Howard's maths page

Who I am :

I am an undergraduate. I play piano.I run.I do programming.

Back to the School of Maths home page.

Go to the Smilecrosofthome page.

From various books,articles and people.

Don't worry if you can't do a question, some are very hard. Feel free to email me about any of them.

2. Given 2n numbers, choose n pairs such that the product of the sums of these pairs is a max.

3. Prove that the sum 299+2999+29999+...+2999,999,999,9999 is divisible by 12.

4. In n-dimensional space, given k flats (n-1 dimensional spaces), what is the maximum number of n-dimensional spaces that can be enclosed ?

5. In a circle with n points, what is the maximum number of regions the circle can be divided up into by the lines joining these points ? (Hint: For n=2,3,4,5 the max=2,4,8,16)

6. (Paul's perimeter-area conjecture) Given two regular polygons of the same perimeter and area, show that the number of their sides is the same.

7. Can you come up with a good proof that it is impossible to draw an x in a square without lifting the pencil or drawing over an old line ?

8. Given a scalene triangle ABC with AC=10,AB=15,BC=17.Draw the bisectors of the angles ACB and ABC such that they meet at a point x. Draw a line through x parallel to CB called PQ (The interesepts with the triangle). Find the perimeter of APQ.

9. Sum the series 6+66+666+6666+...+6..n sixes..66.

10. Prove that the sum of two consecutive Fermat numbers is always congruent to 1 mod(7).

11*. Solve A^3+B^3=23C^3 for the natural numbers A,B and C, or show that it has no solutions.

12. Two square buildings are a distance x apart.Two planks rest between them with their ends in oppossite corners. One plank is 20m and the other is 15m long. They meet at a point a height 6m above the ground. What is the distance x ?

13*. Show that it is possible to make a set of arbitrary size of relatively prime natural numbers such that any subset of them sums to a composite number.

14. Given a graph of n vertices, find the general maximum number of edges that can be added without getting a triangle.

15.Does there exist an arithmetic sequence of integers such that all of its terms are square free ?

16*. (n-Prime sets) An n-prime set is a set of n integers such that each element can be combined in some way with 1,2,3...(n-1) other elements such that their sum is a prime. Can you form an n-prime set for any n ?

17. Show that it is impossible to have a graph where each vertex has a different number of edges connected to it.

18. Find numbers which when reversed and assumed in a different base, have the same value.

19. (Paul's divisiblity problem) Prove that n does not divide (n+1)^k-n^k for all n,k (Natural numbers).

20*. Prove that the maximum number of unit distances in a convex polygon is O(n) (i.e. A constant times n).

21. Is it possible to add enough points to a non-convex polygon so that the resulting polygon is convex ?

22.(Politician problem) If given a set of an odd number of people where every two people have exactly one common friend, show that someone (The politician) knows everyone.

23. Given any four consecutive integers, show that at least one cannot be written as the sum of two squares.

24. Let a,b,c be integer sides of a right-angled triangle, where a < b< c. Show that a*b(b^2-a^2) is divisible by 84.

25. Prove that the sum of all the n-digit integers (n>2) is: 49499...(n-3) 9's..95500...(n-2) 0's..0. (Sorry if that's unclear)

26. Prove that in the Fibonacci numbers there are infitely many multiples of every Fibonacci number.Is this true for the Lucas numbers ?

27. Prove that there are infinitely many n such that n and d*n+(1-d) are perfect squares.

28. Prove that it the sum of the inverses of the divisors of a number is 2 , then that number is perfect.

29. (Talebanfard) Given n integers whose sum is 2n-2, construct a graph with n vertices and these n integers as the degrees.

30. (Icecube's extension of 26) Does there exist an integer which divides no Fibonacci numbers ?

31. Find the set of 3 digit numbers such that the 3 digits at the end of square of that number are the original number.

32. Are there pairs of integers such that two pairs have the same product and sum ? i.e. (a,b),(c,d) where a+b=c+d and ab=cd.

33. Find the positive integer n such that 5x+7y=n has exactly three positive integer solutions.

34. (Erdos to Posa) Given a graph with an infinite number of edges and an infinite number of vertices, prove that it is always possible to choose an infinite subset such that all its vertices are connected to each other or are all disconnected.

