Questions asked by Cesareo R.

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• What is the graphic of #f(x) = sqrt(x+sqrt(x+sqrt(x+sqrt(x+...))))# for #x ge 0#?
  
• Given #p_1=(1,1), p_2=(6,3), p_3=(4,5)# and the straight #y = 1.5-(x-4)# what is the point #p=(x,y) # pertaining to the straight, minimizing #norm(p_1-p) + norm(p_2-p)+norm(p_3-p)#?
  
• Given #C_1->y^2+x^2-4x-6y+9=0#, #C_2->y^2+x^2+10x-16y+85=0# and #L_1->x+2y+15=0#, determine #C->(x-x_0)^2+(y-y_0)^2-r^2=0# tangent to #C_1,C_2# and #L_1#?
  
• Given #L_1->x+3y=0#, #L_2=3x+y+8=0# and #C_1=x^2+y^2-10x-6y+30=0#, determine #C->(x-x_0)^2+(y-y_0)^2-r^2=0# tangent to #L_1,L_2# and #C_1#?
  
• What is the inverse function of #f(x) = cosh(x+a/cosh(x+a/cosh(x+cdots)))# with domain and range?
  
• Given #f(x) = cosh(x+a/cosh(x+a/cosh( cdots)))# and #g(x)# its inverse, what is the minimum distance between then for #a > 0#?
  
• If #sinx+siny=a# and #cosx+cosy=b# how do you find #cos(x-y)# ?
  
• If #sinx+siny=a# and #cosx+cosy=b# how do you find #x,y# ?
  
• #lim_(n->oo)(1/n((n+1)(n+2)(n+3)cdots(2n))^(1/n))?#
  
• If #(x+sqrt(x^2+1))(y+sqrt(y^2+1))=1# what is #x+y#?
  
• #lim_(n->oo)((sqrt(1)+sqrt(2)+sqrt(3)+cdots+sqrt(n))/(n sqrt(n)))# ?
  
• #lim_(n->oo)(1/(1 xx 2)+1/(2 xx 3)+1/(3 xx4) + cdots + 1/(n(n+1)))#?
  
• What is the greater: #1000^(1000)# or #1001^(999)#?
  
• Find the real solution(s) of #sqrt(3-x)-sqrt(x+1) > 1/3# ?
  
• Given the sequence #a_0=1,a_1=2, a_(k+1)=a_k+(a_(k-1))/(1+a_(k-1)^2), k > 1# for what value of #k# occours #52 < a_k < 65# ?
  
• #alpha, beta# are real numbers such that #alpha^3-3alpha^2+5 alpha - 17=0# and #beta^3-3beta^2+5beta+11 = 0#. What is the value of #alpha+beta# ?
  
• Given #a in RR^+, a ne 1# and #n in NN, n > 1# Prove that #n^2 < (a^n + a^(-n)-2)/(a+a^(-1)-2)#?
  
• Given #{r,s,u,v} in RR^4# Prove that #min {r-s^2,s-u^2,u-v^2,v-r^2} le 1/4#?
  
• Find the real solutions for #{(x^4-6x^2y^2+y^4=1),(4x^3y+4xy^3=1):}#?
  
• Given the sequence #a_1=sqrt(y),a_2=sqrt(y+sqrt(y)), a_3 = sqrt(y+sqrt(y+sqrt(y))), cdots# determine the convergence radius of #sum_(k=1)^oo a_k x^k# ?
  
• Prove that for #n > 1# we have #1 xx 3 xx 5 xx 7 xx cdots xx(2n-1) < n^n#?
  
• What is the value of #log_2(Pi_(m=1)^2017Pi_(n=1)^2017(1+e^((2 pi i n m)/2017)))# ?
  
• Is #sqrt(2)^(sqrt(2))# rational ? And #sqrt(2)^(sqrt(2)^sqrt(2))#?. And #sqrt(2)^(sqrt(2)^(sqrt(2)^cdots))#?
  
• Prove that #3^x-1=y^4# or #3^x+1=y^4# have not integer positive solutions. ?
  
• Determine all integer pairs #(x,y)# with #x < y# such that the sum of all the integers strictly contained between then is equal #2016#?
  
• Given #{(p(x)=x^4+a x^3+b x^2+c x+1),(q(x)=x^4+c x^3+b x^2+a x + 1):}# find the conditions for #a, b, c, (a ne c)# such that #p(x)# and #q(x)# have two common roots, then solve #p(x)=0# and #q(x) = 0#?
  
• The equations #{(y = c x^2+d, (c > 0, d < 0)),(x = a y^2+ b, (a > 0, b < 0)):}# have four intersection points. Prove that those four points are contained in one same circle ?
  
