Mathematics and Youth Magazine Problems - Jul 2015, Issue 457

  1. Find all natural numbers $n$ satisfying \[2.2^{2}+3.2^{3}+4.2^{4}+\ldots+n.2^{n}=2^{n+34}.\]
  2. Find integers $a,b,c$ such that \[|a-b|+|b-c|+|c-a|=2014^{a}+2015^{a}.\]
  3. Suppose that $f(x)$ is a polynomial with integral coefficients and $f(1)=2$. Show that $f(7)$ is not a perfect square.
  4. Given an acute triangle $ABC$ with altitudes $AH,BK$. Let $M$ be the midpoint of $AB$. The line through $CM$ intersect $HK$ at $D$. Draw $AL$ perpendicular to $BD$ at $L$. Prove that the circle containning $C,K$ and $L$ is tangent to the line going through $BC$.
  5. Solve the following system of equations \[\begin{cases} 9x^{3}+2x+(y-1)\sqrt{1-3y} & =0\\ 9x^{2}+y^{2}+\sqrt{5-6x} & =6 \end{cases}\] for $x,y\in\mathbb{R}$.
  6. Suppose that $f(x)$ is a polynomial of degree $3$ and its leading coefficient is equal to $2$. Also assume that $f(2014)=2015$, $f(2015)=2016$. Find $f(2016)-f(2013)$.
  7. Let $S_{tp}$ and $V$ respectively be the surface area and the volume of the tetrahedron $ABCD.$ Prove that \[\left(\frac{1}{6}S_{tp}\right)^{3}\geq\sqrt{3}V^{2}.\]
  8. Given an $n$-sided convex polygon ($n\geq4$) $A_{1}A_{2}\ldots A_{n}$. Prove that $$\begin{align*} & n+\sin A_{1}+\sin A_{2}+\ldots+\sin A_{n}\\ \leq & 2\left(\cos\frac{A_{1}-A_{2}}{4}+\cos\frac{A_{2}-A_{3}}{4}+\ldots+\cos\frac{A_{n}-A_{1}}{4}\right). \end{align*}$$ When does the equality happen?.
  9. Find all triples $(x,y,p)$ where $x$ and $y$ are positive integers and $p$ is a prime number satisfying $p^{x}-y^{p}=1$.
  10. Let $k$ be a real number which is greater than $1$. Consider the following sequence \[x_{1}=\frac{1}{2}\sqrt{k^{2}-1},\quad x_{2}=\sqrt{\frac{k^{2}-1}{4}+\frac{1}{2}\sqrt{k^{2}-1}},\ldots,\] \[x_{n}=\underset{n\text{ square root symbols}}{\underbrace{\sqrt{\frac{k^{2}-1}{4}+\sqrt{\frac{k^{2}-1}{4}+\ldots+\sqrt{\frac{k^{2}-1}{4}+\frac{1}{2}\sqrt{k^{2}-1}}}}}}.\] Prove that $\left\{ x_{n}\right\} $ converges and find $\lim_{n\to\infty}x_{n}.$ 
  11. For each positive integer $n$, put \[\psi(n)=\sum_{d|n}d^{2}.\] a) Prove that $\psi(n)$ is multiplicative, i.e. \[\psi(ab)=\psi(a)\psi(b)\text{ if }(a,b)=1.\]
    b) Suppose the $l$ is an odd positive integer. Prove that there are only finitely many positive integers $n$ such that $\psi(n)=\psi(n+l)$.
  12. Given a triangle $ABC$ with the circumscribed circle $(O)$ and the inscribed circle $(I)$. The tangent lines to $(O)$ at $B$ and $C$ intersect at $T$. Let $M$ be the midpoint of $BC$ and $D$ be the midpoint of the the arc $BC$ which does not contain $A$. Suppose that $AM$ intersects $(O)$ at $E$ and $AT$ intersects the side $BC$ at $F$. Let $J$ be the midpoint of $IF$. Prove that $\widehat{AEI}=\widehat{ADJ}$.

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