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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.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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FE with a lot of terms
MrHeccMcHecc   1
N 11 minutes ago by jasperE3
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ $$f(x)f(y)+f(x+y)=xf(y)+yf(x)+f(xy)+x+y+1$$
1 reply
MrHeccMcHecc
2 hours ago
jasperE3
11 minutes ago
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APMO 2016: Great triangle
shinichiman   26
N 32 minutes ago by ray66
Source: APMO 2016, problem 1
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.

Senior Problems Committee of the Australian Mathematical Olympiad Committee
26 replies
shinichiman
May 16, 2016
ray66
32 minutes ago
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Chessboard pattern
BR1F1SZ   2
N an hour ago by Ubfo
Source: 2018 Argentina TST P3
In a $100 \times 100$ board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no $2 \times 2$ square within the board has all four squares of the same color. Prove that the board contains a $2 \times 2$ square colored like a chessboard.
2 replies
BR1F1SZ
Dec 27, 2024
Ubfo
an hour ago
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IMO ShortList 2001, geometry problem 2
orl   48
N an hour ago by legogubbe
Source: IMO ShortList 2001, geometry problem 2
Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.
48 replies
orl
Sep 30, 2004
legogubbe
an hour ago
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Inequalities
sqing   2
N an hour ago by DAVROS
Let $a,b$ be real numbers such that $ a^2+b^2+a^3 +b^3=4 . $ Prove that
$$a+b \leq 2$$Let $a,b$ be real numbers such that $a+b + a^2+b^2+a^3 +b^3=6 . $ Prove that
$$a+b \leq 2$$
2 replies
sqing
Yesterday at 1:10 PM
DAVROS
an hour ago
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Might be the first equation marathon
steven_zhang123   33
N 2 hours ago by eric201291
As far as I know, it seems that no one on HSM has organized an equation marathon before. Click to reveal hidden textSo why not give it a try? Click to reveal hidden text Let's start one!
Some basic rules need to be clarified:
$\cdot$ If a problem has not been solved within $5$ days, then others are eligible to post a new probkem.
$\cdot$ Not only simple one-variable equations, but also systems of equations are allowed.
$\cdot$ The difficulty of these equations should be no less than that of typical quadratic one-variable equations. If the problem involves higher degrees or more variables, please ensure that the problem is solvable (i.e., has a definite solution, rather than an approximate one).
$\cdot$ Please indicate the domain of the solution to the equation (e.g., solve in $\mathbb{R}$, solve in $\mathbb{C}$).
Here's an simple yet fun problem, hope you enjoy it :P :
P1
33 replies
steven_zhang123
Jan 20, 2025
eric201291
2 hours ago
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Inequalities
hn111009   6
N 2 hours ago by Arbelos777
Let $a,b,c>0$ satisfied $a^2+b^2+c^2=9.$ Find the minimum of $$P=\dfrac{a}{bc}+\dfrac{2b}{ca}+\dfrac{5c}{ab}.$$
6 replies
hn111009
Today at 1:25 AM
Arbelos777
2 hours ago
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Congruence
Ecrin_eren   2
N Today at 8:42 AM by Ecrin_eren
Find the number of integer pairs (x, y) satisfying the congruence equation:

3y² + 3x²y + y³ ≡ 3x² (mod 41)

for 0 ≤ x, y < 41.

2 replies
Ecrin_eren
Apr 3, 2025
Ecrin_eren
Today at 8:42 AM
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Eazy equation clap
giangtruong13   1
N Today at 5:54 AM by iniffur
Find all $x,y,z$ satisfy that: $$\frac{x}{y+z}=2x-1; \frac{y}{x+z}=3y-1;\frac{z}{x+y}=5z-1$$
1 reply
giangtruong13
Yesterday at 4:03 PM
iniffur
Today at 5:54 AM
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Olympiad
sasu1ke   3
N Today at 1:00 AM by sasu1ke
IMAGE
3 replies
sasu1ke
Yesterday at 11:52 PM
sasu1ke
Today at 1:00 AM
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How to judge a number is prime or not?
mingzhehu   1
N Yesterday at 11:14 PM by scrabbler94
A=（10X1+1）(10X+1），X1，X∈N+
B=（10 X1+3）（10X+7），X∈N，X1∈N
C=(10 X1+9)(10X+9), X∈N，X1∈N
D=(10 X1+1)(10X+3), X1∈N+，X∈N
E=（10 X1+7）(10X+9），X∈N，X1∈N
F=（10 X1+1）（10X+7），X1∈N+，X∈N
G=（10 X1+3）（10X+9），X∈N，X1∈N
H=（10 X1+1）10X+9），X1∈N+，X∈N
I=（10 X1+3）（10X+3），X1∈N，X∈N
J=( 10X1+7)(10X+7),X∈N，X1∈N