• Proof that #N = (45+29 sqrt(2))^(1/3)+(45-29 sqrt(2))^(1/3)# is a integer ?
  
• Find the solutions #x > 0 in RR# for #2^x + 2^(1+1/sqrt(x))=6#?
  
• Solve for #x in RR# the equation #sqrt(x+3-4sqrt(x-1))+sqrt(x+8-6sqrt(x-1))=1# ?
  
• Let be #N# the smallest integer with 378 divisors. If #N = 2^a xx 3^b xx 5^c xx 7^d#, what is the value of #{a,b,c,d} in NN# ?
  
• Find #{x,y} in NN# such that #(1-sqrt(2)+sqrt(3))/(1+sqrt(2)-sqrt(3))=(sqrt(x)+sqrt(y))/2#?
  
• What is the ellipse which has vertices at #v_1 = (5,10)# and #v_2=(-2,-10)#, passing by point #p_1=(-5,-4)#?
  
• What is the smallest integer #n# such that #n! = m cdot 10^(2016)#?
  
• How to factor #a^8+b^8# ?
  
• If #f_0(x)=1/(1-x)# and #f_k(x)=f_0(f_(k-1)(x))# what is the value of #f_(2016)(2016)#?
  
• If #f(7+x)=f(7-x), forall x in RR# and #f(x)# has exactly three roots #a,b,c#, what is the value of #a+b+c#?
  
• Let #f# such that #f:RR->RR# and for some positive #a# the equation #f(x+a)=1/2+sqrt(f(x)+f(x)^2)# holds for all #x#. Prove that the function #f(x)# is periodic?
  
• Given the equation #sin(sin(sin(sin(sin(x)))))=x/3#. How many real solutions it have?
  
• Suppose that #f:RR->RR# has the properties #(a)# #|f(x)| le 1, forall x in RR# #(b)# #f(x+13/42)+f(x)=f(x+1/6)+f(x+1/7), forall x in RR# Prove that #f# is periodic?
  
• Given #f:[0,1]->RR# an integrable function such that #int_0^1f(x)dx=int_0^1 xf(x)dx= 1# prove that #int_0^1f(x)^2dx ge 4#?
  
• Let #P(x)=x^n+5x^(n-1)+3# where #n > 1# is an integer. Prove that #P(x)# cannot be expressed as the product of two polynomials, each of which has all its coefficients integers and degree at least #1#?
  
• Consider the polynomial #f(x)=x^4-4ax^3+6b^2x^2-4c^3x+d^4# where #a,b,c,d# are positive real numbers. Prove that if #f# has four positive distinct roots, then #a > b > c > d#?
  
• Find all polynomials #P(x)# with real coefficients for which #P(x)P(2x^2)=P(2x^3+x)#?
  
• Find the polynomial #P(x)# with real coefficients such that #P(2)=12# and #P(x^2)=x^2(x^2+1)P(x)# for each #x in RR#?
  
• If #sum_(i=1)^n theta_i= pi# with #theta_i ge 0# what is the maximum value for #sum_(i=1)^n sin^2theta_i#?
  
• What is the value of #S=sqrt(6+2sqrt(7+3sqrt(8+4sqrt(9+cdots))))#?
  
• Let #a=root(2016)(2016)#. Which of the following two numbers is greater #2016# or #a^(a^(a^(a^(vdots^a))))#?
  
• Prove that #sum_(k=1)^n 1/(sin2^kx)=cot x - cot 2^nx# for every #x ne (kpi)/2^k, x in RR, n in NN^+#?
  
• Prove that the fraction #(21n+4)/(14n+3)# is irreducible for every #n in NN#?
  
• Given the system #{(x+y+z=a),(x^2+y^2+z^2=b^2),(xy=z^2):}# determine the conditions over #a,b# such that #x,y,z# are distinct positive numbers?
  
• What is the solution for #cos^nx-sin^nx=1# witn #n in NN^+#?
  
• Given #{a,b,c} in [-L,L]# What is the probability that the roots of #a x^2+b x + c = 0# be real?
  
• There are #77# right-angled blocks of dimensions #3xx3xx1#. Is it possible to place all these blocks in a closed rectangular block of dimensions #7xx9xx11#?
  
• If #a_k in RR^+# and #s = sum_(k=1)^na_k#. Prove that for any #n > 1# we have #prod_(k=1)^n(1+a_k) < sum_(k=0)^n s^k/(k!)#?
  
• Can you solve this problem in Mechanics?
  
