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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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
Yesterday at 3:18 PM
0 replies
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D1019 : Dominoes 2*1
Dattier   4
N 30 minutes ago by Dattier
I have a 9*9 grid like this one:

IMAGE

We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:

IMAGE

Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:

IMAGE
4 replies
Dattier
Mar 26, 2025
Dattier
30 minutes ago
J
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thanks u!
Ruji2018252   0
an hour ago
find all $f: \mathbb{R}\to \mathbb{R}$ and
\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]
0 replies
Ruji2018252
an hour ago
0 replies
J
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Functional equations
hanzo.ei   11
N an hour ago by GreekIdiot
Source: Greekldiot
Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
11 replies
hanzo.ei
Mar 29, 2025
GreekIdiot
an hour ago
J
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D1018 : Can you do that ?
Dattier   1
N 2 hours ago by Dattier
Source: les dattes à Dattier
We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
1 reply
Dattier
Mar 24, 2025
Dattier
2 hours ago
J
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Solve the equetion
yt12   5
N Today at 6:20 AM by KevinKV01
Solve the equetion:$\sin 2x+\tan x=2$
5 replies
yt12
Mar 31, 2025
KevinKV01
Today at 6:20 AM
J
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Easiest functional equation?
ZETA_in_olympiad   28
N Today at 4:07 AM by jkim0656
Here I want the users to post the functional equations that they think are the easiest. Everyone (including the one who posted the problem) are able to post solutions.
28 replies
ZETA_in_olympiad
Mar 19, 2022
jkim0656
Today at 4:07 AM
J
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Polynomial optimization problem
ReticulatedPython   2
N Yesterday at 12:38 PM by Mathzeus1024
Let $$p(x)=-ax^4+x^3$$, where $a$ is a real number. Prove that for all positive $a$, $$p(x) \le \frac{27}{256a^3}.$$
2 replies
ReticulatedPython
Mar 31, 2025
Mathzeus1024
Yesterday at 12:38 PM
J
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Inequalities (Please help me!!!)
yt12   6
N Apr 1, 2025 by lamhihi1234
Let $a,b,c$ be reals with $a+b+c=1$and $ a,b,c \ge \frac{-3}{ 4}$. Prove that
$$\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{ c^2+1} \le \frac{9}{ 10}$$
6 replies
yt12
Mar 4, 2023
lamhihi1234
Apr 1, 2025
J
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Functional Equation
ab_xy123   4
N Apr 1, 2025 by millennium2k
Find all solutions to the functional equation $f(1-x) = f(x) + 1 - 2x$
4 replies
ab_xy123
Mar 16, 2020
millennium2k
Apr 1, 2025
J
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Three 3-digit numbers
miiirz30   3
N Apr 1, 2025 by henryli3333
Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board.

Proposed by Giorgi Arabidze, Georgia
3 replies
miiirz30
Mar 31, 2025
henryli3333
Apr 1, 2025
J
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k functional equation
Tony_stark0094   4
N Mar 31, 2025 by jasperE3
solve for $f:R \rightarrow R$ such that
$$f(x+f(y))=y+f(x+1)$$
4 replies
Tony_stark0094
Mar 31, 2025
jasperE3
Mar 31, 2025
J
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Inequality
lgx57   1
N Mar 31, 2025 by sqing
Let $x,y,z \ge 0$ and $xyz=1$. Prove that

$$\sum \frac{1}{x^2+x+1} \ge 1$$
1 reply
lgx57
Mar 31, 2025
sqing
Mar 31, 2025
J
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functional equation in R2
jasperE3   2
N Mar 31, 2025 by alexheinis
Find all functions $f:\mathbb R\times\mathbb R\to\mathbb R$ such that:
$a)\enspace f(x+z,y+z)=f(x,y)+z$
$b)\enspace f(xw,yw)=wf(x,y)$
both hold $\forall w,x,y,z\in\mathbb R,w\ne0$.
2 replies
jasperE3
Mar 27, 2021
alexheinis
Mar 31, 2025
J
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f(x^2-1998x)-f^2(2x-1999)>=1/4 (VI Soros Olympiad 1990-00 R1 9.4)
parmenides51   1
N Mar 31, 2025 by jasperE3
Is there a function $f(x)$, which satisfies both of the following conditions:

a) if $x \ne y$, then $f(x)\ne f(y)$

b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?
1 reply
parmenides51
May 27, 2024
jasperE3
Mar 31, 2025
J
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J
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Thanks u!
Ruji2018252   3
N Yesterday at 8:29 PM by Primeniyazidayi
Let $x,y,z,t\in\mathbb{R}$ and $\begin{cases}x^2+y^2=4\\z^2+t^2=9\\xt+yz\geqslant 6\end{cases}$.
$1,$ Prove $xz=yt$
$2,$ Find maximum $P=x+z$
3 replies
Ruji2018252
Mar 30, 2025
Primeniyazidayi
Yesterday at 8:29 PM
J
Thanks u!
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Reveal topic tags
inequalities
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G H BBookmark kLocked kLocked NReply
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Ruji2018252
381 posts
Ruji2018252
#1 h  Mar 30, 2025, 11:07 AM • 1 Y
Y by Bet667
Let $x,y,z,t\in\mathbb{R}$ and $\begin{cases}x^2+y^2=4\\z^2+t^2=9\\xt+yz\geqslant 6\end{cases}$.
$1,$ Prove $xz=yt$
$2,$ Find maximum $P=x+z$
Z K Y
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Primeniyazidayi
37 posts
Primeniyazidayi
#2 h  Mar 30, 2025, 12:24 PM
Y by
Note that the first part is just an application of Cauchy-Schwarz and its equality conditions.
For the second part the observation 36=(xz-yt)^2 + (xt+yz)^2 finishes the question easily
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sqing
41401 posts
sqing
#3 h  Mar 31, 2025, 3:50 AM
Y by
Let $x,y,z,t\in\mathbb{R}$ and $\begin{cases}x^2+y^2=4\\z^2+t^2=9\\xt+yz\geqslant 6\end{cases}$. Prove that $$- \sqrt{13}\leq x+z\leq \sqrt{13}$$
*
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Primeniyazidayi
37 posts
Primeniyazidayi
#4 h  Yesterday at 8:29 PM
Y by
min/max x = -+sqrt{2} and min/max z = -+sqrt{9/2}
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