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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
J
H
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
H
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
J
H
Find all the natural numbers a and b such that: if c and d are real that
bo_ngu_toan   4
N 18 minutes ago by imnotgoodatmathsorry
Find all the natural numbers a and b such that: if c and d are real that x²+ax+1=c and x²+bx+1=d have roots then x²+(a+b)x +1 = cd have root
4 replies
bo_ngu_toan
May 14, 2023
imnotgoodatmathsorry
18 minutes ago
J
H
NC State Math Contest Wake Tech Regional Problems and Solutions
mathnerd_101   3
N 26 minutes ago by mathnerd_101
Problem 1: Determine the area enclosed by the graphs of $$y=|x-2|+|x-4|-2, y=-|x-3|+4.$$ Hint
Solution to P1

Problem 2: Calculate the sum of the real solutions to the equation $x^\frac{3}{2} -9x-16x^\frac{1}{2} +144=0.$
Hint
Solution to P2



Problem 3: List the two transformations needed to convert the graph $\frac{x-1}{x+2}$ to $\frac{3x-6}{x-1}.$
Hint
Solution to P3

Problem 4: Let $a,b$ be positive integers such that $a^2-b^2=20,$ and $a^3-b^3=120.$ Determine the value of $a+\frac{b^2}{a+b}.$
Hint
Solution for P4

Problem 5: Eve and Oscar are playing a game where they roll a fair, six-sided die. If an even number occurs on two consecutive rolls, then Eve wins. If an odd number is immediately followed by an even number, Oscar wins. The die is rolled until one person wins. What is the probability that Oscar wins?
Hint
Solution to P5

Problem 6: In triangle $ABC,$ $M$ is on point $\overline{AB}$ such that $AM = x+32$ and $MB=x+12$ and $N$ is a point on $\overline{AC}$ such that $MN=2x+1$ and $BC=x+22.$ Given that $\overline{MN} || \overline{BC},$ calculate $MN.$
Hint
Solution to P6

Problem 7: Determine the sum of the zeroes of the quadratic of polynomial $Q(x),$ given that $$Q(0)=72, Q(1) = 75, Q(3) = 63.$$
Hint

Solution to Problem 7

Problem 8:
Hint
Solution to P8

Problem 9:
Find the sum of all real solutions to $$(x-4)^{log_8(4x-16)} = 2.$$ Hint
Solution to P9

Problem 10:
Define the function
\[f(x) = \begin{cases} x - 9, & \text{if } x > 100 \\ f(f(x + 10)), & \text{if } x \leq 100 \end{cases}\]
Calculate \( f(25) \).

Hint

Solution to P10

Problem 11:
Let $a,b,x$ be real numbers such that $$log_{a-b} (a+b) = 3^{a+b}, log_{a+b} (a-b) = 125 \cdot 15^{b-a}, a^2-b^2=3^x. $$Find $x.$
Hint

Solution to P11

Problem 12: Points $A,B,C$ are on circle $Q$ such that $AC=2,$ $\angle AQC = 180^{\circ},$ and $\angle QAB = 30^{\circ}.$ Determine the path length from $A$ to $C$ formed by segment $AB$ and arc $BC.$

Hint
Solution to P12

Problem 13: Determine the number of integers $x$ such that the expression $$\frac{\sqrt{522-x}}{\sqrt{x-80}} $$is also an integer.
Hint

Solution to Problem 13

Problem 14: Determine the smallest positive integer $n$ such that $n!$ is a multiple of $2^15.$

Hint
Solution to Problem 14

Problem 15: Suppose $x$ and $y$ are real numbers such that $x^3+y^3=7,$ and $xy(x+y)=-2.$ Calculate $x-y.$
Funnily enough, I guessed this question right in contest.

Hint
Solution to Problem 15

Problem 16: A sequence of points $p_i = (x_i, y_i)$ will follow the rules such that
\[ p_1 = (0,0), \quad p_{i+1} = (x_i + 1, y_i) \text{ or } (x_i, y_i + 1), \quad p_{10} = (4,5). \]How many sequences $\{p_i\}_{i=1}^{10}$ are possible such that $p_1$ is the only point with equal coordinates?

Hint
Solution to P16

Problem 18: (Also stolen from akliu's blog post)
Calculate

$$\sum_{k=0}^{11} (\sqrt{2} \sin(\frac{\pi}{4}(1+2k)))^k$$
Hint
Solution to Problem 18

Problem 19: Determine the constant term in the expansion of $(x^3+\frac{1}{x^2})^{10}.$

Hint
Solution to P19

Problem 20:

In a magical pond there are two species of talking fish: trout, whose statements are always true, and \emph{flounder}, whose statements are always false. Six fish -- Alpha, Beta, Gamma, Delta, Epsilon, and Zeta -- live together in the pond. They make the following statements:
Alpha says, "Delta is the same kind of fish as I am.''
Beta says, "Epsilon and Zeta are different from each other.''
Gamma says, "Alpha is a flounder or Beta is a trout.''
Delta says, "The negation of Gamma's statement is true.''
Epsilon says, "I am a trout.''
Zeta says, "Beta is a flounder.''

