Let \( A = \int_{0}^{1} \sin x \, dx \), \( B = \int_{0}^{1} \sin^2 x \, dx \), \( C = \int_{0}^{1} \cos x \, dx \), \( D = \int_{0}^{1} \cos^2 x \, dx \). Order the integrals.

Let \( n \) be a positive integer. Which of the following statements are always true?

I. \( n^3 – n \) is divisible by \( 6 \)

II. \( n^3 – n \) is divisible by \( 4 \)

III. \( n^3 – n \) is never prime

Find a necessary and sufficient condition on \( a \), such that
\[
\sqrt{a – \sqrt{a – \sqrt{a – \cdots}}}
=
\frac{1}{a – \frac{1}{a – \frac{1}{a – \cdots}}}.
\]

A student attempts to solve the following equation:
\[
\frac{x^2 – 5x + 6}{x^2 + x + 1} = \frac{x^2 – 5x + 6}{2x^2 – 3x – 2},
\]
by using the following steps:

\begin{align*}
\text{a)}\ & \frac{1}{x^2 + x + 1} = \frac{1}{2x^2 – 3x – 2} \\
\text{b)}\ & x^2 + x + 1 = 2x^2 – 3x – 2 \\
\text{c)}\ & x^2 + 2x – 3 = 0 \\
\text{d)}\ & x = -3, 1
\end{align*}

Which of the following best describes the solution?

Evaluate the sum
\[
\left(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \right)
+ \left(\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \right)
+ \left(\frac{1}{9} + \frac{1}{27} + \cdots \right) + \cdots .
\]

Consider the statement: “If \( n \) is an integer and \( n^2 \) is divisible by 4, then \( n \) is divisible by 4”. How many counterexamples are there to this in the range \( 50 \leq n \leq 100 \)?

Which of the following are necessary and sufficient conditions for the equations
\[
y = x – 4
\]
and
\[
x^2 – 2y^2 = a
\]
to have solutions

Let 𝐼 and 𝐼𝐼 be two statements. You are asked to show that 𝐼 if and only
if 𝐼𝐼. Which of the following does not prove the statement?

What is the coefficient of \( x^2 \) in the expansion of
\[
(1 + x)^2 \left( \frac{2}{x^2} – 3x^2 \right)^4 \, ?
\]

Let
\[
f(x) = ax^7 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + gx + h,
\]
with \( a, b, c, d, e, f, g, h \) real constants, and \( a > 0 \). Which of the following is possible?

Let 𝐼,𝐼𝐼,𝐼𝐼𝐼,𝐼𝑉 be some statements. Suppose that 𝐼 → 𝐼𝐼 → 𝐼𝐼𝐼 and 𝐼𝑉 →
𝑁𝑜𝑡 𝐼𝐼𝐼 and 𝑁𝑜𝑡 𝐼 → 𝐼𝐼, where 𝑎 → 𝑏 means if a is true, then b is true.
𝑛𝑜𝑡 𝑎 is just the opposite to a, so if a is true, 𝑛𝑜𝑡 𝑎 is false and vice versa.
Suppose 𝐼𝐼 is a true statement. What can we say about the rest of the
statements?

If p is a prime number, which of the following must be true?

I    p is odd
II   p is not divisible by 5
III  p is not divisible by 6

Let \( f(x) \) be a function defined over all real \( x \). You are given that
\[
\int_{0}^{6} 2f(2x)\,dx = 1,
\]
and that \( f(x) \) is antisymmetric in the line \( \frac{3}{2} \), i.e.,
\[
f(3 – x) = -f(x).
\]
Calculate
\[
\int_{2}^{3} \left(f(x) + 1\right)\,dx.
\]

Which of the following statements are true?
I    being a square is a sufficient condition for being a rectangle.
II   being less than 12 is a sufficient condition for being less than 20.
III  being a rectangle is a necessary condition for being a square.
IV  having 4 equal length sides is necessary and sufficient for being a square.
V   an integer being less than 19.5 is sufficient, but not necessary for being
less than 20.

The polynomial
\[
x^3 + (a – 3)x^2 + (b – 3a)x – 3b
\]
has exactly two roots. Which of the following is true?

A large circular room has 2020 light bulbs attached to the edge. Each light
bulb has a switch below it, that controls the state of the two adjacent light
bulbs to it. Given that all the light bulbs start off, how many can be turned
on at once?

A geometric series has first term \( a = \sqrt{32} \) and sixth term \( \frac{1}{a^2} \). Find the sum to infinity.

Let \( f \) be a function satisfying the following condition for all \( x_1, x_2 \) and for \( 0 \leq t \leq 1 \):
\[
f(tx_1 + (1 – t)x_2) \leq t f(x_1) + (1 – t) f(x_2).
\]

Which of the following is a necessary condition for this to hold?

Calculate the derivative of
\[
(1 + 4x)^3 (2x)^{-\frac{1}{2}}.
\]

It is given that a certain equation \( f(x) = 0 \) has \( n \) roots. Which of the following must be true?

I \( f(x + 1) \) has \( n \) roots
II \( 2f(2x + 2) \) has \( n \) roots
III \( f(x) + 1 \) has \( n \) roots
IV \( 2^{f(x)} – 1 = 0 \text{ has } n \text{ roots} \)