AoPS Academy
be a positive integer. Find the smallest value of such that:
is divisible by 
.
?
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 
.
, where 
always occurs at 
, 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
.)
:
Square again and subtract to get 
Thus, 
Roots are 
Negatives are bad because lhs and rhs would have differing signs, and 
fails. Thus, all are bogus except 
. Now observe that for 
, 
, so we must have 
. So, 
, giving 
.
for can be proven by casework.)
:
.
and 
Prove that
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 
because the LHS is always positive, so the RHS is also always positive. Now we split it into cases:
This gives 
solving gives 
This gives 
so 
again.