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.?