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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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Very Cute Functional Equation :)
YLG_123   2
N 13 minutes ago by bin_sherlo
Source: Olimphíada 2021 - Problem 6
Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for all $m, n \in \mathbb{Z}_{>0 }$:
$$f(mf(n)) + f(n) | mn + f(f(n)).$$
2 replies
YLG_123
Jul 9, 2023
bin_sherlo
13 minutes ago
J
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The order of colors
Entei   0
28 minutes ago
There are $3n$ balls, with $n$ red, $n$ green, and $n$ blue balls, randomly arranged in a row. Two observers, one at the front and one at the back, each record the order of the first appearance of each color. What is the probability that both observers record the same order of colors?

For example, the sequence RGGBRB would be read as RGB for the front observer and BRG for the back observer.
0 replies
Entei
28 minutes ago
0 replies
J
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Romanian National Olympiad 1997 - Grade 10 - Problem 4
Filipjack   1
N 34 minutes ago by MS_asdfgzxcvb
Source: Romanian National Olympiad 1997 - Grade 10 - Problem 4
Let $a_0,$ $a_1,$ $\ldots,$ $a_n$ be complex numbers such that [center]$|a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0| \le 1,$ for any $z \in \mathbb{C}$ with $|z|=1.$[/center]

Prove that $|a_k| \le 1$ and $|a_0+a_1+\ldots+a_n-(n+1)a_k| \le n,$ for any $k=\overline{0,n}.$
1 reply
Filipjack
an hour ago
MS_asdfgzxcvb
34 minutes ago
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Geometry
youochange   3
N 36 minutes ago by Double07
m:}
Let $\triangle ABC$ be a triangle inscribed in a circle, where the tangents to the circle at points $B$ and $C$ intersect at the point $P$. Let $M$ be a point on the arc $AC$ (not containing $B$) such that $M \neq A$ and $M \neq C$. Let the lines $BC$ and $AM$ intersect at point $K$. Let $P'$ be the reflection of $P$ with respect to the line $AM$. The lines $AP'$ and $PM$ intersect at point $Q$, and $PM$ intersects the circumcircle of $\triangle ABC$ again at point $N$.

Prove that the point $Q$ lies on the circumcircle of $\triangle ANK$.
3 replies
youochange
Today at 11:27 AM
Double07
36 minutes ago
J
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Putnam 2017 A6
Kent Merryfield   9
N 3 hours ago by imzzzzzz
The $30$ edges of a regular icosahedron are distinguished by labeling them $1,2,\dots,30.$ How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?
9 replies
Kent Merryfield
Dec 3, 2017
imzzzzzz
3 hours ago
J
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Matrices and combinatorics
KAME06   1
N 4 hours ago by Rainbow1971
Source: Ecuador National Olympiad OMEC level U 2024 P1 Day 1
Let $n \in \mathbb{Z}$. A matrix is n-national if its size is $2 \times 2$ and their entries belong to the set $\{2, 2^2, 2^3, ..., 2^n\}$. For example:
$$\begin{bmatrix} 2 & 8 \\ 16 & 4 \end{bmatrix}, \begin{bmatrix} 4 & 4 \\ 8 & 8 \end{bmatrix}, \begin{bmatrix} 8 & 2 \\ 16 & 8 \end{bmatrix}$$For all $n \in \mathbb{Z}$, find the number of invertible n-national matrices.
1 reply
KAME06
Yesterday at 7:59 PM
Rainbow1971
4 hours ago
J
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Putnam 2000 A6
ahaanomegas   15
N 4 hours ago by Levieee
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
15 replies
ahaanomegas
Sep 6, 2011
Levieee
4 hours ago
J
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Polynomial meets geometry
chirita.andrei   1
N 5 hours ago by AndreiVila
Source: Own. Proposed for Romanian National Olympiad 2025.
(a) Let $A,B,C$ be collinear points (in order) and $D$ a point in plane. Consider the disc $\mathcal{D}$ of center $D$ and radius $kBD$, for some $k\in(0,1)$. Prove that $\mathcal{D}\cap [AC]$ is either the empty set or a segment of length at most $2kAC$.
(b) Let $n$ be a positive integer and $P(X)\in\mathbb{C}[X]$ be a polynomial of degree $n$. Prove that \[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]
1 reply
chirita.andrei
Apr 2, 2025
AndreiVila
5 hours ago
J
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real analysis
ay19bme   1
N 5 hours ago by alexheinis
.................
1 reply
ay19bme
Today at 10:04 AM
alexheinis
5 hours ago
J
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Romanian National Olympiad 1997 - Grade 11 - Problem 3
Filipjack   1
N 5 hours ago by MS_asdfgzxcvb
Source: Romanian National Olympiad 1997 - Grade 11 - Problem 3
Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$

a) Prove that there exist nonconstant functions in $\mathcal{F}.$

b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.
1 reply
Filipjack
Today at 10:11 AM
MS_asdfgzxcvb
5 hours ago
J
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Maximizing absolute value of directional derivative of a scalar function
adityaguharoy   1
N 6 hours ago by Mathzeus1024
Source: own but possibly well known
Consider the function $f : \mathbb{R}^3 \to \mathbb{R}$ given by $f(x,y,z) = x + ye^z.$ Show that $\nabla f$ exists everywhere and find the direction along which the absolute value of the directional derivative is maximized at the point $(0,1,0).$


