MathDB

2015 Taiwan TST Round 3

Part of TST Round 3

Subcontests

(2)
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2015 Taiwan TST Round 3 Quiz 3 Problem 2

In a scalene triangle ABCABC with incenter II, the incircle is tangent to sides CACA and ABAB at points EE and FF. The tangents to the circumcircle of triangle AEFAEF at EE and FF meet at SS. Lines EFEF and BCBC intersect at TT. Prove that the circle with diameter STST is orthogonal to the nine-point circle of triangle BICBIC.
Proposed by Evan Chen

collinear

Let OO be the circumcircle of the triangle ABCABC. Two circles O1,O2O_1,O_2 are tangent to each of the circle OO and the rays AB→,AC→\overrightarrow{AB},\overrightarrow{AC}, with O1O_1 interior to OO, O2O_2 exterior to OO. The common tangent of O,O1O,O_1 and the common tangent of O,O2O,O_2 intersect at the point XX. Let MM be the midpoint of the arc BCBC (not containing the point AA) on the circle OO, and the segment AA′‾\overline{AA'} be the diameter of OO. Prove that X,MX,M, and A′A' are collinear.

2015 Taiwan TST Round 3 Quiz 2 Problem 2

Consider the permutation of 1,2,...,n1,2,...,n, which we denote as {a1,a2,...,an}\{a_1,a_2,...,a_n\}. Let f(n)f(n) be the number of these permutations satisfying the following conditions: (1)a1=1a_1=1 (2)∣ai−ai−1∣≤2,i=1,2,...,n−1|a_i-a_{i-1}|\le2, i=1,2,...,n-1 what is the residue when we divide f(2015)f(2015) by 44 ?
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