39th Eötvös Competition Problems 1935



1.  x1, x2, ... , xn is any sequence of positive reals and yi is any permutation of xi. Show that ∑ xi/yi ≥ n.
2.  S is a finite set of points in the plane. Show that there is at most one point P in the plane such that if A is any point of S, then there is a point A' in S with P the midpoint of AA'.



3.  Each vertex of a triangular prism is labeled with a real number. If each number is the arithmetic mean of the three numbers on the adjacent vertices, show that the numbers are all equal.


Solutions

Problem 1
x1, x2, ... , xn is any sequence of positive reals and yi is any permutation of xi. Show that ∑ xi/yi ≥ n.
Solution
By AM/GM (∑ xi/yi)/n ≥ geometric mean of xi/yi which is 1 since ∏ xi = ∏ yi.

Problem 2
S is a finite set of points in the plane. Show that there is at most one point P in the plane such that if A is any point of S, then there is a point A' in S with P the midpoint of AA'.
Solution
The points must form disjoint pairs (A,A') with P the midpoint of each pair. But P is the centroid of each pair and hence the centroid of all the points, but that is a uniquely defined point. 

Problem 3
Each vertex of a triangular prism is labeled with a real number. If each number is the arithmetic mean of the three numbers on the adjacent vertices, show that the numbers are all equal.
Solution
We have 3a1 = a2 + a3 + b1, 3a2 = a1 + a3 + b2, 3a3 = a1 + a2 + ba2 3. Adding gives a1 + a2 + a3 = b1 + b2 + b3. Hence (a1 - b1) = -(a2 - b2) - (a3 - b3) (*).
We also have 3b1 = b2 + b3 + a1. Subtracting from the first equation above, 4(a1 - b1) = (a2 - b2) + (a3 - b3). Adding (*) gives 5(a1 - b1) = 0. Hence a1 = b1. Similarly a2 = b2 and a3 = b3. But now the first two equations give 2a1 = a2 + a3 and 2a2 = a1 + a3, subtracting, 3(a1 - a2) = 0, so a1 = a2. Similarly a2 = a3. Hence all the numbers are equal.


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