How many of these fish are trout?

Hint
Solution to P20
SHORT ANSWER QUESTIONS:
1. Five people randomly choose a positive integer less than or equal to $10.$ The probability that at least two people choose the same number can be written as $\frac{m}{n}.$ Find $m+n.$

Hint
Solution to S1

2. Define a function $F(n)$ on the positive integers using the rule that for $n=1,$ $F(n)=0.$ For all prime $n$, $F(n) = 1,$ and for all other $n,$ $F(xy)=xF(y) + yF(x).$ Find the smallest possible value of $n$ such that $F(n) = 2n.$

Hint
Solution to S2

3. How many integers $n \le 2025$ can be written as the sum of two distinct, non-negative integer powers of $3?$
Huge shoutout to OTIS for teaching me how to solve problems like this.

Hint

Solution to S3

4. Let $S$ be the set of positive integers of $x$ such that $x^2-5y^2=1$ for some other positive integer $y.$ Find the only three-digit value of $x$ in $S.$
Hint
Solution to S4

5. Let $N$ be a positive integer and let $M$ be the integer that is formed by removing the first three digits from $N.$ Find the value of $N$ with least value such that $N = 2025M.$
Hint

Solution to S5
3 replies
mathnerd_101
5 hours ago
mathnerd_101
26 minutes ago
J
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Factorise (x+1)(x+2)(x+3)(x+4)-3
Idiot_of_the64squares   4
N 29 minutes ago by SomeonecoolLovesMaths
On expansion the expression becomes:
$ x^4+10x^3+35x^2 +50x+ 21 $
I cannot solve it further
4 replies
Idiot_of_the64squares
3 hours ago
SomeonecoolLovesMaths
29 minutes ago
J
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floor of (an+b)/(cn+d) is surjective
Miquel-point   1
N 30 minutes ago by Rohit-2006
Source: Romanian NMO 2021 grade 10 P2
Let $a,b,c,d\in\mathbb{Z}_{\ge 0}$, $d\ne 0$ and the function $f:\mathbb{Z}_{\ge 0}\to\mathbb Z_{\ge 0}$ defined by
\[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]Prove that the following are equivalent:
[list=1]
[*] $f$ is surjective;
[*] $c=0$, $b<d$ and $0<a\le d$.
[/list]

Tiberiu Trif
1 reply
Miquel-point
Apr 15, 2023
Rohit-2006
30 minutes ago
J
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pls help me
Leo1608   4
N 38 minutes ago by imnotgoodatmathsorry
If a, b, c are integers and a-b is divisible by c then ab is divisible by $c^2$. Prove that a is divisible by c and b is divisible by c
4 replies
Leo1608
Aug 21, 2024
imnotgoodatmathsorry
38 minutes ago
J
H
The greatest length of a sequence that satisfies a special condition
EmersonSoriano   1
N 43 minutes ago by atdaotlohbh
Source: 2018 Peru Southern Cone TST P9
Find the largest possible value of the positive integer $N$ given that there exist positive integers $a_1, a_2, \dots, a_N$ satisfying
$$ a_n = \sqrt{(a_{n-1})^2 + 2018 \, a_{n-2}}\:, \quad \text{for } n = 3,4,\dots,N. $$
1 reply
EmersonSoriano
Apr 2, 2025
atdaotlohbh
43 minutes ago
J
H
Inspired by giangtruong13
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ 61\leq a+b+2c+d\leq \frac{265}{3}$$$$- \frac{2121}{2}\leq ab+bc-2cd+da\leq \frac{14045}{12}$$$$\frac{519506-7471\sqrt{7471}}{27}\leq ab+bc-2cd+3da\leq 33620$$
2 replies
sqing
Today at 2:57 AM
sqing
an hour ago
J
H
Interesting inequalities
sqing   5
N an hour ago by sqing
Source: Own
Let $ a,b>0 $ and $ a^2+b^2 +ab+a+b=5 $ . Prove that$$ \frac{1}{ a+kb }+ \frac{1}{ b+ka }+ \frac{1}{ab+k } \geq \frac{3}{ k+1 }$$Where $k\geq 0. $
5 replies
sqing
2 hours ago
sqing
an hour ago
J
H
Function equations that I can't solve yet (first post)
GhostVN   1
N an hour ago by Nuran2010
Given $f:R+->R+$ that satisfied
$f$( $\frac{f(x)}{y}$ )$=yf(y)f(f(x))$ for all x,y >0
1 reply
GhostVN
an hour ago
Nuran2010
an hour ago
J
H
Nepal TST 2025 Day 1 Problem 3
Bata325   1
N an hour ago by pco
Source: Nepal TST 2025, Problem 3
Find all functions \( f : \mathbb{R}^+ \to \mathbb{R}^+ \) such that
\[ f(f(x)) + x f(xy) = x + f(y) \]for all positive real numbers \( x \) and \( y \).(Andrew Brahms, USA)
1 reply
Bata325
3 hours ago
pco
an hour ago
J
H
ineq.trig.
wer   17
N an hour ago by arqady
If a, b, c are the sides of a triangle, show that: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{r}{R}\le2$
17 replies
wer
Jul 5, 2014
arqady
an hour ago
J
H
Abelkonkurransen 2025 3b
Lil_flip38   1
N an hour ago by jrpartty
Source: abelkonkurransen
An acute angled triangle \(ABC\) has circumcenter \(O\). The lines \(AO\) and \(BC\) intersect at \(D\), while \(BO\) and \(AC\) intersect at \(E\) and \(CO\) and \(AB\) intersect at \(F\). Show that if the triangles \(ABC\) and \(DEF\) are similar(with vertices in that order), than \(ABC\) is equilateral.
1 reply
Lil_flip38
Mar 20, 2025
jrpartty
an hour ago
J
H
angle wanted, right ABC, AM=CB , CN=MB
parmenides51   4
N an hour ago by Tsikaloudakis
Source: 2022 European Math Tournament - Senior First + Grand League - Math Battle 1.3
In a right-angled triangle $ABC$, points $M$ and $N$ are taken on the legs $AB$ and $BC$, respectively, so that $AM=CB$ and $CN=MB$. Find the acute angle between line segments $AN$ and $CM$.
4 replies
parmenides51
Dec 19, 2022
Tsikaloudakis
an hour ago
J
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R+ Functional Equation
Mathdreams   2
N 2 hours ago by pco
Source: Nepal TST 2025, Problem 3
Find all functions $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that \[f(f(x)) + xf(xy) = x + f(y)\]for all positive real numbers $x$ and $y$.

