Oxbridge Maths - Interview Questions
How many can you solve?
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There are no answers as they're questions done by Oxbridge Maths applicants at interviews.
As a bit of a bump (like santa, interviews are coming) and some random thoughts about the first 5 (not answers). Will try and do some more over the next few days.
1.
The representation is a finite continued fraction, and probably the easiest way to differentiate it is to “unwind” it to a normal fraction so num(x)/denom(x) and differentiate that as usual. Continued fractions (infinite) can be used to make the link between algebraic irrational numbers and the corresponding integer coefficient algebraic equations and analyse transcendental irrational numbers like “e”
2.
Can solve it properly using average speed or harmonic mean inequalities, or a simple problem solving way would be to consider a particular headwind which would make the difference clear (assuming the result was then valid for all headwinds). The AM-GM-HM inequalities are relatively famous and have a nice visual interpretation in terms of a semicircle.
3.
The function is y*f(x) = 1 and its derivative y’/y = - f’/f, so 1/x, 1/sin(x), 1/x^2, … Reason about their signs, evaluate f(x) at certain x values, where f=0,1, reason about its asymptotic behaviour and similar for derivatives ...
4.
Fairly standard definite integral, though a quick sketch or local taylor/maclaurin series would highlight that the integrand is approximately quadratic so roughly ex^2 from (0,0) to (1,e). So the definite integral will be ~e/4 so about 0.7. So anything too different from this should be checked.
5.
Fairly straightforward odd / dots argument. Its well known that odd^2 = 1 mod 8, so the question is just asking that, so there are a few ways to go. Squares and mod 3 came up in the "extra mat" recently.
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6. “Standard” (crops up on a few such lists) derivative question so y=x^x and take logs then implicit/chain rule .. .or write the rhs as e^ln(#) …
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7. How the previous argument for 6) can be adapted for x^x^x and made recursive/inductive for x^x^x^x …
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8. Well known examples correspond to trying to solve x^x^x^... = 2 or = 4. For the (infinite) tetration problem, as the previous examples should illustrate, convergence is important. A rough proof of the domain might correspond to finding the largest x for which the standard substitution is valid.
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9. Easy to google the basic problem, though worthwhile trying to analyse using conditional
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10. Think about factors (how you generate a number abcabc) or a remainder (mod) argument…
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11. Logistics map, could write it as a completed square to emphasise the shape of the parabola, though noting the roots and the gradient (linear with gradient -2k, zero at x=0.5) at the fixed points are things of interest here. Sketch the parabola and y=x line to iterate graphically. So relevant domain is (0,1) and k=4 maps that interval back to itself. What are fixed points, what happens when you start close to those values…. Sample k at 2, 4 say, and think about the different results.
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12. Think about the triangle inequalities for the diagonals.
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13. Think about adapting the usual sqrt(2) argument.
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14. Think about the number of 5s you have, and are the number of 2s relevant?
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15. Question a bit vague, but assuming its making the usual assumptions, choose the colour of the first portion then what about the other two? How many are repeated (rotations)? For 3 portions, no maths is really required, but what about if it was 10?
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16.Sketching an exponential and rotating a line which passes through the origin should be straightforward. A simple, but worth doing, calculation gives the value of k which is tangent to the curve so forms the boundary (1 solution) between 2 and 0 solutions.
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17.Simple mod proof
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18.Well known counter example which has some interesting properties.
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19.Question deals with exponentiation so taking logs is a natural thing to do and it can be rearranged in the form of the given expression. Sketch the given function, why are there 2 solutions (for a,b>1), what is the maximum of the given expression and what can you then say about the intervals for a and b, assuming a and b, assuming a<b.* If there is an integer solution, what must the value of a be and therefore can you determine b?* The solution to such equations can be expressed using the lambert W function, but this is outside elementary analysis, hence the exploratory / analysis nature of the question.* Note the similarity with Q8 for the log(x)/x part.
