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Rmo 2005 (Orissa)

This document contains a 10 question practice exam for the RMO (Orissa) exam from 2005. The exam is 100 marks and lasts 3 hours. Calculators and protractors are not allowed, but rulers and compasses are. Candidates are instructed to answer as many questions as possible. The questions cover a range of math topics including: finding the maximum sum of integers whose product is 1000; determining the area of a regular octagon that is a rectangle; summing 3-digit numbers formed from cyclic permutations of 1-9; proving properties of numbers divided by 27; solving functional equations; solving systems of equations involving 3 variables; finding the maximum value of a ratio involving the perimeter, area, and circumradius of a triangle

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13 views1 page

Rmo 2005 (Orissa)

This document contains a 10 question practice exam for the RMO (Orissa) exam from 2005. The exam is 100 marks and lasts 3 hours. Calculators and protractors are not allowed, but rulers and compasses are. Candidates are instructed to answer as many questions as possible. The questions cover a range of math topics including: finding the maximum sum of integers whose product is 1000; determining the area of a regular octagon that is a rectangle; summing 3-digit numbers formed from cyclic permutations of 1-9; proving properties of numbers divided by 27; solving functional equations; solving systems of equations involving 3 variables; finding the maximum value of a ratio involving the perimeter, area, and circumradius of a triangle

Uploaded by

drssagrawal
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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RMO QUESTIONS

RMO(ORISSA) 2005
Max Mark 100
Time 3 Hrs
Instructions:
Calculators (in any form) and protractors are not allowed.
Ruler and compass allowed.
Answer as many questions as you can.
1.
(a) If product of 1000 positive integers is 1000, then find the maximum sum of that 1000 positive
integers.
(b) In the figure what part of Area of regular octagon is the area of rectangle ABCD.

2.
(a) 1, 2, 3, , 9 are arranged in a circle. If we take 3 continuous numbers in cyclic order we will
get 9 three digit numbers. Find the sum of all the 9 three digit numbers.
(b) X is 6 digit number. By taking left three digit of X and kept right side and make it Y (Example
X = 385467, Y = 467385). Prove that when X anf Y divided by 27 they give the same remainder.
3.
(a) Find all function f such that f ( x ) f ( y ) f ( xy ) x y .

n2
, where x is a real number and x 1 .
4
4. Find all real numbers x, y and z such that xy z x y, xz y x z , yz x y z .
5. Perimeter of a triangle ABC is P and area is A and radius of its circum circle is R, then find the
P. A
maximum value of 3 .
R
(b) n 1 is an integer. Prove that (1 x ) n 1

4n 4n 2 1
, find value of f (1) f (2) ...... f (40) .
2n 1 2 n 1
7. If S n (3 5 ) n (3 5 ) n for n 0,1,2,3...... then prove that S n is an integer. Also prove that

6. For any positive integer n, f (n)

integer just greater than (3 5 ) n is divisible by 2 n .


8. A Quadrilateral ABCD is circumscribed by a circle of radius 1 and AB.BC.CD.DA 4 . Prove that
ABCD is a square.
9. M is an interior point of triangle ABC. Perimeter of triangle ABC is P. Prove that
P
AM MB MC P .
2
1
10. Find the sequence of real numbers x0 , x1 x2 ........ such that 0 x0 1, 0 xn1 2 .
xn
******
****
Dr. Shyam Sundar Agrawal

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