NT sum FE

by CHESSR1DER, Apr 14, 2025, 9:32 PM

Find all functions $f$ $N \rightarrow N$ .
For every $n > N$ for some positive integer $N$ there exist unique pair $(a,b):a>b$ so the following conditions holds.
i) $n = a+b$
ii) $f(a) = f(b)$

number theory

EGMO magic square

by Lukaluce, Apr 14, 2025, 11:03 AM

In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$ , and the sum of the numbers in each column is equal to $1$ . Define $r_i$ to be the largest value in row $i$ , and let $R = r_1 + r_2 + ... + r_{2025}$ . Similarly, define $c_i$ to be the largest value in column $i$ , and let $C = c_1 + c_2 + ... + c_{2025}$ .
What is the largest possible value of $\frac{R}{C}$ ?

Proposed by Paulius Aleknavičius, Lithuania

This post has been edited 1 time. Last edited by Lukaluce, Today at 12:16 PM

board

maximum

EGMO

one cyclic formed by two cyclic

by CrazyInMath, Apr 13, 2025, 12:38 PM

Let $ABC$ be an acute triangle. Points $B, D, E$ , and $C$ lie on a line in this order and satisfy $BD = DE = EC$ . Let $M$ and $N$ be the midpoints of $AD$ and $AE$ , respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$ , respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.

geometry

EGMO 2025

Incenter and midpoint geom

by sarjinius, Jul 17, 2024, 12:41 PM

Let $ABC$ be a triangle with $AB < AC < BC$ . Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$ , respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$ . Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$ . Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$ . Let $K$ and $L$ be the midpoints of $AC$ and $AB$ , respectively.
Prove that $\angle KIL + \angle YPX = 180^{\circ}$ .

Proposed by Dominik Burek, Poland

This post has been edited 4 times. Last edited by sarjinius, Jul 17, 2024, 4:20 PM

parallel tangent line

FE but mmmmmm

by Shewalala, Jan 8, 2023, 3:32 PM

find all functions f:IR - - - > IR such that :
f(xy-1)+f(x)f(y)=2xy-1

functional

algebra

functional equation

Functional equation

by Outwitter, May 20, 2020, 4:37 PM

Find all functions $f : R \to R$ satisfying $f(xy - 1) + f(x)f(y) = 2xy - 1$ for all real numbers $x, y$

function

algebra

functional equation

Bisector, orthogonal projection

by mruczek, Apr 28, 2018, 5:55 PM

Bisector of side $BC$ intersects circumcircle of triangle $ABC$ in points $P$ and $Q$ . Points $A$ and $P$ lie on the same side of line $BC$ . Point $R$ is an orthogonal projection of point $P$ on line $AC$ . Point $S$ is middle of line segment $AQ$ . Show that points $A, B, R, S$ lie on one circle.

6 (of 43)

by math_explorer, Nov 6, 2016, 2:15 AM

I posted about this last time, but it's worth repeating. If you join one social community and then another, and there are some things that both communities like to wander off-topic to talk about, it can be very confusing and disorganized.

This post has been edited 1 time. Last edited by math_explorer, Nov 12, 2016, 6:36 PM

topology

Nice problem

by FabrizioFelen, Sep 22, 2016, 7:10 AM

Find all funtions $f:\mathbb R\to\mathbb R$ such that: $f(xy-1)+f(x)f(y)=2xy-1$ for all $x,y\in \mathbb{R}$ .

algebra

functional equation

AZE BMO TST

Turkey NMO 1997 Problem 1, Diophant Equation

by mestavk, Sep 28, 2011, 8:42 AM

Find all pairs of integers $(x, y)$ such that $5x^{2}-6xy+7y^{2}=383$ .

quadratics

modular arithmetic

algebra

number theory proposed

number theory

Rotating segment by 45 degrees and interchanging endpoints.

by Goutham, Feb 9, 2011, 9:58 AM

A needle (a segment) lies on a plane. One can rotate it $45^{\circ}$ round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?