Question 37

Challenge question #37 (14/11/2016)

Junior Question

November 2016
 S 
M 
T 
W 
T 
F 
S 
$1$ 
$2$ 
$3$ 
$4$ 
$5$ 
 $6$ 
$7$ 
$8$ 
$9$ 
$10$ 
$11$ 
$12$ 
 $13$ 
$14$ 
$15$ 
$16$ 
$17$ 
$18$ 
$19$ 
 $20$ 
$21$ 
$22$ 
$23$ 
$24$ 
$25$ 
$26$ 
 $27$ 
$28$ 
$29$ 
$30$ 
Notice that the four numbers 
 $13$ 
$14$ 
 $20$ 
$21$ 
 have the property that
$\ 14\times20-13\times21$
$=$
$\ 280-273$
$=$
$\ 7$
Prove that any four numbers 
 $a$ 
$b$ 
 $c$ 
$d$ 
 in the calendar will have the property $bc-ad=7$.
Hint

Senior Question

Consider a set of $6$ points in the plane which have been joined by $10$ distinct line segments. Show that at least one triangle is formed.
Hint