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2017 Issue 475

  1. Given a natural number $n$. Find all prime numbers $p$ such that the following number $$A=1010n^{2}+2010(n+p)+10^{10^{1954}}$$ can be written as a difference of two perfect squares.
  2. Given a triangle $ABC$ and let $M$ be the midpoint of $BC$. Suppose that $\widehat{ABC}+\widehat{MAC}=90^{0}$. Prove that $ABC$ is either an isosceles triangle or a right triangle.
  3. Solve the equation $$
    \frac{x}{2\sqrt{x}+1}+\frac{x^{2}}{2\sqrt{x}+3}=\frac{\sqrt{x^{3}}+x}{4}.$$
  4. Give a isoceles trapezoid $ABCD$ inscribed in a circle $(O)$ with $AB$ is parallel to $CD$ and $AB<CD$. Let $M$ be the midpoint of $CD$and let $P$ be any point on the side $MD$. Suppose that $AP$ intersects $(O)$ at the second point $Q$, and $BP$ intersects $(O)$ at the second point $R$. Assume that QR intersects $CD$ at $E$. Let $F$ be the symmetry point of $P$ over the point $E$. Suppose that $EA$ is tangent to $(O)$, prove that $AF$ is perpendicular to $AQ$.
  5. Let $x,y$ belong to $(0,1)$. Find the maximum value of the expression $$
    P=x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}+\frac{1}{\sqrt{3}}(x+y).$$
  6. Given $f\left(x\right)=x^{2}-6x+12$. Solve the equation $f(f(f(f(x))))=65539$.
  7. Find all real solutions of the following system of equations:
    $$\begin{cases} \sqrt{2^{2^{x}}} & =2(y+1)\\ 2^{2^{y}} & =2x \end{cases}.$$
  8. Given a uniform triangular prism $ABC,A'B'C'$ (the base $ABC$ is an equilateral triangle). Let $\alpha$ be the angle between $A'B$ and $(ACC'A')$ and let $\beta$ be the angle between $(A'B'C')$ and $(ACC'A')$. Prove that $\alpha<60^{0}$ and $\cot^{2}\alpha-\cot^{2}\beta=\frac{1}{3}$.
  9. Let $p$, $q$ be two coprime numbers. Let $$ T=\left[\frac{p}{q}\right]+\left[\frac{2p}{q}\right]+\ldots+\left[\frac{(q-1)p}{q}\right],$$ where $[x]$ is the gretest integral number, that isn't execeed $x$ and is called the integral part of $x$.
    a) Find $T$.
    b) Find $p$, $q$ such that $T$ is a prime number.
  10. Let $S$ be the number of all the binary strings of length $n$ with the property that the sum of any $3$ consecutive numbers on any of these strings is always positive. Prove that $$2\left(\frac{3}{2}\right)^{n}\leq S\leq\frac{13}{2}\cdot\left(\frac{20}{9}\right)^{n},\quad\forall n\geq3.$$
  11. Find all functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that $$ f\left(\frac{x}{x-y}\right)+f\left(xf(y)\right)=f\left(xf(x)\right),$$ for every $x>y>0$.
  12. Given a triangle $ABC$. The incircle $(I)$ of $ABC$ is tangent to $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let $B_{1}$(resp. $C_{1}$) be the intersection between the lines which go through $AB$ and $DE$ (resp. $AC$ and $DF$). Let $H$ and $K$ respectively be the orthocenter of $ABC$ and $AB_{1}C_{1}$. Prove that the line which goes through $HK$ contains the point $I$.

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Mathematics & Youth: 2017 Issue 475
2017 Issue 475
Mathematics & Youth
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_64.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_64.html
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