For any natural number P∈{P=10N+1，n∈N}，make P=A or B or C
If P can make the roots of function group(ABC) without any root group completely made up of integer, P will be a prime
For any natural number P∈{P=10N+3，n∈N}，make P=D or E
If P can make the roots of function group(DE) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+7，n∈N}，make P=F or G
If P can make the roots of function group(FG) without any root group completely made up
of integer, P will be a prime
For any natural number P∈{P=10N+9，n∈N}，make P=H or I or J
If P can make the roots of function group(GIJ) without any root group completely made up
of integer, P will be a prime
1 reply
mingzhehu
Yesterday at 2:45 PM
scrabbler94
Yesterday at 11:14 PM
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inequality
revol_ufiaw   3
N Yesterday at 2:55 PM by MS_asdfgzxcvb
Prove that that for any real $x \ge 0$ and natural number $n$,
$$x^n (n+1)^{n+1} \le n^n (x+1)^{n+1}.$$
3 replies
revol_ufiaw
Yesterday at 2:05 PM
MS_asdfgzxcvb
Yesterday at 2:55 PM
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What is an isogonal conjugate and why is it useful?
EaZ_Shadow   6
N Yesterday at 2:40 PM by maxamc
What is an isogonal conjugate and why is it useful? People use them in Olympiad geometry proofs but I don’t understand why and what is the purpose, as it complicates me because of me not understanding it.
6 replies
EaZ_Shadow
Dec 28, 2024
maxamc
Yesterday at 2:40 PM
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Any nice way to do this?
NamelyOrange   3
N Yesterday at 2:00 PM by pooh123
Source: Taichung P.S.1 math program tryouts

How many ordered pairs $(a,b,c)\in\mathbb{N}^3$ are there such that $c=ab$ and $1\le a\le b\le c\le60$?
3 replies
NamelyOrange
Apr 2, 2025
pooh123
Yesterday at 2:00 PM
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Tricky NT
Hip1zzzil   1
N Mar 29, 2025 by pooh123
Source: 2024 Korea
find the minimum value of positive integer which when multiplied by 2024, its digits are only 1 or 0.
1 reply
Hip1zzzil
Mar 28, 2025
pooh123
Mar 29, 2025
J
Tricky NT
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Reveal topic tags
number theory
L
G H BBookmark kLocked kLocked NReply
Source: 2024 Korea
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Hip1zzzil
14 posts
Hip1zzzil
#1 h  Mar 28, 2025, 11:10 PM
Y by
find the minimum value of positive integer which when multiplied by 2024, its digits are only 1 or 0.
Z K Y
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pooh123
9 posts
pooh123
#2 h  Mar 29, 2025, 1:35 PM
Y by
Since \( 2024 = 8 \times 253 \), the number must contain at least 3 zeroes at the end.

We will find the smallest number consisting of only 0's and 1's and divisible by 253.

Since 253 is divisible by 11, the number must consist of an even number of 1's.

Also, a number \( a_n a_{n-1} \dots a_0 \) is divisible by 23 if and only if

\[ a_0 \cdot 7^n + a_1 \cdot 7^{n-1} + \dots + a_n \]
is divisible by 23.

Obviously, the number must have 4 or more digits.

By checking manually, we obtain the smallest number divisible by 253 is \( 11111001 \).

So the answer is \( 11111001 \times 1000 = 11111001000 \).
This post has been edited 1 time. Last edited by pooh123, Mar 29, 2025, 1:36 PM
Reason: typo: 23--->253
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