**Bangladesh National Mathematical Olympiad 2007 : Higher Secondary**

**Problem 1:**

In the figure $AB=8,\ BC=7$ and $CA=6.\ \Delta PAB$ is similar to $\Delta PCA$. What is $PC$?

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**Problem 2:**

$WZ$ is the diameter of circle with center $O$. $OY=5$, arc $XY$ creates angle $60^{\circ}$ at the center. If $\angle ZYO=60^{\circ}$, then $XY=?$.

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**Problem 3:**

In $\Delta ABC,\ \angle PAC=\angle PBC$. The perpendicular from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$ respectively. $D$ is the midpoint of $AB$. What is the value of $\frac {DL}{DM}$?

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**Problem 4:**

$(m,n)$ represents the largest common divisor of integers $m$ and $n$. For example $(2,3)=1$ and $(10,15)=5$. Suppose $n(n+1)(n+2)$ is a square, where $n$=integer.

a) What is $(n,n+1)$?

b) What is $(n+1,n+2)$?

c) What is $(n+1,n(n+2))$?

From your answers $a,b,c$ is it possible for $n(n+1)(n+2)$ to be a square?

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**Problem 5:**

If $x_1, x_2$ are the zeros of the polynomial $x^2-6x+1$, then prove that for every nonnegative integer $n$, $x_1^n+x_2^n$ is an integer and not divisible by $5$.

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**Problem 6:**

Writing down all the integers from $19$ to $92$ we make a large integer $N$.\[N=192021\cdots 909192\]If $N$ is divisible by $3^k$ then what is the maximum value of $k$?

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**Problem 7:**

$f(x)=x^6+x^5+\cdots +x+1$Find the remainder when dividing $f(x^7)$ by $f(x)$.

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**Problem 8:**

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 $a$.

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**Problem 9:**

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 $L$.

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**Problem 10:**

Find the area bounded by the curves $y=|x-1|$ and $x^2+y^2=2x$ (where $y\ge 0$)

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**Problem 11:**

Solve the inequality\[2\cos x \le |\sqrt {1+\sin 2x} -\sqrt {1- \sin 2x}|\]

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**Problem 12:**

Find the minima and maxima of $\left(\frac{\sin 10x}{\sin x}\right )^2$ in the interval $[0,\pi]$.

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**Problem 13:**

Sohag and Pias had some coconuts. They sold each coconut at the price which is equal to the number of the total coconuts. Sohag and Pias began to take $20$ Taka each alternately from the obtained money.

Sohag started the process. After a while he found that there was not enough money to take like before. Then he took the remaining money and to make the sharing fair he gave his pen to Pias. If they took $25$ Taka each alternately the situation would be almost same but in that case Sohag Had to give his pencil to Pias. If the pen costs $5$ Taka more than the pencil. What is the price of the pencil? [The costs of pen and pencil are integers]

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**Problem 14:**

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]

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