AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
points in the plane such that all pairwise distances between them are distinct, prove that there exist two points 
,
among them such that the perpendicular bisector of segment 
has at least two points on each of its sides (i.e., each side contains at least one point other than and ).
is a sequence of real numbers with 
such that for all 
Prove that
be natural numbers. As the New Year arrives, it is a cherished tradition among Iranian families to call their relatives and exchange Nowruz greetings. Just moments after the New Year begins, a family with 
members calls a related family with 
members to celebrate the occasion. Since the two families are close, every member of one family wishes to speak with every member of the other family to exchange greetings. Thus, during this single call, all 
distinct one-on-one conversations must take place exactly once.
. Let the perpendicular line from to 
meets 
at points 
, respectively, and the foot from to 
is 
. 
intersects line at 
, 
intersects line at 
, and lines 
intersect at 
.
is tangent to 
.
be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape by playing optimally. (The perimeter of shape is defined as the total length of the boundary segments between a black and a white cell.), assuming both players play optimally?
+1 w
be a positive integer and let 
be an irrational number in 
. Show that for every integer 
, there exists a positive integer such that 
.
is a family of non-empty subsets of , such that any two distinct subsets in the family are disjoint, and the union of all subsets equals . We say that a partition of a set of integers 
is separated if each subset in the partition does not contain consecutive integers. Prove that, for every positive integer 
, the number of partitions of the set 
is equal to the number of separated partitions of the set 
.
is a separated partition of the set 
. On the other hand, 
is a partition of the same set, but it is not separated since 
contains consecutive integers.
, how many digits does 
have?
not 
has the most factors?
has the most factors?
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
lies on diagonal 
.
.
has 
,
,
.
.
such that 
Prove that 
Where 
then find minimal value of 
Prove that
Where 
to 
have the digit 
and are divisible by 
now, we have covered all the numbers in the 7 ones place. (27, if I counted right) now, lets find numbers with tens digit 7 that do NOT have 7 in its one place, since we already counted the _77 in the first case.
this totals up to 25 numbers, so the total is
so 
?
without a calculator.
for 
:
:
.
and 
where the rest has been omitted because of the floor function. Now 
and multiplying by 
yields 
grid is made. 
and 
,
rectangles can be created.
grid, each unit square is created by dividing the sides of lengths 
and 
horizontal lines and 
vertical lines:
that can be formed, we multiply the number of ways to choose the horizontal lines and the ways to choose the vertical lines:
satisfy 
,
and 
are positive integers, what is the probability that 
is divisible by 
?
.
,
represents a distinct digit from 
to 
.
is our answer. (By python, 
does not work and surprisingly, 
is the only permutation number that is divisible by 
8907