### Problem 1:

It’s Ramadan time, and Saad is out buying Iftar! At the store, he can either get 3kg Jilapi and 4kg Halim for 53 Taka or 5kg Jilapi and 2kg Halim for 37 Taka. But Saad really likes Halim and doesn’t like Jilapi at all! How much should it cost him to get 10kg Halim and 1kg Jilapi?

### Solution:

Let, 1kg Halim cost price = x TK. & 1kg Jilapi cost price = y TK.

According to question,

4x + 3y = 53 … … … ( i )

2x + 5y = 37 … … … ( ii )

Solving equation ( i ) & ( ii ), we have

x = 11, y = 3

Therefore, 10kg Halim and 1kg Jilapi cost price = 10 × 11 + 3 = 113 TK.

Problem 2: Soumitra is picking numbers between 1 and 2020, so that no number is picked more than once. How many numbers will Soumitra have to pick (at least) before you can guarantee that there are five of them with sum greater than 1012?

Problem 3: Blue, Mr. Green and Mr. Red go to a restaurant. One of them is wearing a blue shirt, one is wearing a red shirt and the other is wearing a green shirt. Mr. Blue says “Hey, did you notice that we are wearing shirts of a different color from our names?” The person wearing the red shirt says, “Wow Mr. Blue, you’re right!” You must figure out the color of everyone’s shirts. Green = 1, Blue = 2 and Red = 3. Your answer should be 100 × (the color of Mr. Red’s shirt) + 10 × (the color of Mr. Blue’s shirt) + 1 × (the color of Mr. Green’s shirt).

### Problem 4:

ABC is an isosceles triangle. Its two sides AB and AC are equal to one another. Angle ∠BAC = 40 degrees. The baseline BC is extended up to D. How many degrees is the angle ∠ACD?

### Solution:

Here, ABC is an isosceles triangle and AB = AC, ∠BAC = 40^{0}. The baseline extended up to D. ∠ACD =?

In triangle ABC,

∠BAC + ∠ABC + ∠ACB = 180^{0} [Triangle Theorem]

Or, ∠ABC + ∠ACB = 180^{0} – 40^{0} [Isosceles Triangle Property]

Or, ∠ACB + ∠ACB = 140^{0}

Or, 2∠ACB = 140^{0}

Or, ∠ACB = 70^{0}

Now, ∠ACD = ∠BAC + ∠ABC = 40^{0} + 70^{0} = 110^{0}.

Problem 5: Promi is trying to find the smallest positive integer n, such that n is a multiple of 30 and each digit of n is either 0 or 5. What is the smallest number Promi will come up with?

Problem 6: Shopkeeper Rubab is lazy, so he doesn’t track Paisa and rounds prices down to the nearest Taka. (If the total price of an order is 11-taka 30 paisa, Rubab calculates it as 11 TK.) The cost of five chocolates in Rubab’s shop (according to his calculation) is 13 Tk. and the cost of six chocolates is 16 TK. All chocolates have the same cost, which is an integer number of paisa. How many different prices can one chocolate possibly have?

Problem 7: Shakur and Tihar play a game with coins. There are N coins on a table where N is a positive integer. The players take turns removing a number of coins that is an integer power of 2 (such as 1, 2, 4, 8, 16, 32, …). The person who removes the last coin from the table wins. If Shakur goes first and N is a positive integer less than 1000, for how many values of N does Shakur have a winning strategy (if Tihar plays perfectly)?

Problem 8: Brishty writes the numbers 1, 2, 3, …, 9 on a board in that order. In a move she can pick any 3 adjacent numbers and reverse their order. For example, (1, 2, 3, 4, …) can become (3, 2, 1, 4, …). How many distinct sequences can she make using one or more such moves?

Problem 9: If n is a positive integer, then p/q is the fraction n/(n+675) in its lowest terms. What is the sum of all different possible values of (q-p)? Here ‘lowest terms’ means the common factors have been cancelled out, so that the GCD of the numerator and denominator is 1.

Problem 10: There are 2020 points on a piece of paper, no three of which are on the same line. Zawad wants to join as many of them as possible with line segments. But he does not want three points to become vertices of a triangle! What is the maximum number of lines Zawad can draw?

Problem 11: The hour and minute hands on an analog clock is the same size, and so you can’t tell them apart! The hands move continuously. How many times between noon and midnight is the information on the clock not enough to tell the time?