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

## $show=home 1. Find all natural numbers$x,y,z$such that$x!+y!+z!=\overline{xyz}$and$\overline{xyz}$is a three-digit number. Recall that$n!=1.2.3\ldots n$and$0!=1$. 2. Given a triangle$ABC$with$\widehat{BAC}=50^{0}$,$\widehat{ABC}=60^{0}$. On the sides$AB$and$BC$, choose$D$and$E$respectively such that$\widehat{ACD}=\widehat{CAE}=30^{0}$. Find the angle$\widehat{CDE}$. 3. Given three positive numbers$a,b,c$. Find the maximum value of the expression $T=\frac{a+b+c}{(4a^{2}+2b^{2}+1)(4c^{2}+3)}.$ 4. Suppose that$ABC$is an acute triangle. Its altitudes$AD$,$BE$and$CF$are concurrent at the point$H$. Let$K$be a point on the side$DC$and choose$S$on$HK$such that$AS\perp HK$. Let$I$be the intersection between$EF$and$AH$. Prove that$SH$is the angle bisector of the angle$\widehat{DSI}$. 5. Let$a,b,c$be positive numbers satisfying $a^{4}+b^{4}+c^{4}\leq2(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}).$ Prove that at lease one of the following equations has no solution $ax^{2}+2bx+2c=0,$ $bx^{2}+2cx+2a=0,$ $cx^{2}+2ax+2b=0.$ 6. Solve the system of equations $\begin{cases} x^{3}+y^{3} & =1\\ x^{5}+y^{5} & =1 \end{cases}.$ 7. Given a quadrilateral pyramid$S.ABCD$whose base$ABCD$is a parallelogram. A point$M$varies on the side$AB$($M$is different from$A$and$B$) Let$(\alpha)$be a plane which goes through$M$and is parallel to$SA$and$BD$. Determine the cross section of$S.ABCD$determined by$(\alpha)$and find the position for$M$so that the cross section has maximal area. 8. Solve the equation $3\sin^{3}x+2\cos^{3}x=2\sin x+\cos x.$ 9. Find all pairs of positive integers$(a,b)$so that$4a+1$and$4b-1$are coprime and$a+b$is a divisor of$16ab+1$. 10. Given a polynomial $P(x)=a_{n}x^{n}+\ldots+a_{1}x+a_{0}\quad(a_{n}\ne0,\,n\geq2)$ with integer coefficients. Prove that there are infinitely many integers$k$such that$P(x)+k$cannot be factorized as a product of two positive degree polynomials with integral coefficients. 11. Suppose that the funtion$f:\mathbb{N\to\mathbb{N}}$is onto and the funtion$g:\mathbb{N}\to\mathbb{N}$is one-to-one. Assume furthermore that$f(x)\geq g(n)$for all$n\in\mathbb{N}$. Show that $f(n)=g(n),\,\forall n\in\mathbb{N}.$ 12. Given a triangle$ABC$inscribed in a circle$(O)$.$M$and$N$are two fixed points on$(O)$so that$MN\parallel BC$. Let$P$vary on the line$AM$. The line through$P$which is parallel to$BC$intersects$CA$,$AB$at$E,F$respectively. The circumscribed circle of the triangle$NEF$intersects$(O)$at the second point$Q$(different from$N$). Prove that the line$PQ$always goes through a fixed point when$P$moves. ##$type=three$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 453
2015 Issue 453
Mathematics & Youth