2004 Problem of the Week Competition

Final Results

Undergraduate Division

Cuong Dong     1st prize   $50

Efrem Rensi     2nd prize  $30

Stanislav Georgiev     3rd prize   $20 

Johan Johansson     4th prize $15

 Graduate Division

Carlos Urrutia     1st prize   $50

Peter Friedenbach     2nd prize  $30

Cynan Deleon     3rd prize   $20

 San Jose Math Circle division

Edward Luong     1st prize   $50

Akagawa Powell     2nd prize  $30

Sanders Chang     3rd prize   $20

 Prizes can be picked up at the Math Office, MH 308, at the beginning of spring semester (Edward Luong and Cynan Deleon need to give us your student ID numbers before your checks can be printed).  

Problem 13 is the last problem.     

Final Week!                                                          11/17/04

Week #11  Wed. Nov. 17 – Mon. Nov. 29 

 

Problem 13 (25 points):  We are given 3 dice, each with n faces, whose faces are numbered identically with arbitrary integers. If the dice are tossed at random, prove that the probability P that the sum of the numbers on the three bottom faces is divisible by three is greater than or equal to 1/4. 

 

Week #12  Wed.  Nov.10 – Fri. Nov. 19

Problem 12 (20 points)     Suppose that a circle is tangent to a parabola at a point A and intersects the parabola at points B and C. Let AD be the median of the triangle ABC, and let M be the midpoint of AD. Prove that M lies on the axis of the parabola.

 

Week #11  Wed. Nov. 3 – Fri. Nov. 12 

 

Problem 11 (15 points):  Denote by an= lcm{b1,b2,b3, … ,bn}, n ³ 1, the least common multiple of the first n terms of a strictly increasing sequence of positive integers 0 < b1 < b2 < b3 … .  Prove that the series 1/a1 + 1/a2 + 1/a3 + … + 1/an + … converges to a finite sum. 

 

Week #10  Wed.  Oct. 27 – Fri. Nov. 5

Problem 10 (14 points)     Let  k  be a circle with the center at the origin and radius  R, and suppose that exactly 2004 lattice points lie on  k.  (A point of a coordinate plane is called a lattice point if its coordinates are integers.) Prove that either  R  or  RÖ2  is an integer. 

 

Week #9  Wed. Oct. 20 – Fri. Oct. 29 

 

Problem 9 (13 points):  A candidate schedules a series of campaign speeches during n consecutive hours.  Speech A lasts 1 hour, Speech B lasts two hours, and he can also schedule rest periods of 1 or more hours.  Assuming that no more than 3 consecutive hours of speeches can be scheduled, find a recurrence relation for an, the number of different schedules of speeches and rests for n consecutive hours.  How many schedules are there for a period of 24 consecutive hours? 

 

Week #8  Wed.  Oct. 13 – Fri. Oct. 22

 Problem 8 (12 points)          Let  be a triangle, and let AK be the bisector of the angle  (so that the point K  lies on the side BC).  Find the angles of the , if it is known that the center of the circle inscribed in the  and the center of the circle circumscribed about the  coincide.

 

Week #7  Wed. Oct. 6-Fri. Oct. 15

Problem 7 (11 points): Let N = 87654321 be written in decimal notation.  If A is the sum of the digits of N and B is the sum of the digits of A, then what is the sum of the digits of B.

 

 Week #6  Wed.  Sept. 29 – Fri. Oct. 8

  Problem 6 (10 points)     The sum of 10 natural numbers equals 1001. Find the largest possible value of their greatest common divisor. Prove that your answer is correct.

 

Week #5  Wed. Sept. 22 – Fri. Oct. 1 

Problem 5 (9 points):  Prove for any positive integer n that 2196n – 25n – 180n + 13n is divisible by 2004. 

 

Week #4  Wed.  Sept. 15 – Fri. Sept. 24

 Problem 4 (8 points)     Suppose that  x  and  y  are two real numbers such that

Prove that

 

Week #3  Wed. Sept. 8 - Fri. Sept. 17

Problem 3 (7 points): A woman can walk up a moving "up" escalator in 1/2 minute.  She can walk down this moving "up escalator in 1 and 1/2 minutes.  If her walking pace is the same moving up stairs or down stairs, how long would it take her to climb the escalator stairs if it was not moving?  How long would it take her to go up the moving "up" escalator if she stood still?   

 

Week #2  Wed.  Sept. 1 – Fri. Sept. 10

 Problem 2 (6 points)  An alphabet consists of  n  letters.   Find (with a proof) the length of a longest word that satisfies the following two conditions.

(a)             Any two adjacent letters of this word are distinct;

(b)            It is impossible to reduce this word, by means of crossing out any number of letters, to a form  xyxy,  where  x  and  y  are any two distinct letters of the alphabet.

 

Week #1  Wed. Aug. 25 – Fri. Sept. 3 

Problem 1 (5 points):  Between 3:00 and 4:00, Big Ben looked at his watch and noticed that the minute hand was between 5 and 6.  Later, Big Ben looked again and noticed that the hour hand and the minute hand had exchanged places.  What time was it in the second case?