Let be an acute triangle. Points , and lie on a line in this order and satisfy . Let and be the midpoints of and , respectively. Suppose triangle is acute, and let be its orthocentre. Points and lie on lines and , respectively, such that and are concyclic and pairwise different, and and are concyclic and pairwise different. Prove that and are concyclic.

Let be odd positive integers such that and both and are powers of 2. Prove that

How would you solve this without using calculus?

In a mathematics competition with 10 questions, every group of 3 questions is solved by more than half of the participants. No participant has solved any 7 questions. Exactly one participant has solved 6 questions. What could be the number of participants?
1994
2000
2013
2048
None of the above

Let and . Prove that

Let triangle $ABC$ be inscribed in a circle with center $O$. A line $d$ intersects sides $AB$ and $AC$ at points $E$ and $D$, respectively. Let $M$, $N$, and $P$ be the midpoints of segments $BD$, $CE$, and $DE$, respectively. Let $Q$ be the foot of the perpendicular from $O$ to line $DE$.

Prove that the four points $M$, $N$, $P$, and $Q$ lie on a circle.

Let . Prove that

Let and Prove that

Let a, b, c be positive real numbers. What is the minimum value of the following expression?

(18ca + 12bc + ab)² / (36abc² + 2ab²c + 3a²bc)

I got a difficult equation for users to solve:
Find all functions f: R to R, so that to all real numbers x and y,
1+f(x)f(y)=f(x+y)+f(xy)+xy(x+y-2) holds.

Let be real numbers so that and Show that