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

## $show=home 1. Find an integer size square whose area is a 4-digit number such that the last rightmost three digits are idnentical. 2. Determine the values of$a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$. Given that$2a_{1}=3a_{2}$,$2a_{3}=4a_{4}$,$5a_{4}=2a_{5}$,$2a_{5}=5a_{6}$,$2a_{6}=3a_{7}$,$2a_{7}=3a_{1}$and $$a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}=400.$$ 3. Find all pairs of integers$(x,y)$such that$x^{2}(x^{2}+y^{2})=y^{p+1}$, where$p$is a prime number. 4. Find the maximum and minimum values of the expression $P=xy+yz+zx-xyz$ where$x,y,z$are non-negative real numbers satisfying $x^{2}+y^{2}+z^{2}=3.$ 5. Triangle$ABC$inscribed in circle$(O)$with$\widehat{BAC}=70^{0}$,$\widehat{ACB}=50^{0}$. The points$M,N,P,Q$and$R$on circle$(O)$are such that$PA=AB=BR$,$QB=BC=CM$and$NC=CA=AN$. Let$S$be the intersection of arc$NQ$and the diameter$PP'$of$(O)$. Prove that$\Delta NRS\backsim\Delta NQR$. 6. Solve the equation $x^{3}-3x=\sqrt{x+2}.$ 7. Find the measure of the angles of a triangle$ABC$such that the expression $T=-3\tan\frac{C}{2}+4(\sin^{2}A-\sin^{2}B)$ is greatest possible. 8. Let$S.ABC$be a triangular pyramid where the sides$SA,SB,SC$are pairwise orthogonal,$SA=a$,$SB=b$,$Sc=c$.$H$is the foot of the perpendicular from$S$onto$ABC$. Prove the inequality $aS_{HBC}+bS_{HAC}+cS_{HAB}\leq\frac{abc\sqrt{3}}{2}.$ 9. Let$p$be an odd prime number, and$x,y$are two positive integers such that$\sqrt{x}+\sqrt{y}\leq\sqrt{2p}$. Find the minimum value of the following expression $A=\sqrt{2p}-\sqrt{x}-\sqrt{y}.$ 10. Does there exist a funtion$f:\mathbb{N}^{*}\to\mathbb{N}^{*}$such that $f(mf(n))=n+f(2013m),\quad\forall m,n\in\mathbb{N}^{*}?.$ 11. The non-negative real numbers$a,b,c$are such that $$\max\{a,b,c\}\leq4\min\{a,b,c\}.$$ Prove the inequality $2(a+b+c)(ab+bc+ca)^{2}\geq9abc(a^{2}+b^{2}+c^{2}+ab+bc+ca).$ 12. Let$ABC$be a traingle inscribed in circle centered at$O$, and let$I$be its incenter.$AI,BI,CI$intersect$(O)$at$A_{1},B_{1},C_{1}$;$A_{1}C_{1},A_{1B_{1}}$meet$BC$at$M,N$;$B_{1}A_{1},B_{1}C_{1}$meet$CA$at$P,Q$;$C_{1}B_{1},C_{1}A_{1}$meet$AB$at$R,S$respectively. Prove that $S_{MNPQRS}\leq\frac{2}{3}S_{A_{1}B_{1}C_{1}}.$ ##$type=three$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: 2013 Issue 429
2013 Issue 429
Mathematics & Youth