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Soal Olimpiade matematika 2007 proof

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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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