AoPS Academy
be expressed as a polynomial in 
with real coefficients?
so that both and 
have odd numbers of divisors.
which is defined as all the natural numbers listed in order, side by side, behind a decimal point, without commas? If so, what is it?
and 
are primes, then can 
be a perfect square?
be a trapezoid such that 
lie on a circle with center 
, and side 
is parallel to side 
. Diagonals 
and 
intersect at point 
, and 
. It is given that 
. It is also known that the difference in length between and can be expressed in the form 
, where 
and 
are positive integers and is square-free. Compute the value of 
.
be a twice differentiable function such that![\[
f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1.
\]](http://latex.artofproblemsolving.com/1/e/1/1e178debf8ae0b6b76d2e60ffca95f5a4c4ad4e5.png)
Prove that there exists a point 
such that![\[
f(\xi) \cdot f'(\xi) + f''(\xi) = 0.
\]](http://latex.artofproblemsolving.com/4/6/b/46b8d103981b838ae0aa5afdee104edfae4e4b8f.png)
, where 
always occurs at 
be a ring with unity in which 
and there is a morphism 
from the group 
to the monoid 
such that for all 
, there is a 
such that 
. Prove that is a field.
, above the xy-plane, such that the pyramid 
is a regular tetrahedron.
be a trapezium in which 
and 
. Let 
be then midpoint of the diagonal 
. If ![$[ABCD] = n \times [CDE]$](http://latex.artofproblemsolving.com/e/9/3/e9347ba2c8e9396adfe716159e711408a3b9dc69.png)
, what is the value of ?![$[t]$](http://latex.artofproblemsolving.com/8/a/8/8a86702dc60b9db916798b1115a6b6822bf828b9.png)
denotes the area of the geometrical figure
.)
are the intersection of the supporting halfplanes.
be the unique supporting line, then there is a unique supporting halfplane 
bounded by . This is the only supporting halfplane. Hence 
.
are parallel then is the region between them (including ). If they intersect, then they determine four parts of the plane. Then should be the union of some of those parts, but this is impossible
is one of those parts.
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 
