Suppose that \( f: [a, b] \to \mathbb{R} \) is a smooth, convex function, and there exists a constant \( t>0 \) such that \( f'(x) \geq t \) for all \( x \in (a, b) \). Prove that
\[
\left| \int_a^b e^{i f(x)} dx \right| \leq \frac{2}{t}.
\]
Let \(\phi = \frac{1+\sqrt{5}}{2}\). Let \(f(1)=1\) and for \(n\geq 1\), let
\[ f(n+1) = \left\{\begin{array}{ll}
f(n)+2 & \text{ if } f(f(n)-n+1)=n \\
f(n)+1 & \text{ otherwise}.
\end{array}\right.\]
Prove that \(f(n) = \lfloor \phi n \rfloor\), and determine when \(f(f(n)-n+1)\neq n\) holds.
Let \(\{x_1, x_2, \ldots, x_{21}\} = \{-10, -9, \ldots, -1, 0, 1, \ldots, 9, 10\}\). What is the largest possible value of \(x_1x_2x_3+x_4x_5x_6+\cdots + x_{19}x_{20}x_{21}\)?
Find all integers \( n \) such that \( n^4 + n^3 + n^2 + n + 1 \) is a perfect square.
Determine the minimum number of hyperplanes in \(\mathbb{R}^n\) that do not contain the origin but they together cover all points in \(\{0,1\}^n\) except the origin.
Let \(f(x)\) be a degree 100 real polynomial. What is the largest possible number of negative coefficients of \((f(x))^4\)?
Suppose \( a_1, a_2, \dots, a_{2023} \) are real numbers such that
\[
a_1^3 + a_2^3 + \dots + a_n^3 = (a_1 + a_2 + \dots + a_n)^2
\]
for any \( n = 1, 2, \dots, 2023 \). Prove or disprove that \( a_n \) is an integer for any \( n = 1, 2, \dots, 2023 \).
There are light bulbs \(\ell_1,\dots, \ell_n\) controlled by the switches \(s_1, \dots, s_n\). The \(i\)th switch flips the status of the \(i\)th light and possibly others as well. If \(s_i\) flips the status of \(\ell_j\), then \(s_j\) flips the status of \(\ell_i\). All lights are initially off. Prove that it is possible to turn all the lights on.
Let \(A\) be an 8 by 8 integral unimodular matrix. Moreover, assume that for each \( x \in \mathbb{Z}^8 \), we have \(x^{\top} A x \) is even. What is the possible number of positive eigenvalues for \(A\)?
Define a sequence \( a_n \) by \( a_1 = 1 \) and
\[
a_{n+1} = \frac{1}{n} \left( 1 + \sum_{k=1}^n a_k^2 \right)
\]
for any \( n \geq 1 \). Prove or disprove that \( a_n \) is an integer for all \( n \geq 1 \).