Calculate

$$
\frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\ldots+\frac{1}{2014 \times 2015}
$$

## Solutions to the exercises
For this exercise, it is useful to remember that $\frac{1}{(k-1) k}=$ $\frac{1}{k-1}-\frac{1}{k}$. Thus

$$
\begin{aligned}
\frac{1}{1 \times 2} & =1-\frac{1}{2} \\
\frac{1}{2 \times 3} & =\frac{1}{2}-\frac{1}{3} \\
\frac{1}{3 \times 4} & =\frac{1}{3}-\frac{1}{4} \\
& \vdots \\
\frac{1}{2013 \times 2014} & =\frac{1}{2013}-\frac{1}{2014} \\
\frac{1}{2014 \times 2015} & =\frac{1}{2014}-\frac{1}{2015}
\end{aligned}
$$

We notice that, when we sum everything, almost everything cancels out and we are left with $1-\frac{1}{2015}=\frac{2014}{2015}$.

## 3 Thursday afternoon, September 21: Logic, Joon Kwon

This logic course for group A repeats the one given to group B on the morning of August 19. See Chapter III, Section 2, Subsection 1 of this course material.

## 2 Group B: Algebra

## 1 Thursday morning, September 21: Jean-Louis Tu

## Polynomials

## Notations and Basic Definitions

A polynomial is an expression of the form $a_{0}+a_{1} X+\cdots+a_{n} X^{n}$ for some integer $n \in \mathbb{N}$ and certain numbers $a_{0}, \ldots, a_{n}$, called the coefficients.

The largest integer $m$ such that $a_{m} \neq 0$ is called the degree of the polynomial. The coefficient $a_{m}$ is called the leading coefficient. A polynomial is said to be monic if its leading coefficient is equal to 1.

To every polynomial is associated a polynomial function $f: x \mapsto a_{0}+a_{1} x+$ $\cdots+a_{n} x^{n}$. A real number $x$ such that $f(x)=0$ is called a root of the polynomial.

Let $P, Q$ be two polynomials. We say that $Q$ divides $P$ if there exists a polynomial $R$ such that $P=Q R$.

## The Quadratic Equation

Consider an equation of the form $a x^{2}+b x+c=0$ in the unknown $x$, with $a, b, c$ fixed and $a \neq 0$. Since

$$
\begin{aligned}
a x^{2}+b x+c & =a\left(x^{2}+\frac{b}{a} x+\frac{c}{a}\right) \\
& =a\left(x^{2}+2 \frac{b}{2 a} x+\left(\frac{b}{2 a}\right)^{2}-\left(\frac{b}{2 a}\right)^{2}+\frac{c}{a}\right) \\
& =a\left(\left(x+\frac{b}{2 a}\right)^{2}-\frac{b^{2}-4 a c}{a^{2}}\right)
\end{aligned}
$$

we are led to define

$$
\Delta=b^{2}-4 a c
$$

We call $\Delta$ the discriminant. The equation $a x^{2}+b x+c=0$ is equivalent to $\left(x+\frac{b}{2 a}\right)^{2}=$ $\frac{\Delta}{a^{2}}$.

1) If $\Delta>0$, then $a x^{2}+b x+c=0$ is equivalent to $x+\frac{b}{2 a}= \pm \frac{\sqrt{\Delta}}{a}$, which simplifies to

$$
x=\frac{-b \pm \sqrt{\Delta}}{2 a}
$$

2) If $\Delta=0$, the equation has one and only one solution $x=-\frac{b}{2 a}$.
3) If $\Delta<0$, the equation has no solution.

The graph of a quadratic polynomial is a parabola, oriented upwards or downwards depending on the sign of $a$.

The sign of $\Delta$ is related to the number of intersection points of the parabola with the x-axis.