Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Specimen Cyprus Seniors Provincial/2nd grade/Problems"

(problems)
 
m (correction)
Line 6: Line 6:
 
b)Area<math>(\Beta\Gamma\Delta)</math>=Area<math>(\Beta '\Gamma '\Delta ')</math>.
 
b)Area<math>(\Beta\Gamma\Delta)</math>=Area<math>(\Beta '\Gamma '\Delta ')</math>.
  
−
[[2006 Specimen Seniors Provincial/2nd grade/Problem 1|Solution]]
+
[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 1|Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
Line 12: Line 12:
 
If <math>\alpha=sinx_{1}</math>,<math>\beta=cosx_{1}</math><math>sinx_{2}</math>, <math>\gamma=cosx_{1}cosx_{2} sinx_{3}</math> and <math>\delta=cosx_{1}cosx_{2}cosx_{3}</math> prove that <math>\alpha^2+\beta^2+\gamma^2+\delta^2=1</math>
 
If <math>\alpha=sinx_{1}</math>,<math>\beta=cosx_{1}</math><math>sinx_{2}</math>, <math>\gamma=cosx_{1}cosx_{2} sinx_{3}</math> and <math>\delta=cosx_{1}cosx_{2}cosx_{3}</math> prove that <math>\alpha^2+\beta^2+\gamma^2+\delta^2=1</math>
  
−
[[2006 Specimen Seniors Provincial/2nd grade/Problem 2|Solution]]
+
[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 2|Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
 
Prove that if <math>\kappa, \lambda, \nu</math> are positive integers, then the equation <math>x^2-(\nu +2)\kappa\lambda x+\kappa^2\lambda^2 = 0</math> has irratioanl roots.
 
Prove that if <math>\kappa, \lambda, \nu</math> are positive integers, then the equation <math>x^2-(\nu +2)\kappa\lambda x+\kappa^2\lambda^2 = 0</math> has irratioanl roots.
  
−
[[2006 Specimen Seniors Provincial/2nd grade/Problem 3|Solution]]
+
[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 3|Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
Line 26: Line 26:
 
b) Calculate the value of: <math>\rho_{1}^{2006} + \rho_{2}^{2006}</math>.
 
b) Calculate the value of: <math>\rho_{1}^{2006} + \rho_{2}^{2006}</math>.
  
−
[[2006 Specimen Seniors Provincial/2nd grade/Problem 4|Solution]]
+
[[Specimen Cyprus Seniors Provincial/2nd grade/Problem 4|Solution]]
  
 
== See also ==
 
== See also ==

Revision as of 07:55, 12 November 2006

Problem 1

Let $\Alpha\Beta\Gamma\Delta$ (Error compiling LaTeX. Unknown error_msg) be a parallelogram. Let $(\epsilon)$ be a straight line passing through $\Alpha$ (Error compiling LaTeX. Unknown error_msg) without cutting $\Alpha\Beta\Gamma\Delta$ (Error compiling LaTeX. Unknown error_msg). If $\Beta ', \Gamma ', \Delta '$ (Error compiling LaTeX. Unknown error_msg) are the projections of $\Beta, \Gamma, \Delta$ (Error compiling LaTeX. Unknown error_msg) on $(\epsilon)$ respectively, show that

a) the distance of $\Gamma$ from $(\epsilon)$ is equal to the sum of the distances $\Beta, \Delta$ (Error compiling LaTeX. Unknown error_msg) from $(\epsilon)$.

b)Area$(\Beta\Gamma\Delta)$ (Error compiling LaTeX. Unknown error_msg)=Area$(\Beta '\Gamma '\Delta ')$ (Error compiling LaTeX. Unknown error_msg).

Solution

Problem 2

Problem

If $\alpha=sinx_{1}$,$\beta=cosx_{1}$$sinx_{2}$, $\gamma=cosx_{1}cosx_{2} sinx_{3}$ and $\delta=cosx_{1}cosx_{2}cosx_{3}$ prove that $\alpha^2+\beta^2+\gamma^2+\delta^2=1$

Solution

Problem 3

Prove that if $\kappa, \lambda, \nu$ are positive integers, then the equation $x^2-(\nu +2)\kappa\lambda x+\kappa^2\lambda^2 = 0$ has irratioanl roots.

Solution

Problem 4

If $\rho_{1}, \rho_{2}$ are the roots of equation $x^2-x+1=0$ then:

a) Prove that $\rho_{1}^3=\rho_{2}^3 = -1$ and

b) Calculate the value of: $\rho_{1}^{2006} + \rho_{2}^{2006}$.

Solution

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Specimen_Cyprus_Seniors_Provincial/2nd_grade/Problems&oldid=11663"