2016 Issue 473

  1. Find the last two digits of the difference $2^{9^{2016}}-2^{9^{1945}}$.
  2. Find natural numbers $n$ such that $$n^{6}+8n^{3}+19n^{2}-33n-90=0.$$
  3. Find positive integers $x$ and $y$ so that $$A=x^{2}+y^{2}+\frac{x^{2}y^{2}}{\left(x+y\right)^{2}}$$ is a perfect square.
  4. Given an acute triangle $ABC$ with two altitudes $BE$ and $CK$. Let $R$ and $S$ respectively be the perpendicular projections of $K$ on $BC$ and $BE$, and $P$ and $Q$ respectively the perpendicular projections of $E$ on $BC$ and $CK$. Prove that the lines $RS,PQ$ and $EK$ are concurrent.
  5. Solve the system of equations $$\begin{cases} 5x^{2}+2y^{2}+z^{2} & =2\\ xy+yz+zx & =1 \end{cases}$$
  6. Find integers $m$ so that the equation $$x^{3}+\left(m+1\right)x^{2}-\left(2m-1\right)x-\left(2m^{2}+m+4\right)=0$$ has an integer solution.
  7. Let $x,y,x$ be real numbers satisfying $$\begin{cases} x+z-yz & =1\\ y-3z+xz & =1 \end{cases}.$$ Find the maximum and minimum values of the expression $T=x^{2}+y^{2}$.
  8. Given a triangle $ABC$. Let $\left(O\right)$ be its circumcircle. The excircle $\left(I\right)$ relative to the vertex $A$ is tangent to $AB$ and $AC$ respectively at $D$ and $E$. Let $A'$ be the intersection between $OI$ and $DE$. Similarly, we construct the points $B'$ and $C'$. Prove that $AA',BB'$ and $CC'$ are concurrent.
  9. Let $a,b,c$ be positive numbers such that $a+b+c=1$. Prove that $$a^{b}b^{c}c^{a}\leq ab+bc+ca.$$
  10. Show that there exist infinitely many positive integers $n$ so that $n^{2}+1$ has a prime divisor $p$ which is greater than $2n+\sqrt{10n}$.
  11. Given two arbitrary positive integers $n$ and $p$. Find the number of functions $$f:\left\{ 1,2,3,\ldots,n\right\} \to\left\{ -p,-p+1,-p+2,\ldots,p\right\}$$ which satisfy the property $\left|f\left(i\right)-f\left(j\right)\right|\leq p$ for any $i,j\in\left\{ 1,2,3,\ldots,n\right\} $.
  12. Given a non-right triangle $ABC$. Let $O$ and $H$ respectively be the circumcenter and the orthocenter of $ABC$. Choose an arbitrary point $M$ on $\left(O\right)$ so that $M$ is different from $A,B$ and $C$. Let $N$ denote the symmetric point of $M$ through $BC$. Let $P$ be the second intersection between $AM$ and the circumcircle of $OMN$. Prove that $HN$ goes through the orthocenter of $AOP$.




Mathematics & Youth: 2016 Issue 473
2016 Issue 473
Mathematics & Youth
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