Order KöMaL! Competitions Portal

English Issue, December 2002

Previous pageContentsNext pageORDER FORM

Preparatory problems for the entrance exam of high school

1. Solve the following equation on the set of real numbers:

$\displaystyle \frac{2x+2}{7}=\frac{(x^2-x-6)(x+1)}{x^2+2x-3}.$

2. For what positive integers a is the value of the following expression also an integer?

$\displaystyle \left(\frac{a+1}{1-a}+\frac{a-1}{a+1}-\frac{4a^2}{a^2-1}\right) :\left(\frac{2}{a^3+a^2}-\frac{2-2a+2a^2}{a^2}\right)$

3. Given that the second coordinates of the points A(1,a), B(3,b), C(4,c) are

$\displaystyle a=-\frac{\sin39^\circ+\sin13^\circ}{\sin26^\circ\cdot\cos13^\circ},\qquad b=\sqrt{10^{2+\log_{10}25}},\qquad c= \left(\frac{1}{\sqrt{5}-2}\right)^3-\left( \frac{1}{\sqrt{5}+2}\right)^3$

determine whether the three points are collinear.

4. What is more favourable:

I. If the bank pays 20% annual interest, and the inflation rate is 15% per year, or
II. if the bank pays 12% annual interest, and the inflation rate is 7% per year?

5. The first four terms of an arithmetic progression of integers are a1,a2,a3,a4. Show that 1.a12+ 2.a22+3.a32+ 4.a42 can be expressed as the sum of two perfect squares.

6. In an acute triangle ABC, the circle of diameter AC intersects the line of the altitude from B at the points D and E, and the circle of diameter AB intersects the line of the altitude from C at the points F and G. Show that the points D, E, F, G lie on a circle.

7. The base of a right pyramid is a triangle ABC, the lengths of the sides are AB=21 cm, BC=20 cm and CA=13 cm. A', B', C' are points on the corresponding lateral edges, such that AA'=5 cm, BB'=25 cm and CC'=4 cm. Find the angle of the planes of triangle A'B'C' and triangle ABC.

8. Let f(x)= 2x6- 3x4+x2. Prove that f(sin $\displaystyle \alpha$)+f(cos $\displaystyle \alpha$)=0.

 Támogatóink: ELTE Morgan Stanley