[/quote]
,
Prove that
,
.
.
.
satisfying 
,
.
,
be the circumcentre, 
be the orthocentre, and 
be the centroid.
satisfy the following condition:
is constant.
be a point outside circle 
centered at 
.
intersects bigger 
arc at 
.
intersects 
at 
.
intersects small 
arc at 
.
is tangent to 
be the set of real numbers. Let 
be a function such that ![\[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]](http://latex.artofproblemsolving.com/5/d/3/5d3476fd9195d085d0030d6036a9fde1cf6756a7.png)
for every 
.
.
for every 
or 
for every 
square is filled with numbers 
.
squares is calculated,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation?
amount of balloons, where 
.
amount of ballons where 
.
?
such that the number 
is
consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
to 
(including 
(product) between each number. Then he writes the rest of the numbers in the denominator and also writes 
prime number, there exists only one positive number 
a perfect square of a positive integer
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
