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

## $show=home 1. Find all the prime numbers$p$so that$2^{p}+p^{2}$is a prime. 2. Given an isosceles right triangle$A B C$with the vertex angle$A$. The point$D$inside$A B C$such that$\widehat{A B D}=\widehat{B C D}=30^{\circ} .$Compute the vertex angle$\widehat{C A D}$. 3. Given positive numbers$a, b, c$such that$a+b+c=1 .$Find the maximum value of the expression $$P=\frac{1}{a+3}+\frac{1}{b+3}+\frac{1}{c+3}-\frac{1}{3(a b+b c+c a)}.$$ 4. Suppose that$A B C D$is a thombus with$\widehat{B A C}=60^{\circ}$. Let$M$be a point on the line segment$B C$($M$is different from$B$and$C$). The line$A M$meets the line$C D$at$N$and then let$E$be the intersection between the lines$D M$and$B N .$Show that the line$B C$is tangent to the circumcircle of the triangle$CEN$. 5. Solve the cquation $$8 x^{2}-11 x+1=(1-x) \sqrt{4 x^{2}-6 x+5}.$$ 6. Given real numbers$x, y, z, t$satisfying$x^{2}+y^{2}+z^{2}+t^{2} \leq 2 .$Find the maximum value of the expression $$P(x, y, z, t)=(x+3 y)^{2}+(z+3 t)^{2} + (x+y+z)^{2}+(x+z+t)^{2}.$$ 7. Find all the real roots of the equation $$\left\{\frac{2 x^{2}-5 x+2}{x^{2}-x+1}\right\}=\frac{1}{2}$$ where$\{a\}$denotes the fractional part of the number$a$. 8. Given a convex quadrilateral$A B C D .$Two diagonals$A C$and$B D$intersect at$O .$Let$M$,$N$,$P$,$Q$respectively be the perpendicular projections of$O$on the lines$A B$,$B C$,$C D$,$D A$. Show that$A C$and$B D$are perpendicular if and only if $$\frac{1}{O M^{2}}+\frac{1}{O P^{2}}=\frac{1}{O N^{2}}+\frac{1}{O Q^{2}}$$ 9. Given positive numbers$a$,$b$,$c$such that$a+b+c=3$. Find the minimum value of the expression $$P=\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{6 a b c}{a b+b c+c a}.$$ 10. Find all the pairs$(p, k)$, where$p$is a prime number and$k$is a positive integer. such that $$k !=\left(p^{3}-1\right)\left(p^{3}-p\right)\left(p^{3}-p^{2}\right).$$ 11. Find all the functions$f: \mathbb{Z} \rightarrow \mathbb{Z}$satisfying $$f(2 f(a)+f(b))=2 a+b-4,\, \forall a, b \in \mathbb{Z}.$$ 12. Given a triangle$A B C$inscribed in a circle$(O)$where$B$and$C$are fixed and$A$can vary. There altitudes$A D$,$B E$,$C F$meet at$H$. Let$\left(C_{1}\right)$,$\left(C_{2}\right)$,$\left(C_{3}\right)$be the circles with diameters$A B$,$B C$,$CA$respectively. The line$A H$intersects$\left(C_{2}\right)$at$M$. ($M$belongs to the line scgment$A H$,$B H$intersects$\left(C_{3}\right)$at$N$belongs to the line segment$B H$),$CH$intersects$\left(C_{1}\right)$at$P$($P$belongs to the line segment$C H$),$A N$intersects$\left(C_{1}\right)$at$S$which is different from$A$,$A P$intersects$\left(C_{3}\right)$at$T$which is different from$A .$The lines$N P$intersects$\left(C_{1}\right)$at$X$which is different from$P$, intersects$\left(C_{3}\right)$at$Y$which is different from$N$.$X S$meets$Y T$at$Z$. The line passing through$A$and perpendicular to$N P$intersects$\left(C_{1}\right)$at$J$and intersect$\left(C_{3}\right)$at$L$. a) The circle with center$A$and radius$A N$meets$\left(C_{1}\right)$at$V,$meets$\left(C_{3}\right)$at$U$. Show that$V$,$N$,$I$are collinear and$U$,$P$,$L$collinear. b) Show that the incenter of the triangle$X Y Z$is a fixed point. ##$type=three$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: 2020 Issue 517
2020 Issue 517
Mathematics & Youth