8/16/2019 220_Papiro 03 - Matemática

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Professor Eurico Dias 

Papiro Especial IME 

1

 

1ª QUESTÃO Valor 1,0 

(a) A committee has met 40 times, with 10 members at every meeting. No two people havemet more than once at committee meetings. Prove that there are more than 60 people on the

committee.

(b) Prove that you cannot make more than 30 subcommittees of 5 members from a

committee of 25 members with no two subcommittees having more than one common

member.

2ª QUESTÃO Valor 1,0 

AB=BC and M is the midpoint of AC. H is chosen on BC so that MH is perpendicular to

BC. P is the midpoint of MH. Prove that AH is perpendicular to BP. 

3ª QUESTÃO Valor 1,0 

Which is greater: 3111 or 1714? [No calculators allowed!]

4ª QUESTÃO Valor 1,0 

p1 < p2 < ... < p15 are primes forming an arithmetic progression, show that the differencemust be a multiple of 2·3·5·7·11·13. 

5ª QUESTÃO Valor 1,0 

Show that n(n + 1)(n + 2)(n + 3) + 1 is a square for n = 1, 2, 3, ... .

6ª QUESTÃO Valor 1,0 

Papiro 03MATEMÁTICA

8/16/2019 220_Papiro 03 - Matemática

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Professor Eurico Dias 

Papiro Especial IME 

2

 

For what real b can we find x satisfying: x2 + bx + 1 = x2 + x + b = 0? 

7ª QUESTÃO Valor 1,0 

8ª QUESTÃO Valor 1,0 

9ª QUESTÃO Valor 1,0 

ABC is an equilateral triangle. S is the circle diameter AB. P is a point on AC such that thecircle center P radius PC touches S at T. Show that AP/AC = 4/5. Find AT/AC.  

10ª QUESTÃO Valor 1,0 

The circumradius R of a triangle with sides a, b, c satisfies a

2

 + b

2

 = c

2

 - R

2

. Find the anglesof the triangle.