Questions asked by Özgür Özer

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• Show that apothem of the equilateral triangle, of which one side length is a, will be #sqrt3/6a#?
  
• Can you show that if #(x_1+x_2)/x_3=(y_1+y_2)/y_3#, then #(x_1,y_1)#, #(x_2,y_2)#, and #(x_3,y_3)# are linear?
  
• Why perpendicular bisectors of any triangle intercept in the center of circumscribed circle?
  
• How do you proof that for #a,b,cinRR#, #a/b=b/c=c/a<=>a=b=c#?
  
• How do you show that the slope of #a^y=b^x# is #log_a b#?
  
• How do you show that the slope of #{(y=at+b),(x=ct+d):}# is #a/c#?
  
• How do you draw a regular hexagon with a ruler and compass?
  
• Let side lengths of a triangle be #a#, #b#, and #c#. Then how do you proof that #a^2<2(b^2+c^2)#?
  
• How do you show that if #a+b=0#, then the slope of #x/a+y/b+c=0# is #1#?
  
• How do you proof that for #a,binRR#, #a < b<=>b > a#?
  
• #43^2+52^2+68^2=#?
  
• #34^2+25^2+86^2=#?
  
• #43^2+58^2+62^2=#?
  
• #34^2+85^2+26^2=#?
  
• #48^2+52^2+63^2=#?
  
• #84^2+25^2+36^2=#?
  
• #48^2+53^2+62^2=#?
  
• #84^2+35^2+26^2=#?
  
• Side lengths of an acute triangle are #sqrtn#, #sqrt(n+1)#, and #sqrt(n+2)#. How do you find #n#?
  
• #a+3/(4b)=2# and #b+3/(4a)=6#. How do you find #b/a#?
  
• How do you find #1/(1+2^-1)xx1/(1+3^-1)xx1/(1+4^-1)xx...xx1/(1+n^-1)#?
  
• How do you solve #x^3=x^2#?
  
• Side lengths of an right triangle are #sqrtn#, #sqrt(n+1)#, and #sqrt(n+2)#. How do you find #n#?
  
• Given #a+3/4b=2# and #b+3/4a=6#. How do you find #b/a#?
  
• How do you show that one of the roots of #x^2+ax+b=0# is #a#, if and only if #b=0#?
  
• Given #x+y=2# and #x^3+y^3=5#, what is #x^2+y^2#?
  
• Given #ainRR#, how do you solve #|x|=a#?
  
• Given #x, y#, and #zinRR^+#, what is the solution to the system of equations #xy=a#, #yz=b#, and #xz=c#?
  
• How do you solve the following linear system: #x/a+y/b=2#, #bx-ay=0#?
  
• How do you solve the following system?
  
• Is #y=x^2+1#, #y=2^(x^2)# a solvable system?
  
• For 0< x<π/2, if #sinx+sin3x=cosx+cos3x# then x = ?
  
• How do you solve #secx-tanx=tan40#?
  
• ABC is a equilateral triangle. If #|BD|=|CF|# and #|CE|=6#, how do you find #|AD|=x#?
  
• How do you solve #x-sinx=1" and "x-sinx=-1# equations?
  
• Is #e^x# the unique function of which derivative is itself? Can you prove it?
  
• Can you prove that if #p_k# is prime number, then #sum_(k=1)^np_k^-1# is not an integer, for any #ninNN#?
  
• How do you show that #lim_(x->\infty)(2^(x+3)-2*3^(x-1))/(2^(x-1)+3^(x-2))=-6#?
  
• Does #a_n=(-1/2)^n# sequence converge or diverge? How do you find its limit?
  
• How do you prove that #lim_(x->5)x^2!=24# using limit definition?
  
• How do you find #(cos5x+cos4x)/(sin3x+sin2x)#, if #14x=3pi#?
  
• How do you prove that #lim_(x->1)1/(x-1)# doesnot exist using limit definition?
  
• How do you find #lim_(x->0^+)(sinx)^x#?
  
• How do you find #lim_(x->∞)(x+2sinx-1)/(x+3cosx+1)#?
  
• How do you calculate #sum_(n=1)^(∞)(3/2)^(1-2n)#?
  
• How do you calculate #sum_(n=1)^(∞)(4/3)^(2-n)#?
  
• How do you find the equation of the circle passing through the points #P(x_1,y_1), Q(x_2,y_2), R(x_3,y_3)#?
  
• Is it true that the difference of two matrices is equal to a squared (multiplied by itself) matrice subtracted from another squared matrice? Why?