Found problems: 8
2020 Brazil Team Selection Test, 4
Let $n$ be an odd positive integer. Some of the unit squares of an $n\times n$ unit-square board are colored green. It turns out that a chess king can travel from any green unit square to any other green unit squares by a finite series of moves that visit only green unit squares along the way. Prove that it can always do so in at most $\tfrac{1}{2}(n^2-1)$ moves. (In one move, a chess king can travel from one unit square to another if and only if the two unit squares share either a corner or a side.)
[i]Proposed by Nikolai Beluhov[/i]
2020 Brazil Team Selection Test, 1
Consider an $n\times n$ unit-square board. The main diagonal of the board is the $n$ unit squares along the diagonal from the top left to the bottom right. We have an unlimited supply of tiles of this form:
[asy]
size(1.5cm);
draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0));
[/asy]
The tiles may be rotated. We wish to place tiles on the board such that each tile covers exactly three unit squares, the tiles do not overlap, no unit square on the main diagonal is covered, and all other unit squares are covered exactly once. For which $n\geq 2$ is this possible?
[i]Proposed by Daniel Kohen[/i]
2020 Brazil Team Selection Test, 8
Let $a_1, a_2,\dots$ be an infinite sequence of positive real numbers such that for each positive integer $n$ we have \[\frac{a_1+a_2+\cdots+a_n}n\geq\sqrt{\frac{a_1^2+a_2^2+\cdots+a_{n+1}^2}{n+1}}.\]
Prove that the sequence $a_1,a_2,\dots$ is constant.
[i]Proposed by Alex Zhai[/i]
2020 Brazil Team Selection Test, 5
There are $2020$ positive integers written on a blackboard. Every minute, Zuming erases two of the numbers and replaces them by their sum, difference, product, or quotient. For example, if Zuming erases the numbers $6$ and $3$, he may replace them with one of the numbers in the set $\{6+3, 6-3, 3-6, 6\times 3, 6\div 3, 3\div 6\}$ $= \{9, 3, 3, 18, 2, \tfrac 12\}$. After $2019$ minutes, Zuming writes the single number $-2020$ on the blackboard. Show that it was possible for Zuming to have ended up with the single number $2020$ instead, using the same rules and starting with the same $2020$ integers.
[i]Proposed by Zhuo Qun (Alex) Song[/i]
2020 Brazil Team Selection Test, 2
Let $f(x) = 3x^2 + 1$. Prove that for any given positive integer $n$, the product
$$f(1)\cdot f(2)\cdot\dots\cdot f(n)$$
has at most $n$ distinct prime divisors.
[i]Proposed by Géza Kós[/i]
2020 Brazil Team Selection Test, 7
Each of the $n^2$ cells of an $n \times n$ grid is colored either black or white. Let $a_i$ denote the number of white cells in the $i$-th row, and let $b_i$ denote the number of black cells in the $i$-th column. Determine the maximum value of $\sum_{i=1}^n a_ib_i$ over all coloring schemes of the grid.
[i]Proposed by Alex Zhai[/i]
2020 Brazil Team Selection Test, 6
Find all integers $n\geq 3$ for which the following statement is true: If $\mathcal{P}$ is a convex $n$-gon such that $n-1$ of its sides have equal length and $n-1$ of its angles have equal measure, then $\mathcal{P}$ is a regular polygon. (A [i]regular [/i]polygon is a polygon with all sides of equal length, and all angles of equal measure.)
[i]Proposed by Ivan Borsenco and Zuming Feng[/i]
2020 Brazil Team Selection Test, 3
Let $ABC$ be a triangle such that $AB > BC$ and let $D$ be a variable point on the line segment $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$, lying on the opposite side of $BC$ from $A$ such that $\angle BAE = \angle DAC$. Let $I$ be the incenter of triangle $ABD$ and let $J$ be the incenter of triangle $ACE$. Prove that the line $IJ$ passes through a fixed point, that is independent of $D$.
[i]Proposed by Merlijn Staps[/i]