Problem Bank

An ellipse $C$ has eccentricity $\dfrac{1}{2}$ and foci $F_1(-1, 0)$, $F_2(1, 0)$.

1. Find the equation of $C$. 2. Let $M_0(1, 4)$. Prove that the perpendicular bisector of segmen…

$F_1$ and $F_2$ are the foci of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$. A line through $F_1$ meets the ellipse at $P$ and $Q$. Given

In an acute triangle $ABC$, $\tan A=\dfrac 12$, and $D$ is a point on $BC$ such that $\triangle ABD$ and $\triangle ACD$ have areas $2$ and $4$ respectively. Drop perpendiculars $D…

An arithmetic sequence $\{a_n\}$ has its first term equal to its common difference, and its partial sums $S_n$ satisfy $S_{10} = 55$. A sequence $\{b_n\}$ satisfies $b_1 = 1$ and, …

Prove that for all $x > 0$ with $x \ne 1$,

Let

If every zero of $f(x)$ lies in an interval $[a, b]…

Let $P_1, P_2, P_3, P_4$ be four points on a conic $\Gamma \colon g(x, y) = 0$. The lines $P_1P_3$ and $P_2P_4$ meet at $Q(x_0, y_0)$, and the line $P_1P_2$ has equation $f(x, y) =…

Call a positive integer monotone if it has one digit, or its digits are strictly increasing or strictly decreasing left to right (e.g. $3$, $23578$, $87620$ — but not $88$, $7434…

Let $f(x) = ax^2 + bx + c$ with $a > 0$. Prove that there are at most two integers $s, t$ for which

A set of numbers $A = \{a_1, a_2, \dots, a_n\}$ with $1 = a_1 < a_2 < \cdots < a_n$ and $n \geqslant 2$ has property $P$ if for every $k$ with $2 \leqslant k \leqslant n$ there e…

Let

1. Discuss the monotonicity of $f(x)$ on the interval $[0, \pi]$. 2. Determine, with proof, the number of zeros of $y = …

In $\triangle ABC$, points $M$ and $N$ lie on the sides $AB$ and $AC$ respectively, and satisfy

Prove that the segment $MN$ passes through the ce…