**Problem 4:**

(a) Can two consecutive numbers $n$ and $n-1$ both be divisible by $3$?

(b) Determine the smallest integer $n > 1$ such that $n^2(n-1)$ is divisible by $1971$. Note:

$1971 = 3^3 \times 73$.

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**Problem 8:**

In $\bigtriangleup ABC$, the perpendicular bisector of $AB$ and $AC$ meet at $O$. $AO$ meets $BC$ at $D$. Now, $OD$ = $BD$ = $\dfrac {1}{3}BC$. Find the angles of $\bigtriangleup ABC$.

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**Problem 9:**

$ABCD$ is a square. Circle with diameter $AB$ and circle with center $C$ and radius $CB$ meet inside the square at $P$. Prove that $DP = \sqrt2 AP$.

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**Problem 10:**

Whenever Avik gets a sequence, he multiplies every two distinct terms of that sequence, and then sums up these products to get the Hocus-pocus sum of the sequence. For example, the Hocus-pocus sum for the sequence $a, b, c, d$ is $ab + bc + ac + ad + bd + cd$. If Avik gets a sequence of $100$ terms, where each term is either $2$ or $-1$, what is the minimum Hocus-pocus sum of that sequence?

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