RMO–1993

1. Let ABC be an acute-angled triangle and CD be the altitude through C. If AB = 8 and

CD = 6, find the distance between the mid-points of AD and BC.

2. Prove that the ten’s digit of any power of 3 is even. [e.g. the ten’s digit of 3⁶ = 729 is 2].

3. Suppose A₁A₂ . . .A₂₀ is a 20-sided regular polygon. How many  non- isosceles (scalene) 

triangles can be formed whose vertices are among the vertices of the polygon but whose sides

are not the sides of the polygon?

4. Let ABCD be a rectangle with AB = a and BC = b. Suppose r₁ is the radius of the circle

passing through A and B and touching CD; and similarly r₂ is the radius of the circle passing

through B and C and touching AD. Show that r₁ + r₂>= (5/8)*(a + b).

5. Show that 19⁹³ − 13⁹⁹ is a positive integer divisible by 162.

6. If a, b, c, d are four positive real numbers such that abcd = 1, prove that 

(1 + a)(1 + b)(1 + c)(1 + d) >=16.

7. In a group of ten persons, each person is asked to write the sum of the ages of all the other

9 persons. If all the ten sums form the 9-element set {82, 83, 84, 85, 87, 89, 90, 91, 92} find 

the individual ages of the persons (assuming them to be whole numbers of years).

8. I have 6 friends and during a vacation I met them during several dinners. I found that I dined

with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days;

with  every  3  of  them  on  4  days;  with every 2 of them on 5 days. Further every friend was 

present at  7  dinners and  every  friend was absent at 7 dinners. How many dinners did I have 

alone?