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**Need Help 12 Difficult Math questions i need answered by next friday!!!**

Problem 1:

The cockle shells that grow in Mary's garden need exactly ten litres of water every day and they can be watered only once a day. She has two jugs of nine litres and eleven litres capacity respectively and a pool full of water. Can she water her cockle shells as required?

Problem 2:

The sides BC and AD of a quadrilateral ABCD are parallel. M is the midpoint of AB. The area of ABCD is S. Find the area of the triangle DMD in terms of S.

Problem 3:

In triangle ABC, X is a point on AC such that AX = 15m. XC = 5m, angle AXB = 60 degrees, and angle ABC is two times bigger than angle AXB. Find the length of BC.

Problem 4:

Find all four-digit numbers which are a perfect square and in which the first two digits are equal and the last two digits are also equal.

Problem 5:

The diagonals AC and BD of a quadrilateral ABCD are perpendicular. Find the length of AB if BC = 5cm, DC = 4cm, and AD = 3cm.

Problem 6:

A triangle has the following properties:

It is scalene;

It does not contain a right angle;

It has integer length sides;

Its area is an integer.

Find the triangle with these properties which has the least perimeter. (Use Heron's Formula for the area of a triangle with sides a, b and c: Area = square root (s(s - a)(s - b)(s - c), where s is the semi-perimiter, i.e. s = perimeter divided by 2.

Problem 7:

Mr A decided to invest $530. He bought shares in four different companies, each share worth $18, 23$, 52$, and 69$ respectively. He spent exactly the sum he intended and bought 20 shares altogether. How many shares of each value did he buy?

Problem 8:

At a parade, 200 students are arranged so that they form 10 rows and 20 columns. When the tallest student is chosen in each row, Andrew is the shortest of them. When the shortest student in each column is chosen, Bruce is the tallest of them. Show that Bruce cannot be taller than Andrew.

Problem 9: A school is planning to form a School Council. The Council will consist of four groups of students from years 9, 10, 11 and 12 respectively. Each group will comprise not less than two students from the same year and the total number of students in the council to be 9. In how many ways can the School council be formed if there are three, four, five and six candidates from years9, 10, 11 and 12 respectively?

Problem 10:

Show that 27 X 23^n + 12 X 10^2n is divisible by 11 for all positive integers n.

Problem 11:

After a week of hard calculations John figured out 3^10000.

Then he added up all its digits and thus obtained a new number. Next he added up all the digits of this new number and obtained another number. He continued doing this. Eventually, he obtained a one-digit number. What was that number?

Problem 12:

A group of tourists were offered seats in a number of buses so that there were the same number of tourists in each bus. First the organisers tried to seat 22 tourists in each bus. But it turned out that one of the tourists was left unseated. Then one of the buses went empty and the tourists occupied seats in the remaining buses so that there were the same number of tourists in each of the remaining buses. Find the original number of buses and number of tourists if each bus cannot carry more than 44 people.