Obscure Set Problem

by oVlad, Apr 12, 2025, 9:59 AM

Let $n\geqslant 3$ be a positive integer and $\mathcal F$ be a family of at most $n$ distinct subsets of the set $\{1,2,\ldots,n\}$ with the following property: we can consider $n$ distinct points in the plane, labelled $1,2,\ldots,n$ and draw segments connecting these points such that points $i$ and $j$ are connected if and only if $i{}$ belongs to $j$ subsets in $\mathcal F$ for any $i\neq j.$ Determine the maximal value that the sum of the cardinalities of the subsets in $\mathcal{F}$ can take.

combinatorics

set theory

Unusual Hexagon Geo

by oVlad, Apr 12, 2025, 9:47 AM

Let $ABCDEF$ be a convex hexagon, such that the triangles $ABC$ and $DEF$ are equilateral and the diagonals $AD, BE$ and $CF$ are concurrent. Prove that $AC\parallel DF$ or $BE=AD+CF.$

geometry

hexagon

Equilateral Triangle

Almost Squarefree Integers

by oVlad, Apr 12, 2025, 9:35 AM

A positive integer $n\geqslant 3$ is almost squarefree if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.

number theory

squarefree

NEPAL TST DAY 2 PROBLEM 2

by Tony_stark0094, Apr 12, 2025, 8:37 AM

Kritesh manages traffic on a $45 \times 45$ grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the $45 \times 45$ grid.

Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove $k$ cars from the $45 \times 45$ grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.

What is the minimum value of $k$ that guarantees that Kritesh's job is possible?

combinatorics

Hard cyclic inequality

by JK1603JK, Apr 12, 2025, 4:36 AM

Prove that $\frac{a-1}{\sqrt{b+1}}+\frac{b-1}{\sqrt{c+1}}+\frac{c-1}{\sqrt{a+1}}\ge 0,\quad \forall a,b,c>0: a+b+c=3.$

inequalities

Nepal TST DAY 1 Problem 1

by Bata325, Apr 11, 2025, 1:21 PM

Consider a triangle $\triangle ABC$ and some point $X$ on $BC$ . The perpendicular from $X$ to $AB$ intersects the circumcircle of $\triangle AXC$ at $P$ and the perpendicular from $X$ to $AC$ intersects the circumcircle of $\triangle AXB$ at $Q$ . Show that the line $PQ$ does not depend on the choice of $X$ .(Shining Sun, USA)

This post has been edited 2 times. Last edited by Bata325, Yesterday at 1:23 PM
Reason: title

geometry

NT function debut

by AshAuktober, Apr 9, 2025, 3:53 PM

Let $f$ be a function taking in positive integers and outputting nonnegative integers, defined as follows:
$f(m)$ is the number of positive integers $n$ with $n \le m$ such that the equation $an + bm = m^2 + n^2 + 1$ has an integer solution $(a, b)$ .
Find all positive integers $x$ such that $f(x) \ne 0$ and $f(f(x)) = f(x) - 1.$ (Adit Aggarwal, India.)

This post has been edited 1 time. Last edited by AshAuktober, Apr 10, 2025, 2:52 AM

number theory

Number Theory Chain!

by JetFire008, Apr 7, 2025, 7:14 AM

I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1

This post has been edited 1 time. Last edited by JetFire008, Apr 7, 2025, 7:14 AM

number theory

marathon

A geometry about a parallelogram ABCD

by nAalniaOMliO, Mar 28, 2025, 8:22 PM

On the side $CD$ of parallelogram $ABCD$ a point $E$ is chosen. The perpendicular from $C$ to $BE$ and the perpendicular from $D$ to $AE$ intersect at $P$ . Point $M$ is the midpoint of $PE$ .
Prove that the perpendicular from $M$ to $CD$ passes through the center of parallelogram $ABCD$ .
Matsvei Zorka

This post has been edited 2 times. Last edited by nAalniaOMliO, Apr 9, 2025, 8:25 AM

geometry

parallelogram