(Andrew Brahms, USA)
2 replies
+1 w
Mathdreams
3 hours ago
pco
2 hours ago
J
H
J
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Inequalities
sqing   4
N Apr 4, 2025 by sqing
Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=11.$ Prove that
$$a+ab+abc\leq\frac{49}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=10.$ Prove that
$$a+ab+abc\leq\frac{169}{24}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=14.$ Prove that
$$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=32.$ Prove that
$$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$$
4 replies
sqing
Apr 1, 2025
sqing
Apr 4, 2025
J
Inequalities
G H J
Reveal topic tags
inequalities
L
G H BBookmark kLocked kLocked NReply
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sqing
41534 posts
sqing
#1 h  Apr 1, 2025, 2:59 PM
Y by
Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=11.$ Prove that
$$a+ab+abc\leq\frac{49}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=10.$ Prove that
$$a+ab+abc\leq\frac{169}{24}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=14.$ Prove that
$$a+ab+abc\leq\frac{63+5\sqrt 5}{6}$$Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=32.$ Prove that
$$a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$$
Z K Y
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DAVROS
1645 posts
DAVROS
#2 h  Apr 2, 2025, 6:18 AM
Y by
sqing wrote:
Let $ a, b,c\geq 0 $ and $ 2a+3b+ 4c=32.$ Prove that $a+ab+abc\leq48+\frac{64\sqrt{2}}{3}$
solution
Z K Y
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sqing
41534 posts
sqing
#3 h  Apr 2, 2025, 7:41 AM
Y by
Very very nice.Thank DAVROS.
Z K Y
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sqing
41534 posts
sqing
#4 h  Apr 4, 2025, 11:27 AM
Y by
Let $ a,b\geq 0 $ and $a(a+2b)=1.$ Prove that
$$\frac{a^2b-2}{a^2b-1}\ge \frac{2(a^3+a^2b+ab^2+2)}{a^4+a^3b+ab^3+2} $$$$\frac{a^2b-2}{a^2b-1}\ge \frac{3(a^3+a^2b+ab^2+1)}{a^4+a^3b+ab^3+2} $$$$\frac{a^2b-1}{a^2b-2}\ge \frac{7(84-31\sqrt 3)(a^3+a^2b+ab^2+1)}{321(a^4+a^3b+ab^3+2)} $$$$\frac{a^2b-2}{a^2b-1}\ge \frac{ (618-70\sqrt 3)(a^3+a^2b+ab^2+2)}{429(a^4+a^3b+ab^3+1)} $$$$\frac{a^2b-1}{a^2b-2}\ge \frac{4(327-71\sqrt 3)(a^3+a^2b+ab^2+2)}{3531(a^4+a^3b+ab^3+1)} $$
This post has been edited 1 time. Last edited by sqing, Apr 4, 2025, 2:32 PM
Z K Y
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sqing
41534 posts
sqing
#5 h  Apr 4, 2025, 3:28 PM
Y by
Let $ a,b\geq 0 $ and $\frac{3a^2b+1}{a^2b+1}\le \frac{3(a^3+a^2b+ab^2+1)}{2(a^4+a^3b+ab^3+2)}.$ Prove that
$$\frac{5+\sqrt[3]{431-18\sqrt{417}}+\sqrt[3]{431+18\sqrt{417}}}{12}\geq a(a+2b)\geq 1$$
This post has been edited 1 time. Last edited by sqing, Apr 5, 2025, 1:39 AM
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