**Bangladesh National Mathematical Olympiad 2007 :Junior**

**Problem 1 :**

If $\dfrac {10}{2}=4$,then $5\times 2=?$

[The base of a number may not be $10$ ]

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**Problem 2 :**

In this figure there is a square inscribed in another square.Using this figure derive the Pythagorean Theorem.

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**Problem 3:**

A man has $4$ childern.The age of the first child is a square number.By multiplying the digits of this square number you will get the age of second child and by summing up the digits you will get the age of third child.If you add the digits of the age of the second child,you will get the age of fourth child.If the difference of age of two consequetive childern is not more than $25$ years,then find the ages if $4$ children.

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**Problem 4:**

If $log_{(x+3)}(x^2+15)=2$ then $x=?$

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**Problem 5:**

Find the difference between the non shaded area ($ABDF$ & $FCE$) of thre triangles.Instead of triangles,if there were circles or any other shape,what would be the result -comment on that.

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**Problem 6:**

Mathematics,English and Bangla classes started on the very first day of a month.Mathematics class schedule is $1,3,5,7,9,...$ the schedule for English is $1,4,7,10,13,...$.and for Bangla it is $1,5,9,13,17...$ .In next $3$ months how many time you have to attend all three of these classes at same day?Suppose all the months are of $30$ days.

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**Problem 7:**

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?

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**Problem 8:**

Draw a square which has area that is three times the area of another given square.

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**Problem 9:**

$\sqrt {-1}$ is called the imaginary number '$i$'.Using this can you find the value of $\dfrac {1+i}{1-i}$?

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**Problem 10:**

Find the sum of first $20$ terms of the series $4+7+13+25+...$

What is the sum of first $n$ terms?

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**Problem 11:**

If $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.

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**Problem 12:**

Two circles of equal radius intersect each other at point $C$ and $D$.The centers of the two circles are poont $A$ and $B$ respectively.If their radius is $10$ and area of $ABC$ is $40$.

Then find the distance $x$ between $A$ and $B$. viewtopic.php?f=13&t=5649

**Problem 13:**

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.]

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