Problem Bank

Let $a_i, b_i > 0$ for $1 \leqslant i \leqslant n+1$, and suppose $b_{i+1} - b_i \geqslant \delta > 0$ for $i = 1, 2, \dots, n$, where $\delta$ is a constant. If $\displaystyle\sum…

Let $f(x) = ax^2 + bx + c$ and $g(x) = cx^2 + bx + a$, and set

If $|g(x)| \leqslant 1$ for all $x \in [-1, 1]$, find the maximum poss…

In $\triangle ABC$, the altitudes to sides $a, b, c$ have lengths $h_a, h_b, h_c$, and

1. If $S$ is the area o…

Suppose $x_1, x_2$ are the two real roots of the quadratic equation $x^2 + bx + c = 0$, and $|x_1| + |x_2| = 2|k|$ for some integer $k$. Then the equation is called an *even-type q…

For $0 \le x, y \le 1$, prove that

Consider the parabola $C \colon y^2 = 2px$ (with $p > 0$). The line $x - 2y + 1 = 0$ meets $C$ at points $A$ and $B$ with $|AB| = 4\sqrt{15}$.

1. Find $p$. 2. Let $F$ be the focus…

Let $a, b, c$ be the sides of a triangle. Prove that

Let $f(x)=x^3+ax+\dfrac 14$ and $g(x)=-\ln x$.

1. For what value of $a$ is the $x$-axis tangent to the curve $y=f(x)$? 2. Writing $\min\{m,n\}$ for the smaller of $m,n$, let $h(x)…

Let $M$ be a set with $6$ elements, and define

Find the nu…

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has left and right foci $F_1, F_2$ and eccentricity $e$. A point $P$ on the ellipse is such that t…

Let $f(x)$ be an odd differentiable function on $\mathbb{R}$ with $f(x) < f'(x)$ for $x > 0$ and $f(1) = e$. Let $M$ be the solution set of

The graph of the function $f(x)=ax^3+bx^2+cx+d$ is shown. Determine the sign (positive or negative) of each of $a$, $b$, $c$,…