34th Eötvös Competition Problems 1930

1.  How many integers (1) have 5 decimal digits, (2) have last digit 6, and (3) are divisible by 3?

2.  A straight line is drawn on an 8 x 8 chessboard. What is the largest possible number of the unit squares with interior points on the line?

3.  An acute-angled triangle has circumradius R. Show that any interior point of the triangle other than the circumcenter is a distance > R from at least one vertex and a distance < R from at least one vertex. 

Solutions

Problem 1

How many integers (1) have 5 decimal digits, (2) have last digit 6, and (3) are divisible by 3?

Solution
We can write the number as d₄d₃d₂d₁6. There are 10 choices for each of d₁, d₂, d₃. Now if the digit sum of the other digits is 0, d₄ must be 3, 6 or 9. If it is 1, d₄ must be 2, 5 or 8. If it is 2, d₄ must be 1, 4, or 7. So in any case we have three choices for d₄. Hence 3000 possibilities in all. 

Problem 2

A straight line is drawn on an 8 x 8 chessboard. What is the largest possible number of the unit squares with interior points on the line?

Solution

There are 14 internal grid lines. The line can cross each one at most once, so it can make a total of at most 14 crossings. But one crossing is required each time it changes unit square, so at most 1+14 = 15 squares. But 15 is certainly possible - take a line parallel to a main diagonal and close to it. 

Problem 3

An acute-angled triangle has circumradius R. Show that any interior point of the triangle other than the circumcenter is a distance > R from at least one vertex and a distance < R from at least one vertex.

Solution

Since the triangle is acute-angled, the circumcenter O must lie inside the triangle. So it must lie in (or on) one of the triangles PAB, PBC, PCA. Suppose it lies in PAB. Then PA + PB > OA + OB = 2R, so either PA or PB >R. Similarly, P must lie inside one of the triangles OAB, OBC, OCA and so one of the distances <R.