MathDB

2011 El Salvador Correspondence

Part of El Salvador Correspondence

Subcontests

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2011 El Salvador Correspondence / Qualifying NMO XI

p1. In the month of January of a certain year, there were exactly four Mondays and four Fridays. What day of the week was February 1 1 ?
p2. Determine all positive integers nn that have the following property: "Among the positive divisors of nn different than 1 1 and nn, the largest is 3535 times the smallest."
p3. Vecindad Island has 20112011 inhabitants, divided into three types: the innocent, the wicked and the capricious, the innocent always tell the truth, the wicked always lie, and the capricious alternately tell lies one day and truths the next. A reporter visits the island for two days. On the first day, the reporter interviews all the inhabitants: ∙\bullet The first one says: "There is exactly one villain on the island." ∙\bullet The second says: "There are exactly two villains on the island." ∙\bullet And so on until he reaches inhabitant number 20112011, who says: "There are exactly 20112011 villains on the island." On the second day, the reporter interviews everyone again in the same order: ∙\bullet The first says: "There is exactly one innocent on the island." ∙\bullet The second says: "There are exactly two innocents on the island." ∙\bullet And so on until he reaches inhabitant number 20112011, who says: "There are exactly 20112011 innocents on the island." How many capricious are there on the island?
p4. Let ABCDEABCDE be a regular pentagon such that the star ACEBDACEBD has area 1 1. Let PP be the intersection point of ACAC and BEBE, let QQ be the intersection point of BDBD and CECE. Determine the area of quadrilateral APQD. https://cdn.artofproblemsolving.com/attachments/0/3/dcb0609bc85699b600b61ed97f6345a6b2b832.png
p5. Determine all positive integers a,b,c,da, b, c, d with a<b<c<da <b <c <d, such that 1a+1b+1c+1d\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}+ \frac{1}{d} an integer.