MathDB

Problem 12

Part of 2019 LIMIT Category B

Problems(2)

system of inequalities

Source: LIMIT 2019 CBS1 P12

4/28/2021
The system of inequalities a−b2≥14a-b^2\ge\frac14b−c2≥14b-c^2\ge\frac14c−d2≥14c-d^2\ge\frac14d−a2≥14d-a^2\ge\frac14where a,b,c,da,b,c,d are real numbers has <spanclass=′latex−bold′>(A)</span> no solutions<span class='latex-bold'>(A)</span>~\text{no solutions} <spanclass=′latex−bold′>(B)</span> exactly one solution<span class='latex-bold'>(B)</span>~\text{exactly one solution} <spanclass=′latex−bold′>(C)</span> exactly two solutions<span class='latex-bold'>(C)</span>~\text{exactly two solutions} <spanclass=′latex−bold′>(D)</span> infinitely many solutions<span class='latex-bold'>(D)</span>~\text{infinitely many solutions}
inequalities
rational points on circles

Source: LIMIT 2019 CBS2 P12

4/28/2021
Find the number of rational solutions of the following equations (i.e., rational xx and yy satisfy the equations) x2+y2=2x^2+y^2=2x2+y2=3x^2+y^2=3<spanclass=′latex−bold′>(A)</span> 2 and 2<span class='latex-bold'>(A)</span>~2\text{ and }2 <spanclass=′latex−bold′>(B)</span> 2 and 0<span class='latex-bold'>(B)</span>~2\text{ and }0 <spanclass=′latex−bold′>(C)</span> 2 and infinitely many<span class='latex-bold'>(C)</span>~2\text{ and infinitely many} <spanclass=′latex−bold′>(D)</span> Infinitely many and 0<span class='latex-bold'>(D)</span>~\text{Infinitely many and }0
number theory