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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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Real variables inequality
JK1603JK   0
6 minutes ago
Let a,b,c\in R then prove that \frac{15}{2}\cdot\frac{a^2+b^2+c^2}{(a+b+c)^2}+\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2}\ge 4
0 replies
JK1603JK
6 minutes ago
0 replies
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School Math Problem
math_cool123   2
N 16 minutes ago by wuwang2002
Find all ordered pairs of nonzero integers $(a, b)$ that satisfy $$(a^2+b)(a+b^2)=(a-b)^3.$$
2 replies
math_cool123
Yesterday at 5:03 AM
wuwang2002
16 minutes ago
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2024 COMC B1
QueenArwen   2
N an hour ago by EVKV
For any positive integer number $k$, the factorial $k!$ is defined as a product of all integers between $1$ and $k$ inclusive: $k!=k\times{(k-1)}\times\dots\times{1}$.
Let $s(n)$ denote the sum of the first $n$ factorials, i.e.
$$s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}$$Find the remainder when $s(2024)$ is divided by $8$
2 replies
QueenArwen
Nov 4, 2024
EVKV
an hour ago
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Easy FE; source unknown
NamelyOrange   3
N 3 hours ago by Mathdreams
Find (with proof) all $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x)\ge x$ and $f(f(x)) = x$.
3 replies
NamelyOrange
4 hours ago
Mathdreams
3 hours ago
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Polynomial meets geometry
chirita.andrei   0
Yesterday at 5:42 PM
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 \[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]
0 replies
chirita.andrei
Yesterday at 5:42 PM
0 replies
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Integral inequality with differentiable function
Ciobi_   1
N Yesterday at 3:35 PM by MS_asdfgzxcvb
Source: Romania NMO 2025 12.2
Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]
1 reply
Ciobi_
Yesterday at 2:29 PM
MS_asdfgzxcvb
Yesterday at 3:35 PM
J
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On coefficients of a polynomial over a finite field
Ciobi_   0
Yesterday at 2:59 PM
Source: Romania NMO 2025 12.4
Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.
0 replies
Ciobi_
Yesterday at 2:59 PM
0 replies
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On non-negativeness of continuous and polynomial functions
Ciobi_   0
Yesterday at 2:51 PM
Source: Romania NMO 2025 12.3
a) Let $a\in \mathbb{R}$ and $f \colon \mathbb{R} \to \mathbb{R}$ be a continuous function for which there exists an antiderivative $F \colon \mathbb{R} \to \mathbb{R} $, such that $F(x)+a\cdot f(x) \geq 0$, for any $x \in \mathbb{R}$, and$ \lim_{|x| \to \infty} \frac{F(x)}{e^{|\alpha \cdot x|}}=0$ holds for any $\alpha \in \mathbb{R}^*$. Prove that $F(x) \geq 0$ for all $x \in \mathbb{R}$.
b) Let $n\geq 2$ be a positive integer, $g \in \mathbb{R}[X]$, $g = X^n + a_1X^{n-1}+ \dots + a_{n-1}X+a_n$ be a polynomial with all of its roots being real, and $f \colon \mathbb{R} \to \mathbb{R}$ a polynomial function such that $f(x)+a_1\cdot f'(x)+a_2\cdot f^{(2)}(x)+\dots+a_n\cdot f^{(n)}(x) \geq 0$ for any $x \in \mathbb{R}$. Prove that $f(x) \geq 0$ for all $x \in \mathbb{R}$.
0 replies
Ciobi_
Yesterday at 2:51 PM
0 replies
J
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Proving AB-BA is singular from given conditions
Ciobi_   0
Yesterday at 2:04 PM
Source: Romania NMO 2025 11.4
Let $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $A+B=AB+BA$. Prove that:
a) if $n$ is odd, then $\det(AB-BA)=0$;
b) if $\text{tr}(A)\neq \text{tr}(B)$, then $\det(AB-BA)=0$.
