Level 2
Day 1
On a table, there are $16$ weights of the same appearance, which have all the integer weights from $13$ to $28$ grams, that is, they weigh $13, 14, 15, \dots, 28$ grams. Determine the four weights that weigh $13, 14, 27, 28$ grams, using a two-pan balance at most $26$ times.
We say that a set of positive integers is regular if, for any selection of numbers from the set, the sum of the chosen numbers is different from $1810$. Divide the set of integers from $452$ to $1809$ (inclusive) into the smallest possible number of regular sets.
Given a polygon, a triangulation is a division of the polygon into triangles whose vertices are the vertices of the polygon. Determine the values of $n$ for which the regular polygon with $n$ sides has a triangulation with all its triangles being isosceles.
Day 2
Find all positive integers $a$ such that $4x^2 + a$ is prime for all $x = 0, 1, \dots, a - 1$.
We have a convex quadrilateral $ABCD$ with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and $Q$ on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Determine the value of the radius of the circle that passes through the vertices of the triangle $DPQ$.
In the governor elections, there were three candidates: $A$, $B$, and $C$. In the first round, $A$ received $44\%$ of the votes that were cast between $B$ and $C$. No candidate obtained the majority needed to win in the first round, and $C$ was the one who received the least votes of the three, so there was a runoff between $A$ and $B$. The voters for the runoff were the same as in the first round, except for $p\%$ of those who voted for $C$, who chose not to participate in the runoff; $p$ is an integer, $1 \leqslant p \leqslant 100$. It is also known that all those who voted for $B$ in the first round also voted for him again in the runoff, but it is unknown what those who voted for $A$ in the first round did. A journalist claims that, knowing all this, one can infer with certainty who will win the runoff. Determine for which values of $p$ the journalist is telling the truth. Note: The winner of the runoff is the one who receives more than half of the total votes cast in the runoff.