This course is worth 2 units of course credit, and the assessment consists of the weekly homework assignments.  Students who wish to participate for credit should fill out a Math 199 enrollment form from the Math Department office, and present it to Geoff Mess for signing.

• Question 1: Find a way to divide a square into a finite number of acute-angled triangles, or show that no such division is possible.
• Question 2: Suppose that every point in the plane R² is colored either red, white, or blue.  Show that there exists two points in the plane which are exactly 1 inch apart and have the same color.

• Question 1: Show that for any set of five integers, we can always choose three of these integers whose sum is a multiple of 3.
• Question 2: Let n be a positive integer.  Prove that 2^n-1 divides n! if and only if n is a power of 2 (i.e. n = 2^k for some non-negative integer k).

• Question 1: Find the last digit of 2^(3^(4^5)).  (Here we use the notation x^y to denote the exponential x^y).
• Question 2: 126² = 15876 and 116²=13456 are perfect squares which end in strings of consecutive digits (876 and 3456 respectively).  What is the longest such string that you can find at the end of a perfect square?

• Question 1: Let ABC be a triangle.  Let D be the point on AB such that AD = AB/3, let E be the point on BC such that BE = BC/3, and let F be the point on CA such that CF = CA/3.  The line segments CD, AE, and BF divide ABC into three outer triangles, three quadrilaterals, and one inner triangle.  Show that the inner triangle has area equal to one seventh of the area of ABC.
• Question 2: Define a lattice point to be any point in the plane whose co-ordinates are both integers.  Let ABC be a triangle whose vertices are lattice points, but whose edges and interior do not contain any other lattice points.  Show that ABC has area exactly 1/2.

• Question 1: Let S be a collection of 16 distinct integers, each from 1 and 30 inclusive.  Show that there must exist two distinct elements in S which differ by exactly 3.
• Question 2: Let S be a collection of 16 distinct integers, each from 1 to 30 inclusive.  Show that there must exist two distinct elements in S which are coprime.  (Two numbers are coprime if they have no common factors other than 1).
• Question 3: Let S be a collection of 16 distinct integers, each from 1 to 30 inclusive.  Show that there must exist two distinct elements in S such that one divides the other.
• Question 4: Let a₁, a₂, ..., a₃₀ be a sequence of thirty integers.  Show that there must exist a consecutive subsequence a_i, a_i+1, ..., a_i+j of this sequence whose sum is divisible by 30.
• Question 5: Let 0 < a < 1 be a real number.  Show that there must exist a natural number n such that the fractional part of (n * sqrt(2)) lies between 0 and a.
• Question 6: Let 0 < a < b < 1 be real numbers.  Show that there must exist a natural number n such that the fractional part of (n * sqrt(2)) lies between a and b.

• Question 1: Find all real solutions to the equation 4x² - 40[x] + 51 = 0.   Here [x] denotes the greatest integer less than or equal to x, e.g. [-3.5] = -4.
• Question 2: Let ABC be an equilateral triangle of altitude 1.  A circle with radius 1 rolls along the segment AB, with the center of the circle always staying on the same side of AB as C.  Prove that the length of the arc of the circle which is inside the triangle ABC remains constant as the circle rolls along AB.
• Question 3:  Determine all positive integers n with the property that n = d(n)².  Here d(n) denotes the number of positive divisors of n.
• Question 4:  Suppose a₁, a₂, ..., a₈ are eight distinct integers, each from 1 to 17 inclusive.  Show that there is a positive integer k such that the equation a_i - a_j has at least three different solutions.  Also, show that the statement of this problem fails when "a₁, a₂, ..., a₈ are eight distinct integers" is replaced by "a₁, a₂, ..., a₇ are seven distinct integers".
• Question 5: Let x,y,z be non-negative real numbers such that x+y+z=1.  Show that x² y + y² z + z² x is less than or equal to 4/27.  When can x² y + y² z + z² x be exactly equal to 4/27?