Test Report: Math for Olympiad from MUMS 5

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These collection of math questions were collected from Melbourne University Mathematics & Statistics Society. 
http://ms.unimelb.au.edu/~mums/

1.  The numbers 1,2,…,10 are placed in order so that each number is either strictly greater than all the preceding numbers or else strictly less than all preceding numbers. In how many ways can this be done?

A. 123

B. 543

C. 512

D. 681

E. 340

2.  The points A,B,C, and D lie on a circle with centre O and radius r. AB = 6, CD = 8 and BC = DA = . Find r.

A. 8

B. 6

C. 90

D. 5

E. 15

3.  The sum of the cubes of two positive integers is 6293. What is the sum of their squares?

A. 129

B. 345

C. 211

D. 425

E. 143

4.  Three jolly professors, Tim, Swarup and Kris, are gambling by a billabong. They start with sums of money in the ratio 7 : 6 : 5 and finish with sums of money in the ratio 6 : 5 : 4, in the same order of wealth. One of the professors won $12. How many dollars did he start with?

A. $240

B. $420

C. $440

D. $340

E. $320

5.  Two amateurs were allowed to enter a chess tournament otherwise comprising professionals. Each contestant played once with every other contestant and received one point for a win, half a point for a draw, and zero points for a loss. The two amateurs together gained a total of eight points and all professionals scored the same number of points as each other. Determine the maximum possible number of professionals in the tournament.

A. 14

B. 24

C. 32

D. 18

E. 26

6.  Two different numbers are randomly chosen from the set {1,2,…,12}. Determine the probability that one of the numbers is a multiple of the other. (Write your answer in the form a / b, where a and b have no common factors).

7.  What is the area of the smallest circle in which you can fit six equilateral triangles of area 3 without overlap? (Express your answer exactly).

8.  What is the sum of the first five square numbers ending in a 1?

A. 1005

B. 1013

C. 1007

D. 1050

E. 1070

9.  What is the surface area, in square centimetres, of a cube having volume 343 cm³?

A. 264 cm²

B. 194 cm²

C. 294 cm²

D. 270 cm²

E. 266 cm²

10.  You are asked to form a committee of six people from a group of 10 couples. There are five 'happy' couples and five 'grumpy' couples. A member of a 'happy' couple will only serve on the committee if their beloved is also on the committee, while a member of a 'grumpy' couple refuses outright to work on the committee if their partner is on the committee. In how many ways can you form the committee?

A. 810

B. 812

C. 800

D. 814

E. 780

11.  You are given a 5 × 5 grid and you colour in some of its squares. A colouring is called permissible if for any coloured-in square, either its whole column is coloured in or its whole row is coloured in. How many permissible colourings are there?

A. 962

B. 964

C. 966

D. 968

E. 970

1.    C

2.    D

OA = OB = OC = OD = r. By the converse of Pythagoras, ÐAOD = ÐBOC = 90 ^o. Notice that we can rearrange the four triangles ABO,BCO,CDO,DAO to form a new

quadrilateral WXY Z inside the same circle of radius r with centre O where WX = XY =  and YZ = 6 and ZW = 8. Since ÐWOY = 180^o, WY = 2r is a diameter of the circle

and so ÐWXY = ÐWZY = 90 ^o. Then by Pythagoras in ∆WY Z, 2r = WY = 10 and so r = 5.

3.    D

4.    B

Suppose that the total amount of money of all the professors in dollars is M. Since they started with sums of money in the ratio 7 : 6 : 5, they each started with ,  and , respectively. Since they ended with sums of money in the ratio

6 : 5 : 4, they each ended with ,  and , respectively. Thus, the first professor won , the second professor won and the third professor lost . We are told that one of the professors won $12, so M = 90 x $12 = $1080.

Hence, he started with .

5.    A

6.    A

7.    A

If we inscribe a regular hexagon in a circle and draw its diagonals through the centre, we can see that in a circle of radius r, we can exactly fit six equilateral triangles of side length r. For an equilateral triangle to have area 3, we require the following equation to hold:

Thus the area of the required circle is .

8.    A

Note that for a square number to end in a 1, its square root must end in a 1 or a 9. So the sum of the first five square numbers ending in a 1 is:

1² + 9² + 11² + 19² + 21² = 1 + 81 + 121 + 361 + 441 = 1005

9.    C

Since the cube has a volume of 343 cm³, the side length of the cube must be . Thus, the area of one square face of the cube is 7 cm x 7 cm = 49 cm².

The surface area of the cube is composed of six of these square faces and is 6 x 49 cm² = 294 cm².

10.  A

The number of happy people on the committee must be even; note that 0 is not an option

because 6 grumpy people together is impossible with only 5 grumpy couples, so there must be either 2, 4 or 6 happy people on the committee.

2 happy people: select 1 happy couple from 5, 4 grumpy couples from 5, 1 grumpy person from each included grumpy couple, so 5 × 5 × 24 = 400 possibilities.

4 happy people: select 2 happy couples from 5, 2 grumpy couples from 5, 1 grumpy person from each included grumpy couple, so 10 × 10 × 22 = 400 possibilities.

6 happy people: select 3 happy couples from 5, so 10 possibilities.

Adding them up gives a total of 810 possible committees.

11.  A

Since each coloured square must have a coloured column or row, the problem is equivalent to finding the number of different colourings feasible by colouring when one can only colour in an entire row or an entire column at a time. The number of different colourings of rows is 2⁵ = 32, since each row can be either coloured or not coloured. If all the rows are coloured, then any colouring of the columns will make no difference, so there is only one possibility here. For the remaining 31 cases, all possible colourings of columns will be unique except again for the case when all columns are coloured. Thus, there are 1 + 31² = 962 possible colourings.