Problems in Mathematics and Youth Magazine, 2007 , Issue 9

For Lower Secondary Schools

1. The first 100 positive integer numbers are written consecutively in a certain order. Call the resulting number A. Is A a multiple of 2007?

2. Let ABC be a nonisosceles triangle, where AB is the shortest side. Choose a point D in the opposite ray of BA such that BD=BC. Prove that \angle ACD<90^\circ.

3. Let a,b,c be positive reals such that a+b+c=1. Prove that \left(a+\dfrac{1}{b}\right)\left(b+\dfrac{1}{c}\right)\left(c+\dfrac{1}{a}\right)\geq \left(\dfrac{10}{3}\right)^3.

4.  Solve the equation (x^4+5x^3+8x^2+7x+5)^4+(x^4+5x^3+8x^2+7x+3)^4=16 in \mathbb{R}.

5. Let AH denote the altitude of a right triangle ABC, right angle at A and suppose that AH^2=4AM\cdot AN, where M,N are the feet of the altitude from H to AB and AC, respectively. Find the measures of the angles of triangle ABC.

For Upper Secondary Schools
1. Find all (x,y)\in\mathbb{Z}^2 such that x^{2007}=y^{2007}-y^{1338}-y^{669}+2.

2. Let (x_n) be a sequence given by x_1=5\; , x_{n+1}=x_n^2-2\forall n\geq 1.
Calculate \lim_{n\to\infty}\dfrac{x_{n+1}}{x_1x_2\cdots x_n}.

3. Let a,b,c and denote the three sides of a triangle ABC. Its altitudes are h_a,h_b,h_c and the radius of its three escribed circles are r_a,r_b,r_c. Prove that \dfrac{a}{h_a+r_a}+\dfrac{b}{h_b+r_b}+\dfrac{c}{h_c+r_c}\geq \sqrt{3}.

Toward Mathematical Olympiad
1. In a quadrilateral ABCD, where AD=BC meets at O, and the angle bisector of the angles DAB,CBA meets at I. Prove that the midpoints of AB,CD,OI are colinear.

2. Prove that \left(a-\dfrac{1}{b}\right)\left(b-\dfrac{1}{c}\right)\left(c-\dfrac{1}{a}\right)\geq \\ \left(a-\dfrac{1}{a}\right)\left(b-\dfrac{1}{b}\right)\left(c-\dfrac{1}{c}\right)\forall a,b,c\in [1,+\infty).

3. Find the number of binary strings of length n(n>3) in which the substring 01 occurs exactly twice.

4. Let f:\mathbb{N}\to\mathbb{R} be a function such that f(1)=\dfrac{2007}{6} and

\dfrac{f(1)}{1}+\dfrac{f(2)}{2}+\cdots+\dfrac{f(n)}{n}=\dfrac{n+1}{2}\cdot f(n)\forall n\in\mathbb{N}. Find the limit \lim_{n\to\infty} (2008+n)f(n).

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