BdMO Secondary Problems and Solutions 2020

Problem 1: Let, m be a real number such that the following equation holds:
3m = 4m
Compute the sum of all possible distinct values that (3^3^m)) / m4 can take. [xy = yth power of x. 32 = 9]

Problem 2: A polygon is called beautiful if you can pick three of its vertices to have an angle of 144 degree between them. Compute the number of integers n greater than 8 and no greater than 2024 for which a regular n-gon is beautiful.

Problem 3: In a party of 11 people, certain pairs of people shake hands with each other. In every group of three people, there exists one person who shakes hands with the other two. What is the minimum number of handshakes that can take place in this party?

Problem 4: ABCD is a square. P and Q are two points in segment BC and CD respectively such that ∠APQ = 90°. It is given that AP = 4 and PQ = 1. If we express the length of segment AB as m/n in lowest term, compute m+10n.

Problem 5: Let x_1, x_2, … be real numbers so that for all n > 0,
x_{n+3} = x_{n+2} – 2x_{n+1} + x_n. Suppose x_1 = x_3 = 1 and you’re given that x_{98} = x_{99}. Find the sum x_1 + x_2 + … + x_{100}

Problem 6: A permutation (a_1, a_2, a_3, …, a_n) of the numbers (1, 2, 3, …, n) is called almost-sorted if there exists exactly one i ∈ {1, 2, 3, …, n-1} such that a_i > a_(i+1). What is the number of almost sorted permutations of the numbers (1, 2, 3, …, 13)?

Problem 7: Let f be a function from the set of positive integers to the set of positive integers such that for each positive integer n, if x_1, x_2, …, x_s are all the positive divisors of n, then f(x_1)f(x_2)…f(x_s)=n. Find the sum of all possible values of f(343)+f(3012).

Problem 8: f: Z mapping to Z

f(f(x+y)) = f(x^2) + f(y^2)
f(f(2020)) = 1010.
Find f(2025).
[xy = yth power of x. 32=9]

Problem 9: In triangle ABC, AB= 12, BC=20, CA=16. X and Y are two points in segment AB and AC respectively. K is a point in segment XY, such that XK/KY=7/5. If we let X and Y vary in segment AB and AC, all the positions of K cover a region. If we express the area of that region as m/n in lowest term, compute m + n.

Problem 10: Rahul is at (3,3) on the coordinate plane. In each step he can move one point up or one point to the right. He loves primes, and will never visit a coordinate point where both values are composite. In how many ways can he reach (20,13)?

Problem 11: Urmi is playing a game on a computer. If the computer screen displays the number x, then in the next move, Urmi can do one of the following:
a) Replace x by 4x + 1
b)  Replace x by the largest integer not greater than x/2 Initially the computer screen displays 0. How many different integers less than or equal to 2020 can Urmi achieve through a sequence of moves? It is permitted for the number displayed on the screen to exceed 2020 during the sequence.

 

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