47th Eötvös Competition Problems 1943

1.  Show that a graph has an even number of points of odd degree.

2.  P is any point inside an acute-angled triangle. D is the maximum and d is the minimum distance PX for X on the perimeter. Show that D ≥ 2d, and find when D = 2d.

3.  x₁ < x₂ < x₃ < x₄ are real. y₁, y₂, y₃, y₄ is any permutation of x₁, x₂, x₃, x₄. What are the smallest and largest possible values of (y₁ - y₂)² + (y₂ - y₃)² + (y₃ - y₄)² + (y₄ - y₁)². 

Solutions

Problem 1

Show that a graph has an even number of points of odd degree.

Solution

Let the n points of the graph have degree d₁, d₂, ... , d_n. Then the no. of edges is (∑ d_i)/2. Hence ∑ d_i is even. So an even number of the d_i must be odd. 

Problem 2

P is any point inside an acute-angled triangle. D is the maximum and d is the minimum distance PX for X on the perimeter. Show that D ≥ 2d, and find when D = 2d.

Solution

If X is any point of the segment YZ except its endpoints, then PX < max(PY, PZ), so if the triangle is ABC, then D = max(PA, PB, PC). Also since the triangle is acute, d = min(PR, PS, PT), where R, S, T are the feet of the perpendiculars from P to BC, CA, AB respectively.

At least one of the angles at P must be ≤ 60^o. Suppose it is ∠APS. Then PS = AP cos APS ≤ AP/2. Hence D ≥ d/2. We can get equality only if all the angles are 60^o. But in that case ∠ TAP = ∠SAP = 30^o, so ∠A = 60^o and similarly for the other angles, so ABC is equilateral. 

Problem 3

x₁ < x₂ < x₃ < x₄ are real. y₁, y₂, y₃, y₄ is any permutation of x₁, x₂, x₃, x₄. What are the smallest and largest possible values of (y₁ - y₂)² + (y₂ - y₃)² + (y₃ - y₄)² + (y₄ - y₁)².

Solution

Put [y₁,y₂,y₃,y₄] = (y₁ - y₂)² + (y₂ - y₃)² + (y₃ - y₄)² + (y₄ - y₁)². Obviously, [x₁,x₂,x₃,x₄] = [x₂,x₃,x₄,x₁] = [x₃,x₄,x₁,x₂] = [x₄,x₁,x₂,x₃] and similarly for [x₂,x₁,x₃,x₄] etc, so we need only consider the 6 values: [x₁,x₂,x₃,x₄], [x₁,x₂,x₄,x₃], [x₁,x₃,x₂,x₄], [x₁,x₃,x₄,x₂], [x₁,x₄,x₂,x₃], [x₁,x₄,x₃,x₂]. But equally, [x₁,x₂,x₃,x₄] = [x₁,x₄,x₃,x₂] etc (rotating the other way), so there are only three values to consider: [x₁,x₂,x₃,x₄], [x₁,x₂,x₄,x₃], [x₁,x₃,x₂,x₄].

Now it is easy to check that [x₁,x₂,x₃,x₄] - [x₁,x₂,x₄,x₃] = 2(x₂-x₁)(x₄-x₃) > 0, and [x₁,x₃,x₂,x₄] - [x₁,x₂,x₃,x₄] = 2(x₄-x₁)(x₃-x₂) > 0, so we have [x₁,x₂,x₄,x₃] < [x₁,x₂,x₃,x₄] < [x₁,x₃,x₂,x₄]. Thus [x₁,x₃,x₂,x₄] is the largest possible value and [x₁,x₂,x₄,x₃] is the smallest possible value.