Sums of products of entries in a matrix

by Stear14, May 31, 2025, 10:13 PM

(a) $\$ Each entry of an $\ 8\times 8\$ matrix equals either $\ 1\$ or $\ 2.\$ Let $\ A\$ denote the sum of eight products of entries in each row. Also, let $\ B\$ denote the sum of eight products of entries in each column. Find the maximum possible value of $\ A-B.\$ In other words, find
${\rm max}\ \left[ \sum_{i=1}^8\ \prod_{j=1}^8\ a_{ij} - \sum_{j=1}^8\ \prod_{i=1}^8\ a_{ij} \right]$
(b) $\$ Same question, but for a $\ 2025\times 2025\$ matrix.

combinatorics

linear algebra

matrix

2-var inequality

by sqing, May 27, 2025, 2:15 AM

Let $a,b>0 , a^2+b^2-ab\leq 1 .$ Prove that
$a^3+b^3 -\frac{a^4}{b+1} -\frac{b^4}{a+1} \leq 1$

inequalities proposed

inequalities

Sums of n mod k

by EthanWYX2009, May 26, 2025, 2:48 PM

Given $0<\varepsilon <1.$ Show that there exists a constant $c>0,$ such that for all positive integer $n,$
$\sum_{k\le n^{\varepsilon}}(n\text{ mod } k)>cn^{2\varepsilon}.$ Proposed by Cheng Jiang

number theory

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

a father and his son are skating around a circular skating rink

by parmenides51, May 7, 2020, 9:52 PM

A father and his son are skating around a circular skating rink. From time to time, the father overtakes the son. After the son starts skating in the opposite direction, they begin to meet five times more often. What is the ratio of the skating speeds of the father and the son?

(Tairova)

algebra

geometry

speed

geometry problem

by Medjl, Feb 1, 2018, 2:43 PM

A circle $\omega$ with diameter $AK$ is given. The point $M$ lies in the interior of the circle, but not on $AK$ . The line $AM$ intersects $\omega$ in $A$ and $Q$ . The tangent to $\omega$ at $Q$ intersects the line through $M$ perpendicular to $AK$ , at $P$ . The point $L$ lies on $\omega$ , and is such that $PL$ is tangent to $\omega$ and $L\neq Q$ .
Show that $K, L$ , and $M$ are collinear.

This post has been edited 1 time. Last edited by Medjl, Feb 1, 2018, 3:04 PM

geometry

Parallel lines..

by ts0_9, Mar 26, 2014, 10:58 AM

The triangle $ABC$ is inscribed in a circle $w_1$ . Inscribed in a triangle circle touchs the sides $BC$ in a point $N$ . $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$ , and passing through a point $N$ . Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$ ) respectively. Prove, that lines $AO$ and $JN$ are parallel.

geometry

circumcircle

geometry proposed

Connected, not n-colourable graph

by mavropnevma, Jul 20, 2013, 9:22 PM

The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).
Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.

V. Dolnikov

EDIT. It is confirmed by the official solution that the graph is tacitly assumed to be finite.

induction

algorithm

graph theory

combinatorics proposed

combinatorics

KMN and PQR are tangent at a fixed point

by hal9v4ik, Mar 19, 2013, 5:04 PM

Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$ , and let $AB$ and $CD$ intersect at $K$ . Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$ . Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$ . Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.

Homothety with incenter and circumcenters

by Ikeronalio, Sep 9, 2012, 11:45 AM

Let $I, O$ be the incenter and the circumcenter of triangle $ABC$ , and $D,E,F$ be the circumcenters of triangle $BIC, CIA, AIB$ . Let $P, Q, R$ be the midpoints of segments $DI, EI, FI$ . Prove that the circumcenter of triangle $PQR$ , $M$ , is the midpoint of segment $IO$ .