Difference between revisions of "1977 Canadian MO Problems/Problem 4"
(Created page with "== Problem == Let <cmath>p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0</cmath> and <cmath>q(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots +b_1x+b_0</cmath> be two polynomials with integer coe...") |
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== Solution == | == Solution == | ||
| + | suppose p(x) and q(x) have all even coefficient then as p(x).q(x) involves all the ai(x).bj(x).. all the coefficient of p(x).q(x) must be divisible by 4. which is certainly not the case, hence p(x) or q(x) has atleast one odd coefficient. | ||
| + | Without the loss of generality assume p(x) has an odd coefficient ak(x)^l if q(x) also has an odd coefficient say bt(x)^r the p(x).q(x) will involve the term ak*bt(x)^r+k.. and as both are odd the coefficient is also odd... hence contradiction | ||
| + | sorry ,i dont know latex, so writing in this manner | ||
Latest revision as of 11:16, 11 July 2015
Problem
Let
![\[p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0\]](http://latex.artofproblemsolving.com/c/1/2/c126ff597ee283022b1c7295806fabde784acccf.png)
and
![\[q(x)=b_mx^m+b_{m-1}x^{m-1}+\cdots +b_1x+b_0\]](http://latex.artofproblemsolving.com/d/f/a/dfabcbca27ad4f78679b62c9b3a5618cc10354f9.png)
be two polynomials with integer coefficients. Suppose that all of the coefficients of the product 
are even, but not all of them are divisible by 4. Show that one of 
and 
has all even coefficients
and the other has at least one odd coefficient.
Solution
suppose p(x) and q(x) have all even coefficient then as p(x).q(x) involves all the ai(x).bj(x).. all the coefficient of p(x).q(x) must be divisible by 4. which is certainly not the case, hence p(x) or q(x) has atleast one odd coefficient. Without the loss of generality assume p(x) has an odd coefficient ak(x)^l if q(x) also has an odd coefficient say bt(x)^r the p(x).q(x) will involve the term ak*bt(x)^r+k.. and as both are odd the coefficient is also odd... hence contradiction sorry ,i dont know latex, so writing in this manner
