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

## $show=home 1. For a given prime number$p$, find positive integers$x,y$such that $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}.$ 2. Given an acute triangle$ABC$with the orthocenter$H$. Let$M$be the midpoint of$BC$. The line through$A$parallel to$MH$meets the line through$H$parallel to$MA$at$N$. Prove that $AH^{2}+BC^{2}=MN^{2}.$ 3. Suppose that $\begin{cases} a^{3}-a^{2}+a-5 & =0\\ b^{3}-2b^{2}+2b+4 & =0 \end{cases}.$ Find$a+b$. 4. From a point$M$outside the circle$(O)$draw to tangents$MA,MB$to$(O)$($A,B$are points of tangency).$C$is an arbitrary point on the minor arc$AB$of$(O)$. The rays$AC$and$BC$intersect$MB$and$MA$at$D$and$E$respectively. Prove that the circumcircles of the triangles$ACE$,$BCD$and$OCM$meet at another point which is different from$C$. 5. Find all triples of positive integers$(a,b,c)$such that $(a^{5}+b)(a+b^{5})=2^{c}.$ 6. Solve the following system of equations $\begin{cases} x+y+z+\sqrt{xyz} & =4\\ \sqrt{2x}+\sqrt{3y}+\sqrt{3z} & =\dfrac{7\sqrt{2}}{2}\\ x & =\min\{x,y,z\}\end{cases}.$ 7. Given a diamond$ABCD$with$\widehat{BAD}=120^{0}$. Let$M$vary on the side$BD$. Assume that$H$and$K$are the orthogonal projections of$M$on the lines through$AB$and$AD$respectively. Let$N$be the midpoint of$HK$. Prove that the line through$MN$always passes through a fixed point. 8. Given a triangle$ABC$. Find the maximum value and the minimum value of the expression $P=\cos^{2}2A\cdot\cos^{2}2B\cdot\cos^{2}2C+\frac{\cos4A\cdot\cos4B\cdot\cos4C}{8}.$ 9. Find the smallest$k$such that $S=a^{3}+b^{3}+c^{3}+kabc-\frac{k+3}{6}\left[a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b)\right]\leq0$ for all triples$(a,b,c)$which are the lengths of the sides of a triangle. 10. Given a sequence of polynomials$\{P_{n}(x)\}$satisfying the following conditions$P_{1}(x)=2x$,$P_{2}(x)=2(x^{2}+1)$and $P_{n}(x)=2xP_{n-1}(x)-(x^{2}-1)P_{n-2}(x),\quad n\in\mathbb{N},n\geq3.$ Prove that$P_{n}(x)$is divisible by$Q(x)=x^{2}+1$if and only if$n=4k+2$,$k\in\mathbb{N}$. 11. Consider the funtion $f(n)=1+2n+3n^{2}+\ldots+2016n^{2015}.$ Let$(t_{0},t_{1},\ldots,t_{2016})$and$(s_{0},s_{1},\ldots,s_{2016})$be two permutations of$(0,1,\ldots,2016)$. Prove that there exist two different numbers in the following set $$A=\left\{s_{0}f(t_{0}),s_{1}f(t_{1}),\ldots,s_{2016}f(t_{2016})\right\}$$ such that their difference is divisible by$2017$. 12. Given a triangle$ABC$and an arbitrary point$M$. Prove that $\frac{1}{BC^{2}}+\frac{1}{CA^{2}}+\frac{1}{AB^{2}}\geq\frac{9}{(MA+MB+MC)^{2}}.$ ##$type=three$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: 2015 Issue 458
2015 Issue 458
Mathematics & Youth