BdMO Higher Secondary Problems and Solutions 2020

Problem 1: Lazim rolls two 24-sided dice. From the two rolls, Lazim selects the die with the highest number. N is an integer not greater than 24. What is the largest possible value for N such that there is a more than 50% chance that the die Lazim selects is larger than or equal to N?

Problem 2: How many integers n is there subject to the constraint that 1≤n≤2020 and nn is a perfect square? [By xy, we mean the yth power of x, for example 32=9]

 

Problem 3: Let R be the set of all rectangles centered at the origin and with perimeter 1 (the center of a rectangle is the intersection point of its two diagonals). Let, S be a region that contains all of the rectangles in R (region A contains region B, if B is completely inside of A). The minimum possible area of S has the form πa, where a is a real number. Find 1/a.

Problem 4: 56 lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly 594 points, what is the maximum number of them that could have the same slope?

Problem 5: In triangle ABC, AB = 52, BC = 34 and CA = 50. We split BC into n equal segments by placing n-1 new points. Among these points are the feet of the altitude, median and angle bisector from A. What is the smallest possible value of n?

Problem 6: f is a one-to-one function from the set of positive integers to itself such that f(xy) = f(x) × f(y). Find the minimum possible value of f (2020).

Problem 7: f is a function on the set of complex numbers such that f(z)=1/(z*), where z* is the complex conjugate of z. S is the set of complex numbers z such that the real part of f(z) lies between 1/2020 and 1/2018. If S is treated as a subset of the complex plane, the area of S can be expressed as m×π where m is an integer. What is m? (Note: the complex plane is just like the Cartesian plane, where the x-axis is renamed as the real-axis and the y-axis is renamed as the imaginary axis.) Remember, a complex number is a number of the form a + bi, where a and b are real numbers and i is a number such that i × i = -1. If z = a + bi, the complex conjugate of z is z* = a – bi.

Problem 8: We call a permutation of the numbers 1, 2, 3, …, n ‘kawaii’ if there is exactly one number that is greater than its position. For example: 1, 4, 3, 2 is a ‘kawaii’ permutation (when n=4) because only the number 4 is greater than its position 2. How many kawaii permutations are there if n=14?

Problem 9: Let ABCD be a convex quadrilateral. O is the intersection of AC and BD. AO=3, BO=4, CO=5, DO=6. X and Y are points in segment AB and CD respectively, such that X, O, Y are collinear. The minimum of XB/XA+YC/YD can be written as (a√c)/b, where a/b is in lowest term and c is notd ivisible by any square number greater than 1. What is the value of 10a+b+c?

Problem10: Bristy wants to build a special set A. She starts with A = {0, 42}. At any step, she can add an integer x to the set A if it is a root of a polynomial which uses the already existing integers in A as coefficients. She keeps doing this, adding more and more numbers to A. After she eventually runs out of numbers to add to A, how many numbers will be in A?

Problem 11: ABCD is a convex quadrilateral where BC = CD, AC = AD, ∠BCD = 96° and ∠ACD = 69°. Set P_0 = A, Q_0 = B respectively. We inductively define P_(n+1) to be the center of the incircle of CDP_n, and Q_(n+1) to be the center of the incircle of CDQ_n. If ∠Q_(2024)Q_(2025)P_(2025) – 90° = (2k-1)/2^n, compute k+n.

 

 

 

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