MathDB

p1. Compute

\left(\frac{1}{10}\right)^{\frac12}\left(\frac{1}{10^2}\right)^{\frac{1}{2^4}}\left(\frac{1}{10^3}\right)^{\frac{1}{2^3}} ...

p2. Consider the sequence

1, 2, 2, 3, 3, 3, 4, 4, 4, 4,...,

where the positive integer

m

appears

m

times. Let

d(n)

denote the

n

th element of this sequence starting with

n = 1

. Find a closed-form formula for

d(n)

.
p3. Let

0 < \theta < \frac{\pi}{2}

, prove that

\left( \frac{\sin^2 \theta}{2}+\frac{2}{\cos^2 \theta} \right)^{\frac14}+ \left( \frac{\cos^2 \theta}{2}+\frac{2}{\sin^2 \theta} \right)^{\frac14} \ge (68)^{\frac14}

and determine the value of \theta when the inequality holds as equality.
p4. In

\vartriangle ABC

, parallel lines to

AB

and

AC

are drawn from a point

Q

lying on side

BC

. If

a

is used to represent the ratio of the area of parallelogram

ADQE

to the area of the triangle

\vartriangle ABC

, (i) find the maximum value of

a

. (ii) find the ratio

\frac{BQ}{QC}

when

a =\frac{24}{49}.

https://cdn.artofproblemsolving.com/attachments/5/8/eaa58df0d55e6e648855425e581a6ba0ad3ea6.png
p5. Prove the following inequality

\frac{1}{2009} < \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8}...\frac{2007}{2008}<\frac{1}{40}

PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.Thanks to gauss202 for sending the problems.

Problem Statement