Putnam Exam Practice Problems

The William Lowell Putnam Mathematical Competition, known as the Putnam Exam, is a prestigious annual mathematics competition for undergraduate students in the United States and Canada. It is designed to test originality, technical skill, and knowledge in solving mathematical problems. Here, we will delve into a selection of practice problems that emulate the style and difficulty of the Putnam Exam, focusing on various areas of mathematics. These problems are designed to challenge and stimulate mathematical thought, mirroring the exam’s intent to encourage and recognize excellence in mathematics.

Problem 1: Algebra and Number Theory

Let (n) be a positive integer. Prove that if (n) is not a perfect square, then the equation [ x^2 + y^2 + 1 = nx ] has no integer solutions for (x) and (y). If (n) is a perfect square, give an example of integer solutions for (x) and (y).

Solution 1

To begin solving this problem, we first rearrange the given equation to get it in terms of (x) and (y): [ x^2 - nx + y^2 + 1 = 0 ] For integer solutions to exist, the discriminant of this quadratic equation in terms of (x) must be a perfect square, since (x) is an integer. The discriminant is given by (D = n^2 - 4(1)(y^2 + 1)). Simplifying, we have (D = n^2 - 4y^2 - 4).

For (n) not a perfect square, let’s assume there are integer solutions. Then (n^2 - 4y^2 - 4 = k^2) for some integer (k), implying (n^2 - k^2 = 4y^2 + 4). Factoring the left side gives us ((n+k)(n-k) = 4(y^2 + 1)). Since (y^2 + 1) is always odd (because (y^2) is either 0 or 1 modulo 2), the right-hand side is (4 \times (\text{an odd number})), which is a multiple of 4 but not of 8.

However, for the left-hand side to be a multiple of 4 but not of 8, ((n+k)) and ((n-k)) must both be even, which implies (n) and (k) are either both even or both odd. But in either case, (n^2 - k^2) would be odd, contradicting our premise. Therefore, if (n) is not a perfect square, there are no integer solutions.

If (n) is a perfect square, let’s consider (n = 4), which gives us (x^2 - 4x + y^2 + 1 = 0). By inspection, (x = 2), (y = 1) satisfies this equation, demonstrating an example of integer solutions when (n) is a perfect square.

Problem 2: Geometry and Trigonometry

A circle with center (O) and radius (r) is tangent to two parallel lines, (L_1) and (L_2), which are (2r) apart. Points (A) and (B) are on (L_1) and (L_2), respectively, such that (OA = r\sqrt{2}) and (OB = r\sqrt{5}). Find the distance between (A) and (B).

Solution 2

Given the geometry of the situation, let’s denote the points where the circle is tangent to (L_1) and (L_2) as (C) and (D), respectively. We know (OC = OD = r) because these are radii of the circle. The distance between (L_1) and (L_2) is given as (2r), which also means (CD = 2r).

For point (A) on (L_1), (OA = r\sqrt{2}), implying (AC = r) since (OA^2 = OC^2 + AC^2) by the Pythagorean theorem, and (OC = r).

For point (B) on (L_2), (OB = r\sqrt{5}), which implies (BD = 2r) since (OB^2 = OD^2 + BD^2), and (OD = r).

The distance (AB) can be found by noticing that (AB^2 = (AC + CD + BD)^2 = (r + 2r + 2r)^2 = 25r^2), thus (AB = 5r).

Problem 3: Real and Complex Analysis

Let (f(z)) be an entire function (analytic everywhere in the complex plane) such that (|f(z)| \leq |z|^2 + 1) for all (z). Prove that (f(z)) is a polynomial of degree at most 2.

Solution 3

By Cauchy’s Integral Formula for derivatives, we know that for an entire function (f(z)), its (n)th derivative (f^{(n)}(z)) can be expressed as [ f^{(n)}(z) = \frac{n!}{2\pi i} \int_{C} \frac{f(\zeta)}{(\zeta - z)^{n+1}} d\zeta ] where (C) is a simple closed curve enclosing (z).

Given (|f(z)| \leq |z|^2 + 1), we can choose (C) to be a circle of radius (R) centered at (z). Then for (\zeta) on (C), (|\zeta - z| = R), and we have [ |f^{(n)}(z)| \leq \frac{n!}{2\pi} \int{0}^{2\pi} \frac{|f(z + Re^{i\theta})|}{R^{n+1}} R d\theta ] Given (|f(z)| \leq |z|^2 + 1), for (z = 0) and (R > 1), (|f(Re^{i\theta})| \leq R^2 + 1), we get [ |f^{(n)}(0)| \leq \frac{n!}{2\pi} \int{0}^{2\pi} \frac{R^2 + 1}{R^n} d\theta = \frac{n!}{R^{n-2}} ] For (n > 2), as (R \rightarrow \infty), the right-hand side goes to 0, implying (f^{(n)}(0) = 0) for (n > 2). This means all derivatives of order higher than 2 are zero, implying (f(z)) is a polynomial of degree at most 2.

Conclusion

These problems, ranging from algebra and number theory to geometry and complex analysis, are illustrative of the breadth and depth of mathematical concepts tested in the Putnam Exam. Solving such problems not only enhances mathematical knowledge but also develops critical thinking, creativity, and analytical skills. These attributes are invaluable for any student of mathematics, providing a foundation for further exploration and advancement in the field.

To delve deeper into practice problems similar to those presented in the Putnam Exam, it’s essential to explore resources that offer a wide range of mathematical challenges. This includes textbooks, online forums, and academic journals focused on mathematics. Engaging with these resources and consistently practicing problem-solving will help in developing the skills and strategies necessary to tackle complex mathematical problems.

Moreover, understanding the underlying concepts and theories in mathematics is crucial. This involves not just memorizing formulas and theorems but also grasping the logical flow and the reasoning behind them. The ability to apply mathematical principles to solve real-world problems is a valuable skill that extends beyond the realm of academic competitions, into the broader applications of mathematics in science, technology, engineering, and mathematics (STEM) fields.

In conclusion, the journey to mastering mathematical problem-solving, as exemplified by the challenges of the Putnam Exam, is a rewarding and enriching experience. It demands dedication, persistence, and a genuine passion for mathematics. Through continuous learning, practice, and the application of mathematical concepts to real-world scenarios, individuals can cultivate a deep appreciation for the beauty and utility of mathematics, ultimately contributing to advancements in various fields of study.

By engaging with the mathematical community, participating in competitions, and continuously challenging oneself with complex problems, individuals can cultivate a deeper understanding and appreciation of mathematics. The pursuit of mathematical excellence, as embodied by the Putnam Exam and similar challenges, is a lifelong journey that offers immense personal and professional rewards.