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

## $show=home 1. Do there exist natural numbers$x,y,z$such that $5x^{2}+2016^{y+1}=2017^{z}?.$ 2. Point$O$is chosen in a right triangke$ABC$, right angle at$A$, such that$\widehat{ABO}=30^{0}$and$OA=OC$. Point$E$on side$BC$such that$\widehat{EOB}=60^{0}.$Determine the three angles of traingle$ABC$given that the line$CO$passes through the midpoint$I$of the line segment$AE$. 3. Find all pair of natural numbers$x,y$such that$5^{x}=y^{4}+4y+1$. 4. Solve the system of equations $\begin{cases} x+\sqrt{y-2}+\sqrt{4-z} & =y^{2}-5z+11\\ y+\sqrt{z-2}+\sqrt{4-x} & =z^{2}-5x+11\\ z+\sqrt{x-2}+\sqrt{4-y} & =x^{2}-5y+11 \end{cases}.$ 5. Let$AB=2a$be a line segment with midpoint$O$. Two half-circles, one with center$O$and diagonal$AB$, another with center$O'$and diagonal$AO$are drawn on the same half-plane divided by$AB$. Point$M$, different from$A$and$O$, moves on the half-circle$(O')$.$OM$meets the half-circle$(O)$at$C$. Let$D$be the second intersection point of$CA$and half-circle$(O')$. The tangent line at$C$of half-circle$(O)$meets$OD$at$E$. Find the position of point$M$on$(O')$such that$ME$is parallel to$AB$. 6. Let$ABCD$be a quadrilateral where the diagonals$AC,BD$are equal and perpendicular. The triangles$AMB$,$BNC$,$CPD$,$DQA$, similar in order, are constructed outside the given quadrilateral.$O_{1}$,$O_{2}$,$O_{3}$,$O_{4}$are the midpoints of$MN$,$NP$,$PQ$,$QM$respectively. Prove that the quadrilateral$O_{1}O_{2}O_{3}O_{4}$is s square. 7. Find a formula counting the number of all$2013$-digits natural numbers which are multiple of$3$and all digits are taken from the set$X=\{3,5,7,9\}$. 8. Solve for$x$, $\log_{2}x=\log_{5-x}3.$ 9. The positive integers$a_{1},a_{2},\ldots,a_{2013}$,$b_{1},b_{2},\ldots b_{2013}$where$b_{k}>1$for all$k$are chosen from the set$X=\{1,2,\ldots,2013\}$. Prove that there exists a positive integer$n$satisfying the following two conditions •${\displaystyle n\leq\left(\prod_{i=1}^{2013}a_{i}\right)\left(\prod_{i=1}^{2013}b_{i}\right)+1}$. •$a_{k}b_{k}^{n}+1$is a composite number for every$k\in X$. 10. Let$a,b,c\in\left[0,\frac{1}{2}\right]$be such that$a+b+c=1$. Prove the inequality $a^{3}+b^{3}+c^{3}+4abc\leq\frac{9}{32}.$ 11. Let$ap$be a prime number,$p\equiv1\,(\text{mod }4)$. Determine the sum $\sum_{k=1}^{p-1}\left[\frac{2k^{2}}{p}-2\left[\frac{k^{2}}{p}\right]\right],$ where$[a]$denotes the largest integer not exceeding$a$. 12. Let$ABC$be a triangle inscribed inside circle$(O)$. Point$M$not on lines$BC$,$CA$,$AB$as weel as circle$(O)$;$AM$,$BM$,$CM$intersect$(O)$at$A_{1}$,$B_{1}$,$C_{1}$;$A_{2}$,$B_{2}$,$C_{2}$are the circumcenters of triangles$MBC$,$MCA$,$MAB$respectively. Prove that the lines$A_{1}A_{2}$,$B_{1}B_{2}$,$C_{1}C_{2}$meet at a point on the circle$(O)$. ##$type=three$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 430
2013 Issue 430
Mathematics & Youth