GULF 2019

Let $ABCD$ be a trapezium (trapezoid) with $AD$ parallel to $BC$ and $J$ be the intersection of the diagonals $AC$ and $BD$ . Point $P$ a chosen on the side $BC$ such that the distance from $C$ to the line $AP$ is equal to the distance from $B$ to the line $DP$ .

The following three questions 1, 2 and 3 are independent, so that a condition in one question does not apply in another question.

1.Suppose that $Area( \vartriangle AJB) =6$ and that $Area(\vartriangle BJC) = 9$ . Determine $Area(\vartriangle APD)$ . 2. Find all points $Q$ on the plane of the trapezium such that $Area(\vartriangle AQB) = Area(\vartriangle DQC)$ . 3. Prove that $PJ$ is the angle bisector of $\angle APD$ .

1. Find $N$ , the smallest positive multiple of $45$ such that all of its digits are either $7$ or $0$ .
2. Find $M$ , the smallest positive multiple of $32$ such that all of its digits are either $6$ or $1$ .
3. How many elements of the set $\{1,2,3,...,1441\}$ have a positive multiple such that all of its digits are either $5$ or $2$ ?

Consider the set $S = \{1,2,3, ...,1441\}$ .

1. Nora counts thoses subsets of $S$ having exactly two elements, tbe sum of which is even. Rania counts those subsets of $S$ having exactly two elements, the sum of which is odd. Determine the numbers counted by Nora and Rania.
2. Let $t$ be the number of subsets of $S$ which have at least two elements and the product of the elements is even. Determine the greatest power of $2$ which divides $t$ .
3. Ahmad counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is even. Bushra counts the subsets of $S$ having $77$ elements such that in each subset the sum of the elements is odd. Whose number is bigger? Determine the difference between the numbers found by Ahmad and Bushra.

Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$ , and for $n \ge 7$ : $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$ . For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$ .

For all integers $n \ge 2$ , let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$ . The minus $-$ denotes the ''remove it if it is there'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$ . However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$ .

1. Determine $S_7,S_8,S_9$ and $S_{10}$ .
2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$ . Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced .
3. Is the set $S_{2019}$ well-balanced? Justify your answer.