be an isosceles triangle with 
,
be its incenter. Incircle touches sides 
at 
,
to 
are 
, respectively. Rays 
intersect 
at 
,
touches incircle at 
.
from the positive integers to the positive integers is called Canadian if it satisfies 
for all pairs of positive integers 
and 
.
such that 
for all Canadian functions 
.
on 
respectively. Circumcircle 
of triangle 
intersects 
.
intersects 
at 
cuts 
respectively. Construct parallelogram 
.
are collinear.
-
and 
such that 
.
different coloured marbles on it and another one is empty
The altitudes 
, intersect
.
and 
o
.
b
intersects 
intersects 
at 
lie on the same circle.
intersects the circle 
The line 
.
are collinear.
is an exact 5th power, 
is an exact 6th power, and 
is an exact 7th power.
denote the set of natural numbers and let 
denote 
ordered pairs in 
.
distinct elements in the set 
for 
whose product is a perfect cube.
. Prove that
have real roots and real coefficients with 
.
?
,
.
have roots 
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
is a perfect cube.
digits are distinct?
Solve for 
,
,
,
.
,
be the midpoint of 
.
is a point such that 
is a rectangle, compute the area of 
.
has a sidelength of 
,
,
,
,
,
,
,
are length 
.
.
satisfies for all integers 
If 
and 
,
.
children have black hair, all but 
have red hair, and all but 
have blond hair. How many children are there in the class?
can be partitioned into n Akash subsets.
:
.
Let 
.
Y by
1. The infinite Morse sequence of zeros and ones, 011010011001..., is constructed as follows: start with 0, then at each step, append a block of the same length as the current sequence, obtained by replacing 0 with 1 and vice versa in the existing block. Is this sequence periodic?
2. On an infinite (two-way) tape, a text in Russian is written. It is known that in this text, the number of distinct 15-symbol blocks is equal to the number of distinct 16-symbol blocks. Prove that the text on the tape is periodic in both directions (i.e., bi-infinite and periodic), for example: "...мамамыларамумамамы...".
2. On an infinite (two-way) tape, a text in Russian is written. It is known that in this text, the number of distinct 15-symbol blocks is equal to the number of distinct 16-symbol blocks. Prove that the text on the tape is periodic in both directions (i.e., bi-infinite and periodic), for example: "...мамамыларамумамамы...".
