Difference between revisions of "2016 UMO Problems"
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==Problem 5== | ==Problem 5== | ||
| − | Let <math>a_0,a_1,a_2,\ldots</math> be a sequence of integers (positive, negative, or zero) such that for all nonnegative integers <math>n</math> and <math>k</math>, <math>a_{n+k}^2-(2k + 1)a_{n}a_{n+k} + (k^2 + k)a_n^2 = k^2-k</math>. Find all possible sequences ( | + | Let <math>a_0,a_1,a_2,\ldots</math> be a sequence of integers (positive, negative, or zero) such that for all nonnegative integers <math>n</math> and <math>k</math>, <math>a_{n+k}^2-(2k + 1)a_{n}a_{n+k} + (k^2 + k)a_n^2 = k^2-k</math>. Find all possible sequences (a_n). |
[[2016 UMO Problems/Problem 5|Solution]] | [[2016 UMO Problems/Problem 5|Solution]] | ||
Revision as of 04:20, 20 January 2019
Problem 1
Ada and Otto are engaged in a battle of wits. In front of them is a figure with six dots, and nine sticks are placed between pairs of dots as shown below. The dots are labeled 
. Ada begins the game by placing a pebble on the dot of her choice. Then, starting with Ada and alternating turns, each player picks a stick adjacent to the pebble, moves the pebble to the dot at the other end of the stick, and then removes the stick from the figure. The game ends when there are no sticks adjacent to the pebble. The player who moves last wins. A sample game is described below. If both players play optimally, who will win?
Sample Game
1. Ada places the pebble at B.
2. Ada removes the stick BC, placing the pebble at C.
3. Otto removes the stick CD, placing the pebble at D.
4. Ada removes the stick DE, placing the pebble at E.
5. Otto removes the stick EA, placing the pebble at A.
6. Ada removes the stick AB and wins.
Problem 2
Four fair six-sided dice are rolled. What is the probability that they can be divided into two pairs which sum to the same value? For example, a roll of 
can be divided into 
and 
, each of which sum to 
, but a roll of 
cannot be divided into two pairs that sum to the same value.
Problem 3
Can each positive integer 
be colored either red or blue, such that for all positive integers 
(not necessarily distinct), if 
then are not all the same color?
Problem 4
Equiangular hexagon 
has 
and 
. Segments 
and 
intersect at point 
and segments 
and intersect at point 
. If quadrilateral 
can have a circle inscribed inside of it (meaning there exists a circle that is tangent to all four sides of the quadrilateral), then find 
.
Problem 5
Let 
be a sequence of integers (positive, negative, or zero) such that for all nonnegative integers 
and 
, 
. Find all possible sequences (a_n).
Problem 6
Find all positive integer pairs 
such that 
is divisible by 
.
