Problem List
These are the problems posted on the year 2016. Click to view!

01: Inequality with Tangent  Prove the following inequality for all \(n \in \mathbb{N}\). \[\Large \int_{0}^{\frac{\pi}{2}} \tan^{\sqrt{n}}{x}dx \geq \frac{\pi}{2}\]

02: An Improper Integral  Find the conditions for \(a, b \in \mathbb{R}\) so that the following improper integral converges. \[\Large \int_{0}^\infty \frac{x^a}{1+x^b} dx\]

03: \(\pi\) and Infinite Series  Let \(a_n\) be the \(n\)-th digit below the decimal point of \(\pi\). Do the following series converge?

 (a) \(\Large \sum_{n=1}^{\infty} \frac{a_n}{10}\)

 (b) \(\Large \sum_{n=1}^{\infty} \frac{a_n}{10^n}\)

 (c) \(\Large \sum_{n=1}^{\infty} \frac{a_n}{9^n}\)

04: Different Quadratics  Suppose non-negative integers \(a, b, c\) satisfy \(a+b+c=10\).
 How many different quadratic equations \(ax^2+bx+c=0\) are there?

05: Inequality (1)  For positive real numbers \(a, b, c, d\), prove that \[\Large \frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a} \geq 4\]

06: Sum of Divisors  We start with \(n = 2\). Two players A and B will take turns and add a divisor of \(n\) except \(n\) itself. The game starts with A, and the player who adds a divisor to result in a number greater than \(1990\) wins.
 Who is the winner?

07: System of Trigonometric Equations  Find all solutions \(\left(x, y, z\right)\) to the following system. \[\Large \cos{x} + \cos{y} + \cos{z} = \frac{3\sqrt{3}}{2}\] \[\Large \sin{x} + \sin{y} + \sin{z} = \frac{3}{2}\]

08: A Real-Valued Function  For \(x, y \in \mathbb{R}\), is there a real-valued function \(f(x)\) and \(g(y)\) which satisfies the following equation? \[\Large x^2+xy+y^2=f(x)+g(y)\]  Find the functions, if it exists.

09: Real Solutions of Equation  Find all real solutions to the following equations. \[\Large x+y+z=0, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\]

10: Kempner Series  Show that the Harmonic Series \(\sum_{n=1}^{\infty} \frac{1}{n}\) diverges.
 Now we will modify the series by omitting all terms whose denominator (expressed in base \(10\)) contains a sequence of digits. (ex: \(12\), \(354\) etc.)
 Show that the modified series converges.
 This is known as the Kempner Series.

11: Product of Tangents  Evaluate the following expression. \[\Large \prod_{n=1}^{45}\left(1+\tan{n^{\circ}}\right)\]

12: Factorial and Perfect Squares  Find all \(n\in \mathbb{N}\) such that \[\Large 1!+2!+\cdots+n! = \sum_{k=1}^n k!\] is a perfect square of an integer.

13: Wallis Product  (1) For integers \(n \geq 2\), show that \[\large \int_0^{\frac{\pi}{2}} \sin^{n}{x}dx = \frac{n-1}{n} \int_0^{\frac{\pi}{2}} \sin^{n-2}{x}dx\]  (2) Evaluate \[\large \int_0^{\frac{\pi}{2}} \sin^{3}{x}dx, \int_0^{\frac{\pi}{2}} \sin^{5}{x}dx\]  (3) Show that \[\large \int_0^{\frac{\pi}{2}} \sin^{2n+1}{x}dx = \frac{2\cdot 4\cdot 6\cdot \cdots \cdot (2n)}{3\cdot 5\cdot 7\cdot \cdots \cdot (2n+1)}\] \[\large \int_0^{\frac{\pi}{2}} \sin^{2n}{x}dx = \frac{1\cdot 3\cdot 5\cdot \cdots \cdot (2n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot (2n)} \cdot \frac{\pi}{2}\]
 Now let \(I_n=\int_0^{\frac{\pi}{2}} \sin^{n}{x}dx\).

 (4) Show that \[\large I_{2n+2} \leq I_{2n+1} \leq I_{2n}\] and \[\large \lim_{n \rightarrow \infty} \frac{I_{2n+1}}{I_{2n}} = 1\]  (5) (Wallis Product) Now show that \[\large \prod_{n=1}^{\infty} \left(\frac{2n}{2n-1}\cdot\frac{2n}{2n+1}\right)=\lim_{n \rightarrow \infty} \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5}\cdot\frac{6}{5} \cdot \frac{6}{7} \cdot \cdots \cdot \frac{2n}{2n-1} \cdot \frac{2n}{2n+1} = \frac{\pi}{2}\]

14: A Differentiable Function  For a function \(f(x)\) with the whole real numbers as its domian, suppose that \[\large \forall x, y \in \mathbb{R}, f(x+y)\geq f(x)+f(y)+\left(\tan{x}\tan{y}-1\right)^2\] and \(f(0)\geq 1, f'(0)=1\).
 Find \(f'(3\pi)\).

15: \(f(f(x))=x\)  For a differentiable function \(f: \left[0, 1\right] \rightarrow \mathbb{R}\) suppose that \[\large f(f(x))=x, f(0)=1\] Evaluate \[\Large \int_0^1 \left(x-f(x)\right)^{2016}dx\]