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

## $show=home 1. How many triples of positive integers$(a,b,c)$are there such that $$\text{lcm}(a,b)=1000,\quad \text{lcm}(b,c)=2000,\quad \text{lcm}(a,c)=2000?.$$ 2. Let$ABC$be an isosceles triangle$A$with$\widehat{BAC}=100^{0}$, point$D$on segment$BC$such that$\widehat{CAD}=20^{0}$, point$E$on the ray$AD$such that triangle$ACE$is isosceles at vertex$C$. Determine the measure of all angles of triangle$BDE$. 3. The sum of$m$distinct even positive integers and$n$distinct odd positive integers equal$2014$. Find the greatest possible value of$3m+4n$. 4. Triangle$ABC$is inscribed in circle center at$O$. Parallel lines are drawn through vertices$A,B,C$such that they are not parallel to any of the sides of triangle$ABC$. These parallel lines intersect$(O)$at$A_{1},B_{1},C_{1}$respectively. Prove that the orthocenters of triangles$A_{1}BC$,$B_{1}CA$,$C_{1}AB$are collinear. 5. Solve the system of equations $\begin{cases} (1+x)(1+x^{2})(1+x^{4}) & =1+y^{7}\\ (1+y)(1+y^{2})(1+y^{4}) & =1+x^{7} \end{cases}.$ 6. Find all polynomials with real coefficients$P(x)$such that the following conditions are satisfied $\begin{cases} P(x)-10 & =\sqrt{P(x^{2}+3)}-13\quad(x\geq0)\\ P(2014) & =2024 \end{cases}.$ 7. Let$h_{a},h_{b},h_{c}$and$l_{a},l_{b},l_{c}$denote the altitudes and inner angle-bisectors of a triangle$ABC$. Prove that $\frac{1}{h_{a}h_{b}}+\frac{1}{h_{b}h_{c}}+\frac{1}{h_{c}h_{a}}\geq\frac{1}{l_{a}^{2}}+\frac{1}{l_{b}^{2}}+\frac{1}{l_{c}^{2}}.$ 8. Given that$0<x<\frac{\pi}{2}$. Prove that at least one of the two numbers$\left(\frac{1}{\sin x}\right)^{\frac{1}{\cos^{2}x}}$,$\left(\frac{1}{\cos x}\right)^{\frac{1}{\sin^{2}x}}$is greater than$\sqrt{3}$. ##$type=three$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 441
2014 Issue 441
Mathematics & Youth