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

## $show=home 1. Find all triple of natural numbers$a, b, c$less than$20$such that $$a(a+1)+b(b+1)=c(c+1)$$ where$a$is a prime number and$b$is a multiple of$3 .$2. Let$f(n)=\left(n^{2}+n+1\right)^{2}+1,$where$n$is a positive integer and let $$P_{n}=\frac{f(1) \cdot f(3) \cdot f(5) \ldots f(2 n-1)}{f(2) \cdot f(4) \cdot f(6) \ldots f(2 n)}.$$ Prove the inequality $$P_{1}+P_{2}+\ldots+P_{n}<\frac{1}{2}.$$ 3. Find the maximum value of the expression $$T=\frac{(y+z)^{2}}{y^{2}+z^{2}}-\frac{(x+z)^{2}}{x^{2}+z^{2}},$$ where$x$,$y$,$z$are real numbers such that$x>y$,$z>0$and$z^{2} \geq x y$. 4. Solve for$x$$$\sqrt{14-x^{3}}+x=2\left(1+\sqrt{x^{2}-2 x-1}\right).$$ 5. An acute triangle$A B C$is inscribed in a fixed circle with center at$O$. Let$A I$,$B D$and$C E$denote the altitudes through$A$,$B$and$C$respectively. Prove that the perimeter of the triangle$I D E$does not change when$A, B,$and$C$move on the circle$(O)$such that the area of the triangle$A B C$is always equal to$a^{2}$. 6. Solve the system of equations $$\begin{cases} \sqrt{x^{2}+91} &=\sqrt{y-2}+y^{2} \\ \sqrt{y^{2}+91} &=\sqrt{x-2}+x^{2}\end{cases}.$$ 7. Let$a, b, c$be real numbers such that$4(a+b+c)-9=0 .$Find the maximum value of the sum $$S=\left(a+\sqrt{a^{2}+1}\right)^{b}\left(b+\sqrt{b^{2}+1}\right)^{c}\left(c+\sqrt{c^{2}+1}\right)^{a}.$$ 8. Let$I$and$O$denote respectively the incenter and the circumcenter of a triangle$A B C .$Given that$\widehat{A I O}=90^{\circ},$prove that the area of the triangle$A B C$is less than$\dfrac{3 \sqrt{3}}{4} A I^{2}$. 9. Let$a, b, n$be positive integers,$b>1$and$a$is a multiple of$b^{n}-1 .$Rewritten$a$to the base$b,$prove that the resulting contains at least$n$non-zero digits. 10. Let$x, y, z$be real numbers such that$0<z \leq y \leq x \leq 8$and $$3 x+4 y \geq \max \left\{x y ; \frac{1}{2} x y z-8 z\right\}.$$ Find the maximum value of $$A=x^{5}+y^{5}+z^{5}.$$ 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$such that $$f\left(f(x)+y^{2}\right)=f^{2}(x)-f(x) f(y)+x y+x.$$ 12. Let$K$denote the intersection of the two diagonals of a quadrilateral$A B C D$where$\widehat{A B C}=\widehat{A D C}=90^{\circ} ;$and$A C=A B+A D .$Prove that the radii of the inscribed of the triangles$A B K$and$A D K$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: 2008 Issue 374
2008 Issue 374
Mathematics & Youth