be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
,
and 
intersect at points 
and 
.
and 
intersect in two points 
and 
.
and 
are concurrent.
be positive real numbers for which 
.![\[ \frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}\]](http://latex.artofproblemsolving.com/9/2/f/92fc4af06826d9b973ad96bca902f32d17f71a6e.png)
points on the plane with diameter 
.
and two not necessarily distinct points 
not equal to 
or 
satisfying that![\[|AX-XY|+|BY-XY|\leq \frac{2D}{N-2}.\]](http://latex.artofproblemsolving.com/f/3/6/f3649872c33e1db8880b4aedace38d9439f2b4ae.png)
players, each pair of players plays against each other exactly one match and there are no draws.Show that the players can be arranged in an order 
such that 
defeats 
for all 
.
Y by
A person is locked in a room with a password-protected computer. If they enter the correct password, the door opens and they are freed. However, the password changes every time it is entered incorrectly. The person knows that the password is always a 10-digit number, and they also know that the password change follows a fixed pattern. This means that if the current password is 
and 
is entered, the new password is 
, which is determined by and (naturally, the person does not know or ). Prove that regardless of the characteristics of this computer, the prisoner can free themselves.
Proposed by Reza Tahernejad Karizi
and 
is entered, the new password is 
, which is determined by and (naturally, the person does not know or ). Prove that regardless of the characteristics of this computer, the prisoner can free themselves.Proposed by Reza Tahernejad Karizi
This post has been edited 3 times. Last edited by jannatiar, Mar 10, 2025, 7:30 PM
possible patterns the computer could follow. Let 
be a positive integer. Assume we want to show that there exists a sequence of 
candidates for such a sequence. Each of the 
of these sequences (show this inductively). Thus it suffices to choose a 
And such an 
