MathDB

2021 OMpD

Part of Mathematicians for Fun Olympiad

Subcontests

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Numbers in black and white squares are in a specific ratio

Determine the smallest positive integer nn with the following property: on a board n×nn \times n, whose squares are painted in checkerboard pattern (that is, for any two squares with a common edge, one of them is black and the other is white), it is possible to place the numbers 1,2,3,...,n21,2,3 , ... , n^2, a number in each square, so if BB is the sum of the numbers written in the white squares and PP is the sum of the numbers written in the black squares, then BP=20214321\frac {B}{P} = \frac{2021}{4321}.

Davi Lopes vs Lavi Dopes, a legendary battle

Let nn be a positive integer. Lavi Dopes has two boards n×nn \times n. On the first board, he writes an integer in each of his n2n^2 squares (the written numbers are not necessarily distinct). On the second board, he writes, on each square, the sum of the numbers corresponding, on the first board, to that square and to all its adjacent squares (that is, those that share a common vertex). For example, if n=3n = 3 and if Lavi Dopes writes the numbers on the first board, as shown below, the second board will look like this.
Next, Davi Lopes receives only the second board, and from it, he tries to discover the numbers written by Lavi Dopes on the first board.
(a) If n=4n = 4, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?
(b) If n=5n = 5, is it possible that Davi Lopes always manages to find the numbers written by Lavi Dopes on the first board?
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n-variable inequality with absolute values

Let aa and bb be positive real numbers, with a<ba < b and let nn be a positive integer. Prove that for all real numbers x1,x2,…,xn∈[a,b]x_1, x_2, \ldots , x_n \in [a, b]: ∣x1−x2∣+∣x2−x3∣+⋯+∣xn−1−xn∣+∣xn−x1∣≤2(b−a)b+a(x1+x2+⋯+xn) |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n) And determine for what values of nn and x1,x2,…,xnx_1, x_2, \ldots , x_n the equality holds.

Simple diophantine equation

Determine all pairs of integer numbers (x,y)(x, y) such that: (x−y)2x+y=x−y+6\frac{(x - y)^2}{x + y} = x - y + 6
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Joining P, H, I particles

A Physicist for Fun discovered three types of very peculiar particles, and classified them as PP, HH and II particles. After months of study, this physicist discovered that he can join such particles and obtain new particles, according to the following operations:
• A PP particle with an HH particle turns into one II particle;
• A PP particle with an II particle turns into two PP particles and one HH particle;
• An HH particle with an II particle turns into four PP particles;
Nothing happens when we try to join particles of the same type. It is also known that the physicist has 2222 PP particles, 2121 HH particles and 2020 II particles.
(a) After a finite number of operations, what is the largest possible number of particles that can be obtained? And what is the smallest possible number of particles?
(b) Is it possible, after a finite number of operations, to obtain 2222 PP particles, 2020 HH particles, and 2121 II particles?
(c) Is it possible, after a finite number of operations, to obtain 3434 HH particles and 2121 II particles?

hexagons and areas

Let ABCDEFABCDEF be a regular hexagon with sides 1m1m and OO as its center. Suppose that OPQRSTOPQRST is a regular hexagon, so that segments OPOP and ABAB intersect at XX and segments OTOT and CDCD intersect at YY, as shown in the figure below. Determine the area of the pentagon OXBCYOXBCY.
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Find the bad apple with the minimum number of moves. Beautiful Problem.

Snow White has, in her huge basket, 20212021 apples, and she knows that exactly one of them has a deadly poison, capable of killing a human being hours after ingesting just a measly piece. Contrary to what the fairy tales say, Snow White is more malevolent than the Evil Queen, and doesn't care about the lives of the seven dwarfs. Therefore, she decided to use them to find out which apple is poisonous.
To this end, at the beginning of each day, Snow White prepares some apple salads (each salad is a mixture of pieces of some apples chosen by her), and forces some of the dwarfs (possibly all) to eat a salad each. At the end of the day, she notes who died and who survived, and the next day she again prepares some apple salads, forcing some of the surviving dwarves (possibly all) to eat a salad each. And she continues to do this, day after day, until she discovers the poisoned apple or until all the dwarves die.
(a) Prove that there is a strategy in which Snow White manages to discover the poisoned apple after a few days.
(b) What is the minimum number of days Snow White needs to find the poisoned apple, no matter how lucky she is?

angle wanted, perpendicular on angle bisector, summetric point, circumcircle

Let ABCABC be a triangle with ∠BAC>90o\angle BAC > 90^o and with AB<ACAB < AC. Let rr be the internal bisector of ∠ACB\angle ACB and let ss be the perpendicular, through AA, on rr. Denote by FF the intersection of rr and s s, and denote by EE the intersection of ss with the segment BCBC. Let also DD be the symmetric of AA with respect to the line BFBF. Assuming that the circumcircle of triangle EACEAC is tangent to line ABAB and D D lies on rr, determine the value of ∠CDB\angle CDB.
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