1. (Chosen by VS) We know that:
sin54° = cos(90° − 54°) = cos36°
Using this fact, show that: sin18° = ((5) − 1)/(4) and cos18° = ((10 + 2(5)))/(2) .
Reminder: sin(2α) = 2\sinα\cosα; cos(2α) = 1 − 2sin2α = 2cos2α − 1; sin(3α) = 3\sinα − 4sin3α; cos(3α) = 4cos3α − 3\cosα.


2. (Chosen by VS) Show that:
(cos18° + sin18°)/(cos18° − sin18°) = tan63°


3. (Chosen by RA) In the given 5 × 5 square, the numbers in the each column and row form an arithmetic sequence. You are given some of the numbers. Find x.
        18
  43      
    40  
         
x     26  


4. (Chosen by RA) There is a swarm of bees. The square root of the half the number of bees in that swarm have moved to a shrub of jasmine. A little later, eight-ninths of the whole swarm also moved away to another jasmine shrub. A female bee is confined in a lotus attracted to it by its fragrance in the night. A male bee is humming to that female bee in the lotus. Young lady, how many bees were there in the beginning? (Source: Lilavati, 12th century, AD)
(Many thanks to Ramana for translating it from Sanskrit.)


5. (Chosen by MB) Find a, b and c so that the quadrilateral is a parallelogram with area equal to 80 square units.
figure P5_f1.png


6. (Chosen by MB) Two boats on opposite banks of a river start moving towards each other. They first pass each other 1400 meters from one bank. They each continue to the opposite bank, immediately turn around and start back to the other bank. When they pass each other a second time, they are 600 meters from the other bank. We assume that each boat travels at a constant speed all along the journey. Find the width of the river?


7. (Chosen by KK) For any two distinct positive integers a, b, F(a, b) is defined as following:
F(a, b) = (a2 + b2)/(1 + ab).
  1. Verify that F(30,  112) = 4.
  2. Find another four pairs of positive integers (a, b), such that F(a, b) = 4.
(Hint: Use either properties of quadratic equations, or the properties of proportion: If (p)/(q) = (r)/(s), then (p±r)/(q±s) = k.


8. (Chosen by KK) ABCD is a square with each side 8 cm. A quarter circle and a semi-circle are drawn as shown in the diagram. P is any point on the quarter circle AC, and the line joining DP meets the semi-circle at Q. If the area of the triangle PAB is 12  cm2, find the measure of the line segment DQ.
figure P8_1.jpg


9. (Chosen by AP) In the trapezoid ABCD, AB is parallel to CD and B = ∠C = 90°. We know that the lengths of the sides AB, BC and CD for a geometric sequence. Find, with proof, the angle formed by the diagonals AC and BD.


10. (Chosen by AP) The numbers log(a3b7), log(a5b12) and log(a8b15) are the first three terms of an arithmetic sequence, and the 12th term of the sequence is log(bn). What is n?


11. (Chosen by AP) Two cubical dice each have removable numbers 1 through 6. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two top faces are added. What is the probability that the sum is 7?


12. (Chosen by AP) Suppose that f(x) = a(x − b)(x − c) is a quadratic function, where a, b and c are distinct positive integers less than 10. For each choice of a, b and c , the function f(x) has a minimum value. What is the minimum of these possible minimum values?