Point D is lies on side BC of triangle ABC such that CD=2BD. It turned out that angle ADC=60 and angle ABC=45. Find angle BAC.
A bug is crawling North along a wire when it comes to a wire frame in the shape of a cube. The top and bottom faces of this cube are perfectly level with the bug arriving at one of the lower level corners labelled A. Corners B, C, and D are also on the lower level naming counterclockwise. Upper level corners labelled E, F, G, and H are directly above A, B, C, and D respectively. The only connections to the outside world are at A and C. The wire exiting the cube at C is headed North.
Thus the cube represents a type of temporary trap for the poor bug. The bug travels along the wires taking 1 second to go from one corner to another. When it gets to either A or C there is a chance it will exit, and continue walking North from C or South from A. Once the bug decides to go in the exit direction, it is instantly free (do not count any time for the exiting move)
1. The bug never travels back along the same wire (and cannot reverse direction on a wire).
2. When encountering multiple choices which are all level, each path has an equal chance of being chosen.
3. The bug prefers going up to going level; and prefers level to down. Each is a 2:1 preference.
4. At a corner, the bug first decides between up/level or level/down; then if the choice was "level", it applies Rule 2.
What is the probability the bug will exit North?
What is the expected value of the number of seconds spent on the cube?
Same two questions if there were no up/level and level/down preference, and all directions at a corner (except returning back the same way) were equally likely?
Let ABC be a triangle with integral side lengths such that angle A=3 * angle B. Find the minimum value of its perimeter.