### 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: Let *O *be the centre of a circle. A line intersects two parallel lines at *X *and *Y *and goes through the point *O*. Given that, *OX > OY *and *X *is inside the circle, prove that *Y *is also inside the circle.

Problem 3: At some math Olympiad, 72 medals were handed out. Afterwards, it was found that Sumon had lost the receipt! He only remembers that the total price of the medals was a five-digit number, and the three middle digits were all 9. If the price of all the medals were the same integer, what was the amount spent for each medal?

Problem 4: 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 5:

In this figure *ABCD *is a rectangle and *ACEF *is a square. Area of the square is 625 and perimeter of the rectangle *ABCD *is 62. What is the difference between two sides of rectangle?

### Solution:

Let, the length and width of the rectangle are l and w respectively. AB = CD = l, BC = DA = w.

Now, the perimeter of rectangle, 2 (l + w) = 62 or, l + w = 31

The area of square ACEF = 625. That implies AC = CE = EF = AF = 25

Use Pythagoras theorem, AC^{2} = AB^{2} + BC^{2}

Or, l^{2} + w^{2 }= 25^{2}

Or, (l + w)^{2} – 2lw = 25^{2}

Or, 2lw = (l + w)^{2} – 25^{2}

Or, 2lw = 31^{2} -25^{2}

Or, 2lw = 336

Or, lw = 168

Therefore, (l – w)^{2 }= (l + w)^{2} – 4lw

or, (l – w)^{2 }= 31^{2} – 4×168

or, (l – w)^{2 }= 961 – 672

or, (l – w)^{2 }= 289

or, l – w = 17

Hence the difference between two sides of rectangle is 17.

Problem 6: When Chamok drives to office from his home, he drops Barnomala at school and his wife Bohni at her university. Then, he goes to a park to walk for 20 (Yeah, he walks in a formal attire). Finally, he goes to his office. Let’s draw a simple map of his route.

Home → 4 ways → School → 3 ways → University → 5 ways → Park→ 2 ways → Office

So, you can calculate in how many ways Chamok can drive to office, right? But he has this forgetting habit. He might forget something at home. If he forgets something like this, he will remember it at a destination (say, at the school) and then drive back to collect it. Once he has collected the thing, he starts on his journey once again from the start. Chamok forgets at most one thing in a day, and if he has reached the office, he won’t get back to bring the thing. Now, calculate in how many different ways he might go to office under these conditions.

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

Problem 8: 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?