Problem Bank

Joey, Chloe, and their daughter Zoe share a birthday. Joey is one year older than Chloe, and Zoe turns exactly $1$ today. Today Chloe's age is an integer multiple of Zoe's age, and…

In an acute triangle $ABC$, the sides opposite $A, B, C$ are $a, b, c$. Given

find the maximum va…

Let $z$ be a complex number with $|z| = 1$. Find the maximum and minimum values of

The function $f$ is of the form $f(x) = \dfrac{k}{x}$, and the curve $g(x) = f(x)\cos x + b$ has tangent line $y = -\dfrac{6}{\pi}x + 2$ at $x = \dfrac{\pi}{2}$.

1. Find $g(x)$. 2…

In triangle $ABC$, the sides opposite angles $A, B, C$ are $a, b, c$ respectively. Given that

and the area of triangle $ABC$ is $\d…

Let $f(x) = m(x - 2m)(x + m + 1)$ and $g(x) = e^x - 1$. Determine, with proof, which of the following are true:

1. when $m = 1$, the equation $f(x) = g(x)$ has exactly one real so…

Color $1, 2, \dots, n$ with $k$ colors so that no three distinctly colored numbers form an arithmetic progression. Let $f(n)$ be the largest such $k$. Prove that

A family of curves $\Gamma_m$ is called triangle-preserving if for every triangle $ABC$, some curve in the family contains three points $P,Q,R$ with $\triangle PQR$ congruent to …

Real numbers $a, b$ satisfy

Find the maximum value of $ab$.

The circle $G:(x-2)^2+y^2=r^2$ is the incircle of triangle $ABC$, which is inscribed in the ellipse $\dfr…

Points $A, B, C$ lie on a circle $O$ of radius $1$, with $AB$ a diameter, and $P$ is a point in the closed disc. Find the range of possible values of

Let $a > 1$ and $b > 2$. Find the minimum value of