# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 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 $$\sqrt{2x^{3}-2x^{2}+x}+2\sqrt{3x-2x^{2}}=x^{4}-x^{3}+3.$$ 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 numbera\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$FKL$. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2017 Issue 476
2017 Issue 476
Mathematics & Youth