Problem Bank

Sets $A,B,C$ with $A\cap B=\varnothing$ and $A\cup B=C$ are said to give a partition $(A,B)$ of $C$.

1. Let $A_1=\left{x\mid\tan\left(\dfrac{\pi x}2+\dfrac{\pi}4\right)=-1,\ x\…

Let $I$ be the incenter of $\triangle ABC$ with $AB = 3$, $AC = \sqrt{13}$, and $\angle ABC = 60^\circ$. Find $\overrightarrow{AI}\cdot\overrightarrow{BC}$.

Suppose there exists $x \in \mathbb{R}$ such that

where $a \in \mathbb{Z}$. Find the sum of all integers $a$ satisfying this condition.

Suppose there exists $a > 0$ such that

Find the maximum possible value of $b$.

Through the left focus $F_1$ of the ellipse $W:\dfrac{x^2}{2}+y^2=1$, a line $l_1$ meets the ellipse at points $A$ and $B$, where $A(0,1)$. A second line $l_2$ through $F_1$ meets …

Let

1. If the equation $f(x) = 0$ has exactly $3$ real solutions, find the range of possibl…

In the plane, segments $AB = 3$ and $AC = 5$ make an angle $\angle BAC = \dfrac{\pi}{6}$. As the line $l$ varies over all lines in the plane, find the range of possible values of t…

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…

Consider the inequality

Prove that it holds for all $n$ when $C = \dfrac23$, and fails for some $n…

Let

Determine, with proof, which of the following claims are true.

1. When $a=1$, $f(x)$ has exactly one zero.…

Let $a$ and $b$ be positive numbers with $\dfrac{1}{a} + \dfrac{1}{b} = 1$, and let $n \in \mathbb{N}^*$. Prove that

Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ whose terms are all nonzero reals. For a given integer $n > 1$ and positive number $M$, the sequence…