Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that = 2∠AMP.
Source: IMO LongList, Vietnam 1, IMO 1977, Day 1, Problem 2
In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.
Centroid, altitudes and medians, and concyclic points
BR1F1SZ3
N2 hours ago
by EeEeRUT
Source: Austria National MO Part 1 Problem 2
Let be an acute triangle with . Let be the centroid of triangle and let be the foot of the perpendicular from to side . The median intersects the circumcircle of triangle at a second point . Let be the point where intersects . The line intersects the circle at a point , such that lies between and . Prove that the points and lie on a circle.
Source: USA December TST for the 56th IMO, by Linus Hamilton
A physicist encounters atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon to any other usamon . (This connection is directed.) When she does so, if usamon has an electron and usamon does not, then the electron jumps from to . In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?
Source: IMO ShortList 2001, combinatorics problem 3, HK 2009 TST 2 Q.2
Define a -clique to be a set of people such that every pair of them are acquainted with each other. At a certain party, every pair of 3-cliques has at least one person in common, and there are no 5-cliques. Prove that there are two or fewer people at the party whose departure leaves no 3-clique remaining.