is 
, find the values of a and b.:Inter School Mathematisc Contest 2003 Individual
1. In triangle ABC, produce BC to D so that CD=2BC. Let E be a point on AC and AE=2EC. The extension of DE cuts AB at F. Find DE/EF.
2. In any convex quadrilaterals with unit areas, find the smallest value of the sum of four sides and two diagonals.
3. Find all the positive integers solution(s) of x4-y4-20x2+28y2=107.
4. Let A be a positive integer such that A<100 and A3+23 is divisible by 24. How many integer(s) satisfy the above conditions?
5. Find the sum of the coefficients of the odd powers of x in the expansion of (x4-x3-5x2+4)10.
6. Given that the solution of is , find the values of a and b.
7. A piece of graph paper is folded once so that (0,2) is matched with (4,0) and (7,3) is matched with (m,n), find m+n.
8. Solve the following system of equations:
9. In a football league, every team has to compete against all the other teams once. For each match, the winning team can get 2 points, losing team gain no points and both teams gain 1 point for draw game. One team gains the highest score among all the teams but htis team won the least number of matches. At least how many teams are there?
10. An integral point (x,y) in the first quadrant satisfying 
and 
, how many such points (x,y) are there?
11. Evaluate 
12. Find the sum of all fractions in form of 
where n and m are natural numbers.
13. There are n different positive numbers such that any two among them (let them be x and y) satisfy 
. Find the largest possible value of n.
14. Given that triangle ABC is inscribed in a circle and angle BAC=60o. The tangent to the circle passing through A cuts the extension of CB at P. The bisector of angle APC intersect AB and AC at D and E respectively. Given that AD=15, AC=40. Find BC.
15. Three identical circles with radcii r can cover a circle with radius R. Find the smallest possible value of r.
16. Solve the equation 3x3-[x]=3 where [x] represents the greatest integer not exceeding x.
17. Let x=sinA, y=sinB, z=sin(A+B). Express cos(A+B) in the form of the quotient of two polynomials with integral coefficients in x,y and z.
18. A 9x9x9 cube is composed of twenty-seven 3x3x3 cubes. The big cube is tunneled as follows. Firstly the six 3x3x3 cubes which make up the center of each face as well as the cedntre 3x3x3 cubes which make up the centre of each face as well as the centre 3x3x3 cube are removed. Secondly, each of the twenty remaining 3x3x3 cube is tunneled in the same way. That is, the centre facial unit cubes as well as each centre cube are removed. Find the surface area of the final figure.
19. f1(x), f2(x), f3(x) and f4(x) are polynomials such that
f1(x4)+xf2(x4)+x2f3(x4)=(1+x+x2+x3)f4(x).
Find a common divisor of f1(x), f2(x), f3(x) and f4(x).
20. In a coordinate plane, the coordinates of points O, A and B are (0,0), (1,0), (1,2). C is a point lying on OB with coordinates (x, 2x) where 0
