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2017 Issue 476

  1. Consider the sum in the following form $$\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{n+9}=\frac{p}{q}$$ where $n$ is a natural number and $\frac{p}{q}$ cannot be simplidied. Find the smallest $n$ such that $q$ is divisible by $2006$.
  2. Given an isosceles right triangle $ABC$ with the right angle $A$. Let $M$ be the point which is inside the triangle such that $BM=BA$ and $\widehat{ABM}=30^{0}$. Prove that $MA=MC$.
  3. Given positive numbers $a,b,c$ such that $a+b+c=3$. Find the maximum value of the expression $$P=2(ab+bc+ca)-abc.$$
  4. Given a semicircle $O$ with the fixed diameter $AB$. Let $Ax$ be the ray such that $Ax$ is tangent to the semicircle at $A$ and $Ax$ and the semicircle are on the same half-plane determined $AB$. A point $M$, which is different from $A$, varies on the ray $Ax$. Assume that $MB$ intersects the semicircle at the second point $K$ which is different from $B$. On the ray $AB$ choose $N$ such that $AN=AM$. Prove that when $M$ varies, the line which goes through $K$ and is perpendicular to $KN$ always goes through a fixed point.
  5. Solve the equation $$\sqrt{2x^{3}-2x^{2}+x}+2\sqrt[4]{3x-2x^{2}}=x^{4}-x^{3}+3.$$
  6. Solve the system of equations $$\begin{cases} 3^{x}+2^{y} & =5\\ 3^{y}+2^{x} & =5 \end{cases}$$
  7. Given $n$ positive numbers $a_{1},a_{2},\ldots a_{n}$ ($n\geq2$). Let $$S=a_{1}^{n}+a_{2}^{n}+\ldots+a_{n}^{n}, \quad P=a_{1}a_{2}\cdots a_{n}.$$ Prove that $$\frac{1}{S-a_{1}^{n}+P}+\frac{1}{S-a_{2}^{n}+P}+\ldots+\frac{1}{S-a_{n}^{n}+P}\leq\frac{1}{P}.$$
  8. Given a triangle $ABC$. Show that $$\begin{align*} & 2(\sin A+\sin B+\sin C)\left(\cos\frac{A}{2}+\cos\frac{B}{2}+\cos\frac{C}{2}\right)\\ \leq & \frac{45}{4}+(\cos A+\cos B+\cos C)\left(\sin\frac{A}{2}+\sin\frac{B}{2}+\sin\frac{C}{2}\right).\end{align*}$$
  9. Find $$\min_{a,b}\left\{ \max_{-1\leq x\leq1}\left|ax^{2}+bx+1\right|\right\} ,\quad\min_{a,b}\left\{ \max_{-1\leq x\leq1}\left|ax^{2}+x+b\right|\right\} .$$
  10. Given a natural number $a\geq2$. Prove that there exists infinitely many natural numbers $n$ such that $a^{n}+1$ is divised by $n$.
  11. Let $a$ be a real number which is different from $0$ and $-1$. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(f(x)+ay)=(a^{2}+a)x+f(f(y)-x),\forall x,y\in\mathbb{R}.$$
  12. Given an acute triangle $ABC$ with $AB<AC$ and let $AD$, $BE$ and $CF$ be its altitude. Assume that $EF$ intersects $BC$ at $G$. Let $K$ be the perpendicular projection of $C$ on $AG$. Suppose that $AD$ intersects $CK$ at $H$ and $AC$ intersects $HF$ at $L$. Prove that $A$ is the incenter of the triangle $FKL$.

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Mathematics & Youth: 2017 Issue 476
2017 Issue 476
Mathematics & Youth
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_77.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2017/06/mathematics-and-youth-magazine-problems_77.html
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