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*20.Expand as a geometric series (infinite tree) or think about a recursive way of representing p(A) (simple tree with a cycle) or think about a simple ratio argument for each turn. As should be expected, p(A) is slightly greater than ½ as A rolls first. Three players is similar,
22. Classic max area of rectangle with perimeter constraint. Can simply sub equality constraint into the area expression to remove one variable and simple to maximize the resultant quadratic. The solution is well known. Could also note its simply the am-gm inequality, though the approach is very specific to this problem and overkill.
23 Presume this is square numbers as opposed to perfect numbers, as the latter isnt a well defined question.
24. Its not easy to prove in its current form (induction or …) but you can divide through the inequality by n^n and make it both simpler and a more usual binomial expression on the right, and you should note how its related to “e”. Dividing through by the common term to make the expressions simpler is an obvious transformation (with hindsight) or taking logs (increasing transformation so ok to do on the inequality) and simplifying/do an equivalent transformation. Then induction is easier.
25. Easy to just chug through the relevant numbers (mod 10). Similar questions can be asked about squares mod 3, 4, 5, 8.
22. Classic max area of rectangle with perimeter constraint. Can simply sub equality constraint into the area expression to remove one variable and simple to maximize the resultant quadratic. The solution is well known. Could also note its simply the am-gm inequality, though the approach is very specific to this problem and overkill.
23 Presume this is square numbers as opposed to perfect numbers, as the latter isnt a well defined question.
24. Its not easy to prove in its current form (induction or …) but you can divide through the inequality by n^n and make it both simpler and a more usual binomial expression on the right, and you should note how its related to “e”. Dividing through by the common term to make the expressions simpler is an obvious transformation (with hindsight) or taking logs (increasing transformation so ok to do on the inequality) and simplifying/do an equivalent transformation. Then induction is easier.
25. Easy to just chug through the relevant numbers (mod 10). Similar questions can be asked about squares mod 3, 4, 5, 8.
thanks for all these btw.
27. This function occurs in Q19) and Q8). When x=1 it equals zero its >0 for x>1 and it tends to zero for large x (denominator grows faster than numerator). There is a single stationary point / maximum (the previous argument would be ok, or differentiate again) which can be found using calculus.
28. Divisibility by 9 rule (digit sum). Then reason about how the 5s/0s can be arranged.
29. Should recognise as a normal pdf / bell curve, but to sketch it could reason about its behaviour/values when x= 0,1,2,3, its even (reflection in the y-axis), exponential taylor/maclaurin series gives the local (quadratic) behaviour (about 0, how does it locally compare to cos(x)) and a decaying (to zero) exponential e^-# gives the asymptotic behaviour. Could also differentiate to get the gradient and point of inflection if necessary.
30. In hindsight, the integral can be simply written down, but assuming you dont spot it, the denominator is a product and partial fractions or by parts dont give, so probably something to do with the chain rule and the 1/x is likely due to ln(), so try a substitution u = ln(x).
32. The same surface area refers to the outside of the original cube, so think about dividing up the top face into equal areas which correspond to slices with the given properties..
33. A bit of thought / write down a few terms and see if you can spot/analyse/predict the pattern.
34. Could compute first 5 say factorials to get an idea about numbers, then think how they can be used to show its not a square for n>3
35.For x greater than 1, it should be fairly clear, but for x~0 think about differences in odd/even number of terms in the exponent stack (induction - start with x, then x^x, then x^x^x ,,,, you evaluate right to left and take the limit as x->0 even approximately). You could differentiate, Q6) and Q7) to find any stationary points etc.
37. Standard reverse chain rule or by parts integration question
38. Fairly straightforward piecewise linear graph and area calculation
39. Should be well known, but writing down and thinking about a few factor pairs should give the answer.
40. Few ways you could go (and you have to make sure youre proving the correct thing - maybe try a few triples, and multiples), but a mod argument is one thing to try for showing there are no solutions to integer squares problems. Note its obviously related to pythagorean triples which goes back to Euclid (and even babylonian times) and for primitive triples you can show additional things about a,b,c.
42. Should be fairly straightforward, simply reason about the behaviour of a cubic (define what a cubic is) for each of the cases, and make the arguments valid for both positive and negative cubic coefficients.
