[14.] (10).  There are 33 students in the class and the sum of their ages is 430 years. Is it true that one can find 20 students in the class  such that the sum of their ages is greater 260?

[15.] (15). Prove that from any  5 integer numbers one can choose  3 such that their sum is divisible by 3.

[16.] (15)   Is it possible to write 1974 as a difference of squares of two natural numbers?

[18.] (10) Find 4 natural numbers such that the product of any three of them plus 1 is divisible by the forth number.

[19.] (15) Find n different integer numbers such that the product of any  n-1  of them is divisible by the number which is left.

[20.] (15) Can the sum of digits of a  complete square be equal to 1970?

i) they use only three identical digits?
ii) every other digit is greater than previous?
iii) sum of digits equal to 9?
 

[23.] (15) What is the greatest number of apartments with the same sum of digits in 100 apartments building?

[24.] (15) The clock show 1pm. At what time  the long hand will be above the short one?

[25.] (10). Halloween problem. The virus reproduced itself in the plastic container.  Every day it doubles its size. It took for the virus 50 days to fill up the whole container. How long it took for the virus to fill up a quarter of the container?

[26.] (15)  What  are the integer numbers such that if you delete the last digit it become smaller integer number of times? Find them all.

[27.] (10) In some class the number of students which are absent is 1/6 of the number which are present. After one student left the number of absent students becomes 1/5  of the number of present students. How many students in the class?

[28.] (10) Find two integer numbers such that their sum is three times greater than their difference and twice less than their product.
 

[30.] (15) Two students tall and short left the same house at the same time. They both walk to school. The short  has a step $20\%$ shorter then the tall one, but he makes at the at the same time $20\%$ more steps. Who will get to school faster?

[31.] (10) On a given line find a point such the distance from this point to two other is the smallest. The two other points are not on the line.

[32.] (15) There are 552 weight  1 pound, 2 pounds, ....., 552 pound. Split these weights in three different groups of the same weight.

[33.] (15) A bus went from A to B with the speed 50 miles per hour. It went back with the speed 30 miles per hour. What is the average speed?

[34.] (10) A train goes over 450 meters  bridge in 45 seconds. It pass the electric pole in 15 seconds. How long is the train and what is its speed?

[35.] (10) Cancel 100 digits in the number 123456.......5657585960 such that the result will be

i) the largest
ii) the smallest possible?

[36.] (15) How many four digit numbers divisible by 45 and such that second and third are 97.

[37.] (10) Is there a polygon with 1997 sides and such that there is a line which crosses every side?

[38.] (15) Find all n between 1 and 100 and such that (n-1)! is not divisible by n.

[39.] (10)  A train moving 45 miles per hour meets and is passed by a train moving 36 miles per hour. A passenger in the first train sees the second train takes 6 seconds to pass him. How long in the second train?
 

[40.] (15) The boy has as many sisters as brothers, but each sister has only half as many sisters as brothers. How many brothers and sisters are there in the family?
 
 

[41.] (10)  The International congress of mathematicians in 22 century  is
attended by infinitely many mathematicians. Every mathematician reserved separate
room in the super-hotel in advance. By accident they all came with a friend which
also need a separate room. Is there way to put them all in the super-hotel?
 

[42.] (15) Every point of the circle is colored by at least  one out of three different colors. Show that always there is an arc of 120 degrees such that its ends are colored by the same color. They may be colored also by other colors.

[43.] (10) What fraction is larger

or

Find the arguments which do not use a calculations.
 
 
 
 

[44.] (15) The plane is covered by 43 half planes. Prove that one can always choose  three half planes so they still cover the whole plane.
 
 

[45.] (10) Tom got 20 problems as a homework assignment. He got 8 points for each correct solution and minus 5 points for each wrong solution. He gets no points if he did not tried to solve the problem. His total is 13 points. How many problems he tried to solve?

[46.] (10) Can one make a hole in the standard piece of paper such that a grown man can go through it?

[47.] (10) In the spring Oblomoff lost 25% of his weight. Than he gained 20% in the summer but in the fall he again lost weight. Namely 10%. In the winter he gained again 20%. Did he lost or gain weight that year?

[48.] (15)  New chess figure "Giraffe" make moves 4 squares in one direction and
5 in perpendicular. How many Giraffes one can put on the chess board so they can
not attack each other after any number of moves.
 
 

[49.] (10) In the equation

one number is replaced by dots. Find the number if it is known that one of the roots of the equation is 1.

[50.] (15) For numbers were split in six possible groups of two. It is known that four smallest sums are 1, 5, 8, and 9. What are the original numbers?

[51.] (10)  Tom spends  1/3 part of his time on school,  1/4  on soccer,  1/5 on listening to CD, 1/6 on watching TV, 1/7 on mathematical seminar. Can he live like that?
 
[52.] (15) Four girls Marry, Jill, Ann  and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang 8 songs, more then anybody.   Susan sang 5 songs less then all other girls. How many different songs were performed at the concert?