Let \(f_1(x)=x^2+a_1x+b_1\) and \(f_2(x)=x^2+a_2x+b_2\) be polynomials with real coefficients. Prove or disprove that the following are equivalent.
(i) There exist two positive reals \(c_1, c_2\) such that \[ c_1f_1(x)+ c_2 f_2(x) > 0\] for all reals \(x\).
(ii) There is no real \(x\) such that \( f_1(x)\le 0\) and \( f_2(x)\le 0\).
Does there exist a (possibly \(n\)-dependent) constant \( C \) such that
\[
\frac{C}{a_n} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2 \leq \frac{a_1+ \dots + a_n}{n} - \sqrt[n]{a_1 \dots a_n} \leq \frac{C}{a_1} \sum_{1 \leq i < j \leq n} (a_i-a_j)^2
\]
for any \( 0 < a_1 \leq a_2 \leq \dots \leq a_n \)?
Let \(f:\mathbb R\to\mathbb R\) be a function such that \[ -1\le f(x+y)-f(x)-f(y)\le 1\] for all reals \(x\), \(y\). Does there exist a constant \(c\) such that \( \lvert f(x)-cx\rvert \le 1\) for all reals \(x\)?
Let
\[
f(x) = 1 + \left( \frac{1}{2} \cdot x \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot x^2 \right)^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot x^3 \right)^2 + \dots
\]
Prove that
\[
(\sin x) f(\sin x) f'(\cos x) + (\cos x) f(\cos x) f'(\sin x) = \frac{2}{\pi \sin x \cos x}.
\]
Suppose that we are given 12 points evenly spaced on a circle. Starting from a point in the 12 o’clock position, a particle P will move to one of the adjacent positions with equal probably, 1/2. P stops if it visits all 12 points. What is the most likely point that P stops for the last?
For \( a > b > 0 \), find the value of
\[
\int_0^{\infty} \frac{e^{ax} – e^{bx}}{x(e^{ax}+1)(e^{bx}+1)} dx.
\]
Find the minimum \(m\) (if it exists) such that every convex function \(f:[-1,1]\to[-1,1]\) has a constant \(c\) such that \[ \int_{-1}^1 \lvert f(x)-c\rvert \,dx \le m.\]
Let \( n \) be a positive integer. Suppose that \( a_1, a_2, \dots, a_n \) are non-zero integers and \( b_1, b_2, \dots, b_n\) are positive integers such that \( (b_i, b_n) = 1 \) for \( i = 1, 2, \dots, n-1 \). Prove that the Diophantine equation
\[
a_1 x_1^{b_1} + a_2 x_2^{b_2} + \dots + a_n x_n^{b_n} = 0
\]
has infinitely many integer solutions \( (x_1, x_2, \dots, x_n) \).
Suppose that the edges of a graph \(G\) can be colored by 3 colors so that there is no monochromatic cycle. Prove or disprove that \(G\) has two planar subgraphs \(G_1,G_2\) such that \(E(G)=E(G_1)\cup E(G_2)\).
Assume that \( x \in \mathbb{R}^n \) with at least \( k \) non-zero entries \( ( k> 0 ) \). Let
\[
A = \{ y \in \{-1, 1\}^n : y \cdot x = 0 \}.
\]
Prove that \( |A| \leq k^{-1/2} 2^n \).