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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