HMMT 十一月 2010 · 团队赛 · 第 30 题

rd 3 Annual Harvard-MIT November Tournament Sunday 7 November 2010 Team Round Polyhedron Hopping

1. [ 3 ] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he’s currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started?
2. [ 6 ] In terms of k , for k > 0 how likely is he to be back where he started after 2 k minutes?
3. [ 3 ] While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherry to be one edge away from where she started?
4. [ 6 ] In terms of k , for k > 0, how likely is it that after k minutes Sherry is at the vertex opposite the vertex where she started? Circles in Circles
5. [ 4 ] Circle O has chord AB . A circle is tangent to O at T and tangent to AB at X such that AX = 2 XB . AT What is ? BT
6. [ 6 ] AB is a diameter of circle O . X is a point on AB such that AX = 3 BX . Distinct circles ω and 1 ω are tangent to O at T and T and to AB at X . The lines T X and T X intersect O again at S 2 1 2 1 2 1 T T 1 2 and S . What is the ratio ? 2 S S 1 2 ◦
7. [ 7 ] ABC is a right triangle with ∠ A = 30 and circumcircle O . Circles ω , ω , and ω lie outside 1 2 3 ABC and are tangent to O at T , T , and T respectively and to AB , BC , and CA at S , S , and S , 1 2 3 1 2 3 ′ ′ ′ respectively. Lines T S , T S , and T S intersect O again at A , B , and C , respectively. What is 1 1 2 2 3 3 ′ ′ ′ the ratio of the area of A B C to the area of ABC ? Linear? What’s The Problem? A function f ( x , x , . . . , x ) is said to be linear in each of its variables if it is a polynomial such that 1 2 n no variable appears with power higher than one in any term. For example, 1 + x + xy is linear in x 2 2 and y , but 1 + x is not. Similarly, 2 x + 3 yz is linear in x , y , and z , but xyz is not. 1
8. [ 4 ] A function f ( x, y ) is linear in x and in y . f ( x, y ) = for x, y ∈ { 3 , 4 } . What is f (5 , 5)? xy 1
9. [ 5 ] A function f ( x, y, z ) is linear in x , y , and z such that f ( x, y, z ) = for x, y, z ∈ { 3 , 4 } . What is xyz f (5 , 5 , 5)? 1
10. [ 6 ] A function f ( x , x , . . . , x ) is linear in each of the x and f ( x , x , . . . , x ) = when 1 2 n i 1 2 n x x ··· x 1 2 n x ∈ { 3 , 4 } for all i . In terms of n , what is f (5 , 5 , . . . , 5)? i