Problems Collection
This is a page where you can share the problems you made (try not to use past exams).
Contents
[hide]AMC styled
AIME styled
1. There is one and only one perfect square in the form
where and
are prime. Find that perfect square.
2. and
are positive integers. If
, find
.
3.The fraction,
where and
are side lengths of a triangle, lies in the interval
, where
and
are rational numbers. Then,
can be expressed as
, where
and
are relatively prime positive integers. Find
.
4. Suppose there is complex values and
that satisfy
Find .
5. Suppose
Find the remainder when is divided by
.
6. Suppose that there is rings, each of different size. All of them are placed on a peg, smallest on the top and biggest on the bottom. There are
other pegs positioned sufficiently apart. A
is made if
ring changed position (i.e., that ring is transferred from one peg to another)
No rings are on top of smaller rings.
Then, let be the minimum possible number
that can transfer all
rings onto the second peg. Find the remainder when
is divided by
.
7. Let be a 2-digit positive integer satisfying
. Find the sum of all possible values of
.
8. Suppose is a
-degrees polynomial. The Fundamental Theorem of Algebra tells us that there are
roots, say
. Suppose all integers
ranging from
to
satisfies
. Also, suppose that
for an integer . If
is the minimum possible positive integral value of
.
Find the number of factors of the prime in
.
9. is an isosceles triangle where
. Let the circumcircle of
be
. Then, there is a point
and a point
on circle
such that
and
trisects
and
, and point
lies on minor arc
. Point
is chosen on segment
such that
is one of the altitudes of
. Ray
intersects
at point
(not
) and is extended past
to point
, and
. Point
is also on
and
. Let the perpendicular bisector of
and
intersect at
. Let
be a point such that
is both equal to
(in length) and is perpendicular to
and
is on the same side of
as
. Let
be the reflection of point
over line
. There exist a circle
centered at
and tangent to
at point
.
intersect
at
. Now suppose
intersects
at one distinct point, and
, and
are collinear. If
, then
can be expressed in the form
, where
and
are not divisible by the squares of any prime. Find
.
Someone mind making a diagram for this?
10. Suppose where
and
are relatively prime positive integers. Find
.
Proofs
11. In with
,
is the foot of the perpendicular from
to
.
is the foot of the perpendicular from
to
.
is the midpoint of
. Prove that
is perpendicular to
.
See also
Here's the source for the problems:
1,2,3,4,5,6,8,9,10,11: Ddk001, credits given to Ddk001
7: SANSKAR'S OG PROBLEMS, credits given to SANSGANKRSNGUPTA