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!