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Some materials for problem-solving sessions — modular arithmetics, induction, pigeonhole principle, combinatorics, graphs, and miscellanea 1. Show that the product of every three consecutive integers is (a) divisible by 3; (b) divisible by 6. 2. Show that the product of every four consecutive integers is divisible by 24. 3. What is the last decimal digit of 133133 ? 4. (a) Prove that an integer is divisible by 9 if and only if sum of its decimal digits is divisible by 9. (b) Formulate and prove a similar statement for divisibility by 11. 5. The product of three integers is equal to 10987654321. Can their sum be equal to 124816? 6. A computer program takes an integer and computes the sum of its decimal digits, then the sum of decimal digits of the result etc. until the result is a single digit. If the input of the program is 199199 , what is its output going to be? 7. Show that in a class of 30 students there will always be three students whose birthdays are all in the same month. 8. Show that in a row of 1000 integers it is always possible to find several consecutive ones (maybe just one) whose sum is divisible by 1000. 9. Show that from a group of seven people whose (integer) ages add up to 332 one can select three people with the total age at least 142. 10. Among the integers from 1 to 1, 000, 000, are there more those containing 1 in their decimal notation or those whose decimal notation does not contain any 1’s? The same question for integers from 1 to 10, 000, 000. 11. How many ways are there to elect the secretary, the treasurer, and the librarian from the 100 members of a student society? How many ways are there to elect three executive officers from the 100 society members? Why are the answers to these two questions different? r q p √ 12. Consider real numbers of the form 2 + 2 + 2 + . . . + 2, in other words, √ √ a1 = 2, an+1 = 2 + an . Prove that all these numbers are less than 2. 13. Prove that 3n > n3 for all integers n > 3. 14. Show that every positive integer n > 8 is equal to the sum of two positive integers, one being a multiple of 3 and the other — a multiple of 5, n = 3x + 5y, where x and y are positive integers. (Hint: the induction basis consists of the cases n = 8, 9, 10; the step of induction is 11 = 8 + 3, 12 = 9 + 3 etc.) 15. Recall that the sequence of Fibonacci numbers is defined as follows: f0 = 0, f1 = 1, fn+2 = fn+1 + fn . Prove that fn+m = fn−1 fm + fn fm+1 for all n and m. (Hint: fix n and use induction on m, or vice versa.) 16. Given that a real number x is such that x + x1 is an integer, prove that for each n the number xn + x1n is an integer. 17. A convex polygon with n vertices is dissected into triangles by its diagonals. Prove that the number of triangles is equal to n − 2 and is independent of the choice of diagonals that dissect the polygon. 1 18. The number of ways to select k elements from n is denoted by nk . Show that n(n−1)···(n−k+1) n n! = k!(n−k)! . (k! = 1 · 2 · · · k.) = k! k n 19. Show that (a + b) = an + nan−1 b + n2 an−2 b2 + · · · + nk an−k bk + · · · + bn . 20. Compute the sums n X n , k k=0 21. Show that n X n k , k k=0 n 2 X n k=0 k and n X n k k k=0 2 2n = . n 22. Compute the sums n X k=0 k 2 , n X k k·2 , k=0 and n X k 2k k=0 23. Prove the identity (for a, b, c not equal to one another): (x − c)(x − a) (x − a)(x − b) (x − b)(x − c) + + =1 (c − a)(c − b) (a − b)(a − c) (b − c)(b − a) 24. Suppose p(x) is a polynomial with integer coefficients such that p(0) = p(1) = 2009. Show that p(x) has no integer zeros. 25. The numbers 1, 2, 3, 4, 5, 6 are written (in this order) at the vertices of a regular hexagon. You are allowed to add 1 to two adjacent numbers, and to repeat this operation several times. Is it possible to end up with the configuration where all numbers are the same? 26. All entries of a (a) 4 × 4; (b) 3 × 3 matrix are equal to −1, with the exception of the entry in the top left corner which is equal to 1. You are allowed to multiply all elements of one of the rows by −1, or to do the same to all elements of one of the columns, and to repeat these operations several times. Is it possible to make all the matrix entries the same? 27. Prove that there is no group of 2009 people where each person knows exactly 1981 other persons from this group. 28. Prove that in a volleyball tournament at every moment it is possible to find two teams which played the same number of games. 29. Prove that in each group of 6 people there exist either 3 persons who all know each other (“mutual acquaintances”), or a group of 3 people where no one knows anyone else (“mutual strangers”). 30. Prove that in each group of 10 people there exist either 3 mutual acquaintances, or 4 mutual strangers. 