MathDB

p1. Find the smallest positive integer divisible by exactly

2013

different positive integers.
p2. Find all positive integers

x

such that

x^2 + x + 6

is a perfect square.
[url=https://artofproblemsolving.com/community/c1068820h2935494p26269394]p3. Hannibal and Clarice are still at a barbecue and there are three anticuchos left, each of which it has

10

pieces. Of the

30

total pieces, there are

29

chicken and one meat, the which is at the bottom of one of the anticuchos. To decide who to stay with the piece of meat, they decide to play the following game: they alternately take out a piece of one of the anticuchos (they can take only the outer pieces) and whoever wins the game manages to remove the piece of meat. Clarice decides if she starts or Hannibal starts. What should she decide?
p4. Find all pairs of positive prime numbers

p

and

q

such that

p^q - q^p = 1

.
[url=https://artofproblemsolving.com/community/c4h1845734p12426577]p5. Four points

A, B, C, D

move in space so that always is satisfied

(AB) = (AC) =(DB) =(DC) = 1

, - What is the greatest value that the sum

(AD)+(BC)

can become? - Under what conditions is this sum maximized?
[url=https://artofproblemsolving.com/community/c1068820h2935497p26269412]p6. Juan must pay

4

bills. He goes to an ATM, but doesn't remember the amount of the bills. Just know that a) Each account is a multiple of

1,000

and is at least

4,000

. b) The accounts total 2

00, 000

. What is the least number of times Juan must use the ATM to make sure he can pay the bills with exact change without any excess money? The cashier has banknotes of

2, 000

5, 000

10, 000

, and

20,000

. Juan can decide how much money he asks the cashier each time, but you cannot decide how many bills of each type to give to the cashier.
PS. Problems 3 and 6 were also proposed as Seniors P3 and P6.

Problem Statement