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Cím:	Preparatory problems for the entrance exam of high school
Szerző(k):	László Számadó
Füzet:	2002/decemberi melléklet, 45. oldal	PDF \| MathML
Témakör(ök):	Felvételi előkészítő feladatsor

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

\begin{matrix} \frac{2 x + 2}{7} = \frac{(x^{2} - x - 6) (x + 1)}{x^{2} + 2 x - 3} . \end{matrix}

2. For what positive integers

a

is the value of the following expression also an integer?

\begin{matrix} (\frac{a + 1}{1 - a} + \frac{a - 1}{a + 1} - \frac{4 a^{2}}{a^{2} - 1}) : (\frac{2}{a^{3} + a^{2}} - \frac{2 - 2 a + 2 a^{2}}{a^{2}}) \end{matrix}

3. Given that the second coordinates of the points

A (1, a)

B (3, b)

C (4, c)

are

\begin{matrix} a = - \frac{sin 39^{\circ} + sin 13^{\circ}}{sin 26^{\circ} \cdot cos 13^{\circ}}, b = \sqrt[]{10^{2 + log_{10} 25}}, c = {(\frac{1}{\sqrt[]{5} - 2})}^{3} - {(\frac{1}{\sqrt[]{5} + 2})}^{3} \end{matrix}

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

a_{1}, a_{2}, a_{3}, a_{4}

. Show that

1 \cdot a_{1}^{2} + 2 \cdot a_{2}^{2} + 3 \cdot a_{3}^{2} + 4 \cdot a_{4}^{2}

can be expressed as the sum of two perfect squares.

6. In an acute triangle

A B C

, the circle of diameter

A C

intersects the line of the altitude from

B

at the points

D

and

E

, and the circle of diameter

A B

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

A B C

, the lengths of the sides are

A B = 21 cm

B C = 20 cm

and

C A = 13 cm

A'

B'

C'

are points on the corresponding lateral edges, such that

A A' = 5 cm

B B' = 25 cm

and

C C' = 4 cm

. Find the angle of the planes of triangle

A' B' C'

and triangle

A B C

8. Let

f (x) = 2 x^{6} - 3 x^{4} + x^{2}

. Prove that

f (sin α) + f (cos α) = 0