43. Transformation of y (exponent 1/n) so local taylor/maclaurin of sin() to get the local behaviour when x~0 (or simple squaring, cubing of numbers <1) and reason how that applies to other cases. Note that sin(x) is odd, so what happens when you square, cube, … it.
44. Classic Euclid book 1 proposition 1 problem, so connect centers, intersection points and it should be clear.
45. Could use a gcse recurring decimal argument or one based on a truncated version of 0.999… and argue about the limiting value of the gap between them.
47. Google if necessary
48. Standard a level proof (Euclid)
49. Probably easier to first prove there are an infinite number of primes of the form 4n+3 (modified Euclid), then do 4n+1 which is similar, but a bit harder to justify the important step.
50. Should be reasonably well known, but note the obvious similarity/differences with a circle. If unsure, axis crossing points, scaling, … are usual things to start with.
52. log(x) sketch should be well known, then just think how its modified by taking logs again and again, … Putting a few key values in helps to think about growth rate of the various functions.
53. Typo at the start as it should be “Show (x-a)^2 + (x-b)^2 = 0 …”. Sketching the curves helps, thinking about whether theyre increasing/decreasing, algebra for the simple cases, …
54. Well known continued fraction expansion, and you can get it by doing a substitution for the infinite part which gives a recursive definition and hence a quadratic to solve.
55. Sketching the original function helps to give insight. Similarly, what is the maclaurin for an exponential and why does the extra transformation cause problems, …. Evaluating the derivatives isnt hard, but we got thinking abit about other functions which have a finite value but infinite/undefined derivative at a point. With a bit of thought about inverse elementary functions, it occurs surprisingly often.
57. Obv factorise and reason about the individual factors. The division by 24 could be “confirmed” by calculating a few terms and the justification should be relatively straightforward.
58. Fairly standard binomial working.
59. Relatively straightforward calculations if you break the calculation up into the different winning scenarios and analyse each like an adapted binomial. Plotting the probably of winning game against p (probability of winning a point) should show that if youre slightly more likely to win a point, then youre “much” more likely to win a game, as you may expect.
60. Possibly a bit beyond the scope of what might be asked, but think about a standard exponential series with a matrix A instead of x, …..
62. You could work through a particular n with subsets of different sizes to spot a pattern and maybe make links to binary representation, pascals triangle, …
63. Could try some small numbers instead of 50 to get some insight, but whatever proof is used, it will have to cover all possible arrangements of shaking hands so …
64. Should be fairly easy to show the desired property by subbing a value in, though its not the only possible answer and as a follow on, does an exponential satisfy the given property and if so can you use the property to show e^x >=0 for all x.
65. To prove something like squares (or sum of) cannot be a particular value, a mod argument is worth considering. Especially if the important (trailing) digit is 3.
62. You could work through a particular n with subsets of different sizes to spot a pattern and maybe make links to binary representation, pascals triangle, …
63. Could try some small numbers instead of 50 to get some insight, but whatever proof is used, it will have to cover all possible arrangements of shaking hands so …
64. Should be fairly easy to show the desired property by subbing a value in, though its not the only possible answer and as a follow on, does an exponential satisfy the given property and if so can you use the property to show e^x >=0 for all x.
65. To prove something like squares (or sum of) cannot be a particular value, a mod argument is worth considering. Especially if the important (trailing) digit is 3.
you are a god thank you so much for them so far
Tom Bowler (of the booklet fame) has a few more on twitter
https://twitter.com/Ridermeister
67. Look it up
68. Should be straightforward in polar form, though could practice it by solving in rectangular form so (x+iy)^2 = i.
69. Think about all permutations then symmetries and divide
70. Obviously has some similarities with a circle, so see how far that gets you. What effect does the x^2,y^2 terms have (quadrants), axis crossing points, ….
71. Not too sure about the question
72. If you think it doesnt (try a few examples), its a square so a mod argument might give.
73. Harmonic series obviously tends to infinity, so can that proof be adapted to the finite case/inequality?
74. Maybe evaluate a few simple cases, is there a pattern for the resulting factions you can spot / prove to argue about lack of divisibility.
75. Similar
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