Let $x$ and $y$ be positive reals such that \[ x^3 + y^3 + (x + y)^3 + 30xy = 2000. \] Show that $x + y = 10$.

Click for solution $[x^3+y^3-10^3+30xy]+[(x+y)^3-1000]=0$ And factorise,

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer. Bulgaria

Click for solution Tiks wrote: Hi shyong,do you now what means RHS>LHS : I think he means for $n>3$ , $2n<3^x-3^y$ is always true . since we know that $min\{3^x-3^y\}=3^{p+1}-3^p=3^p\cdot 2$ where $2p+1=n$ But bernoulii inequality tell us that for $p\geq 1$ $3^p=(1+2)^p\geq 1+2p = n$ the equality holds when $p=1$ which is $n=3$ . So for any $n>3$ we always have $3^x-3^y>2n$ , and there is no solution for the equality to hold for $n>3$ .

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. Albania

Click for solution If you consider $A'$ intersection of $EQ$ and $FP$ and $A''$ intersection of $EP$ and $FQ$ then $A'A''$is perpendicular to $BC$. And $A$ is midpoint of $A'A''$

At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is knowns that there were no draws. Find $n$. Serbia

Click for solution in fact there are 2n boys and n girls playing at a tenning tournament. the problem is extremely simple: the boys won all the games they played between them namely n(2n-1) and the also won x games with the girls. the girls won the games between then, namely $\displaystyle \frac {n(n-1)}2 + 2n^2-x$, and all the games between them and the boys which the boys didn't win (obviously). we thus obtain $5n(2n-1) + 5x = \displaystyle 7\frac {n(n-1)}2 + 14n^2-7x$, and we get $x$ out of this and we obtain that $x$ is an integer iff $n$ gives residues 5 or 3 modulo 8 (or something like that). I see no combinatorics in this problem ... do you?