## Bangladesh Booklet of Math Problems, 2015

booklet15

This is the booklet containing the problems used for training and selection purposes.

## Floor and Ceiling

Floor function of ${x}$ where ${x}$ is a real number, denoted by ${\lfloor x\rfloor}$ is the largest integer less than or equal to ${x}$. For example, ${\lfloor 3.1415\rfloor=3}$, ${\lfloor-2\rfloor=-2}$ and ${\lfloor5\rfloor=5}$. Is it true that, for integer ${x}$, we have ${\lfloor x\rfloor =x}$? Do you think ${\lfloor x\rfloor\leq x<\lfloor x\rfloor+1}$ is correct? You can proceed if your answers to both questions are yes. It is obvious that ${x\geq\lfloor x\rfloor}$, and we say that ${\{x\}=x-\lfloor x\rfloor}$ is the fractional part of ${x}$. For example, ${\{3.14\}=3.14-3=.14}$. Clearly, ${0\leq\{x\}<1}$.
Problem 1: Prove that for real ${x}$ and integer ${n}$, we have ${\lfloor x+n\rfloor=\lfloor x\rfloor+n}$.

Problem 2: Prove that for real ${x,y}$, we have ${\lfloor x+y\rfloor\geq\lfloor x\rfloor+\lfloor y\rfloor}$.

Solution 3: We can prove this easily this way. Write ${x=\lfloor x\rfloor+\{x\}}$ and ${y=\lfloor y\rfloor+\{y\}}$. $\displaystyle \begin{array}{rcl} \lfloor x+y\rfloor & = & \lfloor \lfloor x\rfloor+\{x\}+\lfloor y\rfloor+\{y\}\rfloor\\ & = & \lfloor x\rfloor+\lfloor y\rfloor+\lfloor\{x\}+\{y\}\rfloor\\ &\geq&\lfloor x\rfloor+\lfloor y\rfloor \end{array}$

because ${\lfloor x\rfloor+\lfloor y\rfloor}$ is an integer. See the second exercise in the link above. This inequality can be treated as the triangle inequality of floor function.

Now, we will discuss Ceiling function. Ceiling of a real number ${x}$, ${\lceil x\rceil}$ is the smallest integer greater than or equal to ${x}$. So, ${\lceil3.14\rceil=4}$ and ${\lceil4\rceil=4}$ as well. But ${\lceil-4.1\rceil=-4}$ whereas ${\lceil4.1\rceil=5}$. It is clear that if ${x}$ is an integer, then ${\lceil x\rceil=\lfloor x\rfloor=x}$. Which one is true?

1. ${\lceil x\rceil\geq\lfloor x\rfloor}$
2. ${\lceil x\rceil\leq\lfloor x\rfloor}$

The answer should be quite obvious! More precisely, if ${x}$ is not an integer then ${\lceil x\rceil=1+\lfloor x\rfloor}$. Make sense of the following identities: $\displaystyle \lfloor \lfloor x\rfloor\rfloor=\lfloor x\rfloor$ $\displaystyle \lceil \lceil x\rceil\rceil=\lceil x\rceil$ $\displaystyle \{\{x\}\}=\{x\}$

You should do the following exercises to make sure you understand these functions well.
Problem 4: Is it true that ${\lceil x+n\rceil=\lceil x\rceil+n}$ for an integer ${n}$? If so, why?

Problem 5: Is it true that ${\{ x+n\}=\{ x\}}$ for an integer ${n}$? If so, why?

Problem 6: Prove that, if ${x}$ is an integer, ${\lfloor x\rfloor+\lfloor -x\rfloor=0}$, otherwise ${-1}$.

Problem 7: If ${x}$ is an integer, ${\lceil x\rceil+\lceil -x\rceil=0}$, otherwise ${1}$.

Problem 8: If ${x}$ is an integer, ${\{ x\}+\{ -x\}=0}$, otherwise ${1}$.

## Euler Equation

Euler’s identity is regarded as the most beautiful equation in mathematics, since it combines five very important numbers ${e,\pi,i,0,1}$. The identity is: $\displaystyle \begin{array}{rcl} e^{i\pi} +1 & = & 0 \end{array}$

The identity is a special case of the following general one: $\displaystyle \begin{array}{rcl} e^{ix} & = & \cos x+i\sin x \end{array}$

It has many proofs. Here is an elegant one. Let ${y=\cos x+i\sin x}$. Then $\displaystyle \begin{array}{rcl} \dfrac{dy}{dx} & = &-\sin x+i\cos x\\ & = & i^2\sin x+i\cos x\\ & = & i(\cos x+i\sin x)\\ & = & iy\\ \dfrac{dy}y & = idx \end{array}$

Now let’s integrate both sides. As we know ${\int \frac{1}x=\ln x}$ and ${\int dx = x}$, we have $\displaystyle \begin{array}{rcl} \int \dfrac{dy}y & = & \int idx\\ \ln y & = & ix+C \end{array}$

for some constant ${C}$ (since it is indefinite integration). We get $\displaystyle \begin{array}{rcl} e^{ix+C} & = & y\\ & = & \cos x+i\sin x \end{array}$

We need to find the value of ${C}$, so let’s set ${x=0}$, and we get $\displaystyle \begin{array}{rcl} e^{C} & = & 1 \end{array}$

So ${C=0}$ and we have $\displaystyle \begin{array}{rcl} e^{ix} & = & \cos x+i\sin x \end{array}$

Set ${x=\pi}$ and we have $\displaystyle \begin{array}{rcl} e^{i\pi} +1 & = & 0 \end{array}$

DONE!