1. Show that if n is a composite number then n divides (n-1)!
2. Find the last two digits of the number 5^5^10000.
3. Is log_10 2 rational or irrational?
4. How many zeros are there at the end of 100!
5. Can the number 111111111....1 (a string of n ones, where n >1) be a perfect square?
Monday, November 15, 2010
Monday, November 1, 2010
An interesting geometry problem
This is not a Putnam problem, but it could be one. I have been struggling to prove this. If you can try and
let me know if have any success, that would be great.
Consider an acute angled triangle ABC. (This is a traingle in which all 3 angles are less than 90 degrees. ) Let PQR be a light triangle inside ABC. That is, it is a triangle inscribed by a light ray that starts on a point P on AB and then hits a point Q on BC, gets reflected and hits a point R on CA, and finally returns to a starting position A after a second reflection. These rays satisfy the usual property (angle of incidence = angle of reflection) of light at each of the points P, Q, and R. Show that PRQ is an altitude triangle. That is, P, Q and R are the feet of the perpendiculars drawn from C, B and A (respectively) to to their opposite sides.
let me know if have any success, that would be great.
Consider an acute angled triangle ABC. (This is a traingle in which all 3 angles are less than 90 degrees. ) Let PQR be a light triangle inside ABC. That is, it is a triangle inscribed by a light ray that starts on a point P on AB and then hits a point Q on BC, gets reflected and hits a point R on CA, and finally returns to a starting position A after a second reflection. These rays satisfy the usual property (angle of incidence = angle of reflection) of light at each of the points P, Q, and R. Show that PRQ is an altitude triangle. That is, P, Q and R are the feet of the perpendiculars drawn from C, B and A (respectively) to to their opposite sides.
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