CONO-SUR 2008

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.

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.