* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Solutions
Survey
Document related concepts
Law of large numbers wikipedia , lookup
Numbers (TV series) wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Infinitesimal wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Positional notation wikipedia , lookup
Location arithmetic wikipedia , lookup
Bernoulli number wikipedia , lookup
Real number wikipedia , lookup
Large numbers wikipedia , lookup
Transcript
Supported by JMO mentoring scheme answers Charitable Trust November 2011 paper 1 Ans : 7 2 Ans : 4 The worst case is if she picks three different colours on the first three selections. The fourth sock is bound to be one of the three colours. 3 Ans : 6937 9 (unit numbers) + 2 ´ 90 (numbers in the tens) + 3 ´ 900 (numbers in the hundreds) + 4 ´ 1012. 4 Ans : 8 values Let the digits be x, y, z : (100x + 10y + z) - (100z + 10y + x) = 100(x - z) + (z - x) = 99(x - z). Assuming that 1 £ z < x £ 9, this gives 8 possible values for x - z, namely 1, 2, ..., 8. So there are 8 possible answers, namely 99k with k = 1, 2, ..., 8. 5 Ans : 2 If Katrina turns round to face each corner in turn, the numbers can appear in three orders, d being opposite a different number in each case. Choosing u, v, x, y as the distances to the edges so that d² = x² + y², we can write down equations using Pythagoras’s rule. We don’t need to find the values of u, v, x, y individually nor even their square values. We find that d² is one of 49 + 81 - 36, 81 + 36 - 49, or 36 + 49 - 81 and must be the last one as the others do not yield integer answers. 6 If two numbers are even, then 4 is a factor. Otherwise, five numbers must be odd and one even, so one of the numbers is 9, and so 9 is a factor. A more general method arises from thinking this way. There are 4 prime numbers between 1 and 10. The product of these four with 1 itself as a fifth number can not be divided by a square number greater than 1. Including a sixth number will add a further factor which is composed with one of these prime numbers again. Can you generalise the question? 7 Ans : 3Ö3. Preliminary : any 30°, 60°, 90° triangle has sides in the ratio 1 : Ö3 : 2 which can be worked out by splitting an equilateral triangle with sides of length 2 into two equal right-angled triangles. Learn this! Either (i) : because ÐDCE = 60 ° and because CD : CE = 1 : 2, DCDE is right-angled at D and DAEF is congruent to DCDE. Thus AF = 4, BF = 2 and so DBFD is also congruent to these two triangles. Or (ii) : if M is the midpoint of BC so that ÐAMB = 90 °, then DAEF is similar to DBMA. Because AE : BM = 2 : 3, AF = 2 ´ AB ¸ 3 = 4. Again BF = 2 and DAEF, DBFD, DCDE congruent. Finally : the quickest way to proceed from here is to see that DDEF is equilateral and that it has sides of length DE = Ö12 = 2Ö3. We can now calculate the height of DDEF from the same ratio 1 : Ö3 : 2 or by using Pythagoras's rule. It comes to 3. Thus the area of DDEF = 3 ´ 2Ö3 ¸ 2. A more extended calculation would find the area DCDE + DAEF + DBFD and subtract from the area of DABC. Other routes can also be taken to get this result.