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 $x$.

There are $n$ cities in a country. Between any two cities there is at most one road. Suppose that the total number of roads is $n.$ Prove that there is a city such that starting from there it is possible to come back to it without ever travelling the same road twice.

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.

$\triangle ABC$ is an acute angled triangle. Perpendiculars drawn from its vertices on the opposite sides are $AD$, $BE$ and $CF$. The line parallel to $ DF$ through $E$ meets $BC$ at $Y$ and $BA$ at $X$. $DF$ and $CA$ meet at $Z$. Circumcircle of $XYZ$ meets $AC$ at $S$. Given, $\angle B=33 ^\circ.$ find the angle $\angle FSD $ with proof.

Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended $AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y. AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\angle XOP.$

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 $m-1$.