   0
36 minutes ago
A function f : \mathbb{R} \to \mathbb{R} satisfies f(xy) = \frac{f(x)}{y^2} for all positive real numbers x and y . Given that f(25) = 48, what is f(100) ?

Answer: Click to reveal hidden text

Solution: Click to reveal hidden text
0 replies
wonderboy807
36 minutes ago
0 replies
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Indonesian Junior MO (Nationals) 2018, Day 2
somebodyyouusedtoknow   1
N an hour ago by Rayholr123
P6. It is given the integer $Y$ with
$Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.$
Determine the sum of all the digits of such $Y$. (It is implied that $Y$ is written with a decimal representation.)

P7. Three groups of lines divides a plane into $D$ regions. Every pair of lines in the same group are parallel. Let $x, y$ and $z$ respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is ....

P8. It is known a frustum $ABCD.EFGH$ where $ABCD$ and $EFGH$ are squares with both planes being parallel. The length of the sides of $ABCD$ and $EFGH$ respectively are $6a$ and $3a$, and the height of the frustum is $3t$. Points $M$ and $N$ respectively are intersections of the diagonals of $ABCD$ and $EFGH$ and the line $MN$ is perpendicular to the plane $EFGH$. Construct the pyramids $M.EFGH$ and $N.ABCD$ and calculate the volume of the 3D figure which is the intersection of pyramids $N.ABCD$ and $M.EFGH$.

P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number $19$ is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number $26$ is on the 3rd row, 5th column, and 7th diagonal.

(Image should be placed here, look at attachment.)

a) Determine the position of the number $2018$ based on its row, column, and diagonal.
b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to $2018$.

P10. It is known that $A$ is the set of 3-digit integers not containing the digit $0$. Define a gadang number to be the element of $A$ whose digits are all distinct and the digits contained in such number are not prime, and (a gadang number leaves a remainder of 5 when divided by 7. If we pick an element of $A$ at random, what is the probability that the number we picked is a gadang number?
1 reply
somebodyyouusedtoknow
Nov 11, 2021
Rayholr123
an hour ago
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Indonesian Junior MO 2018 (Nationals), Day 1
somebodyyouusedtoknow   6
N an hour ago by Rayholr123
The problems are really difficult to find online, so here are the problems.

P1. It is known that two positive integers $m$ and $n$ satisfy $10n - 9m = 7$ dan $m \leq 2018$. The number $k = 20 - \frac{18m}{n}$ is a fraction in its simplest form.
a) Determine the smallest possible value of $k$.
b) If the denominator of the smallest value of $k$ is (equal to some number) $N$, determine all positive factors of $N$.
c) On taking one factor out of all the mentioned positive factors of $N$ above (specifically in problem b), determine the probability of taking a factor who is a multiple of 4.

I added this because my translation is a bit weird.
Indonesian Version

P2. Let the functions $f, g : \mathbb{R} \to \mathbb{R}$ be given in the following graphs.
Graph Construction Notes
Define the function $g \circ f$ with $(g \circ f)(x) = g(f(x))$ for all $x \in D_f$ where $D_f$ is the domain of $f$.
a) Draw the graph of the function $g \circ f$.
b) Determine all values of $x$ so that $-\frac{1}{2} \leq (g \circ f)(x) \leq 6$.

P3. The quadrilateral $ABCD$ has side lengths $AB = BC = 4\sqrt{3}$ cm and $CD = DA = 4$ cm. All four of its vertices lie on a circle. Calculate the area of quadrilateral $ABCD$.

P4. There exists positive integers $x$ and $y$, with $x < 100$ and $y > 9$. It is known that $y = \frac{p}{777} x$, where $p$ is a 3-digit number whose number in its tens place is 5. Determine the number/quantity of all possible values of $y$.

P5. The 8-digit number $\overline{abcdefgh}$ (the original problem does not have an overline, which I fixed) is arranged from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Such number satisfies $a + c + e + g \geq b + d + f + h$. Determine the quantity of different possible (such) numbers.

