MathDB

p1. Compute

\lim_{x\to 0} \frac{\tan x - x -\frac{x^3}{3}}{x}

.
p2. For how many integers

n

\frac{n}{20-n}

equal to the square of a positive integer.
p3. Find all possible solutions

(x_1, x_2, ..., x_n)

to the following equations, where

x_1 =\frac12 \left(x_n +\frac{x^2_{n-1}}{x_n}\right)

x_2 =\frac12 \left(x_1 +\frac{x^2_{n}}{x_1}\right)

x_3 =\frac12 \left(x_2 +\frac{x^2_{1}}{x_2}\right)

x_4 =\frac12 \left(x_3 +\frac{x^2_{2}}{x_3}\right)

x_5 =\frac12 \left(x_4 +\frac{x^2_{3}}{x_4}\right)

...

x_n =\frac12 \left(x_{n-1} +\frac{x^2_{n}}{x_1}\right)= 2010

.
p4. Let

ABCDEF

be a convex hexagon, whose opposite angles are parallel, satisfying

AF = 3

BC = 4

, and

DE = 5

. Suppose that

AD

BE

CF

intersect in a point. Find

CD

.
p5. Rank the following in decreasing order:

A =\frac{1\sqrt1 + 2\sqrt2 + 3\sqrt3}{\sqrt1 + \sqrt2 + \sqrt3}

B =\frac{1^2 + 2^2 + 3^2}{1 + 2 + 3}

C =\frac{1 + 2 + 3}{3}

D =\frac{\sqrt1 + \sqrt2 + \sqrt3}{\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\frac{1}{\sqrt3}}

.
p6. What is the least

m

such that for any

m

integers we can choose

6

integers such that their sum is divisible by

6

?
p7. Find all positive integers

n

such that

\phi(n) = 16

, where

\phi(n)

is defined to be the number of positive integers less than or equal to

n

that are relatively prime to

n

.
p8. Suppose that for an infinitely differentiable function

f

\lim_{x\to 0}\frac{f(4x) + af(3x) + bf(2x) + cf(x) + df(0)}{x^4}

exists. Find

1000a + 100b + 10c + d

.
p9. Minimize

x^3 + 4y^2 + 9z

under the constraints that

xyz = 1

x, y, z \ge 0

.
p10. Positive real numbers

x, y

, and

z

satisfy the equations

x^2 + y^2 = 9

y^2 +\sqrt2yz + z^2 = 16

z^2 +\sqrt2zx + x^2 = 25

. Compute

\sqrt2xy + yz + zx

.
p11. Find the volume of the region given by the inequality

|x + y + z| + |x + y - z| + |x - y + z| + | - x + y + z| \le 4.

p12. Suppose we have a polyhedron consisting of triangles and quadrilaterals, and each vertex is shared by exactly

4

triangles and one quadrilateral. How many vertices are there?
p13. Rank the following in increasing order:

A =\frac{\sqrt{2011} + \sqrt{2009}}{2}

B =\frac{2011\sqrt{2011} - 2009\sqrt{2009}}{3}

C =\frac{\sqrt{2011} + 2\sqrt{2010} + \sqrt{2009}}{4}

D =\sqrt{2010}

E =\sqrt{2011} -\frac{1}{2\sqrt{2011}}

p14. Suppose

f

and

g

are continuously differentiable functions satisfying

f(x + y) = f(x)f(y) - g(x)g(y)

g(x + y) = f(x)g(y) + g(x)f(y)

Also suppose that

f'(0) = 1

and

g'(0) = 2

. Find

f(2010)^2 + g(2010)^2

.
p15. Find the number of

n

-tuples

(a_1, ..., a_n)

that maximize

a_1a_2a_3 + a_2a_3a_4 + ... + a_{n-2}a_{n-1}a_n

under the constraints that

n \ge 3

and

a_1 + a_2 + ...+ a_n = 3m

for a fixed integer

m

, where

a_i

are positive integers.
Note: 8 problems were common with [url=https://artofproblemsolving.com/community/c4h2765069p24208072]2010 Rice Math Tournament: p1-3, p5, p8, p10-12
PS. You had better use hide for answers.

Problem Statement