# BdMO Junior Problems and Solutions 2019

### Problem 1:

Farhan shows his test score to Muaz, Bristy and Mursalin, but everyone else keeps it hidden. Muaz thinks, “At least two of us get same scores” Bristy thinks, “I didn’t get the lowest score.” Mursalin thinks, “I didn’t get the highest score.”

a) Who got the highest marks?

b) Who got the lowest marks?

### Solution:

Here, Farhan shows his score to Muaz, Brisy and Mursalin. According to question,

Farhan score = Muaz score, Bristy score > Farhan score, Mursalin score < Muaz score.

That implies, Bristy score > Farhan score = Muaz Score > Mursalin score.

a) Bristy got the highest marks.

b) Mursalin got the lowest marks.

Problem 2: Six friends compete in a dart-throwing contest. Dart is played by throwing darts at a circular board, with your score increasing based on which region of the board you hit. The board has 12 regions, with score values ranging through the integers from 1 to 12. Each of our six friends threw two darts, and each dart hits the target in a region with a different value. The scores are:

 Tiham 16 points Dipto 4 points Samiur 7 points Sabbir 11 points Ashraful 21 points

a) What is Mahi’s score?

b) Who hits the region worth 9 points?

Problem 3: In the figure, ABCD is a rectangle and EFGH is a parallelogram. DI is perpendicular to EF and BK is perpendicular to HG. Using the measurements given in the figure, find the value of DI. Problem 4: n is a positive integer such that 2019 + n! is a square number. Find all such values of n. Here, n! = n × (n – 1) × (n – 2) × … × 2 × 1. For example, 4! = 4 × 3 × 2 × 1 = 24.

Problem5: 2, 3, 5, 6, 7, 10, 11, 12, 13, … … is the sequence of integers without all square and cube numbers. What is the 2019th number?

Problem 6: In this figure ABCD is a trapezium where AB||CD. P, Q, R, S are the midpoint of AB; BD; DC; CA respectively. AC and BD intersect at point O. Area of △AOB = 2019 and area of △COD = 2020.What is the area of quadrilateral PQRS? Problem 7: A chess rook can only travel horizontally or vertically, but not diagonally. We color the bottom of a chess rook red. So, when it makes a move, it paints all the squares it travels over red. Prove that, a rook will need at least 2n – 1 moves to every square of an n × n chess board red.

Problem 8: M and N are two positive integers where M is not equal to N. LCM of (M and N) = M2 – N2 + MN. Show that MN is a perfect cubic number.

Problem 9: In the cartesian coordinate system, four points (0, 0), (20, 0), (20, 19) and (0, 19) are used as vertices to draw a rectangle. At first, a ball with negligible size is at the (0, 0) point. It then started to move towards the point (2, 1). Every second, the ball passes the amount of distance between (0, 0) to (2, 1). If it collides with one side of the rectangle, it follows the law of reflection and comes back to the rectangle. If it collides with a corner, it again follows the law of reflection and comes back in the direction it went in. Until the 2019th second, how many times will the ball collide with a corner point?

Problem 10: In figure, the small and big circles have a radius of 1 and 3 respectively. If the small circle revolves round the big circle according to the figure from left to right, what portion of the circumference of the big circle it will cover? 