are there such that 
and 
?
, where 
is a real number. Prove that for all positive 
then prove
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
is tangent to 
to be a point such that it lies on both the circles 
and 
are perpendicular.
weights, each weighing 
and another 
weights with 
each. Write a algorithm that determines the ways in which a scale can be balanced with a weight 
on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers 
.
with orthocenter 
.
to meet 
.
meets the midpoint of 
.
,
.
with side length of 
.
is selected on the diagonal 
such that 
.
as 
.
Y by
There is a paper regular tetrahedron. Petya and Vasya take turns doing cuts with a razor blade along the edges of the tetrahedron until one of the possible sweeps will turn out. Petya makes the first move. Petya wants the resulting paper to have the shape of a triangle, and Vasya wants it to have the shape of a parallelogram. Which player has a strategy to get it done?
(Chernyavskaya I.A.)
(Chernyavskaya I.A.)
This post has been edited 2 times. Last edited by parmenides51, Apr 13, 2021, 2:13 PM
Reason: Peter-> Petya
Reason: Peter-> Petya
