Two circles and intersect each other at and . The common tangent to two circles nearer to touch and at and respectively. Let and be the reflection of and respectively with respect to . The circumcircle of the triangle intersect circles and respectively at points and (both distinct from ). Show that the line is the second tangent to and .

A Magician and a Detective play a game. The Magician lays down cards numbered from to face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.

Prove that the Detective can guarantee a win if and only if she is allowed to ask at least questions.

Proposed by Anant Mudgal

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.

Suppose that is the midpoint of the arc , containing , in the circumcircle of , and let be the -excircle's center. Assume that the external angle bisector of intersects at . Prove that is perpendicular to , where is the incenter of .

Consider the isosceles triangle with and the circle of radius centered at Let be the midpoint of The line intersects a second time at Let be a point on such that Let be the intersection of and Prove that

Given acute triangle ABC ,AB=b,AC=c . M is a variable point on side AB. The circle circumscribing triangle BCM intersects AC at N.

a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.

b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.

The kingdom of Anisotropy consists of cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from to is a sequence of roads such that one can move from to along this sequence without returning to an already visited city. A collection of paths is called diverse if no road belongs to two or more paths in the collection.

Let and be two distinct cities in Anisotropy. Let denote the maximal number of paths in a diverse collection of paths from to . Similarly, let denote the maximal number of paths in a diverse collection of paths from to . Prove that the equality holds if and only if the number of roads going out from is the same as the number of roads going out from .

Proposed by Warut Suksompong, Thailand

Consider a scalene triangle with incentre and excentres and , opposite the vertices and respectively. The incircle touches and at and respectively. Prove that the circles and have a common point other than .

As shown in the figure, there are two points and outside triangle such that and . Connect and , which intersect at point . Let intersect at point . Prove that .

Find all functions such that for all integers and ,
Calvin Wang and Zani Xu