Functional Equation

by AnhQuang_67, Apr 4, 2025, 4:50 PM

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $2\cdot f\Big(\dfrac{-xy}{2}+f(x+y)\Big)=xf(y)+y(x), \forall x, y \in \mathbb{R}$

function

algebra

functional equation

Problem 2

by SlovEcience, Apr 4, 2025, 3:52 PM

Let $a, n$ be positive integers and $p$ be an odd prime such that:
$a^p \equiv 1 \pmod{p^n}.$ Prove that:
$a \equiv 1 \pmod{p^{n-1}}.$

number theory

Problem 1

by blug, Apr 4, 2025, 11:46 AM

Find all $(a, b, c, d)\in \mathbb{R}$ satisfying
$\begin{aligned} \begin{cases} a+b+c+d=0,\\ a^2+b^2+c^2+d^2=12,\\ abcd=-3.\\ \end{cases} \end{aligned}$

system of equations

algebra

Geometry :3c

by popop614, Apr 3, 2025, 12:19 AM

Quadrilateral $ABCD$ has an incenter $I$ Suppose $AB > BC$ . Let $M$ be the midpoint of $AC$ . Suppose that $MI \perp BI$ . $DI$ meets $(BDM)$ again at point $T$ . Let points $P$ and $Q$ be such that $T$ is the midpoint of $MP$ and $I$ is the midpoint of $MQ$ . Point $S$ lies on the plane such that $AMSQ$ is a parallelogram, and suppose the angle bisectors of $MCQ$ and $MSQ$ concur on $IM$ .

The angle bisectors of $\angle PAQ$ and $\angle PCQ$ meet $PQ$ at $X$ and $Y$ . Prove that $PX = QY$ .

This post has been edited 1 time. Last edited by popop614, Yesterday at 12:42 AM
Reason: asfdasdf

geometry

Functional equations

by hanzo.ei, Mar 29, 2025, 4:33 PM

Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$

algebra

functional equation

A board with crosses that we color

by nAalniaOMliO, Mar 28, 2025, 8:37 PM

In some cells of the table $2025 \times 2025$ crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least $k$ crosses.
Find the minimal $k$ for which it is always possible to color crosses in two colors such that any balanced set has crosses of both colors.

This post has been edited 1 time. Last edited by nAalniaOMliO, Mar 29, 2025, 10:41 AM

combinatorics

Assisted perpendicular chasing

by sarjinius, Mar 9, 2025, 3:41 PM

In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$ , let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$ . Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$ , and let $F$ be the point on line $AC$ such that $FA = FE$ . Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$ .
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.

Pythagorean journey on the blackboard

by sarjinius, Mar 9, 2025, 3:16 PM

A positive integer is written on a blackboard. Carmela can perform the following operation as many times as she wants: replace the current integer $x$ with another positive integer $y$ , as long as $|x^2 - y^2|$ is a perfect square. For example, if the number on the blackboard is $17$ , Carmela can replace it with $15$ , because $|17^2 - 15^2| = 8^2$ , then replace it with $9$ , because $|15^2 - 9^2| = 12^2$ . If the number on the blackboard is initially $3$ , determine all integers that Carmela can write on the blackboard after finitely many operations.

Pythagorean Triples

number theory

H not needed

by dchenmathcounts, May 23, 2020, 11:00 PM

Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$ ). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.

Robin Son

This post has been edited 2 times. Last edited by v_Enhance, Oct 25, 2020, 6:01 AM
Reason: backdate

$f(xy)=xf(y)+yf(x)$

by yumeidesu, Apr 14, 2020, 10:16 AM

Find $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y)=f(x)+f(y), \forall x, y \in \mathbb{R}$ and $f(xy)=xf(y)+yf(x), \forall x, y \in \mathbb{R}.$

algebra

functional equation

Submit

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math_exploration

by AnhQuang_67, Apr 4, 2025, 4:50 PM

by SlovEcience, Apr 4, 2025, 3:52 PM

by blug, Apr 4, 2025, 11:46 AM

by popop614, Apr 3, 2025, 12:19 AM

by hanzo.ei, Mar 29, 2025, 4:33 PM

by nAalniaOMliO, Mar 28, 2025, 8:37 PM

by sarjinius, Mar 9, 2025, 3:41 PM

by sarjinius, Mar 9, 2025, 3:16 PM

by dchenmathcounts, May 23, 2020, 11:00 PM

by yumeidesu, Apr 14, 2020, 10:16 AM