- a) Let $p$ and $q$ be different prime numbers, and let $a=p^{2}$, $b=p q$ and $c=p^{3} q$. Which of $a b$, $b c$ and $c a$ is a square number? Halle would like to place nine of the ten integers $1,2, \ldots, 9,10$ into the cells of a $3 \times 3$ grid in such a way that the three numbers within each row are written in increasing order from left to right and the product of the three numbers within each row is equal to a square number.
b) Give an example showing that Halle's task is possible.
c) How many different grids can Halle produce? - Twelve points, four of which are vertices, lie on the perimeter of a square. The distance between adjacent points is one unit. Some of the points have been connected by straight lines. $B$ is the intersection of two of those lines, as shown in the diagram.
a) Find the ratio $A B: B C$. Give your answer in its simplest form.
b) Find the area of the shaded region. - a) The expression $10 x y-2 x+5 y-1$ factorises as $(a x+b)(c y+d)$ where $a, b, c, d$ are integers and $a>0$. Find the values of $a, b, c, d$.
b) Using the factorisation above and considering factor pairs of $99$, find all integer pairs $(x, y)$ such that $10 x y-2 x+5 y=100$. Enter $y$ values only on the answer sheet.
c) Find all integer pairs $(x, y)$ such that $10 x y-2 x+5 y=100000001$. Enter $y$ values only on the answer sheet. - Daniel and Alessia each have eight boxes labelled $1$ to $8$. To start, each of them has $n$ balloons, all in their box $1$.
a) Alessia is playing a game with her balloons. She may make the following move: Choose a box $k$ containing at least two balloons. Pop a single balloon in box $k$ and then move another balloon from box $k$ to box $k+1$. Find the value of $n$ such that, after a finite number of moves, all that remains is a single balloon in box $8$.
b) Daniel is playing a different game with his balloons. He may use any combination of the following two moves- Pop two balloons in box $k$ and move a third balloon from box $k$ to box $k+1$.
- Pop a single balloon in each of boxes $k$ and $k+1$ and then move another balloon from box $k+1$ to box $k+2$.
- is a single balloon in box $4$?
- is a single balloon in box $8$?
- Freya creates a sequence with first term 1 and each subsequent term 5 more than the previous term. Hilary creates a different sequence with first term $a$ and each subsequent term $3$ less than the previous term. Both sequences are continued forever.
a) There is at least one number that appears in both sequences. Let $c$ be the smallest of those numbers. Find the possible values of $c$.
b) Given that there are exactly 100 numbers which appear in both sequences, find the possible values of $a$. Enter the following on the answer sheet - How many possible values of $a$ are there?
- The smallest possible value of $a$.
- The largest possible value of $a$.
[Solutions] United Kingdom Mathematical Olympiad For Girls 2020
British
Contest
EGMO
National
UK - Anh
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