**Bangladesh National Mathematical Olympiad 2007 : Secondary**

**Problem 1**

Solve for ($x,y$) in real number where $x^x=y$ and $y^y=y.$

viewtopic.php?f=13&t=4186

**Problem 2**

Writing down all the integers from $1$ to $100$ we make a large integer $N$.

$N=123456789101112...9899100$.

What will be the remainder if we divide $N$ by $3$?

viewtopic.php?f=13&t=5616

**Problem 3**

A square has sides of length $2$. Let $S$ is the set of all line segments that have length $2$ and whose endpoints are on adjacent side of the square. Say $L$ is the set of the midpoints of all segments in $S$. Find out the area enclosed by $L$.

viewtopic.php?f=13&t=592

**Problem 4**

Two parallel chords of a circle have length $10$ and $14$. The distance between them is $6$. The chord parallel to these chords and half way between them has length $\sqrt a$. Find $a$.

viewtopic.php?f=13&t=591

**Problem 5**

A ball is thrown upward vertically to a height of $650$ meters from ground.Each $2$ times it hits the ground it bounces $\dfrac 25$ of the height it fell in the previous stage.How much the ball will travel before it stops?

viewtopic.php?f=13&t=5617

**Problem 6**

What is the area bounded by the region $|x+y|+|x-y|=4$ ?Where $x$ and $y$ are real numbers.

viewtopic.php?f=13&t=5614

**Problem 7**

Find the smallest positive integer $n>1$, such that $\sqrt{1+2+3+...+n}$ is an integer.$(n<10)$.

viewtopic.php?f=13&t=5613

**Problem 8**

If $m +12 = p^a$ and $m -12 = p^b$ where $a,b,m$ are integers and $p$ is a prime number. Find all possible primes $p > 0$ . [Note: $p$ only takes three values]

viewtopic.php?f=13&t=597

**Problem 9**

If $x^2+3x-4$ is a factor of $x^3+bx^2+cx+11$,then find the values of $b$ and $c$.

viewtopic.php?f=13&t=5615

**Problem 10**

A drawer in a darkened room contains $100$ black socks, $80$ blue socks, $60$ red socks and $40$ purple socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? [A pair of socks = $2$ socks of the same color.]

viewtopic.php?f=13&t=5610

**Problem 11**

Find the area of the largest square inscribed in a triangle of sides $5,6$ and $7$.

viewtopic.php?f=13&t=5611

**Problem 12**

Find the remainder on dividing ($x^{100}-2x^{51}+1$) by ($x^2-1$)?

viewtopic.php?f=13&t=4187

**Problem 13**

Prove that if $a$ and $b$ are two integers,then $a\times b=LCM(a,b)\times GCD(a,b)$ .

viewtopic.php?f=13&t=5612