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k a June Highlights and 2025 AoPS Online Class Information
jlacosta   0
Jun 2, 2025
Congratulations to all the mathletes who competed at National MATHCOUNTS! If you missed the exciting Countdown Round, you can watch the video at this link. Are you interested in training for MATHCOUNTS or AMC 10 contests? How would you like to train for these math competitions in half the time? We have accelerated sections which meet twice per week instead of once starting on July 8th (7:30pm ET). These sections fill quickly so enroll today!

[list][*]MATHCOUNTS/AMC 8 Basics
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[*]AMC 10 Problem Series[/list]
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Summer camps are starting this month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have a transformative summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
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0 replies
jlacosta
Jun 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
J
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Calvin needs to cover all squares
Rijul saini   4
N 2 minutes ago by SimplisticFormulas
Source: India IMOTC 2025 Day 2 Problem 1
Consider a $2025\times 2025$ board where we identify the squares with pairs $(i,j)$ where $i$ and $j$ denote the row and column number of that square, respectively.

Calvin picks two positive integers $a,b<2025$ and places a pawn at the bottom left corner (i.e. on $(1,1)$) and makes the following moves. In his $k$-th move, he moves the pawn from $(i,j)$ to either $(i+a,j)$ or $(i,j+a)$ if $k$ is odd and to either $(i+b,j)$ and $(i,j+b)$ if $k$ is even. Here all the numbers are taken modulo $2025$. Find the number of pairs $(a,b)$ that Calvin could have picked such that he can make moves so that the pawn covers all the squares on the board without being on any square twice.

Proposed by Tejaswi Navilarekallu
4 replies
Rijul saini
Yesterday at 6:35 PM
SimplisticFormulas
2 minutes ago
J
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inequality
SunnyEvan   3
N 4 minutes ago by SunnyEvan
Let $ x,y \geq 0 ,$ such that : $ \frac{x^2}{x^3+y}+\frac{y^2}{x+y^3} \geq 1 .$
Prove that : $$ x^2+y^2-xy \leq x+y $$$$ (x+\frac{1}{2})^2+(x+\frac{1}{2})^2 \leq \frac{5}{2} $$$$ (x+1)^2+(y+1)^2 \leq 5 $$$$ (x+2)^2+(y+2)^2 \leq 13 $$
3 replies
SunnyEvan
an hour ago
SunnyEvan
4 minutes ago
J
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Different scores possible in interview
Rijul saini   3
N 7 minutes ago by Adywastaken
Source: India IMOTC Practice Test 2 Problem 1
In a job interview, the candidates are asked questions in a sequence. The initial score is $0$. The candidate's score is calculated as follows:

$\bullet$ after a correct answer, the score is increased by $1$;
$\bullet$ after a wrong answer, the score is divided by $2$.

If the candidate is asked $n$ questions and answers all of them, how many different scores are possible?

Note: Two different response sequences of the same length can result in the same score: the sequences $RRW$ and $WWR$ with the same length, where $R$ denotes the correct answer and $W$ denotes the wrong answer, both result in the same score of 1.

Proposed by S. Muralidharan
3 replies
Rijul saini
Yesterday at 6:54 PM
Adywastaken
7 minutes ago
J
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Concurrence
LiamChen   0
7 minutes ago
Source: MOP1998
Problem:
0 replies
LiamChen
7 minutes ago
0 replies
J
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set of points, there exist two lines containing n points
jasperE3   1
N 8 minutes ago by ririgggg
Source: 2004 Brazil TST Test 2 P1
Find the smallest positive integer $n$ that satisfies the following condition: For every finite set of points on the plane, if for any $n$ points from this set there exist two lines containing all the $n$ points, then there exist two lines containing all points from the set.
1 reply
jasperE3
Apr 5, 2021
ririgggg
8 minutes ago
J
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XY is tangent to a fixed circle
a_507_bc   2
N 11 minutes ago by math-olympiad-clown
Source: Baltic Way 2022/15
Let $\Omega$ be a circle, and $B, C$ are two fixed points on $\Omega$. Given a third point $A$ on $\Omega$, let $X$ and $Y$ denote the feet of the altitudes from $B$ and $C$, respectively, in the triangle $ABC$. Prove that there exists a fixed circle $\Gamma$ such that $XY$ is tangent to $\Gamma$ regardless of the choice of the point $A$.
2 replies
a_507_bc
Nov 12, 2022
math-olympiad-clown
11 minutes ago
J
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Super easy problem
M11100111001Y1R   6
N 14 minutes ago by sami1618
Source: Iran TST 2025 Test 2 Problem 1
The numbers from 2 to 99 are written on a board. At each step, one of the following operations is performed:

$a)$ Choose a natural number \( i \) such that \( 2 \leq i \leq 89 \). If both numbers \( i \) and \( i+10 \) are on the board, erase both.

$b)$ Choose a natural number \( i \) such that \( 2 \leq i \leq 98 \). If both numbers \( i \) and \( i+1 \) are on the board, erase both.

By performing these operations, what is the maximum number of numbers that can be erased from the board?
6 replies
M11100111001Y1R
May 27, 2025
sami1618
14 minutes ago
J
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Beware the degeneracies!
Rijul saini   7
N 17 minutes ago by Adywastaken
Source: India IMOTC 2025 Day 1 Problem 1
Let $a,b,c$ be real numbers satisfying $$\max \{a(b^2+c^2),b(c^2+a^2),c(a^2+b^2) \} \leqslant 2abc+1$$Prove that $$a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2) \leqslant 6abc+2$$and determine all cases of equality.

Proposed by Shantanu Nene
7 replies
Rijul saini
Yesterday at 6:30 PM
Adywastaken
17 minutes ago
J
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13th PMO Area Part 1 #17
scarlet128   1
N 20 minutes ago by scarlet128
Source: https://pmo.ph/wp-content/uploads/2014/08/13thPMO-Area_ver5.pdf
The number x is chosen randomly from the interval (0, 1]. Define y = floor of (log base 4(x)). Find the sum of the lengths of all subintervals of (0, 1] for which y is odd.
1 reply
scarlet128
35 minutes ago
scarlet128
20 minutes ago
J
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Romanian Geo
oVlad   3
N 24 minutes ago by NuMBeRaToRiC
Source: Romania TST 2025 Day 1 P2
Let $ABC$ be a scalene acute triangle with incentre $I{}$ and circumcentre $O{}$. Let $AI$ cross $BC$ at $D$. On circle $ABC$, let $X$ and $Y$ be the mid-arc points of $ABC$ and $BCA$, respectively. Let $DX{}$ cross $CI{}$ at $E$ and let $DY{}$ cross $BI{}$ at $F{}$. Prove that the lines $FX, EY$ and $IO$ are concurrent on the external bisector of $\angle BAC$.

David-Andrei Anghel
3 replies
1 viewing
oVlad
Apr 9, 2025
NuMBeRaToRiC
24 minutes ago
J
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IMO 2011 Problem 5
orl   86
N 26 minutes ago by bjump
Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$.

Proposed by Mahyar Sefidgaran, Iran
86 replies
orl
Jul 19, 2011
bjump
26 minutes ago
J
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11th PMO Nationals, Easy #5
scarlet128   1
N 28 minutes ago by Mathzeus1024
Source: https://pmo.ph/wp-content/uploads/2020/12/11th-PMO-Questions.pdf
Solve for x : 2(floor of x) = x + 2{x}
1 reply
scarlet128
2 hours ago
Mathzeus1024
28 minutes ago
J
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Cute Geometry
EthanWYX2009   0
36 minutes ago
In triangle \( X_AX_BX_C \), let \( X \) and \( Y \) be a pair of isogonal conjugate points. The line \( XX_A \) intersects \( X_BX_C \) at \( P \), and the line \( XY \) intersects \( X_BX_C \) at \( Q \). Let the circumcircle of \( XX_BX_C \) and the circumcircle of \( XPQ \) intersect again at \( R \) (other than \( X \)). Prove that the line \( RX \) bisects \( \angle PRX_A \).
IMAGE
0 replies
EthanWYX2009
36 minutes ago
0 replies
J
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Interior point of ABC
Jackson0423   0
38 minutes ago
Let D be an interior point of the acute triangle ABC with AB > AC so that ∠DAB = ∠CAD. The point E on the segment AC satisfies ∠ADE = ∠BCD, the point F on the segment AB satisfies ∠F DA = ∠DBC, and the point X on the line AC satisfies CX = BX. Let O1 and O2 be the circumcenters of the triangles ADC and EXD, respectively. Prove that the lines BC, EF, and O1O2 are concurrent
0 replies
Jackson0423
38 minutes ago
0 replies
J
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J
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Troll inequality
B1t   7
N Apr 27, 2025 by sqing
:-D