• A parabola is drawn on the plane. Build its axis of symmetry with the help of a compass (drawing tool) and a ruler.?
  
• Find the solutions of #x^2=2^x#?
  
• Can you solve this problem on Mechanics?
  
• Another problem on Mechanics?
  
• More on Mechanics?
  
• How to solve this exercise in Mechanics?
  
• Solve this exercise in Mechanics?
  
• #lim_(n->oo)(1^alpha+2^alpha+cdots+n^alpha)/n^(alpha+1) =#?
  
• Given the surface #f(x,y,z)=y^2 + 3 x^2 + z^2 - 4=0# and the points #p_1=(2,1,1)# and #p_2=(3,0,1)# determine the tangent plane to #f(x,y,z)=0# containing the points #p_1# and #p_2#?
  
• How to determine an ellipse passing by the four points #p_1 = {5, 10},p_2 = {-2, -10};p_3 = {-5, -4};p_4 = {5, -5};#?
  
• Given #ABC# a triangle where #bar(AD)# is the median and let the segment line #bar(BE)# which meets #bar(AD)# at #F# and #bar(AC)# at #E#. If we assume that #bar(AE)=bar(EF)#, show that #bar(AC)=bar(BF)#?.
  
• What is the value of #lim_(n->oo)sum_(k=0)^n(2n+1)/(n+k+1)^2#?
  
• How does the profile of the ground have to be, to be able to drive with triangular wheels (equilateral) and without bumps?
  
• How do you solve for #x in RR# the equation #x! = e^x# ?
  
• Do there exist real numbers #a,b# such that (1) #a+b# is rational and #a^n+b^n# is irrational for each natural #n ge 2# (2) #a+b# is irrational and #a^n+b^n# is rational for each natural #n ge 2#?
  
• Given the integer #N >0# there are exactly #2017# ordered pairs #{x,y }# of positive integers satisfying #1/x+1/y=1/N#. Prove that #N# is a perfect square ?
  
• In Cuba there are banknotes with values of 3, 10 and 20 pesos. Using only these banknotes, what is the largest amount that can not be formed?
  
• With #abs c gt absa+absb# calculate #lim_(x->oo)1/x int_0^x (dt)/(a sint+ bcost+c)#?
  
• Fill in the circles with numbers from 1 to 8 in such a way, that connected circles does not contain consecutive numbers.?
  
• Find the positive integer #n# such that #sum_(k=1)^n floor(log_2 k) = 2018#?
  
• Solve #cos(cos(cos(x)))=sin(sin(sin(x)))# ?
  
• Let #N# be the positive integer with #2018# decimal digits, all of them 1: that is #N = 11111cdots111#. What is the thousand digit after the decimal point of #sqrt(N)#?
  
• Evaluate #lim_(n->oo) 1/n^4 prod_(j=1)^(2n) (n^2+j^2)^(1/n)#?
  
• Solve the diophantine equation #x-y^4=4# where #x# is a prime?
  
• Find all triples #(x,y,p)# where #x# and #y# are positive integers and #p# is a prime, satisfying the equation #x^5+x^4+1 = p^y#?
  
• Find the positive integer solutions to the equation #x^3-y^3=x y+61#?
  
• What is the least natural number #n# for which the equation #floor(10^n/x)=2018# has an integer solution?
  
• Watering a garden?
  
• Find the remainder when #9 xx 99 xx 999 xx \underbrace{99 cdots 9}_{2017" " 9's}# is divided by #1000#?
  
• Determine the #a# value #0 < a < 1# in #1/16 log_a256^("colog"_(a^2)256^(log_(a^4)256^{cdots^("colog"_(a^(2^64))256)}))="Im"[z]# where #z# is the solution for #2^4033 z^2-2^2017 z+1 = 0# #(A)1/4, (B)1/8, (C)1/16, (D)1/32, (E) 1/64# ?
  
• Find the triangles with angles #A, B, C# and correspondingly opposite sides #a,b.c# such that #(aA+bB+cC)/(a+b+c)# has a minimum. Does this expression have a maximum?
  
• For any point #P# inside a given triangle #ABC#, denote by #x, y#, and #z# the distances from #P# to the lines #[BC], [AC]#, and #[AB]#, respectively. Find the position of #P# for which the sum #x^2 + y^2 + z^2# is a minimum.?
  
• If #(x+1)^n = sum_(k=0)^n c_k x^k# then show #sum_(k=0)^n 3^k c_k = 2^(2n)# ?
  
• Given #a = 1+sqrt2# find #lim_(x->0)((a+x)^a/a^(a+x))^(1/x)# Try not to use the L'Hopital method.?