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 .
Olympiad styled
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