The WORD PROBLEMS of SAINT PETERSBURG (Russia)

2001 MATHEMATICAL OLYMPIAD

(the source: Russian page)

Second Round: 6th grade, 7th grade, 8th grade

6th  grade


    29. There are 10 male cats and 19 female cats sitting in a row on a cat exhibition. For every female cat, there is a bigger male cat sitting near her. Prove that for every male cat there is a smaller female cat sitting near by. 
                                                                                 (S. Berlov) 


     30. Prove that, if the digits of a 10 digit number are written in the reversed order, the resulting number would not be equal to 3 times the original number.

(A. Khrabrov) 


     31. There are 10 coins. Two of them are fake. In one step, Detector R7 checks  three coins and points to one of the coins. It is known that the Detector can not point to a real coin, if one of the testing coins is fake. How can you find the two fake coins in 6 steps?

(R. Ismailov, V. Frank, O.Vanushina) 


     32. There are five 2-digit positive integers written on a board. Wilbur can add 1 to all the numbers or 2 to all the numbers. After this, Charlotte can erase a number, divisible by 13, or a number with the sum of the digits, divisible by 7 (if exists). Prove that Charlotte can play in such a way that Wilbur can not write a 6-digit number.

(K. Kohas') 


     33. There were 150 street lamps on a one side of the Street of Broken Lamps.
There was at least one broken lamp in every group of three subsequent lamps. After electrician Petrov has repaired several lamps, it is at most one broken lamp in every group of four side by side lamps. Prove that Petrov has repaired at least 25 lamps. 

(K. Kohas') 


     34. There are two sets of weights on a balance with two cups. Daria's cup has a set of 1g, 3g, ....., 2001g. Katie's cup has a set of 2g, 4g, ....., 2000g. Daria plays first. She takes a weight off from her cup one by one until her cup is lighter than Katie's one. Then Katie takes a weight off from her cup until her cup is lighter than Daria's one. Then it is Daria's turn to take weights off from her cup. Then Katie's. And so on. 
The winner is a girl who first can take all the weights off from her cup.
Who will be  the winner if both girls do the best? 

(Y. Lifschitz) 

7th grade

 
    35. The floor of the reception room of Baron Munchausen is paved by square stones of the same size. Baron contends that his new carpet (one piece) covers exactly 24 stones so that every horizontal row and every vertical column has exactly 4 stones  covered by the carpet.
Isn't  Baron Munchausen joking? 
(I. Katzev) 


     36. Sasha wrote the first million of positive integers, not divisible by 4. Roma counted the sum of 1000 subsequent numbers from Sasha's row. Can the sum be equal to 20012002? 

(A. Golovanov) 


     37. There are five 2-digit positive integers written on a board. Every minute Wilbur adds 1 to all the numbers or 2 to all the numbers. After this, Charlotte erases a number, divisible by 13, or a number with the sum of the digits, divisible by 7 (if exists). Prove that no matter how Wilbur plays, Charlotte will be able to erase all numbers from the board. 

(K. Kohas') 


     38. Let D be the center of the base AC of an isosceles triangle ABC.  Let E be the base of the perpendicular droped from D on BC. AE intersects  BD at point F. Which segment is longer: BF or BE? 

(F. Bakharev) 


     39. A 6-digit number, divisible by 9,  was multiplied by 111111. Prove that there is at least one digit 9 in base 10 expansion of the product. 

(A. Khrabrov) 
    40. Problem 34.

    41. A black-white board 12x12 is painted like a chess board. In a step one can repaint any two side by side squares: black into green, green into white, white into black.  What is the minimum number of steps needed to repaint the board into the "opposite" white-black board? 

(K. Kohas') 
 

8th grade

 
    42. Is there a 20-digit positive integer such that the number written in reversed order is 3 times bigger than the integer? 
(A. Khrubrov) 


     43. Problem 38.

    44. Let positive integer u and v be so that for any positive integer k, the numbers ku +2 and kv + 3 have a common integer divisor more than 1. What values does the ratio u/v take? 

(A. Golovanov, D. Karpov, A. Pastor) 


     45. Problem 34.

    46. In a triangle ABC,  BC =2AC. There is a point D on the side BC so that the angle CAD  equals the angle CBA. The line AD intersects the bisector of the external angle C in a point E. Prove that AE = AB.

(S. Berlov) 


     47. There are 2001 towns in a country. Every two towns are connected by a direct bus line or a direct railroad line. It is impossible to visit 16 towns, each town only once, and return back taking only one type of transportation. Prove that it is impossible to visit 17 towns, each town only once, and return back using only one type of transportation.

(A. Golovanov, D. Karpov, inaccurate translation from Hungarian) 


     48. A pack of cards with numbers from 0 to 78 is given to a spectator. He shuffles the cards, chooses 40 cards, gives them to a juggler, and  takes the remaining cards. The first juggler chooses 2 cards from the cards given to him and returns them to the spectator. The spectator adds 1 card from his 38 cards to those 2 cards. He shuffles these 3 cards and gives them to a second juggler. The second juggler shows which of the cards were added by the spectator. Explain how this trick can be shown.

(K. Kohas') 
Translated from Russian by
Daria, Ekaterina, Alexander and Olga Merkurjev
February, 2002 
Los Angeles