Let ℤ be the set of integers. Determine all functions f : ℤ → ℤ such that, for all integers a and b,
f(2a) + 2f(b) = f(f(a + b)).
In triangle ABC, point A1 lies on side BC and point B1 lies on side AC. Let P and Q be points on segments AA1 and BB1, respectively, such that PQ is parallel to AB. Let P1 be a point on line PB1, such that B1 lies strictly between P and P1, and ∠PP1C = ∠BAC. Similarly, let Q1 be a point on line QA1, such that A1 lies strictly between Q and Q1, and ∠CQ1Q = ∠CBA.
Prove that points P, Q, P1, and Q1 are concyclic.
A social network has 2019 users, some pairs of whom are friends. Whenever user A is friends with user B, user B is also friends with user A. Events of the following kind may happen repeatedly, one at a time:
Three users A, B, and C such that A is friends with both B and C, but B and C are not friends, change their friendship statuses such that B and C are now friends, but A is no longer friends with B, and no longer friends with C. All other friendship statuses are unchanged.
Initially, 1010 users have 1009 friends each, and 1009 users have 1010 friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.
Find all pairs (k, n) of positive integers such that
k! = (2n – 1)(2n – 2)(2n – 4) … (2n – 2n–1).
The Bank of Bath issues coins with an H on one side and a T on the other. Harry has n of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly k > 0 coins showing H, then he turns over the kth coin from the left; otherwise, all coins show T and he stops. For example, if n = 3 the process starting with the configuration
T HT would be T HT → HHT → HTT → TTT, which stops after three operations.
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
(b) For each initial configuration C, let L(C) be the number of operations before Harry stops. For example, L(T HT ) = 3 and L(TTT ) = 0. Determine the average value of L(C) over all 2n
possible initial configurations C.
Let I be the incentre of acute triangle ABC with AB ≠ AC. The incircle ꙍ of ABC is tangent to sides BC, CA, and AB at D, E, and F, respectively. The line through D perpendicular to EF meets ꙍ again at R. Line AR meets ꙍ again at P. The circumcircles of triangles PCE and PBF meet again at Q.
Prove that lines DI and PQ meet on the line through A perpendicular to AI.