0 replies
Ciobi_
Yesterday at 2:04 PM
0 replies
J
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Equivalent definition for C^1 functions
Ciobi_   0
Yesterday at 1:54 PM
Source: Romania NMO 2025 11.3
Prove that, for a function $f \colon \mathbb{R} \to \mathbb{R}$, the following $2$ statements are equivalent:
a) $f$ is differentiable, with continuous first derivative.
b) For any $a\in\mathbb{R}$ and for any two sequences $(x_n)_{n\geq 1},(y_n)_{n\geq 1}$, convergent to $a$, such that $x_n \neq y_n$ for any positive integer $n$, the sequence $\left(\frac{f(x_n)-f(y_n)}{x_n-y_n}\right)_{n\geq 1}$ is convergent.
0 replies
Ciobi_
Yesterday at 1:54 PM
0 replies
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RREF of some matrices
tommy2007   3
N Yesterday at 1:51 PM by tommy2007
for $\forall n \in \mathbb{N},$
what is the maximum integer that appears in one of the Reduced Row Echelon Forms of $n \times n$ matrices which has only $-1$ and $1$ for their entries?
3 replies
tommy2007
Yesterday at 6:57 AM
tommy2007
Yesterday at 1:51 PM
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Finding pairs of functions of class C^2 with a certain property
Ciobi_   0
Yesterday at 1:31 PM
Source: Romania NMO 2025 11.1
Find all pairs of twice differentiable functions $f,g \colon \mathbb{R} \to \mathbb{R}$, with their second derivative being continuous, such that the following holds for all $x,y \in \mathbb{R}$: \[(f(x)-g(y))(f'(x)-g'(y))(f''(x)-g''(y))=0\]
0 replies
Ciobi_
Yesterday at 1:31 PM
0 replies
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Inverse of absolute value function
MetaphysicalWukong   2
N Yesterday at 12:32 PM by paxtonw
how does the function have an inverse for k= 101, 203, 305, 509, 611 and 713?

how do we deduce this without graphing software?
2 replies
MetaphysicalWukong
Yesterday at 7:21 AM
paxtonw
Yesterday at 12:32 PM
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Unique global minimum points
chirita.andrei   0
Yesterday at 11:06 AM
Source: Own. Proposed for Romanian National Olympiad 2025.
Let $f\colon[0,1]\rightarrow \mathbb{R}$ be a continuous function. Suppose that for each $t\in(0,1)$, the function \[f_t\colon[0,1-t]\rightarrow\mathbb{R}, f_t(x)=f(x+t)-f(x)\]has an unique global minimum point, which we will denote by $g(t)$. Prove that if $\lim\limits_{t\to 0}g(t)=0$, then $g$ is constant zero.
0 replies
chirita.andrei
Yesterday at 11:06 AM
0 replies
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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
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functional equation in R2
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Reveal topic tags
algebra
functional equation
function
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jasperE3
11149 posts
jasperE3
#1 h  Mar 27, 2021, 10:18 PM
Y by
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 $\forall w,x,y,z\in\mathbb R,w\ne0$.
This post has been edited 1 time. Last edited by jasperE3, Mar 27, 2021, 10:20 PM
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jasperE3
11149 posts
jasperE3
#2 h  Mar 31, 2025, 6:58 AM
Y by
Bump (this problem is easy enough)
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alexheinis
10515 posts
alexheinis
#3 h  Mar 31, 2025, 9:14 AM
Y by
Taking $x=y=0,w=2$ we find $f(0,0)=0$. Define $g(x):=f(x,y)$ then with a) we find $f(p,q)=q+g(p-q)$.
Substituting this in b) we find $g((x-y)w)=wg(x-y)$ when $w\not=0$. Also true for $w=0$ hence $g(tw)=wg(t)$ for all $t,w$.
With $t=1$ we find $g(w)=cw$. Finally we find $f(x,y)=y+c(x-y)$ and these linear maps work.
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