functional equation

by hanzo.ei, Apr 6, 2025, 6:08 PM

Find all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying the equation
$(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.$

functional equation

algebra

Two Functional Inequalities

by Mathdreams, Apr 6, 2025, 1:34 PM

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x) \le x^3$ and $f(x + y) \le f(x) + f(y) + 3xy(x + y)$ for any real numbers $x$ and $y$ .

(Miroslav Marinov, Bulgaria)

functional inequalities

algebra

functional equation

Geometry

by youochange, Apr 6, 2025, 11:27 AM

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 $\triangle ANK$ .

This post has been edited 1 time. Last edited by youochange, Yesterday at 11:28 AM
Reason: Y

geometry

Pythagorean new journey

by XAN4, Apr 6, 2025, 3:41 AM

The number $4$ is written on the blackboard. Every time, Carmela can erase the number $n$ on the black board and replace it with a new number $m$ , if and only if $|n^2-m^2|$ is a perfect square. Prove or disprove that all positive integers $n\geq4$ can be written exactly once on the blackboard.

number theory

number theory unsolved

Squence problem

by AlephG_64, Apr 5, 2025, 1:19 PM

Francisco wrote a sequence of numbers starting with $25$ . From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was $8$ , what is the fifth term of the sequence?

algebra

sqrt(2) and sqrt(3) differ in at least 1000 digits

by Stuttgarden, Mar 31, 2025, 1:09 PM

We write the decimal expressions of $\sqrt{2}$ and $\sqrt{3}$ as $\sqrt{2}=1.a_1a_2a_3\dots\quad\quad\sqrt{3}=1.b_1b_2b_3\dots$ where each $a_i$ or $b_i$ is a digit between 0 and 9. Prove that there exist at least 1000 values of $i$ between $1$ and $10^{1000}$ such that $a_i\neq b_i$ .

number theory

Spain

Something nice

by KhuongTrang, Nov 1, 2023, 12:56 PM

Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$

This post has been edited 2 times. Last edited by KhuongTrang, Nov 19, 2023, 11:59 PM

inequalities

combinatorics and number theory beautiful problem

by Medjl, Feb 1, 2018, 3:16 PM

A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that
$i$ of each $a$ consecutive integers at least one is coloured red;
$ii$ of each $b$ consecutive integers at least one is coloured blue;
$iii$ of each $c$ consecutive integers at least one is coloured green;
$iiii$ of each $d$ consecutive integers at least one is coloured purple.
Determine all good quadruples with $a = 2.$

number theory

beautiful functional equation problem

by Medjl, Feb 1, 2018, 3:10 PM

Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that :
$i)$ $f(p)=1$ for all prime numbers $p$ .
$ii)$ $f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$
find the smallest $n \geq 2016$ such that $f(n)=n$

This post has been edited 2 times. Last edited by Medjl, Feb 1, 2018, 3:16 PM

function

number theory

prime numbers

50 points in plane

by pohoatza, Jun 28, 2007, 12:30 PM

Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.