2016 Issue 472

  1. Find all $3$ digit perfecy squares such that if we pick any of those numbers, say $n^{2}$, and then interchange the tens digit and the units digit of that one, we will get $\left(n+1\right)^{2}$.
  2. Consider the following sequence $1$; $3+5$; $7+9+11$; $13+15+17+19$; $\ldots$. Prove that each term of the sequence is a cube of some positive integer.
  3. Find integers $x$ and $y$ such that $x^{2}-2x=27y^{3}$.
  4. Given a triangle $ABC$ with $AB^{2}-AC^{2}=\frac{BC^{2}}{2}$. Suppose futhermore that the angle $BAC$ is obtuse. Let $D$ be the point on the side $AB$ such that $BC=2CD$. The line which goes through $D$ and is perpendicular to $AB$ intersects the line through $AC$ at $E$. Let $K$ be the intersection between the line through $CD$ the line through $BE$. Prove that $K$ is the midpoint of $BE$.
  5. Given positive numbers $a$ and $b$. Prove that $$\frac{a+b}{1+ab}+\left(\frac{1}{1+a}+\frac{1}{1+b}\right)ab+\frac{a+b+2ab}{\left(1+a\right)\left(1+b\right)ab}\geq3.$$ When does the equality happen?.
  6. Solve the equation $$\sqrt[3]{2x^{3}+6}=x+\sqrt{x^{2}-3x+3}.$$
  7. Find the integral solutions of the inequality $$x^{6}-2x^{3}-6x^{2}-6x-17<0.$$
  8. Given a scalene triangle $ABC$ with the orthocenter $H$. The incircle $\left(I\right)$ of the triangle if tangent to the sides $BC$, $CA$, $AB$ respectively at $A_{1}$, $B_{1}$, $C_{1}$. The line $d_{1}$ which goes through $I$ and is paralled to $BC$ intersects the line through $\left(B_{1}C_{1}\right)$ at $A_{2}$. Similarly, we construct the points $B_{2}$, $C_{2}$. Prove that $A_{2}$, $B_{2}$, $C_{2}$ are colinear and the line through these points is perpendicular to $IH$.
  9. Given the equation $$x^{n}+a_{1}x^{n-1}+\ldots+a_{n-1}x+a_{n}=0$$ where the coefficients satisfy $$\begin{align*}
    \sum_{i=1}^{n}a_{i} & =0,\\
    \,\sum_{i=1}^{n-1}\left(n-1\right)a_{i} & =2-n,\\
    \sum_{i=1}^{n-2}\left(n-i\right)\left(n-i-1\right)a_{i} & =3-n\left(n-1\right).
    \end{align*}$$ Suppose that $x_{1},x_{2},\ldots,x_{n}$ are $n$ solutions of the above equation. Compute the value of the following expressions $$\begin{align*}
    E & =\sum_{1\leq i<j\leq n}\frac{1}{\left(1-x_{i}\right)\left(1-x_{i}\right)},\\
    F & =\sum_{i=1}^{n}\frac{1}{\left(1-x_{i}\right)^{2}}.
  10. There are $20$ stones and we divide them into $3$ piles. Then we start to moving stones from one pile to another with the follow rule. Each time we can move half of a pile with even number of stones to any pile. Prove that nomatter how we divide the stones into $3$ piles, after finite of moves, we will get a pile with $10$ stones.
  11. For each positive integer $t$, let $\phi\left(t\right)$ denote the numbers of positive integers which are not greater than $t$ and are relatively prime to $t$ (Euler phi function). Now assume that $n,k$ are positive integers and $p$ is an odd prime number. Prove that there exists a positive integer $a$ such that $p^{k}$ is a divisor of all numbers $\phi\left(a\right),\phi\left(a+1\right),\ldots$ and $\phi\left(a+n\right)$.
  12. Given an acute triangle $ABC$ with the altitudes $AD$, $BE$ and $CF$. The circles with the diameters $AB$ and $AC$ respectively intersect the rays $DF$ and $DE$ at $Q$ and $P$. Let $N$ be the cicumcenter of the triangle $DEF$. Prove that
    a) $AN\perp PQ$.
    b) $AN$, $BP$ and $CQ$ are concurrent.




Mathematics & Youth: 2016 Issue 472
2016 Issue 472
Mathematics & Youth
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