A functional equation

by super1978, May 26, 2025, 2:02 AM

Find all functions $f: \mathbb R \to \mathbb R$ such that: $f(f(x)(y+f(y)))=xf(y)+f(xy)$ for all $x,y \in \mathbb R$

Geometric mean of squares a knight's move away

by Pompombojam, May 26, 2025, 12:38 AM

Each square of an $8 \times 8$ chessboard has a real number written in it in such a way that each number is equal to the geometric mean of all the numbers a knight's move away from it.

Is it true that all of the numbers must be equal?

Chessboard

combinatorics

ded
comment

Three operations make any number

by awesomeming327., May 25, 2025, 9:24 PM

The number $3$ is written on the board. Anna, Boris, and Charlie can do the following actions: Anna can replace the number with its floor. Boris can replace any integer number with its factorial. Charlie can replace any nonnegative number with its square root. Prove that the three can work together to make any positive integer in finitely many steps.

factorial

algebra

Another right angled triangle

by ariopro1387, May 25, 2025, 4:13 PM

Let $ABC$ be a right angled triangle with $\angle A=90$ . Point $M$ is the midpoint of side $BC$ And $P$ be an arbitrary point on $AM$ . The reflection of $BP$ over $AB$ intersects lines $AC$ and $AM$ at $T$ and $Q$ , respectively. The circumcircles of $BPQ$ and $ABC$ intersect again at $F$ . Prove that the center of the circumcircle of $CFT$ lies on $BQ$ .

geometry

Iranian TST

2025

JBMO TST Bosnia and Herzegovina 2023 P4

by FishkoBiH, May 25, 2025, 1:38 PM

Let $n$ be a positive integer. A board with a format $n*n$ is divided in $n*n$ equal squares.Determine all integers $n$ ≥3 such that the board can be covered in $2*1$ (or $1*2$ ) pieces so that there is exactly one empty square in each row and each column.

combinatorics

A sharp one with 3 var

by mihaig, May 13, 2025, 7:20 PM

Let $a,b,c\geq0$ satisfying
$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$ Prove
$ab+bc+ca+abc\geq4.$

inequalities

Prove that IMO is isosceles

by YLG_123, Oct 12, 2024, 6:10 PM

Let $ABC$ be an acute-angled scalene triangle with circumcenter $O$ . Denote by $M$ , $N$ , and $P$ the midpoints of sides $BC$ , $CA$ , and $AB$ , respectively. Let $\omega$ be the circle passing through $A$ and tangent to $OM$ at $O$ . The circle $\omega$ intersects $AB$ and $AC$ at points $E$ and $F$ , respectively (where $E$ and $F$ are distinct from $A$ ). Let $I$ be the midpoint of segment $EF$ , and let $K$ be the intersection of lines $EF$ and $NP$ . Prove that $AO = 2IK$ and that triangle $IMO$ is isosceles.

IMO 2017 Problem 4

by Amir Hossein, Jul 19, 2017, 4:30 PM

Let $R$ and $S$ be different points on a circle $\Omega$ such that $RS$ is not a diameter. Let $\ell$ be the tangent line to $\Omega$ at $R$ . Point $T$ is such that $S$ is the midpoint of the line segment $RT$ . Point $J$ is chosen on the shorter arc $RS$ of $\Omega$ so that the circumcircle $\Gamma$ of triangle $JST$ intersects $\ell$ at two distinct points. Let $A$ be the common point of $\Gamma$ and $\ell$ that is closer to $R$ . Line $AJ$ meets $\Omega$ again at $K$ . Prove that the line $KT$ is tangent to $\Gamma$ .

Proposed by Charles Leytem, Luxembourg

This post has been edited 1 time. Last edited by djmathman, Jun 16, 2020, 4:13 AM

Circumcircle of ADM

by v_Enhance, Jul 19, 2012, 8:12 PM

Triangle $ABC$ is inscribed in circle $\Omega$ . The interior angle bisector of angle $A$ intersects side $BC$ and $\Omega$ at $D$ and $L$ (other than $A$ ), respectively. Let $M$ be the midpoint of side $BC$ . The circumcircle of triangle $ADM$ intersects sides $AB$ and $AC$ again at $Q$ and $P$ (other than $A$ ), respectively. Let $N$ be the midpoint of segment $PQ$ , and let $H$ be the foot of the perpendicular from $L$ to line $ND$ . Prove that line $ML$ is tangent to the circumcircle of triangle $HMN$ .

four points lie on a circle

by pohoatza, Jun 28, 2007, 6:32 PM

Let $ABCD$ be a trapezoid with parallel sides $AB > CD$ . Points $K$ and $L$ lie on the line segments $AB$ and $CD$ , respectively, so that $AK/KB=DL/LC$ . Suppose that there are points $P$ and $Q$ on the line segment $KL$ satisfying $\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.$ Prove that the points $P$ , $Q$ , $B$ and $C$ are concyclic.

Proposed by Vyacheslev Yasinskiy, Ukraine

This post has been edited 2 times. Last edited by wu2481632, Dec 10, 2018, 2:24 AM

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by super1978, May 26, 2025, 2:02 AM

by Pompombojam, May 26, 2025, 12:38 AM

by awesomeming327., May 25, 2025, 9:24 PM

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by YLG_123, Oct 12, 2024, 6:10 PM

by Amir Hossein, Jul 19, 2017, 4:30 PM

by v_Enhance, Jul 19, 2012, 8:12 PM

by pohoatza, Jun 28, 2007, 6:32 PM