Problem Bank

Consider the line $l \colon kx - y - k + 1 = 0$ and

The line $l$ meets the grap…

Each side of a convex $2019$-gon is colored red, green, or blue, with exactly $673$ sides of each color. Prove that one can draw $2016$ diagonals of the polygon, pairwise non-inter…

The function $f(x) = \ln x + \dfrac{m}{x} - 3$ has two zeros.

1. Find the range of possible values of $m$. 2. Let $a, b$ be the two zeros of $f(x)$. Prove that

For a bounded planar region, the largest distance between any two of its points is called the region's *diameter…

A moving line $l$ through the point $M(4,3)$ meets the positive $x$-axis at a point $A$ and the positive $y$-axis at a point $B$. Among all such lines, take the one for which the a…

Let $m$ be a positive integer, and let $a_1, a_2, \dots, a_{4m+2}$ be an arithmetic sequence with nonzero common difference. If after deleting two terms $a_i$ and $a_j$ ($i < j$) t…

As shown in the figure, in the quadrilateral $ABCD$ the areas of the triangles $ABD$, $BCD$, and $ABC$ are in the…

Let $f(x) = x^2 + (a-2)x - a\ln x$ with $a \in \mathbb{R}$.

1. If $a = 1$, find the extreme values of $f(x)$. 2. Discuss the monotonicity of $f(x)$. 3. For $n \in \mathbb{N}^*$, p…

Positive reals $a, b, c$ satisfy $a + b + c = 1$. Let $m$ be the maximum value of

Determine which of $1$, $\dfrac{5}{4}$, $\dfrac{3}{2}$, $\dfrac{…

Let $M(x_0,y_0)$ be a point on the hyperbola $C:\dfrac{x^2}2-y^2=1$, with foci $F_1$, $F_2$. If $\overrightarrow{MF_1}\cdot\overrightarrow{MF_2}<0$, find the range of $y_0$.

Solve the inequality

On the side $BC$ of an equilateral triangle $ABC$ of side length $1$, take $n$ equally spaced interior points ($n\geqslant 2$), labeled $P_1,P_2,\dots,P_{n-1}$ in the direction of …