[/quote]
,
be positive real numbers such that 
and 
,
.
and 
lie on segments 
and 
,
and 
intersect at point 
.
and 
intersect at a second point 
.
intersect at a second point 
.
lies on segment 
such that 
.
and 
are similar.
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
,
and 
could be chosen to satisfy 
and 
.
-
and 
Prove that
.
are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers 
and 
are on the board, then 
,
as 
,
as 
.
Prove that
,
and 
.
satisfies the following property:![\[
f\left( \lambda x + (1 - \lambda)x',\; \lambda y + (1 - \lambda)y' \right) > \min\{f(x, y),\; f(x', y')\}
\]](http://latex.artofproblemsolving.com/3/b/d/3bd9fbf82b0b1d98a3c68745bb3f159cd9e2529a.png)
and for all 
.
be a real numbers such that 
T
segments 
are parallel. Angle bisectors of 
and 
meet at 
and 
meet at 
is cyclic
is even , 
is odd and 
is divisible by 
.Prove that there exists a prime number 
such that 
and 
,![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
Y by
The numbers 
are written in a row in that exact order. Igor and Ruslan take turns inserting the signs 
between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by 
, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves?
are written in a row in that exact order. Igor and Ruslan take turns inserting the signs 
between them, starting with Igor. Each turn consists of placing one sign. Once all signs are placed, the value of the resulting expression is computed. If the value is divisible by 
, Igor wins; otherwise, Ruslan wins. Which player has a winning strategy regardless of the opponent’s moves?This post has been edited 1 time. Last edited by egxa, Apr 18, 2025, 5:24 PM
