BdMO Junior Problems and Solutions 2018

Problem 1: The area of a rectangle is 240. All the lengths of the sides of this rectangle are integer, what can be the lowest possible perimeter of this rectangle?

Problem 2: If the average of first n positive integers is 2018, then find the value of n?

Problem 3: Shoumo, Oindry and Esha take turns counting from 1 to one more than the last number said by the last person. Shoumo starts by saying “1”, so Oindry follows by saying “1, 2”, Esha follows by saying “1, 2, 3”. Shoumo then says “1, 2, 3, 4”, and so on. What is the 50th number said? (For example, the numbers 3,4 said by Shoumo last time here are ninth and tenth numbers).

Problem 4: The squares of three positive numbers add up to 2018. The biggest of these three numbers is the sum of the smaller two. If the difference between the smaller two numbers is 2, what is the difference between the cubes of the smaller two numbers?

Problem 5: 𝟑𝒙𝟐 + 𝒚𝟐 = 𝟏𝟎𝟖; determine all the positive integer values of x and y.

Problem 6: Given 8 lines on a plane and no two of them are parallel. Prove that, at least two of them form an angle less than 23°.

Problem 7: All possible 4-digit numbers are created using 5,6,7,8 and then sorted from smallest to largest. In the same manner, all possible 4-digit numbers are created using 3,4,5,6 and then sorted from smallest to largest. Then first number of the second type is subtracted from first number of the first type, second number of the second type is subtracted from second number of the first type and so on. What will be the summation of these difference (subtraction results)?

Problem 8: In triangle ABC, AB=10, CA=12. The bisector of ∠𝐁 intersects CA at E, and the bisector of ∠𝐂 intersects AB at D. AM and AN are the perpendiculars to CD and BE respectively. If MN=4, then find BC.

Math Olympiad problems

Problem 9: Find the number of positive integers that are divisors of at least one of 1010, 1212, 1515.

Problem 10: Points D and E divide equal sides AC and AB of an equilateral triangle ABC according to the ratio of 𝑨𝑫: 𝑫𝑪 = 𝑩𝑬: 𝑬𝑨 = 𝟏: 𝟐. Edges BD and CE meet at point O. Find ∠𝐀𝐎𝐂.