Geo problem

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0 replies

jlacosta

Apr 2, 2025

0 replies

Two circles, a tangent line and a parallel

Valentin Vornicu 103

N 30 minutes ago by zuat.e

Source: IMO 2000, Problem 1, IMO Shortlist 2000, G2

Two circles $G_1$ and $G_2$ intersect at two points $M$ and $N$ . Let $AB$ be the line tangent to these circles at $A$ and $B$ , respectively, so that $M$ lies closer to $AB$ than $N$ . Let $CD$ be the line parallel to $AB$ and passing through the point $M$ , with $C$ on $G_1$ and $D$ on $G_2$ . Lines $AC$ and $BD$ meet at $E$ ; lines $AN$ and $CD$ meet at $P$ ; lines $BN$ and $CD$ meet at $Q$ . Show that $EP = EQ$ .

103 replies

Valentin Vornicu

Oct 24, 2005

zuat.e

30 minutes ago

Inequalities

idomybest 3

N an hour ago by damyan

Source: The Interesting Around Technical Analysis Three Variable Inequalities

The problem is in the attachment below.

3 replies

idomybest

Oct 15, 2021

damyan

an hour ago

Function on positive integers with two inputs

Assassino9931 2

N an hour ago by Assassino9931

Source: Bulgaria Winter Competition 2025 Problem 10.4

The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$ . Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$ . Determine all possible values of $f(2025, 2025)$ .

2 replies

Assassino9931

Jan 27, 2025

Assassino9931

an hour ago

Normal but good inequality

giangtruong13 4

N an hour ago by IceyCold

Source: From a province

Let $a,b,c> 0$ satisfy that $a+b+c=3abc$ . Prove that: $\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5}$

4 replies

giangtruong13

Mar 31, 2025

IceyCold

an hour ago

No more topics!

Geo problem

Pirkuliyev Rovsen 3

N Nov 29, 2010 by jgnr

Source: Baltic Way 1995

Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$ and let $H$ be the foot of the altitude from $B$ . Let $P$ and $Q$ be orthogonal projections of $A$ and $C$ on the bisector of the angle $B$ . Prove that the four points $H,P,M$ and $Q$ lie on the same circle.

3 replies