For positive real numbers \( x, y > 0 \), define
\[ E(x,y) = x^{-\!y} + y. \]
1. If \( x, y \geq 1 \), prove that
\[ E(x,y) + E(y,x) \geq 4. \]
2. If \( 0 < x, y < 1 \), prove that
\[ \frac{2}{1+xy} + x + y < E(x,y) + E(y,x) < 2 + \frac{x}{y} + \frac{y}{x}. \]
7 replies
B1t
Apr 27, 2025
sqing
Apr 27, 2025
J
Troll inequality
G H J
Reveal topic tags
inequalities
L
G H BBookmark kLocked kLocked NReply
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B1t
25 posts
B1t
#1 h  Apr 27, 2025, 1:44 PM
Y by
:-D





For positive real numbers \( x, y > 0 \), define
\[ E(x,y) = x^{-\!y} + y. \]
1. If \( x, y \geq 1 \), prove that
\[ E(x,y) + E(y,x) \geq 4. \]
2. If \( 0 < x, y < 1 \), prove that
\[ \frac{2}{1+xy} + x + y < E(x,y) + E(y,x) < 2 + \frac{x}{y} + \frac{y}{x}. \]
This post has been edited 4 times. Last edited by B1t, Apr 28, 2025, 2:08 PM
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NO_SQUARES
1141 posts
NO_SQUARES
#2 h  Apr 27, 2025, 1:49 PM
Y by
Something wrong
This post has been edited 1 time. Last edited by NO_SQUARES, Apr 27, 2025, 2:27 PM
Z K Y
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MR.1
138 posts
MR.1
#3 h  Apr 27, 2025, 1:49 PM
Y by
imo tst?
Z K Y
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B1t
25 posts
B1t
#4 h  Apr 27, 2025, 2:00 PM
Y by
MR.1 wrote:
imo tst?

yeah
Z K Y
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B1t
25 posts
B1t
#5 h  Apr 27, 2025, 2:24 PM • 1 Y
Y by NO_SQUARES
NO_SQUARES wrote:
B1t wrote:
For positive real numbers \( x, y > 0 \), define
\[ E(x,y) = x^{-\!y} + y. \]
1. If \( x, y \geq 1 \), prove that
\[ E(x,y) + E(y,x) \geq 4. \]

For this, I think, we can just use AM-GM:
\[x+y+\frac{1}{x^y}+\frac{1}{y^x} \geqslant 4\sqrt[4]{\frac{1}{x^{y-1}y^{x-1}}} \geqslant 4, \]because $x^{y-1} \geqslant x^0=1$ and, similarly, $y^{x-1} \geqslant 1$.

\[ \frac{1}{x^{y-1} \cdot y^{x-1}} \leq \frac{1}{x^0 \cdot y^0} = 1\ ? \]
This post has been edited 2 times. Last edited by B1t, Apr 27, 2025, 2:25 PM
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B1t
25 posts
B1t
#6 h  Apr 27, 2025, 2:30 PM
Y by
hint
This post has been edited 2 times. Last edited by B1t, Apr 27, 2025, 2:43 PM
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B1t
25 posts
B1t
#7 h  Apr 27, 2025, 2:59 PM
Y by
solution
This post has been edited 3 times. Last edited by B1t, Apr 27, 2025, 3:08 PM
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sqing
42598 posts
sqing
#8 h  Apr 27, 2025, 3:24 PM
Y by
Very good.
This post has been edited 1 time. Last edited by sqing, Apr 27, 2025, 3:25 PM
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