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

## $show=home 1. Let $$S=1\cdot2^{0}+2\cdot2^{1}+3\cdot2^{2}+\ldots+2016\cdot2^{2015}$$ where$2^{0}=1$. Compare$S$and$2015\cdot2^{2016}$. 2. Given a right triangle$ABC$with the right angle$A$and$AB<AC$. On the opposite ray of the ray$AB$choose$D$so that$BD=AC$and so the opposite ray of the ray$CA$choose$E$so that$CE=AD$. The ray$DC$intersects$BE$at$F$. Find the angle$\widehat{CFB}$. 3. Let$a,b,c$be positive integers such that $$\frac{a^{2}+1}{bc}+\frac{b^{2}+1}{ca}+\frac{c^{2}+1}{ab}$$ is also a posotive integer. Prove that $$\left(a,b,c\right)\leq\left[\sqrt{a+b+c}\right],$$ where$\left(a,b,c\right)$is the greatest common divisor of$a,b,c$and$\left[x\right]$is the integer part of$x$. 4. Given a right triangle$ABC$with the right angle$A$with$AB<AC$,$BC=2+2\sqrt{3}$, and the inradius is equal to$1$. Find the angles$B$and$C$. 5. Solve the equation $$\frac{1}{\sqrt{x+1}}+\frac{1}{\sqrt{2x+1}}+\frac{1}{\sqrt{1-2x}}=\frac{4\sqrt{10}}{5}.$$ 6. Solve the system of equations $$e^{x}=y+\sqrt{z^{2}+1}$$ $$e^{y}=z+\sqrt{x^{2}+1}$$ $$e^{z}=x+\sqrt{y^{2}+1}$$ 7. Which number is bigger $$\sin\left(\cos x\right)\text{ or }\cos\left(\sin x\right)?$$ 8. Let$A,B$and$C$be the angles of a triangle. Prove that $$\frac{1}{\sin A}+\frac{1}{\sin B}+\frac{1}{\sin C}\leq\frac{2}{3}\left(\cot\frac{A}{2}+\cot\frac{B}{2}+\cot\frac{C}{2}\right).$$ 9. Find the maximal positive value of$T$such that the following inequality $$\frac{a+b}{b\left(a+1\right)}+\frac{b+c}{c\left(b+1\right)}+\frac{c+a}{a\left(c+1\right)}\geq T$$ always holds for all positive numbers$a,b,c$satisfying$abc=1$. 10. Find all pairs of positive integers$x,y$so that$x^{2}$is divisible by$2xy^{2}-y^{2}+1$. 11. Find all monotonic funtions$f:\left(0,+\infty\right)\to\mathbb{R}$such that $$f\left(x+y\right)=x^{2016}f\left(\frac{1}{x^{2015}}\right)+y^{2016}f\left(\frac{1}{y^{2015}}\right)$$ for all positive numbers$x$and$y$. 12. Given an isosceles triangle$ABC$with the vertex angle$\hat{A}<90^{0}$. Let$CD$be the altitude from$C$. Let$E$be the midpoint of$BD$and let$M$be the midpoint of$CE$. The angle bisector of$\widehat{BDC}$intersects$CE$at$P$. The circle with center at$C$and with radius$\left|CD\right|$intersects$AC$at$Q$. Let$K=PQ\cap AM$. Prove that$CKD$is a right triangle. ##$type=three$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: 2016 Issue 465
2016 Issue 465
Mathematics & Youth