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

1. Solve the following equation on the set of real numbers:
2x+27=(x2-x-6)(x+1)x2+2x-3.
 

2. For what positive integers a is the value of the following expression also an integer?
(a+11-a+a-1a+1-4a2a2-1):(2a3+a2-2-2a+2a2a2)
 

3. Given that the second coordinates of the points A(1,a), B(3,b), C(4,c) are
a=-sin39+sin13sin26cos13,b=102+log1025,c=(15-2)3-(15+2)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 1a12+2a22+3a32+4a42 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=21cm, BC=20cm and CA=13cm. A', B', C' are points on the corresponding lateral edges, such that AA'=5cm, BB'=25cm and CC'=4cm. Find the angle of the planes of triangle A'B'C' and triangle ABC.
 

8. Let f(x)=2x6-3x4+x2. Prove that f(sinα)+f(cosα)=0.