35*. Prove that a list of 101 consecutive integers must contain an increasing or decreasing sequence of 11 integers, no matter what the order.

36. (Lavelle Exponential) Find the function f(x) such that its first derivative is equal to exp(f(x)).

37. (Simple Party problem) What is the minimum number of people who must be present at a party in order for at least 3 people to either be complete strangers or know each other.

38*. Prove that for any n>0, there exist infinitely many k such that 1+(2k+1)*2^n is prime. Note that a proof follows immediately from Dirichlet's theorem, what I want is an independent way.

39. Prove that a euclidean number (i.e. 2x3x5x7...xP + 1, where P is a prime) cannot be a perfect square.

40*.Consider the fibonacci sequence (mod n), n is any number. What is the best upper bound for the length of the repeating pattern of terms ?

41.Prove that (n+1) always divides n^(k) + n^(k+1).

42.Prove that if a sphere has 5 points on its surface, 4 will always be contained in a hemisphere. (This neatly generalises to more points,dimensions and fractions of the spheres).

43.What is the number of subsets n objects can be divided into ? (There are atleast 3 proofs to this).

44. Prove that all even perfect numbers are of the form (2^n)x(2^(n+1)-1), where 2^(n+1)-1 is a prime number.Also note that for the last condition to hold, (n+1) itself must be a prime.

45*. (Dodgy visualisation problem) Why is a 4-dimensional cube bounded by 8 3-dimensional cubes ?

46*. (Simple Happy End problem) Given 9 points on a plane, how many points will always fall in a convex pattern ?

47*. (Erdos) Prove that a product of consecutive integers can never be a perfect square.

48*. Define a subset of the terms of an arithmetic sequence such that all its elements are squarefree.

49*. Considering a convex polyhedron of atleast 5 points, can you always view it in such a way that 5 points appear convex ? If not, what is the bound ?

50*. How many different trees can be made containing n vertices (i.e. Graphs of n indistinguishible vertices and n-1 edges).

51. How many arithmetic sequences have 100 as their sum ?

52.(Taeisinger, 1915) Prove that the sum 1/1+1/2+1/3+...+1/n is never an integer for n>1.

53*.(Icecube) Prove that the product of the primes <= (m+1) is <=4^m.

54*.(Mathsoc)Consider a finite group of n*n matrices and let A be the sum of all its elements.Prove that if the trace of A is zero, A is zero.

55. Find an expression for th nth Fibonacci number in terms of Lucas numbers.

56. If a rectangular box has surface area 22 and sum of lengths of sides 24, what is the length of its diagonal ?

57. Why is the set of reals of cardinality 2^(aleph_null) ?

58*.Given n objects, is there a simple formula for the number of pairs of pairs of pairs ... of pairs of objects ?

59. (Betrothed numbers) Find a pair of integers (m,n) such that the sum of the divisors of m equals the sum of the divisors of n equals m+n+1.

60.Given a planar quadrilateral, prove that (1/3)d^2 < a^2 + b^2 + c^2.

61. What is the maximum number of areas n circles can enclose ?

62. Determine whether 97 divides n^2 - 85, for some n>=1.

63. Given a set of fibonacci numbers spaced according to an arithmetic progression, find a good formula for their sum.

64. Prove that it is impossible for a knight starting in the a corner of a chessboard to move to the oppossite corner and visit all other squares once and only once.

65*. (Erdos, 1932) Prove that the sum of the reciprocals of an arbitrary number of terms of an arithmetic sequence can never be an integer.

66. (Matt's russian question) Prove that the infinite sum 1+1/2+1/3+... less the terms which have a 9 in their denominator, converges. What do they converge to ? What about if we remove 2,3 or any digit numbers ?

67. Sum to the nth term the series 1/2 + 3/4 + 5/8 + .... + (2n-1)/2^n.

68.(Russian) Prove that 2^(2^n)+2^(2^(n-1))+1 has atleast n prime factors, not necessarily distinct.

69. Prove that it is possible to assign colours (From a set of 7) to the vertices of any graph on a torous such that no two vertices with the same colour are adjacent.

