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

## $show=home 1. Find all pair of natural numbers$x, y$such that $$\left(2^{x}+1\right)\left(2^{x}+2\right)\left(2^{x}+3\right)\left(2^{x}+4\right)-5^{y}=11879$$ 2. Let$n$be a positive integer and let$U(n)=\left\{d_{1} ; d_{2} ; \ldots ; d_{m}\right\}$be the set of all positive divisors of$n$. Prove that $$d_{1}^{2}+d_{2}^{2}+\ldots+d_{m}^{2} \leq n^{2} \sqrt{n}$$ 3. Prove that $$\frac{1}{a^{4}(a+b)}+\frac{1}{b^{4}(b+c)}+\frac{1}{c^{4}(c+a)} \geq \frac{3}{2}$$ where$a$,$b$,$c$are three positive numbers satisfying$a b c=1$. 4. Solve the equation $$3 \sqrt{x^{3}+8}=2 x^{2}-6 x+4$$ 5. Let$A B C D$be a square,$M$is a point lying on$C D$($M \neq C$,$M \neq D$). Through the point$C$draw a line perpendicular to$A M$at$HB H$meets$A C$at$K$. Prove that a)$M K$is always parallel to a fixed line when$M$moves on the side$C D$. b) The circumcenter of the quadrilateral$ADMK$lies on a fixed line. 6. Let$a, b, c$be positive real numbers such that$a b c=1$. Prove that $$\frac{1}{\sqrt{a^{3}+2 b^{3}+6}}+\frac{1}{\sqrt{b^{3}+2 c^{3}+6}}+\frac{1}{\sqrt{c^{3}+2 a^{3}+6}} \leq 1$$ 7. Consider all triangles$A B C$where$A<B<C \leq \frac{\pi}{2}$. Find the least value of the expression $$M=\cot ^{2} A+\cot ^{2} B+\cot ^{2} C +2(\cot A-\cot B)(\cot B-\cot C)(\cot C-\cot A)$$ 8. Let$A B C$be a triangle with$B C=a$,$A C=b$,$A B=c$. A line$d$passing through its incenter meets$A B$,$A C$,$B C$respectively at$M$,$N$,$P$. Prove that $$\frac{a}{\overline{B P} \cdot \overline{P C}}+\frac{b}{C N \cdot N A}+\frac{c}{\overline{A M} \cdot \overline{M B}}=\frac{(a+b+c)^{2}}{a b c}$$ 9. Let$x, y, z$be non-zero real numbers such that$x+2 y+3 z=5$and$2 x y+6 y z+3 x z=8$. Prove that $$1 \leq x \leq \frac{7}{3} ; \frac{1}{2} \leq y \leq \frac{7}{6} ; \frac{1}{3} \leq z \leq \frac{7}{9}$$ 10. Solve the system of equations $$\begin{cases} \sqrt{x}+\sqrt{y} &=\sqrt{3(x+y)} \\ 4 x^{3}+6 x^{2}+4 x+1 &=15 y^{4}\end{cases}$$ 11. Find all functions$f: \mathbb{R} \rightarrow \mathbb{R}$satisfy $$f(f(x)+y)=f(x+y)+x f(y)-x y-x+1$$ 12. Suppose that the tetrahedron$A B C D$satisfies the following conditions: All faces are acute triangles and$B C$is perpendicular to$AD$. Let$h_{a}$,$h_{d}$be respectively the lengths of the altitudes from$A$,$D$onto the opposite faces, and let$2 \alpha\left(0^{\circ}<\alpha<45^{\circ}\right)$be the measure of the dihedral angle at edge$B C$,$d$is the distance between$B C$and$A D$. Prove the inequality $$\frac{1}{h_{a}}+\frac{1}{h_{d}} \leq \frac{1}{d \cdot \sin \alpha}$$ ##$type=three$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

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2010 Issue 394
2010 Issue 394
Mathematics & Youth