fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 17)

A number is called "mystic" if it can be expressed as the sum of at least three consecutive positive integers greater than 1, in which the larger parcel is more than twice the smaller parcel. Note that 25 (=3+4+5+6+7) and 44 (=2+3+4+5+6+7+8+9) are mystic numbers (7>2*3 and 9>2*2). Which of the following is true?

I. 124 is mystic
II. 125 is mystic
III. 126 is mystic

(A) I. only
(B) II. only
(C) III. only
(D) Exactly two of them
(E) None of them

$$?\,\,\,:\,\,\,M + \left( {M + 1} \right) + \left( {M + 2} \right) + \ldots + N$$
$$M \ge 2$$
$$N > 2M\,\,\,\,\,\left[ { \Rightarrow \,\,\,\,\,N > 4\,\,\,\,\,\mathop \Rightarrow \limits^{N\,\,{\mathop{\rm int}} } \,\,\,\,\,N \ge 5} \right]$$

$${{N\left( {N + 1} \right)} \over 2}\,\,\,\mathop = \limits^{{\rm{arithm}}{\rm{.}}\,{\rm{seq}}{\rm{.}}} \,\,\,\underbrace {1 + 2 + \ldots + \left( {M - 1} \right)}_{{{M\left( {M - 1} \right)} \over 2}\,\,\,\,\left[ {{\rm{arith}}{\rm{.}}\,{\rm{seq}}{\rm{.}}} \right]} + M + \left( {M + 1} \right) + \ldots + N\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = {{N\left( {N + 1} \right)} \over 2} - {{M\left( {M - 1} \right)} \over 2}$$
$$2\,\, \cdot \,\,?\,\,\, = \,\,\,{N^2} + N - {M^2} + M = \left( {N - M} \right)\left( {N + M} \right) + N + M = \left( {N + M} \right)\left( {N - M + 1} \right)$$

$$\left\{ \matrix{
\,N - M + 1 \ge 3\,\,\,\,\left[ { \ge \,\,{\rm{3}}\,\,{\rm{parcels}}\,\,\left( {{\rm{stem}}} \right)\,\,{\rm{,}}\,\,{\rm{fingers}}\,\,{\rm{trick}}} \right] \hfill \cr
\,N + M > N - M + 1\,\,\,\left[ {M > 0.5} \right] \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\min \left( {N + M,N - M + 1} \right) = N - M + 1 \ge 3\,\,\,\,\left( * \right)$$
$$\left( {**} \right)\,\,\left\{ \matrix{
\,N + M\,\,{\rm{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M = \underbrace {N + M}_{{\rm{even}}} - \underbrace {2M}_{{\rm{even}}}\,\,\,{\rm{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M + 1\,\,{\rm{odd}} \hfill \cr
\,N + M\,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M = \underbrace {N + M}_{{\rm{odd}}} - \underbrace {2M}_{{\rm{even}}}\,\,\,{\rm{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\,N - M + 1\,\,{\rm{even}} \hfill \cr} \right.$$


$$2\,\, \cdot \,\,?\,\,\, = \,\,\,\left( {N + M} \right)\left( {N - M + 1} \right)\,\,\,{\rm{is}}\,\,{\rm{given}}\,\,{\rm{by}}\,\,\,{\rm{even}} \cdot {\rm{odd}}\,\,\left( {**} \right),\,\,{\rm{where}}\,\,\left( * \right)\,\,\,{\rm{odd}} \ge {\rm{3}}\,{\rm{,}}\,\,{\rm{even}}\,\, \ge 4\,\,\,\,\,\left( {***} \right)$$


$${\rm{I}}{\rm{.}}\,\,\,124\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 124 = 248 = {2^3} \cdot 31\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\left\{ \matrix{
\,{\rm{odd}} = 31 \hfill \cr
\,{\rm{even}} = {2^3} \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,N + M = 31 \hfill \cr
\,N - M + 1 = {2^3} \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\, \ldots \,\,\, \Rightarrow \,\,\,\left( {N,M} \right) = \left( {19,12} \right)$$
$$124 = 12 + 13 + \ldots + 18 + 19\,\,\,{\rm{but}}\,\,{\rm{impossible}}\,\,\,\left( {N < 2M} \right)$$

$${\rm{II}}{\rm{.}}\,\,\,125\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 125 = 250 = 2 \cdot {5^3}\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,{\rm{one}}\,\,{\rm{option}}\,\,:\,\,\left\{ \matrix{
\,{\rm{odd}} = {5^2} \hfill \cr
\,{\rm{even}} = 2 \cdot 5 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,N + M = 25 \hfill \cr
\,N - M + 1 = 10 \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\, \ldots \,\,\, \Rightarrow \,\,\,\left( {N,M} \right) = \left( {17,8} \right)$$
$$125 = 8 + 9 + \ldots + 16 + 17\,\,\,{\rm{is}}\,\,{\rm{viable}}!\,\,\,\left( {N > 2M} \right)$$

$${\rm{III}}{\rm{.}}\,\,\,126\,\,\,\, \Rightarrow \,\,\,\,2 \cdot 126 = 252 = {2^2} \cdot {3^2} \cdot 7\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,{\rm{one}}\,\,{\rm{option}}\,\,:\,\,\left\{ \matrix{
\,{\rm{odd}} = 3 \cdot 7 \hfill \cr
\,{\rm{even}} = {2^2} \cdot 3 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\mathop \ldots \limits^{{\rm{Do}}\,\,{\rm{it}}!} \,\,\,\, \Rightarrow \,\,\,\,\left( {N,M} \right) = \left( {16,5} \right)$$
$$126 = 5 + 6 + \ldots + 15 + 16\,\,\,{\rm{is}}\,\,{\rm{viable}}!\,\,\,\left( {N > 2M} \right)$$


YES, this is a very hard problem, but I believe many "pieces" of this solution are VERY useful for candidates aiming really outstanding performances!


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.