Imo Prelim (2001-2002)

1.

2.A carpenter sells armchairs, bookcases and cabinets. A person buys 8 armchairs, 2 bookcases and 5 cabinets and pays $343. Find the total cost of 1 armchair, 1 bookcase and 1 cabinet.(1 mark)

3.For any positive integer n , let F(n) be the rightmost digit of n in base 10 and let T_n=F(n²)-F(n), find the sum T₁+T₂+¡K+T₂₀₀₂.(1 mark)

4.Find the last two digits of the integer 2²⁰⁰²(2²⁰⁰³-1)in base 10.(1 mark)

5.Let x,y,z be positive numbers satisfying the equations :

   x²+y²-z²=¡Ô3(xy) and x²-y²+z²=¡Ô2(xz)

Find the ratio y:z(1 mark).

6.Let b be a positive number. It is known that the equation x⁶-2bx²+b²-100=0 has exactly two real roots whose difference is 2. Find the value of b.(1 mark)

7.In£GABC, ¡çBAC=40^o and ¡çABC=60^o. D and E are points on sides AC and AB respectively such that ¡çCBD=40^o and ¡çBCD=70^o. Let BD intersect CE at F and AF intersect BC at G. Find ¡çGFC.(1 mark)

8.ABCD is a square iwth side length 1 unit. E, F, G and H are points on AB, BC, CD and DA repectively such that AE = BF = CG = DH = 20/21. Find the area of the region bounded by AF, BG, CH and DE.(2 marks)

9.Let x, y be real numbers such that x²+4y²-4=0. Find the maximum value of

x²+2xy+4y²+x+2y.(2 marks)

10.A certain number of unit cubes are stuck together to form a cuboid. The six faces of the cuboid, none of which is a square, are painted. If x is the number of unit cubes with no face painted, y is the number of unit cubes with exactly 1 face painted and z is the number of unit cubes with exactly 2 faces painted, then x-y+z=2002. Find the volume of the cuboid.(2 marks)

11.In£GABC, M and Nare two points on BC such that BM¡ÕBN, BM=NC=4 and MN=3. If ¡çBAM=¡çMAN=¡çNAC, find the length of AC.(2 marks)

12.Find the number of 10-digit positive integers such that

 (a)each digit is either 1 or 2, and

 (b)there exist two consecutive 1's.(2 marks)

13.Integers x and y satisfy 5×10⁷>x>y. Suppose x-2001 and y-2001 are respectively the squares of two consecutive integers and d is the greatest common divisor(or highest common factor) of x and y. Find x when d is maximum.(2 marks)

14.Let [x] be the greatest integer less than or equal to x. If ¡Ux+1¡U-1=(x-[x])/(¡Ux-1¡U), find the largest possible value of ¡Ux¡U.(2 marks)

15.Find k such that if P, Q, R, S are points on sides AB, BC, CD, DA rspectively of a convex quadrilateral ABCD and

  then the area of PQRS is 52% of the area of ABCD.(2 marks)

16.(a,b,c,d) and (a',b',c',d') are said to be distinct if and only if a¡Úa' or b¡Úb' or c¡Úc' or d¡Úd'. Find the number of distinct (a,b,c,d) such that a, b,c,d are integers, 1¡Øa<b<c<d¡Ø30 and a+d=b+c.(2 marks)

17.p is a prime number such that 1/p, in decimal notations, is a recurring decimal with a period of 7 digits. For example, 4649 is a possible of p because 1/4649=0.00021510002151¡KFind another possible value of p.(2 marks).

18.A triangular shape paper has sides of lengths ¡Ô2, 4/3, (¡Ô10)/3 units. The paper is folded along a line perpendicular to the side of length 4/3. Find the largest possible overlapped area.(3 marks)