31. At some point during a volleyball tournament, the observers computed the number of teams which played an odd number of games by that moment. Prove that this number is even. 32. In a group of 10 friends, 14 quarrels (between pairs of friends) happened. Prove that there still exist 3 persons from that group who remained friends with each other. 2 33. Mr and Mrs Brown invited four other couples over for a party. During the party, some of them shook hands with the other ones; no two people shook their hands twice, and no person shook their partner’s hand. When it was time to leave, Mr Brown asked everyone how many times they shook someone’s hand, and all the numbers turned out to be different. How many hands did Mrs Brown shake? 34. The triple (a, b, n) is called a Ramsey triple if in each group of n there exist either a mutual acquaintances, or b mutual strangers. Earlier, you proved that (3, 3, 6) is a Ramsey triple, and that (3, 4, 10) is a Ramsey triple. (a) Show that (3, 3, 5) is not a Ramsey triple. (b) Show that (3, 4, 9) is a Ramsey triple. (Hint: analyze the last solution from the lecture, and write down the restrictions on red and blue valencies of vertices for a graph with 9 vertices for which this solution does not work.) 35. Prove that if (a, b − 1, n) is a Ramsey triple, and (a − 1, b, m) is a Ramsey triple, then (a, b, n + m) is also a Ramsey triple. 36. Given that |ai+1 − ai | < 21i for all i, show that the sequence a1 , a2 , a3 , . . . is convergent. √ 37. Compute the limit lim sin((2 + 3)n π). n→∞ 38. A cube in 3d space is positioned in such a way that all its vertices have integer coordinates. Prove that the side of the cube is of integral length. 39. (a) Is it possible to place an uncountable family of circles in the plane, if no circle is allowed to be placed inside another one? (b) Let us call a T-shaped figure in the plane the union of two perpendicular segments, with an endpoint of one of them being the midpoint of the other one. Does there exist an uncountable family of pairwise non-intersecting T-shaped figures? 40. Show that for all positive integers n the numbers 2010n and 2010n + 10n have the same number of digits in the decimal notation. 41. Is the complex number z = 53 + 45 i a root of unity? In other words, does there exist a positive integer n for which zn = 1? 42. Denote by S the figure consisting of all points above the graph y = x2 in the plane. Is it possible to cover the plane by a finite number of figures congruent to S? 43. In Wonderland, they have only two types of coins, namely a and b wonderdollars. Assume that a and b are coprime. (a) Prove that there exists a positive integer N such that any integer number of wonderdollars greater than N can be paid by certain amount of coins without change. (b) Find the maximal number S that cannot be paid without change. (c) Show that if 0 6 s 6 S, and it is impossible to pay s without change, then it is possible to pay S − s without change. 44. Show that for every odd k > 3 there exist positive integers a1 < a2 < . . . < ak with 1 1 1 + + ... + = 1. a1 a2 ak 45. Alice and Bob have an infinite sheet of paper with a square grid on it. They play the following game: taking turns, they paint the sides of the squares (Alice is using the red colour, and Bob is using the blue colour); it is not allowed to paint any side which has been painted already. Alice was the first to make the move. Prove that Bob can play in such a way that Alice could never make a closed red contour. 3 46. A circle is inscribed in a face of a cube of side 1. Another circle is circumscribed about a neighbouring face of the cube. Find the minimal distance between points of these circles. 47. Given that ab = 4 and c2 + 4d2 = 4, show that (a − c)2 + (b − d)2 > 1.6. 48. Let us call two integers m 6= n good if their prime number decompositions involve the same primes, and the same holds for m + 1 and n + 1. (for example, 6 and 48 form a good pair, since 6 = 2 · 3, 48 = 24 · 3, 6 + 1 = 7, 48 + 1 = 49 = 72 ). Is the number of good pairs (m, n) finite or infinite? 49. Let us call an integer n nice if any prime number in its prime number decomposition appears more than once, and the same holds for n + 1 (for example, the number 288 is good, since 288 = 25 · 32 , 288 + 1 = 172 ). Is the number of nice integers finite or infinite? 50. Show that the polynomial (x − 1)(x − 2) . . . (x − 100) − 1 cannot be decomposed into a product of two polynomials of positive degrees with integer coefficients 51. For real numbers a1 6 a2 6 a3 and b1 6 b2 6 b3 , we have a1 +a2 +a3 = b1 +b2 +b3 and a1 a2 + a1 a3 + a2 a3 = b1 b2 + b1 b3 + b2 b3 . Prove that if a1 6 b1 , then a3 6 b3 . 52. Show that if p is a prime number, then the polynomial xn − p cannot be decomposed into a product of two polynomials of positive degrees with integer coefficients 53. How many ways are there to label the faces of a cube by integers 1, 2, . . . , 6? Every integer should be used, two labellings are the same if they cannot be identified rotating the cube. 4

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