Various Olympiad problems

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Problem: Geppetto and Pinocchio play according to the following rules: Geppetto writes on the board six different numbers in a row, and Pinocchio comes up with his own four numbers for them \( x_{1}, x_{2}, x_{3}, x_{4} \) and under each number Geppetto writes respectively any of the sums \( x_{1}+x_{2}, x_{1}+x_{3}, x_{1}+ \) \( x_{4}, x_{2}+x_{3}, x_{2}+x_{4}, x_{3}+x_{4} \) (each only once), after which for each sum, equal to the number under it, Pinocchio gets 3 apples, and for a larger one 1 apple. What is the maximum number of apples Pinocchio can be guaranteed to get?

12.4.1 Various Olympiad problems

5.14 $

Problem: There are \( n \) people gathered, each tow of which either know each other, or have exactly one mutual acquaintance. At the same time, there is no one, who is familiar with everyone. Prove that \( n-1 \) is the square of an integer.

12.4.2 Various Olympiad problems

5.14 $

Problem: In a class of 21 pupils, each three pupils have done either Mathematics or English homework together exactly once. Can we claim that there are four pupils in this class, any three of which have done homework together in the same subject?

12.4.3 Various Olympiad problems

3.85 $

Problem: John, leaving the house, looked at his watch and saw that the line, dividing the angle between the hour and minute hands in half, passes through number 12. When he came to school, the line, dividing the angle between the hour and minute hands in half, passed through the mark, corresponding to 13 minutes. How long did John walk from home to school if it is known that he left after 8:00, and arrived at school before 9:00?

12.4.4 Various Olympiad problems

5.14 $

Problem: On a \( 100 \times 100 \) board there are 2500 kings, not hitting each other. Let's prove that the number of these arrangements is \( \leq 51^{100} \).

12.4.5 Various Olympiad problems

10.28 $

Problem: At the vertices of a 178-gon numbers are written, and the sum of numbers at any successive \( k \) vertices is equal. What is the smallest \( k \) for which all numbers are equal, given the condition \( k>1 \) ?

12.4.6 Various Olympiad problems

5.14 $

Problem: A regular 85-gon is inscribed in a circle, at the vertices of which different natural numbers are written. A pair of non-neighboring vertices of a polygon \( A \) and \( B \) is called interesting if at least on one of the two arcs of \( A B \) at all vertices of the arc there are numbers greater than the numbers, written at vertices \( A \) and \( B \). What is the least number of interesting pairs of vertices that this polygon can have?

12.4.8 Various Olympiad problems

5.14 $

Problem: The expert company assesses \( n \) girls in accordance with 3 criteria: intelligence, beauty and kindness, in points (arbitrary natural numbers, they can be considered real numbers, it's not important). The ratings for each criterion are different for all girls, that is, they can be sorted by their intelligence, beauty or kindness. Prove or disprove that it is possible to choose three such girls, that for any of the unchosen ones, one of these three will exceed her in at least two criteria.

12.4.7 Various Olympiad problems

8.99 $

Problem: In how many ways can the numbers \( 1,2,3, \ldots, 1024 \) be coloured in two colours (blue and red) so that any 2 numbers, the sum of which is a power of two, are coloured in different colours.

12.4.9 Various Olympiad problems

3.85 $

Problem: Find such smallest \( k \) that for any 15-colour colouring of a \( 30 \times k \) table, there are two rows and columns, at the intersection of which there are 4 cells of the same colour.

12.4.10 Various Olympiad problems

7.71 $

Problem: How many integer solutions does the equation \( |a|+|b|+|c|+|d|+|e|=100 \) have?

12.4.11 Various Olympiad problems

3.85 $

Problem: What is the sum of all numbers obtained from the number \( k \) by permuting the digits (including \( k \) ), if \( k=5307447 \) ? Representing 0 at the beginning of the number is not allowed. As an answer, write down the remainder of dividing the obtained sum by 10000 . If the question of the problem allows several answers, then indicate them all in the form of a set.

12.4.12 Various Olympiad problems

5.14 $

Problem: Numbers from 1 to 50 are written on cards. Is it possible to arrange these cards into 11 bags (so that there is at least one card in each bag) so that the product of the numbers on the cards in each bag is divisible by 9 ?

12.4.13 Various Olympiad problems

3.85 $

Problem: Several not necessarily different natural numbers are written on the board in one line from left to right. It is known that each next number, except for the first one, is either greater than the previous one by 1 , or two times smaller than the previous one. a) It is possible that the first number is 12 , and the seventh one is 2 ? b) It is possible that the first number is equal to 1200 , and the \( 25^{\text {th }} \) one is equal to 63 ? c) What is the smallest amount of numbers that can be written on the blackboard if the first number is 1200 and the last number is 5 ?

12.4.14 Various Olympiad problems

7.71 $

Problem: How many ways are there to colour \( n \) balls in 3 colours (different variants are those in which the number of balls of a certain colour differs).

12.4.15 Various Olympiad problems

3.85 $