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

## $show=home 1. Prove that there do not exist two coprime integers$a$and$b$which are greater than$1$and satisfy the fact that$a^{2007}+b^{2007}$is divisible by$a^{2006}+b^{2006}$. 2. Find prime numbers$a,b,c,d,e$such that $$a^{4}+b^{4}+c^{4}+d^{4}+e^{4}=abcde.$$ 3. Given$f\left(x\right)=a^{2016}x^{2}+bx+a^{2016}c-1$where$a,b,c\in\mathbb{Z}$. Suppose that the equation$f\left(x\right)=-2$has two positive integral solutions. Prove that $$A=\frac{f^{2}\left(1\right)+f^{2}\left(-1\right)}{2}$$ is a composite number. 4. Given a triangle$ABC$and$\left(I\right)$is its incircle. Let$D$be the point of tangency between$\left(I\right)$and$BC$. The line which goes through$D$and is perpendicular to$AI$intersects$IB$and$IC$respectively at$P$and$Q$. Prove that all$B,C,P$and$Q$lie on a circle whose center is on$AD$. 5. Find all the real numbers$k$such that for all non-negative numbers$a,b,cwe always have the following inequality $$\left[a+k\left(b-c\right)\right]\left[b+k\left(c-a\right)\right]\left[c+k\left(a-b\right)\right]\leq abc.$$ 6. Find the minimum value of the expression \begin{align*} f= & \sqrt{x^{2}-2x+2}+\sqrt{x^{2}-8x+32}+\sqrt{x^{2}-6x+25}\\ & +\sqrt{x^{2}-4x+20}+\sqrt{x^{2}-10x+26}. \end{align*} 7. How many solutions does the following equations have $$\log_{\frac{5\pi}{2}}x=\cos x.$$ 8. Given a tetrahedronSABC$such that$SA,SB$and$SC$are mutually perpendicular. A moving plane$\left(P\right)$which contains the centroid of the given tetrahedron intersects$SA,SB$and$SC$respectively at$A_{1},B_{1}$, and$C_{1}$. Prove that $$\frac{1}{SA_{1}^{2}}+\frac{1}{SB_{1}^{2}}+\frac{1}{SC_{1}^{2}}\geq\frac{4}{R^{2}},$$ where$R$is the radius of the circumscribed sphere of the tetrahedron$SABC$. 9. Find the integral part of the following number $$T=\frac{2}{1}\cdot\frac{4}{3}\cdot\frac{6}{5}\ldots\frac{2016}{2015}.$$ 10. A positive interger is called a "HV2015'' number if in its decima representation there are$2015$consecutive digits$9$. Let a stricly increasing sequence of positive integers such that the sequence$\left(\dfrac{a_{n}}{n}\right)$,$n=1,2,3,\ldots$is bounded. Prove that there are infinitely many "HV2015'' numbers in the given sequence$\left(a_{n}\right),n=1,2,3,\ldots$. 11. Given a grid of the size$1\times n\left(n>2\right)$. Now we fill in each square with$0$or$1$. A way to fill the grid is called "good'' if any three consecutive squares do not contain the same number. Let$a_{n}$be the number of "good'' ways to fill the grid. Compute$a_{n}$. 12. Given a triangle$ABC$. Prove that $$\sin\frac{A}{2}+\sin\frac{B}{2}+\sin\frac{C}{2}\geq\frac{1}{\sqrt{2}}\left(\sqrt{\sin\frac{A}{2}}+\sqrt{\sin\frac{B}{2}}+\sqrt{\sin\frac{C}{2}}\right).$$ ##$type=three$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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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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Mathematics & Youth: 2016 Issue 474
2016 Issue 474
Mathematics & Youth