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

## $show=home 1. Find all positive integers$a$and$b$such that$b|a+2$and$a|b+3$. 2. Given a right triangle$ABC$with the right angle$A$. Choose$E$on the side$BC$such that$EC=2EB$. Prove that$AC^{2}=3(EC^{2}-EA^{2})$. 3. Solve the following equation $\frac{1}{x+\sqrt{x^{2}-1}}=\frac{1}{4x}+\frac{3x}{2x^{2}+2}.$ 4. Let$BC$be a chord of a circle with center$O$and radius$R$. Assume that$BC=R$. Let$A$be apoint on the major arc$BC$($A\ne B$,$A\ne C$), and$M,N$points on the chord$AC$such that$AC=2AN=\frac{3}{2}AM$. Choose$P$on$AB$such that$MP$is perpendicular to$AB$. Prove that three points$P,O$and$N$are collinear. 5. Assume that equation $ax^{3}-x^{2}+bx-1=0,\quad(a\ne0)$ has three positive real solutions. Find the minimum value of the expression $M=(1-2ab)\frac{b}{a^{2}}.$ 6. Let$x$and$y$be two positive real numbers satisfying$32x^{6}+4y^{3}=1$. Find the maximum value of the expression $P=\frac{(2x^{2}+y+3)^{3}}{3(x^{2}+y^{2})-3(x+y)+2}.$ 7. Given an acute triangle$ABC$($AB>AC$). The heights$BB'$and$CC'$intersect at$H$. Let$M,N$respectively be the midpoints of the sides$AB,AC$and$O$the circumcenter.$AH$intersects$B'C'$at$E$, and$AO$intersects$MN$at$F$. Prove that$EF\parallel OH$. 8. Given three positive numbers$a,b,c$. Find the maximum value of$k$so that the following inequality holds $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-3\geq3\left(\frac{a^{2}+b^{2}+c^{2}}{ab+bc+ca}-1\right).$ 9. Find all positive integers$x,y,z$which form an airthmetic progression and satisfy the following equation $\frac{x^{2}(x+y)(x+z)}{(x-y)(x-z)}+\frac{y^{2}(y+z)(y+x)}{(y-z)(y-x)}+\frac{z^{2}(z+x)(z+y)}{(z-x)(z-y)}=2016+(x+y-z)^{2}.$ 10. Given a$999\times999$table of squares. Each square is colored by white or red. Consider a set of triples of squares$(C_{1},C_{2},C_{3})$which satisfy the following properties: the first two squares$C_{1},C_{2}$are in the same row, the last two squares$C_{2},C_{3}$are in the same column,$C_{1},C_{3}$are white, and$C_{2}$is red. Find the maximum number of elements in such a set. 11. Find all positive integers$n>1$and all primes$p$such that the polynomial$f(x)=x^{n}-px+p^{2}$ca be factorized as a product of two non-constant polynomials with integer coefficients. 12. Assume that$ABC$is an equilateral triangle and$M$is a point which is not on the lines through$BC$,$CA$and$AB$. Prove that the Euler lines of the triangles$MBC$,$MCA$, and$MAB$are either concurrent or parallel. ##$type=three$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: 2014 Issue 450
2014 Issue 450
Mathematics & Youth