70. (12 Weight Problem) Given that real coins weigh something different to a fake coin (Either above or below), use three weighings to determine which one of 12 coins is a counterfeit.

71. A spider has 8 feet, 8 shoes and 8 socks. In how many different orders can he dress his feet in the morning ?

72.(D.J.Newman's Analytic Number Theory) Show without use of Stirling's formula : n!>(n/e)^e.

73.(Same as above) Show that if order counts, the number of possible partitions of n (A natural number) is 2^(n-1).

74. If two 2-dimensional curves have the same line and surface integrals (For any arbitrary vector or scalar field), prove that they must be the same (As far as rotation,translation and reflection go), or prove otherwise.( Under what conditions is it true? What about the obvious extension to n-dimensions ?).

75.(Thue) Consider the substitution sequence: 0->01,1->10 so we get: 1001011001101001011010010110100110010110.... Prove that no word (Any string of digits) appears 3 times in a row or more.

76*.(Erdos) Does there exist a red-blue colouring of the integers such that if we consider the first N multiples of k, the difference between the number of reds and the number of blues is bounded by B (independent of k).

77. Prove that the number of numbers less than n which are relatively prime to n is never n/4. i.e. phi(n) is never n/4.

78. Given 8 points on a circle, what is the probability that they will lie inside some semicircle.

79*. Given an NxN matrix with 1s or -1s as its elements and such that the product of every row and column is -1, what is the number of ways you can make such a matrix ?

80. (Long-Lavelle-Thom) If H(n) is the sum of the first n reciprocals , prove that as n tends to infinity, ln(x)->H(xn)-H(n).

81. Given a positive rational number m, prove that m+1/m can never be an integer.

82. In pascal's triangle, prove that the number of odd numbers in each row is always a power of 2.

83*. Given 7 dwarves of distinct height, how many ways can they be arranged so that the heights alternate (i.e. smaller,larger,smaller,larger...).

84. (Moser) Given 5 great circles on a sphere, no three meeting at a single point, prove that there must be atleast one spherical 5-sided polygon.

85. Given 6 pairs of visually identical coins such that one of each pair weighs 15g and the other 16g, use two balancings to determine which ones weigh what.

86. Given a right angled triangle with the sides in arithmetic progression, prove that the ratio of the sides must be 3:4:5.

87. Given a circle with n points on its circumference and straight lines joining each pair, find the number of triangles formed which do not involve any of the circumference points as any of their vertices.

88*. (Concerning a general problem) Prove that 73!+1 is a prime number.

89. Prove that 7^(2n)-2352n-1 is divisible by 2304.

90. Find the smallest positive integer a such that it leaves remainder of 1 on division by 2, 2 on division by 3,..., 9 on division by 10.

91. Find a combinatorical interpretation for (1+2+...+n)^2=(1+8+27+...+n^3).

92*. Prove that the sum of the first n cubes divides the sum of the first n kth powers (k odd) and that the sum of the first n squares divides the sum of the first n kth powers (k even).

93. Define a_1=1,a_(n+1)=a_n*a_(n-1)*...a_2*a_1. Prove that the terms are pairwise relatively prime and that the sum of the inverses is 2.

94. If the descendents of n are defined to be (n+1) and (n/(n+1)), prove that every rational number is a descendent of 1 in a unique way.

95. Prove that there are only 5 regular convex polyhedrons.

96. Prove that every convex polyhedron must have a polygonal face of at most 5 sides.

97. (Eoin's fresher problem) Is 300!>100^300 ?

98. Given the first n integers, how many ways can they be arranged such that no number is in its correct postion (eg. 3 is correct in 32,1,3,19,12...)? What is the limit as n tends to infinity ?

Any ideas ?

Some good maths books - Burton's 'Elementary number theory',R. Guy's 'Unsolved problems in number theory', '500 Mathematical Challenges' from the MAA, Steuben's '20 Years before the blackboard' and,of course, Courant and Robbins' 'What is Mathematics ?'.

Some good sites for maths Icecubes home page Proofs of pythagorean theorem EMS homepage

Slightly more course related are the following : Terry's site and Stephen Walsh's site

A great site about Fibonacci numbers is Ira

Although I'm not exactly a proponent of mathematical competitions, they do sometimes have good probems so here are some past papers.