Difference between revisions of "Euclid 2019/Problem 5"
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==Problem== | ==Problem== | ||
| − | (a) Determine the two pairs of positive integers <math>(a,b)</math> with <math>a<b</math> that satisfy the equation <math>\sqrt a+\sqrt b=\sqrt 50</math>. | + | (a) Determine the two pairs of positive integers <math>(a,b)</math> with <math>a<b</math> that satisfy the equation <math>\sqrt a+\sqrt b=\sqrt (50)</math>. |
(b) Consider the system of equations: <math>c+d=2000</math> and <math>\frac{c}{d}=k</math>. Determine the number of integers <math>k</math> with <math>k\leq0</math> for which there is at least one pair of integers <math>(c,d)</math> that is a solution to the system. | (b) Consider the system of equations: <math>c+d=2000</math> and <math>\frac{c}{d}=k</math>. Determine the number of integers <math>k</math> with <math>k\leq0</math> for which there is at least one pair of integers <math>(c,d)</math> that is a solution to the system. | ||
==Solution== | ==Solution== | ||
| + | |||
| + | (a) <math>\sqrt(50)=5\sqrt(2)</math>, and <math>5\sqrt(2)</math> can be expressed as <math>\sqrt(2)+4\sqrt(2)</math>, or <math>2\sqrt(2)+3\sqrt(2)</math>. | ||
| + | From the first equation, we get that <math>\sqrt(2)+\sqrt(32)</math>, and from the second of these equations, we have <math>\sqrt(8)+\sqrt(18)</math>. So our solutions for <math>(a,b)</math> are <math>\boxed{(2,32), (8,18)}</math>. | ||
Latest revision as of 14:25, 13 October 2025
Problem
(a) Determine the two pairs of positive integers 
with 
that satisfy the equation 
.
(b) Consider the system of equations: 
and 
. Determine the number of integers 
with 
for which there is at least one pair of integers 
that is a solution to the system.
Solution
(a) 
, and 
can be expressed as 
, or 
.
From the first equation, we get that 
, and from the second of these equations, we have 
. So our solutions for are 
.
