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

## $show=home 1. Find all natural numbers$n$satisfying $2.2^{2}+3.2^{3}+4.2^{4}+\ldots+n.2^{n}=2^{n+34}.$ 2. Find integers$a,b,c$such that $|a-b|+|b-c|+|c-a|=2014^{a}+2015^{a}.$ 3. Suppose that$f(x)$is a polynomial with integral coefficients and$f(1)=2$. Show that$f(7)$is not a perfect square. 4. Given an acute triangle$ABC$with altitudes$AH,BK$. Let$M$be the midpoint of$AB$. The line through$CM$intersect$HK$at$D$. Draw$AL$perpendicular to$BD$at$L$. Prove that the circle containning$C,K$and$L$is tangent to the line going through$BC$. 5. Solve the following system of equations $\begin{cases} 9x^{3}+2x+(y-1)\sqrt{1-3y} & =0\\ 9x^{2}+y^{2}+\sqrt{5-6x} & =6 \end{cases}$ for$x,y\in\mathbb{R}$. 6. Suppose that$f(x)$is a polynomial of degree$3$and its leading coefficient is equal to$2$. Also assume that$f(2014)=2015$,$f(2015)=2016$. Find$f(2016)-f(2013)$. 7. Let$S_{tp}$and$V$respectively be the surface area and the volume of the tetrahedron$ABCD.$Prove that $\left(\frac{1}{6}S_{tp}\right)^{3}\geq\sqrt{3}V^{2}.$ 8. Given an$n$-sided convex polygon ($n\geq4$)$A_{1}A_{2}\ldots A_{n}. Prove that \begin{align*} & n+\sin A_{1}+\sin A_{2}+\ldots+\sin A_{n}\\ \leq & 2\left(\cos\frac{A_{1}-A_{2}}{4}+\cos\frac{A_{2}-A_{3}}{4}+\ldots+\cos\frac{A_{n}-A_{1}}{4}\right). \end{align*} When does the equality happen?. 9. Find all triples(x,y,p)$where$x$and$y$are positive integers and$p$is a prime number satisfying$p^{x}-y^{p}=1$. 10. Let$k$be a real number which is greater than$1$. Consider the following sequence $x_{1}=\frac{1}{2}\sqrt{k^{2}-1},\quad x_{2}=\sqrt{\frac{k^{2}-1}{4}+\frac{1}{2}\sqrt{k^{2}-1}},\ldots,$ $x_{n}=\underset{n\text{ square root symbols}}{\underbrace{\sqrt{\frac{k^{2}-1}{4}+\sqrt{\frac{k^{2}-1}{4}+\ldots+\sqrt{\frac{k^{2}-1}{4}+\frac{1}{2}\sqrt{k^{2}-1}}}}}}.$ Prove that$\left\{ x_{n}\right\} $converges and find$\lim_{n\to\infty}x_{n}.$11. For each positive integer$n$, put$\displaystyle\psi(n)=\sum_{d|n}d^{2}$. a) Prove that$\psi(n)$is multiplicative, i.e. $\psi(ab)=\psi(a)\psi(b)\text{ if }(a,b)=1.$ b) Suppose the$l$is an odd positive integer. Prove that there are only finitely many positive integers$n$such that$\psi(n)=\psi(n+l)$. 12. Given a triangle$ABC$with the circumscribed circle$(O)$and the inscribed circle$(I)$. The tangent lines to$(O)$at$B$and$C$intersect at$T$. Let$M$be the midpoint of$BC$and$D$be the midpoint of the the arc$BC$which does not contain$A$. Suppose that$AM$intersects$(O)$at$E$and$AT$intersects the side$BC$at$F$. Let$J$be the midpoint of$IF$. Prove that$\widehat{AEI}=\widehat{ADJ}$. ##$type=three$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 457
2015 Issue 457
Mathematics & Youth