10 Questions
| Total Attempts: 93

Questions and Answers

- 1.You have 12 balls which look identical, but one of them is either slightly heavier or slightly lighter than the others. You have a set of scales, but no weights, and you are able to weigh any selection of the 12 balls against any selection of the remaining balls. You can also mark individual balls to keep track of them without affecting their weight. What is the minimum number of weighings you need to perform to ensure that you can identify the odd ball and determine whether it is heavier or lighter than the others.
- 2.You are presented with four boxes. One of them contains a prize, the other three contain nothing. Dr Know invites you to guess which box contains the prize and if you guess correctly you win the prize. You make your selection, but before opening the box Dr Know (who knows which box contains the prize) offers you a further option. He opens two of the three other boxes and shows you that they are both empty. He then offers you the option of changing your mind and opting for the fourth box instead of your initial choice. What is the probability that you will win the prize if you accept his offer and now opt to open the fourth box?
- 3.A regular 5-point star is inscribed in another regular 5-point star, as shown. What, in terms of the golden ratio Φ = ½ (1+√5), is the ratio of the area of the smaller star to the area of the larger star?
- 4.A chocolate orange consists of a sphere of smooth uniform chocolate. It has mass M and radius r, sliced into equal wedges by planes through its vertical axis. It stands on a horizontal table held together by a narrow weightless ribbon round its equator. Given that the centre of mass of a segment of angle θ lies at a distance of [3πr sin(θ/2)]/(8θ) from the vertical axis, what is the minimum tension in the ribbon needed to hold the chocolate orange together in a gravitational field of strength G.
- 5.N beetles (points) are located at the corners of a regular polygon with n sides each of length L. Each beetle crawls directly towards the next beetle, all at the same speed and all crawling anti-clockwise with respect to the center of the polygon. They each travel a distance s before meeting. Another beetle starting at a vertex of the polygon and traveling at the same speed as all the others walks in a straight line (starting along one of the sides) and also covers a distance s. If the lone beetle walking in a straight line starts at point A and ends at point T, and if the center of the polygon is point O, what is the angle AOT in radians?
- 6.Let z = w
^{6}+ (1-p^{2})w^{4}+ (p^{2}-p^{4})w^{2}+ p^{4}, where w is complex, and p is a positive real number. The roots of this equation may be written in the form w = x +*i*y, and plotted in the complex plane. Let r be the value of p for which all six roots lie on the edge of a rectangle. Let h be the value of p for which the six roots lie at the vertices of a regular hexagon What is the value of h-r - 7.A cube has edges of unit length. A laser cuts the largest possible hole with square cross-section through the cube, without splitting the cube into pieces. What is the size of the largest possible cube that can pass through the hole (ie what is the length of this cube’s sides)?
- 8.A round hole is drilled through the center of a sphere. Sitting on a flat surface with the hole vertical, the resulting shape has height h. What is its volume in terms of h.
- 9.Consider a square S
_{0}with sides of length L. Now consider a figure S_{1 }created by adding squares with areas of 1/9 of S_{0}externally and centrally to each side of S_{0}_{ }For n>0, the shape S_{n}is created by adding squares with areas of 1/9^{n}of S_{0}centrally and externally to each side of S_{n-1. }(for n ε Z). If A_{n}is the area of S_{n, }what is Lim A_{n }as n approaches infinity. - 10.A relay runner drops her baton on the straight part of an athletics track. It lands in a random position on the track. The running lanes are all of equal width apart and the baton’s length is equal to exactly half this width. Assuming you can ignore the widths of the baton and of the lane dividers, what is the probability that the baton crosses one of the lane dividers

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