MathDB

1

Part of 1978 Romania Team Selection Test

Problems(4)

partition of {1,2,3,4,5,6,7,8,9}

Source: Romanian TST 1978, Day 1, P1

9/28/2018
Prove that for every partition of {1,2,3,4,5,6,7,8,9} \{ 1,2,3,4,5,6,7,8,9\} into two subsets, one of the subsets contains three numbers such that the sum of two of them is equal to the double of the third.
counting
integer xy-net

Source: Romanian TST 1978, Day 2, P1

9/29/2018
Associate to any point (h,k) (h,k) in the integer net of the cartesian plane a real number ah,k a_{h,k} so that a_{h,k}=\frac{1}{4}\left( a_{h-1,k} +a_{h+1,k}+a_{h,k-1}+a_{h,k+1}\right) , \forall h,k\in\mathbb{Z} .
a) Prove that it´s possible that all the elements of the set A:={ah,k∣h,k∈Z} A:=\left\{ a_{h,k}\big| h,k\in\mathbb{Z}\right\} are different. b) If so, show that the set A A hasn´t any kind of boundary.
analytic geometrynumber theorycartesian plane
another problem about trapezoids

Source: Romanian TST 1978, Day 3, P1

9/30/2018
In a convex quadrilateral ABCD, ABCD, let A’,B’ A’,B’ be the orthogonal projections to CD CD of A, A, respectively, B. B.
a) Assuming that BB’≤AA’ BB’\le AA’ and that the perimeter of ABCD ABCD is (AB+CD)⋅BB’, (AB+CD)\cdot BB’, is ABCD ABCD necessarily a trapezoid? b) The same question with the addition that ∠BAD \angle BAD is obtuse.
geometryperimetertrapezoidPure geometry
almost Fermat

Source: Romanian TST 1978, Day 4, P1

9/30/2018
Show that for every natural number a≥3, a\ge 3, there are infinitely many natural numbers n n such that an≡1(modn). a^n\equiv 1\pmod n . Does this hold for n=2? n=2?
number theorymodular arithmetic