BD-BDMO 2013

Higher Secondary P1

A polygon is called degenerate if one of its vertices falls on a line that joins its neighboring two vertices. In a pentagon $ABCDE$ , $AB=AE$ , $BC=DE$ , $P$ and $Q$ are midpoints of $AE$ and $AB$ respectively. $PQ||CD$ , $BD$ is perpendicular to both $AB$ and $DE$ . Prove that $ABCDE$ is a degenerate pentagon.

Higher Secondary P2

Let $g$ be a function from the set of ordered pairs of real numbers to the same set such that $g(x, y)=-g(y, x)$ for all real numbers $x$ and $y$ . Find a real number $r$ such that $g(x, x)=r$ for all real numbers $x$ .

Higher Secondary P3

Let $ABCDEF$ be a regular hexagon with $AB=7$ . $M$ is the midpoint of $DE$ . $AC$ and $BF$ intersect at $P$ , $AC$ and $BM$ intersect at $Q$ , $AM$ and $BF$ intersect at $R$ . Find the value of $[APB]+[BQC]+[ARF]-[PQMR]$ . Here $[X]$ denotes the area of polygon $X$ .

Higher Secondary P4

If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$ , find $\dfrac{a}{b}$ .

Higher Secondary P5

Let $x>1$ be an integer such that for any two positive integers $a$ and $b$ , if $x$ divides $ab$ then $x$ either divides $a$ or divides $b$ . Find with proof the number of positive integers that divide $x$ .

Higher Secondary P7

If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$ , then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.

Higher Secondary P10 $X$ is a set of $n$ elements. $P_m(X)$ is the set of all $m$ element subsets (i.e. subsets that contain exactly $m$ elements) of $X$ . Suppose $P_m(X)$ has $k$ elements. Prove that the elements of $P_m(X)$ can be ordered in a sequence $A_1, A_2,...A_i,...A_k$ such that it satisfies the two conditions: (A) each element of $P_m(X)$ occurs exactly once in the sequence, (B) for any $i$ such that $0<i<k$ , the size of the set $A_i \cap A_{i+1}$ is $m-1$ .