2nd Eötvös Competition Problems 1895

1.  n cards are dealt to two players. Each player has at least one card. How many possible hands can the first player have?

2.  ABC is a right-angled triangle. Construct a point P inside ABC so that the angles PAB, PBC, PCA are equal.

3.  A triangle ABC has sides BC = a, CA = b, AB = c. Given (1) the radius R of the circumcircle, (2) a, (3) t = b/c, determine b, c and the angles A, B, C. 

Solutions

Problem 1

n cards are dealt to two players. Each player has at least one card. How many possible hands can the first player have?

Solution

There are two choices for each card (player A or player B). So 2ⁿ possibilities. But we must exclude the two cases where one player gets all the cards, so 2ⁿ - 2.

Problem 2

ABC is a right-angled triangle. Construct a point P inside ABC so that the angles PAB, PBC, PCA are equal.

Solution

∠APB = 180^o - ∠PAB - ∠PBA = 180^o - ∠PBC - ∠PBA = 180^o - ∠B = 90^o. Similarly, ∠BPC = 180^o - ∠C.

Taking P on the circle diameter AB ensures ∠APB = 90^o. Take X as the intersection of the perpendicular to AC at C and the line AB. Then ∠BXC = ∠C, so if P lies on the minor arc BC, then ∠BPC = 180^o - ∠C.

Problem 3

A triangle ABC has sides BC = a, CA = b, AB = c. Given (1) the radius R of the circumcircle, (2) a, (3) t = b/c, determine b, c and the angles A, B, C.

Solution

As usual we have sin A = a/(2R).

We have tan B = (b sin A)/(c - b cos A) = (t sin A)/(1 - t cos A). Then C = 180^o - A - B. Then a/sin A = b/sin B gives b, and c = t/b.