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

## $show=home 1. Find natural numbers$a$and$b$given that the sum of four numbers$a+b$,$a-b$,$ab$,$a: b$is equal to$1575$. 2. Given triangle$A B C$with$\widehat{A}=120^{\circ}$,$\widehat{B}=40^{\circ}$. On the side$A C$choose the point$M$such that$A B=A M$. On the opposite ray of$A B$choose the point$N$such that$\widehat{A M N}=40^{\circ}$. Find the measurement of the angle$\widehat{B N C}$. 3. Find all pairs of integers$(x ; y)$satisfying$0 \leq x+y \leq 6$and $$x-\frac{1}{x^{3}}=y-\frac{1}{y^{3}}.$$ 4. Given an acute triangle$A B C$inscribed in a circle$(O ; R)$. The altitudes$A D$,$B E$,$C F$meet at$H$. Let$M$be the midpoint of$A H$. Draw$M N$perpendicular to$B M$,$N$is on$A C$. Show that$O N || B C$and$E M=R \cos A$. 5. Solve the equation $$x^{2} \sqrt{2-x^{4}}-x^{4}+x^{3}-1=0.$$ 6. Given real numbers$x, y, z$such that$x^{2}+y^{2}+z^{2}=3$. Prove that $$8(2-x)(2-y)(2-z) \geq(x+y z)(y+x z)(z+x y)$$ 7. Find all triangles$A B C$such that the lengths of all sides are positive integers and the length of$A C$is equal to the length of the interior angle bisector of the angle$A$. 8. Given an acute triangle$A B C$inscribed in a circle$(O)$. Let$I_{a}$,$I_{b}$,$I_{c}$respectively be the centers of the excircles of the angles$A$,$B$,$C$. The line$A I_{a}$intersects$(O)$at$D$which is different from$A$. On$I_{b} D$,$I_{c} D$respectively choose the points$E$,$F$such that$\widehat{A B C}=2 \widehat{I_{a} B E}$,$\widehat{A C B}=2 \widehat{I_{a} C F}$,$E$,$F$are inside the trangle$I_{a} B C$. Show that$E F$intersects$I_{b} I_{c}$at some point on$(O)$. 9. Given a function $$f(x)=\frac{x^{2}+a x+b}{x^{2}+1}$$ with$a, b$are integers. Suppose the range of$f(x)$is the set of$11$integers. Find the maximum and minimum values of the expression$M=a^{2}+b^{2}$. 10. For each positive integer$n$, let$f(n)=\left(n^{2}+n+1\right)^{2}+1$. Find the smallest positive integer$k$such that$$f(n) \cdot f(n+1) \ldots f(n+k-1)$$ is a perfect square for some positive integer$n$. 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$satisfying $$f(f(x)+y)=f(f(x))-y f(x)+f(-y)-2,\, \forall x, y \in \mathbb{R}.$$ 12. Given an isosceles triangle$A B C$with the vertex angle$A$inscribed in a circle$(O)$. Suppose that$A D$is a diameter of$(O)$. The points$E$,$F$respectively on$D C$,$D B$. Let$G$be on$E F$such that$\dfrac{G F}{G E}=\dfrac{F B}{C E}$. Show that$C G$and$A F$meets each other at some point on$(O)$. ##$type=three$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 527
2021 Issue 527
Mathematics & Youth