AoPS Academy
with 
, satisfies the following property:
integers obtained by arranging four digits 
in some order, are all not divisible by 
.
grid, and in each cell, there is a coin with one side white, and other side black. We start by all coin facing white. Alice and Bob executes the following operation:
pairwise distinct cells, and turn over all of the coins in those cells. Next, Bob chooses one row or column, and turn over all of the coins in that row, or column.
with the following property: coins facing black, after 
turns.
be a triangle with integer side lengths. Let 
be points on segment 
such that 
are in this order, 
, and 
.
and 
, passes through . Find the minimum of the perimeter of triangle .
. As shown below, we dessert the board into of equilateral triangle, with side length 
. We call the 
points of 
is good in the figure.
of tiles with side length 
, 
, (thus the tile is right-angled). How many ways are there to fill the board with these tiles such that
Each vertex of the tiles are on good points, and There doesn't exist 
tiles, such that it forms a equilateral triangle of side length .
be two 
complex matrices of the same rank, and let be a positive integer. Prove that 
if and only if 
.
to find the third side 
He uses logarithms as follows. He finds 
and doubles it; adds to that the double of 
subtracts the sum of the logarithms of 
and 
divides the result by 
and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
be a sequence of positive numbers and suppose that 
is divergent. Let further 
. Show that there exists a sequence 
of positive numbers such that 
is convergent and
. (S. 8)
be given real numbers and let 
for all 
. Show that the sequence 
converges and find its limit.
.
be an real matrix with minimal polynomial 
. Prove that the trace of 
is zero.
of positive integers fast-growing if 
for all 
. Also, for a sequence 
of real numbers and a sequence 
of positive integers, let us denote![\[
S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|.
\]](http://latex.artofproblemsolving.com/2/f/c/2fc080e56fa9d8d6b1cf8a0bb2cb404a65d9ec73.png)
, 
such that for every sequence , if all the series![\[
\sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c)
\]](http://latex.artofproblemsolving.com/0/8/a/08aa112d16a9c772145ff03a74fb512b4d02340d.png)
are convergent, then the series 
is also convergent?, , 
such that for every sequence , if all the series![\[
S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d)
\]](http://latex.artofproblemsolving.com/5/8/9/589be33c766fde9e227f56d280d02322585e2ba3.png)
are convergent, then the series is also convergent?
be a ring with 
its centre (
), 
the set of invertible elements and 
.
has the property 
if there exists a subgroup 
of group 
such that 
, 
and 
for any 
. is commutative;
has the property , then 
, for any 
.
and 
are the identity elements for 
and 
does the series 
converge (and absolutely converge)?
greater than 
and 
for 
.
.
for which 
is divisible by 
.
is divisible by 
