Test Report: Math for Olympiad from MUMS 2

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

1.  Circles C₁ and C₂ intersect at points A and B. They have common tangents CD and EF, where C and E lie on C₁, and D and F lie on C₂. The line DA passes through the midpoint of segment CE. If CE = 7 and AB = 12, find the ratio of the radius of C₁ to the radius of C₂. Express your answer in simplest form.

A. 7:42

B. 7:24

C. 7:26

D. 7:62

E. 7:48

2.  Daniel and Joanna, whilst journeying through outer space, came across a spaceship wreck, and with it an injured yet amiable alien who had lost his fingers. The pair of travellers were keen to find the alien's fingers for him, but did not know how many to look for. Just prior to crashing his spaceship, the alien was, naturally, doing some quick calculations on a piece of paper which Daniel rescued from the wreckage. It read:

5x² – 50x + 125 = 0

\x = 5 or x = 8

Joanna quickly realized that the number base in which the alien was working was probably equal to the number of fingers he ought to have. How many alien-fingers were Joanna and Daniel looking for?

A. 11

B. 12

C. 13

D. 14

E. 15

3.  Express   in simplest form.

A. 8

B. 7

C. 11

D. 13

E. 14

4.  Express the value of this sum in simplest form:

(where  is the greatest integer less than or equal to x)

5.  Find all y satisfying 'y + 1' + 2' y – 2' < 6

A. 1 < y < 3

B. -1 < y < 3

C. 1 < y < -3

D. -1 < y < -3

E. -1 < y < 3

6.  Find an integer n such that both n + 17 and n – 20 are both perfect squares.

A. 324

B. 344

C. 364

D. 384

E. 348

7.  Find the number of 4 × 4 arrays with entries from {1, 2, 3, 4} such that the sum of each row is divisible by 4 and the sum of each column is divisible by 4.

A. 8⁹

B. 4⁷

C. 4⁸

D. 4⁹

E. 6⁹

8.  Find the number of three digit numbers whose digit sum is ten.

A. 34

B. 38

C. 54

D. 58

E. 60

9.  Given 6^2x+y = 36⁷ and 6^x+4y = 217⁷, determine xy.

A. 28

B. 30

C. 18

D. 20

E. 24

10.  How many points with positive integer coordinates are there strictly inside the area bounded by the lines x = 0, y = 0 and the graph y = 10/x?

A. 23

B. 24

C. 25

D. 26

E. 27

1.    B

Let P be the midpoint of CE, Q be the midpoint of AB and R be the midpoint of DF. By symmetry, PQR is a straight line, and is perpendicular to all of CE, AB and DF. Also,

PQ = QR (a fact which is well known enough not to be obscure but not quite fitting of the title "well known"). ∆PQB and ∆PRF are similar, so , so FR = 12.

Construct lines from the tangent points to the centres of each circle, and observe that the ratio of the radii is equal to the ratio of the lines connecting the tangent points, namely

CE and DF. Since CE = 7 and DF = 24, the ratio of the radii of C₁ and C₂ is 7 : 24.

Proof of "fact": Let X be the intersection of AB and CD. ÐXCA = ÐABC, from the alternate segment theorem. ÐCXB is common, so ∆XBC and ∆XCA are similar, and hence , and so CX² = AX ·BX. Similarly, DX² = AX ·BX and so CX = DX. Since CE, AB and DF are parallel by symmetry, it follows that PQ = QR.

2.    C

Let b be the base in question, and note that "50" means 5b and "125" means b²+2b+5. The equation then becomes b² +b(2−5x)+5(x² +1) = 0. Substitute in the known roots x =5

and x = 8 to get b²−23b+130 = (b−10)(b−13) = 0 and b²−38b+325 = (b−13)(b−25) = 0.

The only solution consistent with both of these is b = 13.

3.    A

Alternatively, cancelling a factor of 2⁷ on top and bottom gives . The numbers aren't very large now, so evaluating gives

4.    E

Note that, removing the last term, the series can be broken up into segments consisting of sums of equal terms. Note that for a positive integer n, (n + 1)² − n² = 2n + 1, so the number of integers at least n² but smaller than (n + 1)² is 2n + 1. In other words, there are 2n + 1 integers x where . But then, the series just reduces to a sum of

, and there are 9 of them from 1 to 99 . So the sum of the

first 99 terms is 9. Then we only have to add the last term, , to obtain the sum of the series as .

5.    B

6.    B

7.    D

Consider filling out the top 3 × 3 portion of the array, the number of ways of which is obviously 4⁹. Then, the six remaining entries that are not the bottom right corner are determined uniquely, in order to make the row totals and column totals a multiple of 4. Pick the remaining entry to make the 4th column total a multiple of 4; there is again a unique way of doing this. Then, each column has sum a multiple of 4, so the entire array has sum a multiple of 4, and since the three top rows also sum to multiples of 4, then the sum of the fourth row must also be a multiple of 4. Thus, for each selection in the top left 3 × 3 portion of the array, there is exactly one way of filling out the rest of the array, so the total number of possible arrays is 4⁹.

8.    C

First, we will determine all integer triples (a, b, c) with 0 < a < b < c < 9

where a + b + c = 10. Note that a = 0, 1, 2 or 3.

·         If a = 0, then b + c = 10, so the only possible triples are (0, 1, 9), (0, 2, 8), (0, 3, 7), (0, 4, 6) and (0, 5, 5).

·         If a = 1, then b + c = 9, so the only possible triples are (1, 1, 8), (1, 2, 7), (1, 3, 6) and (1, 4, 5).

·         If a = 2, then b + c = 8, so the only possible triples are (2, 2, 6), (2, 3, 5) and (2, 4, 4).

·         If a = 3, then b + c = 7, so the only possible triple is (3, 3, 4).

Thus, we can see that there are 13 possible triples which satisfy the conditions. We can

divide these into the following:

·         All digits are non-zero and distinct – each of these triples corresponds to six distinct three-digit numbers.

·         All digits are non-zero and two coincide – each of these triples corresponds to three distinct three-digit numbers.

·         All digits are distinct and one of them is zero – each of these triples corresponds to four distinct three-digit numbers.

·         Two digits coincide and the other is zero – each of these triples corresponds to two distinct three-digit numbers.

There are four, four, four and one, respectively, of each of these types of triples. Thus, the

total number of three-digit numbers whose digit sum is ten is 4 x 6 + 4 x 3 + 4 x 4 + 1 x 2 = 54.

9.    D

10.  A

A point (x, y) will lie strictly inside the area bounded by the lines x = 0,

y = 0 and the graph y = 10/x if xy < 10, where x and y are positive. The only pairs (x, y)

of integers which satisfy this are the pairs:

=> (1, k) with 1 < k < 9

=> (k, 1) with 1 < k < 9

=> (2, 2), (2, 3), (3, 2), (2, 4), (4, 2) and (3, 3)

There are nine points of the first kind, nine of the second kind and six of the third kind.

However, we have counted the point (1, 1) twice, so there are 9 + 9 + 6 – 1 = 23 points

which lies strictly inside the given area.