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

## $show=home 1. Given integers$a, b, c$satisfying$a=b-c=\dfrac{b}{c}$. Prove that$a+b+c$is a cube of some integer. 2. Given an isosceles$A B C$with the acute vertex angle$A$. Draw the altitude$B H$($H$is on$A C)$. The line through$H$and parallel to$B C$and the line through$C$and parallel to$B H$intersect at$E .$Let$M$be the midpoint of$H E .$Find the value of the angle$\widehat{A M C}$. 3. Solve the equation $$x^{2}-3 x+1=-\frac{\sqrt{3}}{3} \sqrt{x^{4}+x^{2}+1}.$$ 4. Given a half circle with the center$O$and the diameter$A B=2 R$. Let$M$be a point on the opposite ray of the ray$A B$. Draw a secant$M C D$of the circle$(C$is between$M$and$D$). Assume that$A D$intersects$B C$at$I$. Determine the position of$M$given that$M C I O$is a cyclic quadrilateral. 5. Given real numbers$a$,$b$and$c$satisfying$\left(1+a^{2}\right)\left(4+b^{2}\right)\left(9+c^{2}\right) \leq 100$. Show that $$-4 \leq 3 a b+2 a c+b c \leq 16.$$ 6. Solve the system of equations $$\begin{cases}\sqrt{x^{2}+x y+y^{2}}+\sqrt{y^{2}+y z+z^{2}}+\sqrt{z^{2}+z x+x^{2}} &=\sqrt{3}(x+y+z) \\ \sqrt{x y z}-(\sqrt{x}+\sqrt{y}+\sqrt{z})+2 &=0 \end{cases}.$$ 7. For any triangle$A B C$, show that $$\tan \frac{A}{4}+\tan \frac{B}{4}+\tan \frac{C}{4}+\tan \frac{A}{4} \tan \frac{B}{4} + \tan \frac{B}{4} \tan \frac{C}{4}+\tan \frac{C}{4} \tan \frac{A}{4} \leq 3(9-5 \sqrt{3}).$$ When does the equality happen? 8. Given a triangle$A B C$inscribed in a circle$(O)$. The medians$A A_{1}$,$B B_{1}$,$C C_{1}$respectively intersect$(O)$at$A_{2}$,$B_{2}$,$C_{2}$. Let$A B=c$,$B C=a$,$C A=b$. Show that $$\frac{A_{1} A_{2}}{a}+\frac{B_{1} B_{2}}{b}+\frac{C_{1} C_{2}}{c} \geq \frac{\sqrt{3}}{2}.$$ 9. Given non-negative numbers$a$,$b$,$c$satisfying$a+b+c=\dfrac{4}{3}$. Prove that $$3\left[a(a-1)^{2}+b(b-1)^{2}+c(c-1)^{2}\right] \geq a b+b c+c a.$$ When does the equality happen? 10. Find all pairs of positive integers$(n ; k)$so that $$(n+1)(n+2) \ldots(n+k)-k$$ is a complete square. 11. Find all functions$f(x)$which are continuous on$[a ; b]$, are differentiable on$(a ; b)$, and satisfy $$f^{\prime}(x) \leq \frac{f(b)-f(a)}{b-a},\, \forall x \in(a ; b),$$ where$a$,$b$are given real numbers with$a < b$. 12. Given a circle$(O)$and a point$K$lying outside the circle. Draw the tangents$K I$,$K J$to the circle at$I$,$J$. On the opposite ray of$I O$take an arbitrary point$O$. The circle with center$O'$radius$O^{\prime} J$intersects$(O)$at the other point$A$.$A I$intersects$\left(O^{\prime}\right)$at the other point$D$. The line through$K$perpendicular to$O^{\prime}D$meets$\left(O^{\prime}\right)$at$B$and$C$. Show that$I$is the center of the incircle of$A B C$. ##$type=three$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: 2019 Issue 503
2019 Issue 503
Mathematics & Youth