oh you want hard questions? thats easy:
(polynomials)
x^3 - x^2 -ax - b = 0 has 3 positive roots prove that x^3 -x^2 + bx + a = 0 has 1 positive root and 2 complex roots
prove that log_a(x) cannot be written as the quotient of 2 polynomials with real coefficients
find all polynomials with possibly complex coefficients satisfying p(x^2)-p(x)p(x+1) = 0
(conics)
given ellipse x^2/a^2 + y^2/b^2=1 a find the area of the elliptical sector define by the piece bounded by the ellipse and lines joining points (acos(t),bsin(t)) (acos(u),bsin(u)) to the focus S(ae,0)
(Mechanics)
A rocket accelerates itself by shooting out gases at the rate of m kilograms per sectond at a velocity of v relative to the rocket, if the mass of the rocket is M and initially it carries F kilograms of fuel and it's initial velocity is V, find the distance travelled in time T and it's terminal speed after all fuel has been burnt.
At any time, a particle at position (x,y) is subjected to acceleration of magnitude 10/(x^2 + y^2) towards the origin. initially it is at position (5,0) with velocity sqrt(3) upwards, describe the locus of such particle's path
(counting)
Given n colours, how many different ways can you colour the faces of a cube? two colouring are considered identical if one can rotate one into the other.
(inequality)
It is known that for a certain function f,
f(ua + (1-u)b) >= uf(a) + (1-u)f(b) for all u between 0 and 1 and all a,b.
prove that:
f(u_1 a_1 + u_2 a_2 + ... + u_n a_n) >= u_1 f(a_1) + u_2 f(a_2) + ... + u_n f(a_n)
where u_i are positive and u_i sum to 1.
It is known that f(x) = x^a where 0<a<1 and x>0 has the property.
hence derive that for 1/p + 1/q = 1, p,q positive
(a_1 b_1 + a_2 b_2 + ... + a_n b_n) <= (a_1^p + a_2^p + ...+ a_n^p)^(1/p) * (b_1^q + b_2^q + ... +b_n^q)^(1/q)
for positive a's and b's.
hence show that
(a_1^p + a_2^p + ...+ a_n^p)^(1/p) + (b_1^p + b_2^p + ...+ b_n^p)^(1/p) >=
((a_1+b_1)^p + (a_2+b_2)^p + ...+ (a_n+b_n)^p)^(1/p) for p>=1
(geometry)
Show that the altitudes of a triangle are concurrent
do the same for the medians
and the perpendicular bisector os sides.
now show the the 3 points where the lines meet are colinear.
ABC is a triangle. D,E lies on side BC, F,G lies on CA and H,I lies on AB.
angle BAD = angle CAE
angle CBF = angle ABG
angle ACH = andgle BCI
prove that if AD,BF and CH are concurrent, then so are AE BG and CI
I will post some more when you finish these ones
