Soal Olimpiade matematika 2007 proof by eri0518ase

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```									             Georgia Institute of Technology
High School Mathematics Competition 2007
Varsity Proof-Based Test
Problem #1

ID#:

Show that if x, y, z, w are positive real numbers, then

(x2 + x + 1)(y 2 + y + 1)(z 2 + z + 1)(w2 + w + 1)
≥ 81.
xyzw
Georgia Institute of Technology
High School Mathematics Competition 2007
Varsity Proof-Based Test
Problem #2

ID#:

Find the number of paths (that is, moving only vertically or horizontally)
in the following array which spell out the word M AT HEM AT ICIAN .

M
MAM
MATAM
MATHTAM
MATHEHTAM
MATHEMEHTAM
MATHEMAMEHTAM
MATHEMATAMEHTAM
MATHEMATITAMEHTAM
MATHEMATICITAMEHTAM
MATHEMATICICITAMEHTAM
MATHEMATICIAICITAMEHTAM
MATHEMATICIANAICITAMEHTAM
Georgia Institute of Technology
High School Mathematics Competition 2007
Varsity Proof-Based Test
Problem #3

ID#:

Show that in every tetrahedron, there must be at least one vertex at
which each of the face angles is acute.
Georgia Institute of Technology
High School Mathematics Competition 2007
Varsity Proof-Based Test
Problem #4

ID#:

Prove that if α, β and γ are the angles of a triangle, then

tan α + tan β + tan γ = tan α tan β tan γ
Georgia Institute of Technology
High School Mathematics Competition 2007
Varsity Proof-Based Test
Problem #5

ID#:

The square numbers are numbers of the form n2 for some n. The trian-
n(n + 1)
gular numbers are numbers of the form 1 + 2 + 3 + · · · + n =          for
2
some n. Show that there are inﬁnitely many numbers that are both square
and triangular numbers.

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