Day 1
We define $I(n)$ as the result when the digits of $n$ are reversed. For example, $I(123)=321$, $I(2008)=8002$. Find all integers $n$, $1\leq{n}\leq10000$ for which $I(n)=\lceil{\frac{n}{2}}\rceil$. Note: $\lceil{x}\rceil$ denotes the smallest integer greater than or equal to $x$. For example, $\lceil{2.1}\rceil=3$, $\lceil{3.9}\rceil=4$, $\lceil{7}\rceil=7$.
Let $P$ be a point in the interior of triangle $ABC$. Let $X$, $Y$, and $Z$ be points on sides $BC$, $AC$, and $AB$ respectively, such that $<PXC=<PYA=<PZB$. Let $U$, $V$, and $W$ be points on sides $BC$, $AC$, and $AB$, respectively, or on their extensions if necessary, with $X$ in between $B$ and $U$, $Y$ in between $C$ and $V$, and $Z$ in between $A$ and $W$, such that $PU=2PX$, $PV=2PY$, and $PW=2PZ$. If the area of triangle $XYZ$ is $1$, find the area of triangle $UVW$.
Two friends $A$ and $B$ must solve the following puzzle. Each of them receives a number from the set $\{1,2,…,250\}$, but they don’t see the number that the other received. The objective of each friend is to discover the other friend’s number. The procedure is as follows: each friend, by turns, announces various not necessarily distinct positive integers: first $A$ says a number, then $B$ says one, $A$ says a number again, etc., in such a way that the sum of all the numbers said is $20$. Demonstrate that there exists a strategy that $A$ and $B$ have previously agreed on such that they can reach the objective, no matter which number each one received at the beginning of the puzzle.
Day 2
What is the largest number of cells that can be colored in a $7\times7$ table in such a way that any $2\times2$ subtable has at most 2 colored cells?
Let $ABC$ be an isosceles triangle with base $AB$. A semicircle $\Gamma$ is constructed with its center on the segment AB and which is tangent to the two legs, $AC$ and $BC$. Consider a line tangent to $\Gamma$ which cuts the segments $AC$ and $BC$ at $D$ and $E$, respectively. The line perpendicular to $AC$ at $D$ and the line perpendicular to $BC$ at $E$ intersect each other at $P$. Let $Q$ be the foot of the perpendicular from $P$ to $AB$. Show that $\frac{PQ}{CP}=\frac{1}{2}\frac{AB}{AC}$.
A palindrome is a number that is the same when its digits are reversed. Find all numbers that have at least one multiple that is a palindrome.