6 replies
somebodyyouusedtoknow
Nov 11, 2021
Rayholr123
an hour ago
J
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Brilliant Problem
M11100111001Y1R   7
N 2 hours ago by flower417477
Source: Iran TST 2025 Test 3 Problem 3
Find all sequences \( (a_n) \) of natural numbers such that for every pair of natural numbers \( r \) and \( s \), the following inequality holds:
\[ \frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2 \]
7 replies
M11100111001Y1R
May 27, 2025
flower417477
2 hours ago
J
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In Cyclic Quadrilateral ABCD, find AB^2+BC^2-CD^2-AD^2
Darealzolt   1
N 2 hours ago by Beelzebub
Source: KTOM April 2025 P8
Given Cyclic Quadrilateral \(ABCD\) with an area of \(2025\), with \(\angle ABC = 45^{\circ}\). If \( 2AC^2 = AB^2+BC^2+CD^2+DA^2\), Hence find the value of \(AB^2+BC^2-CD^2-DA^2\).
1 reply
Darealzolt
Today at 4:10 AM
Beelzebub
2 hours ago
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Another FE
M11100111001Y1R   3
N 2 hours ago by AndreiVila
Source: Iran TST 2025 Test 2 Problem 3
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all $x,y>0$ we have:
$$f(f(f(xy))+x^2)=f(y)(f(x)-f(x+y))$$
3 replies
M11100111001Y1R
Yesterday at 8:03 AM
AndreiVila
2 hours ago
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Iran TST Starter
M11100111001Y1R   4
N 2 hours ago by flower417477
Source: Iran TST 2025 Test 1 Problem 1
Let \( a_n \) be a sequence of positive real numbers such that for every \( n > 2025 \), we have:
\[ a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i} \]Prove that there exists a natural number \( M \) such that for all \( n > M \), the following holds:
\[ a_n < \frac{1}{1404} \]
4 replies
M11100111001Y1R
May 27, 2025
flower417477
2 hours ago
J
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Inequality with abc=1
tenplusten   10
N 2 hours ago by Adywastaken
Source: JBMO 2011 Shortlist A7
$\boxed{\text{A7}}$ Let $a,b,c$ be positive reals such that $abc=1$.Prove the inequality $\sum\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1}\geq 3$
10 replies
tenplusten
May 15, 2016
Adywastaken
2 hours ago
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A game of cutting
k.vasilev   11
N 2 hours ago by NicoN9
Source: All-Russian Olympiad 2019 grade 10 problem 2
Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?
11 replies
k.vasilev
Apr 23, 2019
NicoN9
2 hours ago
J
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Problem 10
SlovEcience   1
N 3 hours ago by lbh_qys
Let \( x, y, z \) be positive real numbers satisfying
\[ xy + yz + zx = 3xyz. \]Prove that
\[ \sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}. \]
1 reply
SlovEcience
3 hours ago
lbh_qys
3 hours ago
J
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Cup of Combinatorics
M11100111001Y1R   8
N 3 hours ago by sansgankrsngupta
Source: Iran TST 2025 Test 4 Problem 2
There are \( n \) cups labeled \( 1, 2, \dots, n \), where the \( i \)-th cup has capacity \( i \) liters. In total, there are \( n \) liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.

$a)$ Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most \( \frac{4n}{3} \) steps.

$b)$ Prove that in at most \( \frac{5n}{3} \) steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
8 replies
M11100111001Y1R
May 27, 2025
sansgankrsngupta
3 hours ago
J
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2-var inequality
sqing   2
N 3 hours ago by sqing
Source: Own
Let $ a,b> 0 , ab(a+b+1) =3.$ Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{24}{(a+b)^2} \geq 8$$$$ \frac{a}{b^2}+\frac{b}{a^2}+\frac{49}{(a+ b)^2} \geq \frac{57}{4}$$Let $ a,b> 0 , (a+b)(ab+1) =4.$ Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{40}{(a+b)^2} \geq 12$$$$\frac{a}{b^2}+\frac{b}{a^2}+\frac{76}{(a+ b)^2} \geq 21$$
2 replies
sqing
May 25, 2025
sqing
3 hours ago
J
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2-var inequality
sqing   10
N 3 hours ago by sqing
Source: Own
Let $ a,b>0 , a^2+b^2-ab\leq 1 . $ Prove that
$$a^3+b^3 -\frac{a^4}{b+1} -\frac{b^4}{a+1} \leq 1 $$
10 replies
sqing
May 27, 2025
sqing
3 hours ago
J
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J
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function composition with quadratics yields no real roots (Auckland MO 2024 P11)
Equinox8   2
N Mar 12, 2025 by alexheinis
It is known that for quadratic polynomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$ the equation $P(Q(x))=Q(P(x))$ does not have real roots. Prove that $b \neq d$.
2 replies
Equinox8
Mar 12, 2025
alexheinis
Mar 12, 2025
J
function composition with quadratics yields no real roots (Auckland MO 2024 P11)
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Reveal topic tags
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Equinox8
1744 posts
Equinox8
#1 h  Mar 12, 2025, 12:34 AM
Y by
It is known that for quadratic polynomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$ the equation $P(Q(x))=Q(P(x))$ does not have real roots. Prove that $b \neq d$.
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ohiorizzler1434
816 posts
ohiorizzler1434
#2 h  Mar 12, 2025, 1:21 AM
Y by
bruh!! It's sufficient to show that if b=d, then there could be real roots! But if we let P(x)=x^2+2x+c, Q(x)=x^2+x+c, we can do casework on the sign of c to find solutions to P(Q(x))=Q(P(x)) using intermediate value theorem.
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alexheinis
10638 posts
alexheinis
#4 h  Mar 12, 2025, 9:57 AM
Y by
I will write $PQ$ for the composition.
Note that $PQ(x)=x^4+2cx^3+{\rm lot}$ and that $QP(x)=x^4+2ax^3+{\rm lot}$.
Hence if $a\not=c$ then $PQ-QP$ has exact degree 3 and it has a real root, contradiction.
It follows that $a=c$. Then it is clear that $b\not=d$, otherwise we have $P=Q$ and $PQ=QP$.
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