Preparatory problems for the entrance exam of high school
László Számadó
1. Solve the following equation on the set of real numbers:
\(\displaystyle
\frac{2x+2}{7}=\frac{(x^2x6)(x+1)}{x^2+2x3}.
\)
2. For what positive integers a is the value of the following expression also an integer?
\(\displaystyle
\left(\frac{a+1}{1a}+\frac{a1}{a+1}\frac{4a^2}{a^21}\right)
:\left(\frac{2}{a^3+a^2}\frac{22a+2a^2}{a^2}\right)
\)
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 a_{1},a_{2},a_{3},a_{4}. Show that 1^{.}a_{1}^{2}+ 2^{.}a_{2}^{2}+3^{.}a_{3}^{2}+ 4^{.}a_{4}^{2} 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)= 2x^{6} 3x^{4}+x^{2}. Prove that f(sin \(\displaystyle \alpha\))+f(cos \(\displaystyle \alpha\))=0.
