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

## $show=home 1. Find all prime numbers$p$,$q$and positive integers$n$so that $$p(p+1)+q(q+1)=n(n+1).$$ 2. Find all prime numbers$a, b, c, d$satisfying $$\begin{cases}a+b^{2} &=c d \\ c+d^{2} &=17 b \end{cases}.$$ 3. Find all solutions which are prime numbers of the following equation $$x^{y}+y^{x}+(x+y+1)^{3}=x^{3}+y^{3}+z+1.$$ 4. Let$M$be a point inside a square$A B C D$. The rays$A M$,$B M$,$C M$,$D M$respectively intersect the circle circumscribing the square at$E$,$F$,$G$,$H$. The tangents of the circle at$F$and$H$meet at$K$Show that three points$K$,$G$,$E$are collinear. 5. Given positive numbers$a, b, c$satisfying$a+b+c=3$. Show that $$\frac{a^{3}+b^{2}+c^{2}}{a^{2}+1}+\frac{b^{3}+c^{2}+a^{2}}{b^{2}+1}+\frac{c^{3}+a^{2}+b^{2}}{c^{2}+1} \geq \frac{9}{2}.$$ 6. Solve the system of equations $$\begin{cases} x^{z}+y^{z}-2 &=z^{3}-z \\ y^{x}+z^{x}-2 &=x^{3}-x \\ z^{y}+x^{y}-2 &=y^{3}-y\end{cases}$$ where$x, y, z$are integers. 7. Find real solutions of the equation $$x 2^{x^{2}}=2^{2 x+1}.$$ 8. Given a triangle$A B C$inscribed in a circle$(O)$. Let$M$be a point on the arc$B C$which does not contain$A$($M$is different from$B$and$C$). Draw$B E$perpendicular to$A M$($E$is on$A M$). Let$N$be the intersection hetween$A M$and$B C$, and$H$the orthocenter of the triangle$C M N$. Show that the line$H E$always passes through a fixed point when$M$varies. 9. Given$n$positive numbers$x_{1}, x_{2}, \ldots, x_{n}$. Find the minimum value of the expression $$S=\frac{\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{1+2+\ldots+n}}{x_{1} x_{2}^{2} \ldots x_{n}^{n}}.$$ 10. Find all functions$f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$satisfying $$f(x f(y))+f\left(y^{2021} f(x)\right)=x y+x y^{2021},\, \forall x, y \in \mathbb{R}^{+}.$$ 11. Consider the sequence$\left\{a_{n}\right\}$with $$a_{1}=1,\, a_{2}=\frac{1}{2},\quad n a_{n}=(n-1) a_{n-1}+(n-2) a_{n-2} .$$ Find$\displaystyle\lim_{n\to\infty} \frac{a_{n}}{a_{n-1}}$. 12. Given a triangle$A B C$and a point$M$on the line passes through$B$,$C$($M$is different from$B$,$C$). Let$K$,$L$respectively be the second intersections between the circumcircles of the triangles$A M B$,$A M C$and another line which passes through$M$and different from$M A$and$B C$. Let$P$,$Q$respectively be the perpendicular projections of$A$on$B K$,$C L$.$B K$and$C L$meet at$R$,$P O$and$M K$meet at$N$. Show that a)$\dfrac{N P}{N Q}=\dfrac{M B}{M C}$. b) If$M$is the midpoint of$B C$then$A K R L$is a harmonic quadrilateral (a harmonic quadrilateral is a cyclic quadrilateral in which the products of the lengths of opposite sides are equal). ##$type=three$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: 2021 Issue 523
2021 Issue 523
Mathematics & Youth