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.
-deck of cards, i.e. a standard 
-card deck together with two jokers. Ada draws cards from the deck until Ada has drawn an ace, a king, and a queen. How many cards does Ada pick up on average?
given that the repetend has a length of 
? Express the answer as a three-digit number. Include preceding zeros if there are any.
be a convex cyclic hexagon. Suppose that 
and 
Compute the circumradius of 
on positive integers 
Let 
equal the smallest positive integer 
such that 
How many positive integers 
satisfy 
?
be a triangle and the points 
and 
on 
, 
and 
on 
, and 
and 
on 
be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
, respectively. Prove that the lines 
and 
are concurrent.
be fixed points on this order on a line. Let 
be a variable circle through 
and and suppose it meets the perpendicular bisector of at the points 
and 
. Let 
and be the other points of intersection of 
and 
with . Prove that 
passes through a fixed point independent of .
, such that 
, there exists an integer satisfying 
?
the sum of all positive divisors of .
be the set of all lines in the plane and 
be a set of 
points in the plane. For a line 
let 
be the number of distinct points among the orthogonal projections of the points from on 
. Find the maximum possible number of distinct values of (this quantity is computed for any line ) as varies.
be an integer. Alice and Bob play the following game on the vertices of a regular -gon. Alice places her token on a vertex of the n-gon. Afterwards Bob places his token on another vertex of the n-gon. Then, with Alice playing first, they move their tokens alternately as follows for 
rounds: In Alice’s turn on the -th round, she moves her token positions clockwise or anticlockwise. In Bob’s turn on the -th round, he moves his token 
position clockwise or anticlockwise. If at the end of any person’s turn the two tokens are on the same vertex, then Alice wins the game, otherwise Bob wins. Decide for each value of which player has a winning strategy.
be an integer. Prove that 
is not a perfect cube of an integer.
be a positive integer. Prove that there exists a positive integer and distinct primes 
such that if 
denotes the number of positive integers less than or equal to and not divisible by any of , then
be an integer and denote 
. For a function 
we denote Im 
, Fix 
and 
. Find all possible values of 
Im 
+ Fix + 
.
with 
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so that:
with all real numbers 
satisfies 
we marked an arbitrary point , which does not coincide with and 
. The tangent to the circle at point intersects the line at point . Prove that the tangent to the circle at point bisects the segment 
.
we marked an arbitrary point , which does not coincide with and . The tangent to the circle at point intersects the line at point . Prove that the tangent to the circle at point bisects the segment .
be the point of intersection of the tangent at and line .
, which is trivially right. By equal tangents, 
, so 
must be a median of the triangle as desired.
