[/quote]
,
numebr on his list? Write its last two digits as your answer.
grid of square tiles. Robert chooses four of these squares uniformly at random. The probability that the centers of these four squares form a square themselves is 
where 
are coprime naturals. Write the largest 
digit odd factor of 
.
.
which touches one face of the cube and passes through all 
vertices of the face opposite to it. Find the integer closest to 
are distinct numbers in AP such that 
are also in AP. Find the smallest possible value of 
(in degrees).
that satisfy 
is the set of positive real solutions to the system![\[
x^3 + y^3 + z^3 = x + y + z
\]](http://latex.artofproblemsolving.com/7/7/a/77a748a8b55a6b44761bced9bb1031193b6127dc.png)
![\[
x^2 + y^2 + z^2 = xyz
\]](http://latex.artofproblemsolving.com/9/3/a/93a4dffc6fd17b9f19f738d3b3865f56aa0151a9.png)
then find the number of elements in 
and 
are primes, then can 
be a perfect square?
be a trapezoid. By 
we denote the angle bisector of angle 
. Let 
and 
.
is cyclic.
such that 
for all 
?
.
such that for all any 
denote the number of positive integers less than 
that are relatively prime to 
for which the equation 
has at least 
solutions in 
such that for all any 
:
of functions from the set of positive integers to itself that satisfy ![\[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\]](http://latex.artofproblemsolving.com/5/4/8/5489ee420a448756636625801f2bc7dcce6f0ef8.png)
for every positive integer 
means 
.
be a function from the set of integers to the set of positive integers. Suppose that, for any two integers 
is divisible by 
.
,
is divisible by 
.
such that ![\[f(x+f(x)+2f(y))+f(2f(x)-y)=4x+f(y)\]](http://latex.artofproblemsolving.com/4/4/2/4423651bb829b92debd65a23f6cee3d4d4b8530f.png)
holds for all reals 
.
in which each term is either 0 or 1 is called a binary sequence of length 
be the number of binary sequences of length 
be the number of binary sequences of length 
for all positive integers 
![\[ f\left( f(x) y + \frac xy \right) = xyf(x^2+y^2). \]](http://latex.artofproblemsolving.com/b/6/5/b656324b6499f7457b7b9eccbf5eaa01b6ad6e59.png)
,
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 
.
Y by
The midpoint of each side of the base of the quadrangular pyramid is connected with a segment to the intersection point of the medians of the opposite side face. Prove that:
a) these segments intersect and the intersection point are divided in a ratio of
, counting from the side of the foundation;
b) the midpoints of these segments are vertices of parallelogram.
Find the ratio of the area of this parallelogram to the area the base of the pyramid.
a) these segments intersect and the intersection point are divided in a ratio of
, counting from the side of the foundation;b) the midpoints of these segments are vertices of parallelogram.
Find the ratio of the area of this parallelogram to the area the base of the pyramid.
This post has been edited 1 time. Last edited by parmenides51, Jul 11, 2020, 6:15 AM
