BdMO Junior Problems and Solutions 2020

Problem 1: m and n are positive integers such that 1 + 2m = n2. Find the sum of all possible values of 10m+n. [xy = yth power of x. 32 = 9]

Problem 2: Consider rectangle ABCD. Let E be the midpoint of side AD and let F be the midpoint of ED. Let G be the intersection of CE with the line AB and let H be the intersection of BF with line CD. The ratio of areas of the triangle BCG and triangle BCH can be expressed as m/n in lowest term. Compute 10m+10n+mn.

Problem 3: Payel has two 20-sided dice. He rolls them and takes their sum. What number has the highest probability of happening? (A 20-sided dice is a polyhedron (a 3d object) with 20 faces, with the numbers from 1 to 20 on them. Each number has an equal probability of coming up on a roll of the dice)

Problem 4: In a sudoku-tournament, the winner will be selected from play-offs among the top 10 ranked participants. The participants at #10 and #9 of the ranking will challenge each other, the loser will receive 10th prize and the winner will challenge #8. The winner of the first challenge and #8, will challenge #7 and the loser will receive 9th prize. The ultimate winner will be the one who receives the 1st prize. In how many ways these 10 participants may receive the prizes?

Problem 5: For a positive real number x, let [x] be its integer part. For example, [3.14] =3, [5] =5, [6.9] =6. Let z be the largest real number such that [3/z] + [4/z] =5. What is the value of 21z?

Problem 6: Point P lies inside square ABCD such that AP + CP = 27, BP – DP = 17 and ∠DAP = ∠DCP Compute the area of the square ABCD.

Problem 7: Tiham is trying to find 6-digit positive integers PQRSTU (where PQRSTU are not necessarily distinct). But he only wants the numbers where the sum of the 3-digit number PQR, and the 3-digit number STU is divisible by 37. How many such numbers can he find? (Remember, a six-digit number can’t have zero as the first digit!)

Problem 8: Let ABC be a triangle where ∠B = 50 and ∠C = 60. D is the midpoint of BC. The circumcircle of a triangle ABC is defined to be the circle going through the three vertices. The circumcircles of ACD and ABD intersects AB and AC at F and E respectively. The circumcentre of AEF is O. ∠FDO=?

Problem 9: You have 2020 piles of coins in front of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins and so on. So, the 2020th pile contains 2020 coins. A move consists of selected a positive integer k and removing exactly k coins from every pile that contains at least k coins. What is the minimum number of moves required to remove all the coins?

Problem 10: Sakal da is trying to find the largest positive integer n, such that the 7-base representation of n looks like a 10-base number which is exactly 2n. He noticed, one such number is 156, because 156 base 7 is 312. What is the number he came up with? (Writing a number in 10-base means writing it in the decimal system. So, 234 = 200 + 30 + 4 = 2×102 + 3×10 + 3×1. If we go to 7-base, we just change 10 to 7. So, 234 in base 7 would be = 2×72 + 3×7 + 4×1 = 123 in base 10.)

Problem 11:Find the sum of all possible n such that n, n2+10, n2 – 2, n2 – 8, n3 + 6 are all prime numbers. (Hint: there is at least one such n) [xy = yth power of x. 32=9]