Pre RMO practice 2019

1) Determine whether the following functions are well-defined:

1a) f: Q \rightarrow Z defined by f(a/b)=a

1b) f: Q \rightarrow Q defined by f(a/b)=a^{2}/b^{2}

2) Determine whether the function f: R^{+} \rightarrow Z defined by mapping a real number r to the first digit to the right of the decimal point in a decimal expansion of r is well-defined.

3) Apply the Euclidean algorithm to obtain GCD of (57970,10353) and express it as a linear combination of 57970 and 10353.

4) For each of the following pairs of integers a and b, determine their greatest common divisor, their least common multiple, and write their greatest common divisor in the form ax+by for some integers x and y.

(a) a=20, b=13

(b) a=69, b=372

(c) a=792, b=275

(d) a=11391, b=5673

(e) a=1761, b=1567

(f) a=507885, b=60808

5) Prove that if the integer k divides the integers a and b then k divides as+bt for every pair of integers s and t.

6) Prove that if n is composite then there are integers a and b such that a divides ab but n does not divide either a or b.

7) Let a, b and N be fixed integers with a and b non-zero and let d= (a,b) be the greatest common divisor of a and b. Suppose x_{0} and y_{0} are particular solutions to ax+by=N. Prove for any integer r that integers x=x_{0}+\frac{b}{d}t and y=y_{0}-\frac{a}{d}t are also solutions to ax+by=N (this is in fact the general solution).

8) Determine the value \phi(n) for each integer n \leq 30 where \phi denotes the Euler-\phi function.

9) Prove the Well-Ordering Property of integers by induction and prove the minimal element is unique.

10) If p is a prime prove that there do not exist non-zero integers a and b such that a^{2}=pb^{2} (that is, \sqrt{p} is not a rational number).

11) Let p be a prime and n \in Z^{+}. Find a formula for the largest power of p which divides n! (it involves the greatest integer function).

12) Prove for any given positive integer N there exist only finitely many integers n with \phi(n)=N where \phi denotes Euler’s \phi-function.

13) Prove that if d divides n then $latex  \phi(d)$ divides \phi(n) where \phi denotes Euler’s \phi-function.

More later,

Hope this gives you some math meal to churn for the Pre RMO or PRMO or even the ensuing RMO of Homi Bhabha Science Foundation.

Regards,

Nalin Pithwa

 

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