[/quote]
,
be a triangle with incentre 
and circumcircle 
.
and 
be the second intersection points of 
and 
,
meets 
at a point 
,
at a point 
.
be the intersection point of the line through 
and the line through 
.
and 
meet at a point 
.
and 
are either parallel or concurrent.
Prove that
is an odd number, let 
be a positive integer. If 
then 
satisfying
and 
Prove that
be the set of integers. Determine all functions 
such that, for all integers 
and 
,
Proposed by Liam Baker, South Africa
,
that makes the following equation have no integer root 
.
such that for any polynomial 
of degree 
of degree at most 
and 
intersect at exactly 
be a convex quadrilateral. If 
are the midpoints of the segments 
,
,
and 
.
.
that satisfy the following condition: for all 
and 
relatively prime to ![\[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]](http://latex.artofproblemsolving.com/e/8/5/e85c5b249ac8e75b45a525b8786414798a4921fe.png)
by the expression
and 1.
Find the range of 
such that 
is an integer.
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
Y by
An island inhabitants design an emergency response service based on helicopters. Their island is a disc of radius 
km and they plan to purchase helicopters capable of flying 
km/h. Find the minimum number helicopters needed for the emergency service in order to be able to reach every point of the island within 
min. Assume that helicopter’s takeoff and landing times are negligible.
km and they plan to purchase helicopters capable of flying 
km/h. Find the minimum number helicopters needed for the emergency service in order to be able to reach every point of the island within 
min. Assume that helicopter’s takeoff and landing times are negligible.This post has been edited 2 times. Last edited by parmenides51, Jun 21, 2022, 10:05 AM
