p1. Let $AB$ be the diameter of the circle and $P$ is a point outside the circle. The lines $PQ$ and $PR$ are tangent to the circles at points $Q$ and $R$. The lines $PH$ is perpendicular on line $AB$ at $H$ . Line $PH$ intersects $AR$ at $S$. If $\angle QPH =40^o$ and $\angle QSA =30^o$, find $\angle RPS$. p2. There is a meeting consisting of $40$ seats attended by $16$ invited guests. If each invited guest must be limited to at least $ 1$ chair, then determine the number of arrangements. p3. In the crossword puzzle, in the following crossword puzzle, each box can only be filled with numbers from $ 1$ to $9$. Across: 1. Composite factor of $1001$ 3. Non-polyndromic numbers 5. $p\times q^3$, with $p\ne q$ and $p,q$ primes Down: 1. $a-1$ and $b+1$ , $a\ne b$ and $p,q$ primes 2. multiple of $9$ 4. $p^3 \times q$, with $p\ne q$ and $p,q$ primes p4. Given a function $f:R \to R$ and a function $g:R \to R$, so that it fulfills the following figure: Find the number of values of $x$, such that $(f(x))^2-2g(x)-x \in\{-10,-9,-8,…,9,10\}$. p5. In a garden that is rectangular in shape, there is a watchtower in each corner and in the garden there is a monitoring tower. Small areas will be made in the shape of a triangle so that the corner points are towers (free of monitoring and/or supervisory towers). Let $k(m,n)$ be the number of small areas created if there are $m$ control towers and $n$ monitoring towers. a. Find the values of $k(4,1)$, $k(4,2)$, $k(4,3)$, and $k(4,4)$ b. Find the general formula $k(m,n)$ with $m$ and $n$ natural numbers .