Hint
1 reply
adityaguharoy
Jul 27, 2023
Mathzeus1024
6 hours ago
J
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Romanian National Olympiad 1997 - Grade 11 - Problem 1
Filipjack   0
Today at 10:48 AM
Source: Romanian National Olympiad 1997 - Grade 11 - Problem 1
Let $m \ge 2$ and $n \ge 1$ be integers and $A=(a_{ij})$ a square matrix of order $n$ with integer entries. Prove that for any permutation $\sigma \in S_n$ there is a function $\varepsilon : \{1,2,\ldots,n\} \to \{0,1\}$ such that replacing the entries $a_{\sigma(1)1},$ $a_{\sigma(2)2}, $ $\ldots,$ $a_{\sigma(n)n}$ of $A$ respectively by $$a_{\sigma(1)1}+\varepsilon(1), ~a_{\sigma(2)2}+\varepsilon(2), ~\ldots, ~a_{\sigma(n)n}+\varepsilon(n),$$the determinant of the matrix $A_{\varepsilon}$ thus obtained is not divisible by $m.$
0 replies
Filipjack
Today at 10:48 AM
0 replies
J
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Differentiable function with a constant ratio
KAME06   1
N Today at 10:26 AM by Mathzeus1024
Source: Ecuador National Olympiad OMEC level U 2024 P2 Day 1
Let $\alpha >0$ a real number. Given a differentiable function $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$, let $\gamma$ the curve $y=f(x)$ on the XY-plane. For all point $P$ on $\gamma$, the tangent to $\gamma$ on $P$ intersect the x-axis and the y-axis on $A$ and $B$, respectively, such $P \in AB$ and $\frac{BP}{PA}=\alpha$.
If $(20,24)$ belongs to $\gamma$, find all possible functions $f(x)$.
1 reply
KAME06
Yesterday at 8:13 PM
Mathzeus1024
Today at 10:26 AM
J
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Null Traces of 2 Matrices
Saucepan_man02   2
N Today at 10:16 AM by loup blanc
Let $A,B\in \mathcal{M}_2(\mathbb{C})$ two non-zero matrices such that $AB+BA=O_2$ and $\det(A+B)=0$. Prove $A$ and $B$ have null traces.
2 replies
Saucepan_man02
Yesterday at 8:01 AM
loup blanc
Today at 10:16 AM
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J
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3 var inquality
sqing   1
N Apr 2, 2025 by sqing
Source: Own
Let $a,b,c>0$. Prove that
$$ \dfrac{ab}{a^2-ab+3b^2} + \dfrac{bc}{b^2-bc+3c^2} + \dfrac{ca}{c^2-ca+3a^2} \le1$$$$ \dfrac{ab}{a^2-ab+ 2b^2} + \dfrac{bc}{b^2-bc+2 c^2} + \dfrac{ca}{c^2-ca+ 2a^2}\le \dfrac{3}{2}$$$$ \dfrac{ab}{2a^2-ab+3b^2} + \dfrac{bc}{2b^2-bc+3c^2} + \dfrac{ca}{2c^2-ca+3a^2} \le \dfrac{3}{4}$$
1 reply
sqing
Apr 2, 2025
sqing
Apr 2, 2025
J
3 var inquality
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Reveal topic tags
inequalities proposed
inequalities
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Source: Own
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sqing
41446 posts
sqing
#1 h  Apr 2, 2025, 2:04 PM
Y by
Let $a,b,c>0$. Prove that
$$ \dfrac{ab}{a^2-ab+3b^2} + \dfrac{bc}{b^2-bc+3c^2} + \dfrac{ca}{c^2-ca+3a^2} \le1$$$$ \dfrac{ab}{a^2-ab+ 2b^2} + \dfrac{bc}{b^2-bc+2 c^2} + \dfrac{ca}{c^2-ca+ 2a^2}\le \dfrac{3}{2}$$$$ \dfrac{ab}{2a^2-ab+3b^2} + \dfrac{bc}{2b^2-bc+3c^2} + \dfrac{ca}{2c^2-ca+3a^2} \le \dfrac{3}{4}$$
Z K Y
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sqing
41446 posts
sqing
#2 h  Apr 2, 2025, 2:07 PM
Y by
Let $a,b,c>0$. Prove that
$$ \dfrac{c^2+ab}{a^2-ab+ 2b^2} + \dfrac{a^2+bc}{b^2-bc+2 c^2} + \dfrac{b^2+ca}{c^2-ca+ 2a^2} \le3$$$$ \dfrac{c^2+ab}{a^2-ab+ b^2} + \dfrac{a^2+bc}{b^2-bc+ c^2} + \dfrac{b^2+ca}{c^2-ca+ a^2} \le6$$$$ \dfrac{c^2+ab}{a^2-ab+ 4b^2} + \dfrac{a^2+bc}{b^2-bc+4 c^2} + \dfrac{b^2+ca}{c^2-ca+ 4a^2} \le \dfrac{3}{2}$$
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